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Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T15:27:15Z

Qifang: /* 4.4.3 Mechanical Effects */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

=== 4.1.1 Tangential Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, circumference (<math display="inline">C</math>) of the disk can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the tangential velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the small fluctuation of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The disk was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the disk's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the disk's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational fluctuation.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the disk to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and disk imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T15:25:09Z

Qifang: /* 4.2.3 Velocity Uncertainty */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

=== 4.1.1 Tangential Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, circumference (<math display="inline">C</math>) of the disk can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the tangential velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the small fluctuation of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The disk was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the disk's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the disk's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational fluctuation.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the disk to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T15:21:38Z

Qifang: /* 4.2 Group I: Asymmetric Magnet Configuration */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

=== 4.1.1 Tangential Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, circumference (<math display="inline">C</math>) of the disk can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the tangential velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the small fluctuation of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The disk was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the disk's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the disk's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:59:11Z

Qifang: /* 4.1.2 Stability Metric */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

=== 4.1.1 Tangential Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, circumference (<math display="inline">C</math>) of the disk can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the tangential velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the small fluctuation of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:52:02Z

Qifang: /* 4.1.1 Velocity Formula */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

=== 4.1.1 Tangential Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, circumference (<math display="inline">C</math>) of the disk can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the tangential velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:48:43Z

Qifang: /* 4.1 Experimental Process */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:48:12Z

Qifang: /* 4.1 Experimental Process */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The experimental setup employed an SS411P Hall-effect sensor to detect the rotation of a circular disk with a diameter of 8 cm. The sensor was operated at a constant supply voltage of 5 V, corresponding to a current of 0.005 A. To maintain consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass, a vertical separation of 3 cm was kept between the magnets and the Hall sensor.

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter circular disk. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:43:36Z

Qifang: /* 4.1 Experimental Process */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter circular disk. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:39:29Z

Qifang: /* 3.3.2 Role of the Pull-Down Resistor */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:37:27Z

Qifang: /* 3.3.2 Role of the Pull-Down Resistor */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 5.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:30:46Z

Qifang: /* 3.3.1 Wiring Connections */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Breadboard_implementation_of_the_Hall_sensor_circuit.jpeg |thumb|center|300px|'''Fig. 4.''' Breadboard implementation of the Hall sensor circuit.]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:26:06Z

Qifang: /* 3.3.1 Wiring Connections */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.
[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

File:Breadboard implementation of the Hall sensor circuit.jpeg

2026-04-15T14:25:59Z

Qifang:

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-15T14:23:06Z

Qifang: /* 3.3.1 Wiring Connections */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement, as shown in '''Fig. 4.''' The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T10:24:14Z

Qifang: /* 3.1 Apparatus and Setups */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnet × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T10:17:42Z

Qifang: /* 5 Conclusion */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

= 6 Reference =

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T10:14:39Z

Qifang: /* 3.3.2 Role of the Pull-Down Resistor */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T10:11:30Z

Qifang: /* 3.3.2 Role of the Pull-Down Resistor */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform.<ref>Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.</ref>]]

(from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T10:08:46Z

Qifang: /* 3.3.2 Role of the Pull-Down Resistor */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File: Effect_of_the_pull-down_resistor_on_the_output_waveform.jpeg|thumb|center|300px|'''Fig. 4.''' Effect of the pull-down resistor on the output waveform]]

(from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

File:Effect of the pull-down resistor on the output waveform.jpeg

2026-04-14T10:06:03Z

Qifang:

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T10:03:02Z

Qifang: /* 3.1 Apparatus and Setups */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto; text-align:center;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:54:21Z

Qifang: /* 3.1 Apparatus and Setups */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 80mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:42:17Z

Qifang: /* 2.2.3 Functions for each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:41:55Z

Qifang: /* 2.2.3 Functions for Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:41:16Z

Qifang: /* 2.2.3 Functions For Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions for Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:39:52Z

Qifang: /* 3.2 Power Supply Configuration */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:39:16Z

Qifang: /* 3.1 Apparatus and Setups */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in '''Fig. 3. '''

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:37:00Z

Qifang: /* 3.1 Apparatus and Setups */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in Fig. 3.

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system.]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T09:36:36Z

Qifang: /* 3 Experimental Details */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

== 3.1 Apparatus and Setups ==

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets × 2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand × 2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed. The actual experimental setups is shown below in Fig. 3.

[[File:Experimental setups of the entire Hall sensor system.jpeg|thumb|center|500px|'''Fig. 3.''' Experimental setups of the entire Hall sensor system]]

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

File:Experimental setups of the entire Hall sensor system.jpeg

2026-04-14T09:16:26Z

Qifang: Qifang uploaded a new version of File:Experimental setups of the entire Hall sensor system.jpeg

File:Experimental setups of the entire Hall sensor system.jpeg

2026-04-14T09:09:38Z

Qifang:

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:42:41Z

Qifang: /* 3 Experimental Details */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

{| class="wikitable" style="margin:auto;"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm
Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm
Thickness: 5mm
| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| style="text-align:center;" | /
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| style="text-align:center;" | /
| Construct and connect the sensor circuit
|-
| Wires
| style="text-align:center;" | /
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:23:55Z

Qifang: /* 2.2.3 Functions For Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified internal circuit configuration of the SS411P Hall sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:23:13Z

Qifang: /* 2.2.3 Functions For Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.''' Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:22:52Z

Qifang: /* 2.2.3 Functions For Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px|'''Fig. 2.'''Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.<ref>Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor, Product Sheet 005914-1-EN, October 2009.</ref>]]

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:19:27Z

Qifang: /* 2.2.3 Functions For Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px'''Fig. 2.''']]

'''Figure 2.''' Block diagram of the SS411P Hall sensor (from [2]).

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:17:56Z

Qifang: /* 2.2.3 Functions For Each Pin */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File: Simplified_internal_circuit_configuration_of_the_SS411P_Hall_sensor.jpeg|thumb|center|500px]]

'''Figure 2.''' Block diagram of the SS411P Hall sensor (from [2]).

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

File:Simplified internal circuit configuration of the SS411P Hall sensor.jpeg

2026-04-14T08:16:48Z

Qifang:

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:05:18Z

Qifang: /* 2.1 Fundamental Basics - Hall Effect */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in '''Fig. 1.''' A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.</ref>]]

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File:fig2.jpg]

'''Figure 2.''' Block diagram of the SS411P Hall sensor (from [2]).

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T08:01:19Z

Qifang: /* 2.1 Fundamental Basics - Hall Effect */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in the figure. A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|Principle of the Hall effect]]
[[File:Principle of the Hall effect.jpeg|thumb|center|500px|'''Fig. 1.''' Principle of the Hall effect in a rectangular semiconductor plate.<ref>Honeywell, ''SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'', Product Sheet 005914-1-EN, October 2009.</ref>]]

'''Figure 1.''' Principle of the Hall effect used in this experiment (from [1]).

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File:fig2.jpg]

'''Figure 2.''' Block diagram of the SS411P Hall sensor (from [2]).

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T07:49:41Z

Qifang: /* 2.1 Fundamental Basics - Hall Effect */

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in the figure. A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of the Hall effect.jpeg|thumb|center|500px|Principle of the Hall effect]]

'''Figure 1.''' Principle of the Hall effect used in this experiment (from [1]).

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File:fig2.jpg]

'''Figure 2.''' Block diagram of the SS411P Hall sensor (from [2]).

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

Rotational Speed Measurement System Based on Hall-Effect Sensor

2026-04-14T07:32:45Z

Qifang: Created page with "= 1 Introduction = == 1.1 Objectives == As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal. == 1.2 Related Applications..."

= 1 Introduction =

== 1.1 Objectives ==

As a semiconductor device based on the Hall effect, the Hall sensor is capable of converting magnetic-field variation into an electrical response, making it highly suitable for rotational monitoring and related sensing applications. The objective of this experiment is to demonstrate how periodic magnetic variation associated with rotational motion can be translated into a measurable electrical signal.

== 1.2 Related Applications ==

This experiment is representative of a practical approach to non-contact rotational sensing, which is widely applied in systems requiring speed monitoring. Such a principle is particularly relevant to motor-based devices, automated machinery, and rotational control systems, where accurate detection of rotational behavior is essential for performance regulation and operational reliability.

= 2 Background =

== 2.1 Fundamental Basics - Hall Effect ==

The Hall effect could be explained using the rectangular semiconductor plate shown in the figure. A bias current <math display="inline">I_{bias}</math> flows through the Hall element via the current terminals <math display="inline">C_{1}</math> and <math display="inline">C_{2}</math> , while a magnetic field <math display="inline">B</math> is applied perpendicular to the plane of the semiconductor. Under these conditions, the moving charge carriers experience the Lorentz force,

<center><math>\vec{F}=q\vec{v}\times\vec{B}</math>,</center>

which deflects them laterally and causes charge accumulation on opposite sides of the semiconductor. This establishes a transverse electric field <math display="inline">E_{H}</math> , known as the Hall field, and gives rise to a measurable Hall voltage <math display="inline">V_{H}</math> across the sensing terminals <math display="inline">S1</math> and <math display="inline">S2</math>.

At equilibrium, the magnetic deflection is balanced by the electric field within the Hall element. Consequently, the Hall voltage is proportional to the applied magnetic field and the bias current. In a simplified form, it may be expressed as

<center><math display="inline">V_{H} \propto I_{bias}B</math>,</center>

and more specifically as

<center><math display="inline">V_{H} = \frac{I_{bias}B}{nqt}</math>,</center>

where <math display="inline">n</math> is the carrier concentration, <math display="inline">q</math> is the carrier charge, and <math display="inline">t</math> is the thickness of the semiconductor plate. The geometric parameters <math display="inline">l</math>, <math display="inline">w</math>, and <math display="inline">t\ </math>represent the length, width, and thickness of the Hall element, respectively, while <math display="inline">E_{bias}</math> denotes the electric field associated with the applied bias.

This relationship shows that the Hall voltage arises from the interaction between carrier motion and the externally applied magnetic field. In practical Hall sensors, the Hall voltage is further processed by internal circuitry to generate a stable output signal, thereby enabling magnetic-field variation to be converted into an electrical response for rotational speed measurement.

[[File:Principle of Hall Effect.jpeg]

'''Figure 1.''' Principle of the Hall effect used in this experiment (from [1]).

== 2.2 Speed Hall sensor ==

=== 2.2.1 Classification and Selection of Hall Sensors ===

Hall sensors can be classified into different types according to their magnetic response characteristics, such as unipolar, bipolar, omnipolar, and latching sensors. In this experiment, a bipolar Hall sensor (SS411P) was selected because the rotating disk carries two magnets with opposite magnetic polarities, and this sensor is specifically designed to respond to alternating North and South poles, making it more suitable than the other types for generating distinct output states during rotation.

=== 2.2.2 Working Principle ===

Building on this physical basis, the Hall sensor serves as a practical magnetic switching device in the present system. Rather than measuring the Hall voltage directly, the sensor uses its internal circuitry to convert the detected magnetic state into a discrete electrical output. When the rotating disk brings oppositely oriented magnets past the sensing region, the magnetic polarity at the sensor changes periodically, and the SS411P correspondingly switches between two output states. This produces the alternating high- and low-voltage waveform observed on the oscilloscope, from which the signal period can be measured and related to the rotational motion of the disk. In this way, the Hall sensor provides the essential link between magnetic-field variation and electrical speed measurement in a non-contact configuration.

=== 2.2.3 Functions For Each Pin ===

For the SS411P sensor, the three pins are VS, GND, and OUT. The VS pin is connected to the supply voltage and provides the operating power for the sensor, the GND pin serves as the electrical reference and completes the circuit, and the OUT pin delivers the output signal generated in response to the detected magnetic field.

[[File:fig2.jpg]

'''Figure 2.''' Block diagram of the SS411P Hall sensor (from [2]).

= 3 Experimental Details =

{| class="wikitable"
|-
! '''Apparatus'''
! '''Specification'''
! '''Purpose'''
|-
| Circular disk
| Diameter: 90mm Thickness: 3mm
| Mounted on the motor shaft to provide stable rotational motion
|-
| Magnets*2
| Diameter: 5mm

Thickness: 5mm

| Generate alternating magnetic fields for Hall sensor detection
|-
| Electric motor
| DC motor with adjustable rotational speed
| Drive the circular disk to produce periodic motion
|-
| Hall sensor
| SS411P
| Detect the changing magnetic field and generate a digital output signal
|-
| Stand*2
| __
| Support and align the motor, disk, and Hall sensor
|-
| DC Power supply
| 2231A-30-3
| Supply power to the Hall sensor and the motor
|-
| Digital Oscilloscope
| RTB2004

2.5GSa/s

| Display and measure the output waveform from the Hall sensor
|-
| Breadboard
| __
| Construct and connect the sensor circuit
|-
| Wires
| __
| Provide electrical connections between the components
|}

== 3.1 Apparatus and Setups ==

To ensure both mechanical stability and reliable electrical measurement, a rigid circular disk with uniform thickness was mounted on the motor shaft to maintain rotational balance and minimize wobble, then rotated by motor for continuous motion. To improve the mechanical coupling, a central hole was drilled in the disk by soldering machine and subsequently adjusted using a heat gun to achieve a closer fit to the shaft. Two magnets with opposite magnetic directions were attached to the disk, while the Hall sensor was positioned adjacent to the rotating path so that the changing magnetic field could be detected during rotation. The motor and Hall sensor were powered by an external supply, and the Hall sensor was connected on a breadboard for circuit integration. Finally, the sensor output was connected to the oscilloscope, where the resulting digital waveform could be observed.

== 3.2 Power Supply Configuration ==

Specifically, the power supply employed in the present experiment consisted of three independent channels, two of which were used. Channel 3 was dedicated to supplying the Hall sensor. Based on the SS411P datasheet, a supply voltage of 5 V was selected, and the sensor was therefore operated at <math display="inline">V_{CC} = 5\, V</math>. Channel 2 was used to drive the motor. By varying the motor supply voltage, the rotational speed of the disk could be adjusted, thereby allowing the effect of rotational speed on the output signal to be investigated under controlled conditions.

[[File:fig3.jpg]

'''Figure 3.''' Experimental setup of the Hall sensor system

== 3.3 Breadboard Setup ==

=== 3.3.1 Wiring Connections ===

The breadboard connections were arranged according to the three-pin configuration of the Hall sensor and the requirements of signal measurement. The supply terminal of the sensor was connected to the positive power rail, providing the operating voltage required for the Hall sensor. The ground terminal was connected to the ground rail, thereby establishing the common electrical reference for the circuit. The output terminal of the Hall sensor was then routed to a dedicated signal node on the breadboard. This same node was connected to the oscilloscope input so that the electrical response of the sensor could be observed directly during disk rotation.

=== 3.3.2 Role of the Pull-Down Resistor ===

In addition, a pull-down resistor was connected between the output node and ground to establish a defined low-level state and to prevent the output from floating in the absence of active switching. As illustrated in the figure, without a pull-down path the output node may not return immediately to a well-defined voltage level after switching, which can lead to slow recovery and waveform distortion. By introducing the resistor, the output is driven toward a stable low-level state, thereby improving the sharpness and stability of the digital signal. For this reason, a resistor was incorporated into the present circuit so that a clearer and more reproducible waveform could be obtained on the oscilloscope.

[[File:fig4.jpg]

'''Figure 4.''' Effect of the pull-down resistor on the output waveform (from [3]).

[[File:fig5.jpg]

'''Figure 5.''' Breadboard implementation of the Hall sensor circuit

= 4 Experimental Procedures and Results =

== 4.1 Experimental Process ==

The apparatus utilized a SS411P Hall effect sensor to detect the rotation of an 8 cm diameter rotor. The sensor was powered with a constant input of 5V and 0.005A. A vertical distance of 3 cm was maintained between the magnets and the sensor to ensure a consistent magnetic flux change <math display="inline">\mathrm{\Delta}\Phi</math> during each pass.

=== 4.1.1 Velocity Formula ===

The diameter <math display="inline">D = 8cm</math>, rotor circumference (C) can be calculated as:

<math display="block">C = \pi*D \approx 25.13cm</math>

Assuming the sensor receives one pulse for every revolution of the disk, then:

<math display="block">n = f</math>

where <math display="inline">n</math> is the rotational frequency (rev/s).

Assuming each pulse represents one full rotation, the velocity is:

<math display="block">v = f*C = f*25.13cm/s</math>

=== 4.1.2 Stability Metric ===

The Standard Deviation (StdDev, <math display="inline">\sigma_{f}</math>) recorded by the oscilloscope represents the temporal jitter of the pulse triggers. It is utilized as the primary indicator of measurement uncertainty and system instability.

The velocity uncertainty (<math display="inline">\sigma_{v}</math>):

<math display="block">\sigma_{v} = \sigma_{f}*25.13</math>

=== 4.1.3 Linear Regression and Fitting ===

To quantify the relationship between electrical input (x) and velocity (y), the Method of Least Squares is applied:

<math display="block">y = a + bx</math>

Where slope (b) represents the sensitivity of speed to changes in voltage or current, intercept (a) is the theoretical threshold value for the motor to overcome internal friction.

The quality of the linear fit is assessed using the <math display="inline">R^{2}</math> coefficient:

<math display="block">R^{2} = 1 - \frac{{SS}_{res}}{{SS}_{tot}} = 1 - \frac{\sum_{}^{}{(v_{i} - {\widehat{v}}_{i})}^{2}}{\sum_{}^{}{(v_{i} - \overline{v})}^{2}}</math>

Where <math display="inline">{SS}_{res}</math> (Residual Sum of Squares) presents unexplained variance between observed and predicted values, <math display="inline">{SS}_{tot}</math> (Total Sum of Squares) presents total variance of the observed velocity data.

== 4.2 Group I: Asymmetric Magnet Configuration ==

The rotor was equipped with an asymmetric magnet arrangement comprising four magnets of identical diameter: three with uniform thickness and one with a different thickness.

=== 4.2.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 1:

'''Table 1''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.107-0.140
| 0.44499
| 1.55193
| 11.17815
|-
| 0.60
| 0.119-0.145
| 0.82223
| 2.37705
| 20.65442
|-
| 0.80
| 0.129-0.155
| 1.19922
| 2.97037
| 30.12441
|-
| 1.00
| 0.130-0.160
| 1.43416
| 3.96689
| 36.02610
|-
| 1.20
| 0.140-0.165
| 1.76934
| 4.67602
| 44.44582
|-
| 1.40
| 0.154-0.176
| 2.10642
| 68.09750
| 52.91327
|-
| 1.60
| 0.163-0.194
| 2.48317
| 5.73160
| 62.37723
|-
| 1.80
| 0.177-0.214
| 2.90326
| 12.73610
| 72.92989
|-
| 2.00
| 0.186-0.240
| 3.11264
| 7.98737
| 78.18952
|}

Voltage and rotational speed are basically linearly related: <math display="inline">v = 42.19V - 5.20</math>, as shown in Figure 6.

Voltage and error do not follow a linear relationship, as shown in Figure 7, but under standard error they basically show an increasing trend.

[[File:fig6.jpg]

'''Figure 6.''' Tangential velocity vs. Input voltage

[[File:fig7.jpg]

'''Figure 7.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = \ 0.99729</math>). However, at 1.40V, an instability peak appears, with a StdDev reaching 68.10 mHz.

=== 4.2.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 2:

'''Table 2''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.60-0.70
| 0.89290
| 3.09062
| 22.42965
|-
| 0.14
| 0.81-0.89
| 1.33012
| 176.81500
| 33.41261
|-
| 0.15
| 1.02-1.09
| 1.59823
| 9.56639
| 40.14754
|-
| 0.16
| 1.16-1.25
| 2.27975
| 1109.13000
| 57.26732
|-
| 0.17
| 1.33-1.45
| 2.22710
| 170.18700
| 55.94475
|-
| 0.18
| 1.42-1.62
| 2.64184
| 471.37200
| 66.36302
|-
| 0.19
| 1.58-1.73
| 2.54130
| 262.94300
| 63.83746
|-
| 0.20
| 1.74-1.79
| 2.75797
| 375.50600
| 69.28021
|-
| 0.21
| 1.74-1.81
| 2.92263
| 238.11500
| 73.41647
|}

The obtained current has a linear relationship with the rotational speed <math display="inline">v = 613.38I - 50.71</math>, as shown in Figure 8. The voltage and error do not conform to a linear or exponential relationship, as shown in Figure 9.

[[File:fig8.jpg]

'''Figure 8.''' Tangential velocity vs. Input voltage

[[File:jig9.jpg]

'''Figure 9.''' Frequency standard deviation vs. Input voltage

In this mode, the data exhibited significant scatter (<math display="inline">R^{2} = 0.90830</math>), with the fluctuation reaching a peak value of 1109.13 mHz at 0.16 A. This suggests that the combination of magnetic field asymmetry and voltage regulation under current-driven operation led to the rotor's instability.

Also, a higher <math display="inline">R^{2}</math> in constant voltage mode compared to constant current mode indicates that voltage control is more stable for this motor system.

=== 4.2.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 1 and 2 is shown in Table 3.

'''Table 3''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.039
| 0.13
| 0.0777
|-
| 0.60
| 0.0597
| 0.14
| 4.4438
|-
| 0.80
| 0.0747
| 0.15
| 0.2404
|-
| 1.00
| 0.0997
| 0.16
| 27.8755
|-
| 1.20
| 0.1175
| 0.17
| 4.2773
|-
| 1.40
| 1.7115
| 0.18
| 11.8468
|-
| 1.60
| 0.1441
| 0.19
| 6.6085
|-
| 1.80
| 0.3201
| 0.2
| 9.4375
|-
| 2.00
| 0.2007
| 0.21
| 5.9845
|}

<ol style="list-style-type: upper-alpha;">
<li>'''Stability in Constant Voltage (CV) Mode'''</li></ol>

In the CV mode, the uncertainty <math display="inline">\sigma_{v}</math> remained relatively low (typically < 0.35 cm), indicating that voltage control provides a fundamentally stable rotational environment for this motor.

In the 1.40 V Anomaly: A localized peak in uncertainty (<math display="inline">\sigma_{v} = 1.7115cm/s</math>) was observed at 1.40 V. Since Magnet Group 1 utilized magnets of varying thickness, this instability likely indicates a mechanical resonance triggered at a specific angular velocity due to the rotor's mass imbalance.

<ol start="2" style="list-style-type: upper-alpha;">
<li>'''Severe Instability in Constant Current (CC) Mode'''</li></ol>

In the CC mode exhibited significantly higher uncertainty compared to CV mode, with <math display="inline">R^{2}</math> dropping to 0.9083.

The peak uncertainty reached 27.88 cm/s at 0.16 A. This is a result of the asymmetric magnetic field (one magnet being thicker than the others).

In CC mode, the power supply must continuously adjust the voltage to maintain a constant current against a varying load. The asymmetry creates a non-uniform torque requirement during each rotation, causing the supply to "hunt" for the correct voltage, which amplifies rotational jitter.

<ol start="3" style="list-style-type: upper-alpha;">
<li>'''Impact of Magnetic Asymmetry'''</li></ol>

Non-uniform Pulse Timing: Because one magnet had a different thickness, the magnetic field detected by the Hall sensor was non-uniform. This led to variations in the timing of the triggered pulses, directly increasing the StdDev.

Mechanical Imbalance: The mass distribution of the asymmetric magnets caused the rotor to vibrate, especially as speed increased. These vibrations caused the distance between the sensor and magnets to fluctuate slightly, further destabilizing the output signal.

<ol start="4" style="list-style-type: upper-alpha;">
<li>'''Conclusion'''</li></ol>

The analysis of Magnet Group 1 confirms that magnetic asymmetry is a primary driver of system instability. While Constant Voltage mode offers better resilience to these imbalances, the Constant Current mode exacerbates them through continuous electrical feedback. These findings justified the transition to the symmetric configuration (Group 2) as an experimental improvement.

== 4.3 Group II: Symmetric Magnet Configuration ==

To reduce the instabilities noted in Group 1, the improved experiment utilized a symmetric set of four magnets with identical diameter and thickness.

=== 4.3.1 Constant Voltage Mode ===

Input voltage was from 0.40V to 2.00V, and the results are shown in Table 4:

'''Table 4''' Stability metric and versus velocity under constant voltage mode

{| class="wikitable"
|-
! '''Voltage/V'''
! '''Current range/A'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.40
| 0.109-0.115
| 0.42658
| 2.29900
| 10.71569
|-
| 0.60
| 0.112-0.124
| 0.75189
| 1.17105
| 18.88748
|-
| 0.80
| 0.120-0.130
| 1.11022
| 3.74946
| 27.88873
|-
| 1.00
| 0.136-0.147
| 1.48762
| 4.81741
| 37.36901
|-
| 1.20
| 0.147-0.155
| 1.89560
| 2.80534
| 47.61747
|-
| 1.40
| 0.156-0.177
| 2.25674
| 9.29897
| 56.68931
|-
| 1.60
| 0.169-0.197
| 2.57919
| 22.52500
| 64.78925
|-
| 1.80
| 0.174-0.235
| 2.87329
| 37.24640
| 72.17704
|-
| 2.00
| 0.189-0.261
| 3.25731
| 1259.96000
| 81.82363
|}

A strong linear relationship was observed between the input voltage and tangential velocity (<math display="inline">v = 44.79V - 7.30</math>), as illustrated in Figure 10. Furthermore, the measurement error (StdDev) followed a linear correlation with the voltage (<math display="inline">StdDev\ = \ 21.79V\ - \ 13.48</math>), as shown in Figure 11, where the error exhibited a clear upward trend as the voltage increased.

[[File:fig10.jpg]

'''Figure 10.''' Tangential velocity vs. Input voltage

[[File:fig11.jpg]

'''Figure 11.''' Frequency standard deviation vs. Input voltage

In this mode, the velocity exhibited exceptional linearity (<math display="inline">R^{2} = 0.99883</math>). Although stability was superior at most voltage levels, a significant resonance-like fluctuation emerged at 2.00 V (<math display="inline">StdDev\ = \ 1259.96\ mHz</math>). This instability is attributed to the impact of high-speed operation on mechanical imbalances within the system.

=== 4.3.2 Constant Current Mode ===

Input current was from 0.13A to 0.21A, and the results are shown in Table 5:

'''Table 5''' Stability metric and versus velocity under constant current mode

{| class="wikitable"
|-
! '''Current/A'''
! '''Voltage range/V'''
! '''Frequncy/Hz'''
! '''StdDev/mHz'''
! '''Velocity/cm/s'''
|-
| 0.13
| 0.76-0.78
| 1.11181
| 24.26930
| 27.92867
|-
| 0.14
| 0.93-0.97
| 1.42701
| 27.95240
| 35.84649
|-
| 0.15
| 1.12-1.18
| 1.75377
| 136.14300
| 44.05470
|-
| 0.16
| 1.25-1.38
| 2.06477
| 29.71690
| 51.86702
|-
| 0.17
| 1.33-1.55
| 2.27231
| 102.44900
| 57.08043
|-
| 0.18
| 1.47-1.75
| 2.46409
| 121.15900
| 61.89794
|-
| 0.19
| 1.57-1.77
| 2.94815
| 956.48900
| 74.05753
|-
| 0.20
| 1.58-1.91
| 3.04197
| 793.15500
| 76.41429
|-
| 0.21
| 1.62-2.06
| 3.16084
| 544.79300
| 79.40030
|}

As shown in Figure 12, the velocity demonstrated a linear response to current, following the equation <math display="inline">v = 662.71\ I - 56.16</math>. The error, however, did not follow a linear correlation with current (Figure 13), and it displayed a broad increasing trend across the measured range.

[[File:fig12.jpg]

'''Figure 12.''' Tangential velocity vs. Input voltage

[[File:fig13.jpg]

'''Figure 13.''' Frequency standard deviation vs. Input voltage

In this mode, the current range remained identical to Group I (0.13–0.21 A). The implementation of symmetric conditions significantly enhanced the fitting quality, raising the <math display="inline">R^{2}</math> value from 0.90830 in Group 1 to 0.98309. Furthermore, the peak StdDev at 0.16 A plummeted from 1109.13 mHz to 29.72 mHz, underscoring the critical role of magnetic symmetry in maintaining the stable operation of constant-current systems.

=== 4.3.3 Velocity Uncertainty ===

The velocity uncertainty calculated from Tables 4 and 5 is shown in Table 6.

'''Table 6''' Asymmetric magnet speed uncertainty

{| class="wikitable"
|-
! '''CV mode(V)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
! '''CC mode(A)'''
! <math display="inline">\mathbf{\sigma}_{\mathbf{v}}</math>'''(cm/s)'''
|-
| 0.40
| 0.0578
| 0.13
| 0.6099
|-
| 0.60
| 0.0294
| 0.14
| 0.7025
|-
| 0.80
| 0.0942
| 0.15
| 3.4216
|-
| 1.00
| 0.1211
| 0.16
| 0.7469
|-
| 1.20
| 0.0705
| 0.17
| 2.5748
|-
| 1.40
| 0.2337
| 0.18
| 3.045
|-
| 1.60
| 0.5661
| 0.19
| 24.0392
|-
| 1.80
| 0.9361
| 0.2
| 19.9341
|-
| 2.00
| 31.6662
| 0.21
| 13.6921
|}

In the constant current mode, the velocity uncertainty at 0.16 A dropped significantly from 27.88 cm/s (Group 1) to 0.75 cm/s. Symmetry minimizes the torque ripple, allowing the power supply to maintain a stable equilibrium without aggressive voltage compensation.

For abnormal point <math display="inline">\sigma_{v} = 31.66</math> in CV mode suggests that high-speed operation magnifies even microscopic mechanical imbalances, triggering a resonance point in the apparatus.

== 4.4 Error Analysis ==

=== 4.4.1 Frequency Measurement Uncertainty ===

The rotational speed was calculated from the frequency measured by the Hall sensor on the oscilloscope. Therefore, any fluctuation in frequency directly caused uncertainty in the calculated speed. In this experiment, the StdDev shown on the oscilloscope represents the fluctuation of the measured frequency, so it can be used as an indicator of measurement uncertainty.

=== 4.4.2 Hall sensor Limitations ===

One possible source of error is the Hall sensor itself. The output signal depends on the distance and alignment between the sensor and the magnet. If the sensor position was not perfectly fixed(In our experiment, the distance was fixed at 3cm), or if the magnetic field was not uniform(In our experiment, we used different magnets groups as control parameters), the detected pulse timing could vary slightly.

=== 4.4.3 Mechanical Effects ===

Mechanical factors such as motor friction, air resistance, vibration, and rotor imbalance could also affect the results. These factors may cause the rotation speed to fluctuate during operation, especially at higher speeds. This can increase the scatter of the measured frequency and make the fitting less ideal.

=== 4.4.4 Why Constant Voltage Gives a Better Fit ===

The linear fitting under constant voltage was better than under constant current. This is likely because the motor speed is more directly related to the applied voltage, while the current is more affected by load, friction, and torque changes. Under constant-current operation, the power supply has to adjust the voltage continuously, which may introduce extra instability.

=== 4.4.5 StdDev Trend ===

The StdDev does not show a perfectly clear trend, but it generally becomes larger at higher voltage, current, or power. This suggests that the system becomes less stable at higher speed. Since the scatter is quite large, the relation between StdDev and electrical input is only approximately linear.

= 5 Conclusion =

In this experiment, a Hall sensor was used to measure the rotational speed of a motor-driven disk by detecting periodic magnetic signals. The results show that the tangential velocity exhibits a clear linear relationship with both input voltage and input current. However, the linear fitting under constant voltage mode is significantly better than that under constant current mode, indicating that voltage control provides a more stable operating condition for the system.

For the measurement uncertainty, represented by the standard deviation of the frequency, an overall increasing trend was observed as the input increased. Although the data points show noticeable scatter, the general trend can still be approximated as linear. Compared to other possible models, the linear relationship provides a more consistent description across different experimental conditions, suggesting that it is a reasonable approximation for the system behavior.

Furthermore, the comparison between asymmetric and symmetric magnet configurations shows that magnetic symmetry plays an important role in system stability. The symmetric configuration significantly reduced fluctuations and improved the consistency of the measured signals.

Overall, the experiment demonstrates that Hall sensors can be effectively used for non-contact rotational speed measurement, and that both electrical control mode and mechanical symmetry are key factors affecting measurement accuracy and stability.

'''Reference'''

[1] M. Crescentini, S. F. Syeda and G. P. Gibiino, "Hall-Effect Current Sensors: Principles of Operation and Implementation Techniques," in IEEE Sensors Journal, vol. 22, no. 11, pp. 10137-10151, 1 June 1, 2022, doi: 10.1109/JSEN.2021.3119766.

[2] Park, Su-Mi, and Hong-Je Ryoo. "Pulsed power modulator with active pull-down using diode reverse recovery time." IEEE Transactions on Power Electronics 35.3 (2019): 2943-2949.

[3] Honeywell, SS311PT/SS411P Bipolar Hall-Effect Digital Position Sensors with Built-in Pull-up Resistor'','' Product Sheet 005914-1-EN, October 2009.

File:Principle of the Hall effect.jpeg

2026-04-14T07:16:02Z

Qifang:

Main Page

2026-04-10T04:42:36Z

Qifang:

'''Welcome to the wiki page for the course PC5271: Physics of Sensors "(in AY25/26 Sem 2)!'''

This is the repository where projects are documented. You will need to create an account for editing/creating pages. If you need an account, please contact Christian.

'''Logistics''':
Our '''location is S11-02-04''', time slots for "classes" are '''Tue and Fri 10:00am-12:00noon'''.

'''Deadline for final entry to the wiki: 23 April 2026 23:59'''.

==Projects==

===[[Fluorescence Sensor for Carbon Quantum Dots: Synthesis, Characterization, and Quality Control]]===

Group menber: Zhang yiteng, Li Xiaoyue, Peng Jianxi

This project aims to develop a low-cost, repeatable optical sensing system to quantify the quality of Carbon Quantum Dots (CQDs). We synthesize CQDs using a microwave-assisted method with citric acid and urea, and characterize their fluorescence properties using a custom-built setup comprising a UV LED excitation source and a fiber-optic spectrometer. By analyzing spectral metrics such as peak wavelength, intensity, and FWHM, we establish a robust quality control protocol for nanomaterial production.

===[[Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point]]===
Team members: Yuan Siyu; Zhu Ziyang; Wang Peikun; Li Xunyu

We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

'''CK:''' We likely have all the parts for this, but let us know the frequency so we can find the proper amplifier and circuit board.

'''SY:''' Thanks for your confirmation. The operating frequency is around 70-80 kHz.

'''CK:''' Have!

===[[EA Spectroscopy as a series of sensors: Investigating the Impact of Solvent Type on Mobility in Organic Diodes]]===
Team members: Li Jinhan; Liu Chenyang

We will use EA spectroscopy, which will include optical sensors, electrical sensors, and lock-in amplifiers, among other components as a highly sensitive, non-destructive optical sensing platform to measure the internal electric field modulation response of organic diodes under operating conditions, and to quantitatively extract carrier mobility based on this measurement. By systematically controlling the thin film preparation temperature and comparing the EA response characteristics of different samples, the project aims to reveal the influence of film preparation temperature on device mobility and its potential physical origins.

===[[Optical Sensor of Magnetic Dynamics: A Balanced-Detection MOKE Magnetometer]]===
Team members: LI Junxiang; Patricia Breanne Tan Sy

We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

===[[Precision Measurement of Material and Optical Properties Using Interferometry]]===
Team members: Yang SangUk; Zhang ShunYang; Xu Zifang

We will be constructing an interferometer and use it as a tool for precision measurement. One primary objective is determination of the refractive index of solution of different salt concentration by analyzing the resulting shift interference fringes.

===[[Precision Thermocouple Based Temperature Measurement System]]===
Team members: Sree Ranjani Krishnan; Nisha Ganesh ; Burra Srikari

We will design, build, and validate a precision thermocouple-based temperature measurement system using the Seebeck effect. The system will convert the extremely small thermoelectric voltage generated by a thermocouple into accurate, real-time temperature data. Since the output voltage is really small we will be using an instrumentation amplifier to amplify the output voltage and use an Arduino to digitalize the results.
Materials needed: K-type thermocouple/Thermophile;Arduino

===[[Surface EMG Sensor for Muscle Activity Measurement: AFE Design and Signal Processing]]===
Team members: Liu Chenxi; Wang Jingyi; Zhong Baoqi; Hong Jialuo; Zhang Lishang;

Electromyography (EMG) measures the electrical activity generated by skeletal muscles and is widely used in biomedical sensing, rehabilitation, and human–machine interfaces. The electrical signals produced by muscle fibers are typically in the microvolt to millivolt range and are easily corrupted by noise and motion artifacts, making proper signal conditioning essential. In this project, we design and implement an analog front-end (AFE) for surface EMG acquisition, including an instrumentation amplifier, band-pass filtering, and a 50 Hz notch filter to suppress power-line interference. The conditioned signal is then observed and recorded using an oscilloscope for further analysis of muscle activity in both the time and frequency domains.

===[[Humidity Detector Based on Quartz Crystal Oscillator]]===

Group menber: Ma Xiangyi; Li Xukuan; Zhang Yixuan; Zhu Rongqi

This project aims to develop a humidity sensor based on a quartz crystal oscillator. The group will first construct the oscillator circuit on a breadboard. They will then fabricate the sensing device by depositing water-absorbing materials onto the quartz crystal. Humidity detection will be achieved by measuring the frequency change of the circuit caused by the mass variation on the crystal surface.

'''To Prof.''' Several materials are available for use as the water-absorbing layer, such as PVA, polyimide, graphene oxide, and silica gel. We are currently unsure which material would be the most suitable for our device. Could you provide us with some suggestions?

===[[Alcohol Sensor Based on Gas-Sensitive Resistive Materials]]===
Team members: Lyu Jiaxin; Yue Yucheng; Zhang ningxin; Zhong Yihui

This project aims to develop a low-cost alcohol sensing system based on gas-sensitive resistive materials. The presence of alcohol vapor induces changes in electrical resistance, which are measured and analyzed. The sensor is calibrated under different alcohol concentrations, and key parameters such as sensitivity and response time are evaluated to demonstrate reliable alcohol detection.

===[[Rotational Speed Measurement System Based on Hall-Effect Sensor]]===
Team members: Cui Yufan; Guo Chentong; He Qifang; Wang Xingyuan

This project aims to design, construct, and characterize a Hall-effect-based rotational speed measurement system. The system operates by monitoring the periodic magnetic field produced by magnets attached to a rotating disk. A Hall sensor converts this magnetic variation into an electrical signal, whose period is measured using an oscilloscope. From this measured period, the rotational speed can be calculated and used to assess the performance of the experimental setup.

==Resources==
===Books and links===
* A good textbook on the Physics of Sensors is Jacob Fraden: Handbook of Mondern Sensors, Springer, ISBN 978-3-319-19302-1 or [https://link.springer.com/book/10.1007/978-3-319-19303-8 doi:10.1007/978-3-319-19303-8]. There shoud be an e-book available through the NUS library at https://linc.nus.edu.sg/record=b3554643
* Another good textbook: John B.Bentley: Principles of Measurement Systems, 4th Edition, Pearson, ISBN: 0-13-043028-5 or https://linc.nus.edu.sg/record=b2458243 in our library.

===Software===
* Various Python extensions. [https://www.python.org Python] is a very powerful free programming language that runs on just about any computer platform. It is open source and completely free.
* [https://www.gnuplot.info Gnuplot]: A free and very mature data display tool that works on just about any platform used that produces excellent publication-grade eps and pdf figures. Can be also used in scripts. Open source and completely free.
* Matlab: Very common, good toolset also for formal mathematics, good graphics. Expensive. We may have a site license, but I am not sure how painful it is for us to get a license for this course. Ask if interested.
* Mathematica: More common among theroetical physicists, very good in formal maths, now with better numerics. Graphs are ok but can be a pain to make looking good. As with Matlab, we do have a campus license. Ask if interested.

===Apps===
Common mobile phones these days are equipped with an amazing toolchest of sensors. There are a few apps that allow you to access them directly, and turn your phone into a powerful sensor. Here some suggestions:

* Physics Toolbox sensor suite on [https://play.google.com/store/apps/details?id=com.chrystianvieyra.physicstoolboxsuite&hl=en_SG Google play store] or [https://apps.apple.com/us/app/physics-toolbox-sensor-suite/id1128914250 Apple App store].

===Data sheets===
A number of components might be useful for several groups. Some common data sheets are here:
* Photodiodes:
** Generic Silicon pin Photodiode type [[Media:Bpw34.pdf|BPW34]]
** Fast photodiodes (Silicon PIN, small area): [[Media:S5971_etc_kpin1025e.pdf|S5971/S5972/S5973]]
* Photogates:
** reflective, with mounting holes: [[Media:TTElectronics-OPB704WZ.pdf|OPB704.WZ]]
** transmissive, no mounting holes: [[Media:Vishay_TCST1103.pdf|TCST1103]]
* PT 100 Temperature sensors based on platinum wire: [[Media:PT100_TABLA_R_T.pdf|Calibration table]]
* Thermistor type [[Media:Thermistor B57861S.pdf|B57861S]] (R0=10kΩ, B=3988Kelvin). Search for [https://en.wikipedia.org/wiki/Steinhart-Hart_equation Steinhart-Hart equation]. See [[Thermistor]] page here as well.
* Humidity sensor
** Sensirion device the reference unit: [[media:Sensirion SHT30-DIS.pdf|SHT30/31]]
* Thermopile detectors:
** [[Media:Thermopile_G-TPCO-035 TS418-1N426.pdf|G-TPCO-035 / TS418-1N426]]: Thermopile detector with a built-in optical bandpass filter for light around 4μm wavelength for CO2 absorption
** [[Media:Thermopile_TS305_TCPO-033.pdf|TPCO-033 / TS305]]: Thermopile detector with wideband sensitivity 5um-25um
** [[Media:Thermopile_G-TPCO-019_TS105-10L5.5MM.pdf|G-TPCO-019 /TS105-10L5.5MM]]: Thermopile detector with wideband sensitivity 5um-25um and silicon lens (field of view: 10 degree)
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]

* Magnetic field sensors:
** Fluxgate magnetometer [[media:Data-sheet FLC-100.pdf|FCL100]]
** Hall switch 1 (SOT-23 casing): [[media:INFineon_TLE49681KXTSA1.pdf|TLE49681]]
** Hall switch 2 (TO-92 casing): [[media:DiodesInc_AH9246-P-8.pdf|AH9246]]
** Linear Hall sensor (to come)
* Lasers
** Red laser diode [[media:HL6501MG.pdf|HL6501MG]]
* Generic amplifiers
** Instrumentation amplifiers: [[media:Ad8221.pdf|AD8221]] or [[media:AD8226.pdf|AD8226]]
** Conventional operational amplifiers: Precision: [[media:OP27.pdf | OP27]], General purpose: [[media:OP07.pdf | OP07]]
** JFET op-amp, reasonably fast: [https://www.ti.com/document-viewer/tl071/datasheet TL071]
** Transimpedance amplifiers for photodetectors: [[media:AD8015.pdf | AD8015]]

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

==Old wikis==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2425S2 AY2024/25 Sem 2] and [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2].

Main Page

2026-04-10T04:35:10Z

Qifang:

'''Welcome to the wiki page for the course PC5271: Physics of Sensors "(in AY25/26 Sem 2)!'''

This is the repository where projects are documented. You will need to create an account for editing/creating pages. If you need an account, please contact Christian.

'''Logistics''':
Our '''location is S11-02-04''', time slots for "classes" are '''Tue and Fri 10:00am-12:00noon'''.

'''Deadline for final entry to the wiki: 23 April 2026 23:59'''.

==Projects==

===[[Fluorescence Sensor for Carbon Quantum Dots: Synthesis, Characterization, and Quality Control]]===

Group menber: Zhang yiteng, Li Xiaoyue, Peng Jianxi

This project aims to develop a low-cost, repeatable optical sensing system to quantify the quality of Carbon Quantum Dots (CQDs). We synthesize CQDs using a microwave-assisted method with citric acid and urea, and characterize their fluorescence properties using a custom-built setup comprising a UV LED excitation source and a fiber-optic spectrometer. By analyzing spectral metrics such as peak wavelength, intensity, and FWHM, we establish a robust quality control protocol for nanomaterial production.

===[[Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point]]===
Team members: Yuan Siyu; Zhu Ziyang; Wang Peikun; Li Xunyu

We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

'''CK:''' We likely have all the parts for this, but let us know the frequency so we can find the proper amplifier and circuit board.

'''SY:''' Thanks for your confirmation. The operating frequency is around 70-80 kHz.

'''CK:''' Have!

===[[EA Spectroscopy as a series of sensors: Investigating the Impact of Solvent Type on Mobility in Organic Diodes]]===
Team members: Li Jinhan; Liu Chenyang

We will use EA spectroscopy, which will include optical sensors, electrical sensors, and lock-in amplifiers, among other components as a highly sensitive, non-destructive optical sensing platform to measure the internal electric field modulation response of organic diodes under operating conditions, and to quantitatively extract carrier mobility based on this measurement. By systematically controlling the thin film preparation temperature and comparing the EA response characteristics of different samples, the project aims to reveal the influence of film preparation temperature on device mobility and its potential physical origins.

===[[Optical Sensor of Magnetic Dynamics: A Balanced-Detection MOKE Magnetometer]]===
Team members: LI Junxiang; Patricia Breanne Tan Sy

We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

===[[Precision Measurement of Material and Optical Properties Using Interferometry]]===
Team members: Yang SangUk; Zhang ShunYang; Xu Zifang

We will be constructing an interferometer and use it as a tool for precision measurement. One primary objective is determination of the refractive index of solution of different salt concentration by analyzing the resulting shift interference fringes.

===[[Precision Thermocouple Based Temperature Measurement System]]===
Team members: Sree Ranjani Krishnan; Nisha Ganesh ; Burra Srikari

We will design, build, and validate a precision thermocouple-based temperature measurement system using the Seebeck effect. The system will convert the extremely small thermoelectric voltage generated by a thermocouple into accurate, real-time temperature data. Since the output voltage is really small we will be using an instrumentation amplifier to amplify the output voltage and use an Arduino to digitalize the results.
Materials needed: K-type thermocouple/Thermophile;Arduino

===[[Surface EMG Sensor for Muscle Activity Measurement: AFE Design and Signal Processing]]===
Team members: Liu Chenxi; Wang Jingyi; Zhong Baoqi; Hong Jialuo; Zhang Lishang;

Electromyography (EMG) measures the electrical activity generated by skeletal muscles and is widely used in biomedical sensing, rehabilitation, and human–machine interfaces. The electrical signals produced by muscle fibers are typically in the microvolt to millivolt range and are easily corrupted by noise and motion artifacts, making proper signal conditioning essential. In this project, we design and implement an analog front-end (AFE) for surface EMG acquisition, including an instrumentation amplifier, band-pass filtering, and a 50 Hz notch filter to suppress power-line interference. The conditioned signal is then observed and recorded using an oscilloscope for further analysis of muscle activity in both the time and frequency domains.

===[[Humidity Detector Based on Quartz Crystal Oscillator]]===

Group menber: Ma Xiangyi; Li Xukuan; Zhang Yixuan; Zhu Rongqi

This project aims to develop a humidity sensor based on a quartz crystal oscillator. The group will first construct the oscillator circuit on a breadboard. They will then fabricate the sensing device by depositing water-absorbing materials onto the quartz crystal. Humidity detection will be achieved by measuring the frequency change of the circuit caused by the mass variation on the crystal surface.

'''To Prof.''' Several materials are available for use as the water-absorbing layer, such as PVA, polyimide, graphene oxide, and silica gel. We are currently unsure which material would be the most suitable for our device. Could you provide us with some suggestions?

===[[Alcohol Sensor Based on Gas-Sensitive Resistive Materials]]===
Team members: Lyu Jiaxin; Yue Yucheng; Zhang ningxin; Zhong Yihui

This project aims to develop a low-cost alcohol sensing system based on gas-sensitive resistive materials. The presence of alcohol vapor induces changes in electrical resistance, which are measured and analyzed. The sensor is calibrated under different alcohol concentrations, and key parameters such as sensitivity and response time are evaluated to demonstrate reliable alcohol detection.

===[[Design and Implementation of a Hall Sensor-Based Rotational Speed Measurement System]]===
Team members: Cui Yufan; Guo Chentong; He Qifang; Wang Xingyuan

This project aims to design, construct, and characterize a Hall-effect-based rotational speed measurement system. The system operates by monitoring the periodic magnetic field produced by magnets attached to a rotating disk. A Hall sensor converts this magnetic variation into an electrical signal, whose period is measured using an oscilloscope. From this measured period, the rotational speed can be calculated and used to assess the performance of the experimental setup.

==Resources==
===Books and links===
* A good textbook on the Physics of Sensors is Jacob Fraden: Handbook of Mondern Sensors, Springer, ISBN 978-3-319-19302-1 or [https://link.springer.com/book/10.1007/978-3-319-19303-8 doi:10.1007/978-3-319-19303-8]. There shoud be an e-book available through the NUS library at https://linc.nus.edu.sg/record=b3554643
* Another good textbook: John B.Bentley: Principles of Measurement Systems, 4th Edition, Pearson, ISBN: 0-13-043028-5 or https://linc.nus.edu.sg/record=b2458243 in our library.

===Software===
* Various Python extensions. [https://www.python.org Python] is a very powerful free programming language that runs on just about any computer platform. It is open source and completely free.
* [https://www.gnuplot.info Gnuplot]: A free and very mature data display tool that works on just about any platform used that produces excellent publication-grade eps and pdf figures. Can be also used in scripts. Open source and completely free.
* Matlab: Very common, good toolset also for formal mathematics, good graphics. Expensive. We may have a site license, but I am not sure how painful it is for us to get a license for this course. Ask if interested.
* Mathematica: More common among theroetical physicists, very good in formal maths, now with better numerics. Graphs are ok but can be a pain to make looking good. As with Matlab, we do have a campus license. Ask if interested.

===Apps===
Common mobile phones these days are equipped with an amazing toolchest of sensors. There are a few apps that allow you to access them directly, and turn your phone into a powerful sensor. Here some suggestions:

* Physics Toolbox sensor suite on [https://play.google.com/store/apps/details?id=com.chrystianvieyra.physicstoolboxsuite&hl=en_SG Google play store] or [https://apps.apple.com/us/app/physics-toolbox-sensor-suite/id1128914250 Apple App store].

===Data sheets===
A number of components might be useful for several groups. Some common data sheets are here:
* Photodiodes:
** Generic Silicon pin Photodiode type [[Media:Bpw34.pdf|BPW34]]
** Fast photodiodes (Silicon PIN, small area): [[Media:S5971_etc_kpin1025e.pdf|S5971/S5972/S5973]]
* Photogates:
** reflective, with mounting holes: [[Media:TTElectronics-OPB704WZ.pdf|OPB704.WZ]]
** transmissive, no mounting holes: [[Media:Vishay_TCST1103.pdf|TCST1103]]
* PT 100 Temperature sensors based on platinum wire: [[Media:PT100_TABLA_R_T.pdf|Calibration table]]
* Thermistor type [[Media:Thermistor B57861S.pdf|B57861S]] (R0=10kΩ, B=3988Kelvin). Search for [https://en.wikipedia.org/wiki/Steinhart-Hart_equation Steinhart-Hart equation]. See [[Thermistor]] page here as well.
* Humidity sensor
** Sensirion device the reference unit: [[media:Sensirion SHT30-DIS.pdf|SHT30/31]]
* Thermopile detectors:
** [[Media:Thermopile_G-TPCO-035 TS418-1N426.pdf|G-TPCO-035 / TS418-1N426]]: Thermopile detector with a built-in optical bandpass filter for light around 4μm wavelength for CO2 absorption
** [[Media:Thermopile_TS305_TCPO-033.pdf|TPCO-033 / TS305]]: Thermopile detector with wideband sensitivity 5um-25um
** [[Media:Thermopile_G-TPCO-019_TS105-10L5.5MM.pdf|G-TPCO-019 /TS105-10L5.5MM]]: Thermopile detector with wideband sensitivity 5um-25um and silicon lens (field of view: 10 degree)
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]

* Magnetic field sensors:
** Fluxgate magnetometer [[media:Data-sheet FLC-100.pdf|FCL100]]
** Hall switch 1 (SOT-23 casing): [[media:INFineon_TLE49681KXTSA1.pdf|TLE49681]]
** Hall switch 2 (TO-92 casing): [[media:DiodesInc_AH9246-P-8.pdf|AH9246]]
** Linear Hall sensor (to come)
* Lasers
** Red laser diode [[media:HL6501MG.pdf|HL6501MG]]
* Generic amplifiers
** Instrumentation amplifiers: [[media:Ad8221.pdf|AD8221]] or [[media:AD8226.pdf|AD8226]]
** Conventional operational amplifiers: Precision: [[media:OP27.pdf | OP27]], General purpose: [[media:OP07.pdf | OP07]]
** JFET op-amp, reasonably fast: [https://www.ti.com/document-viewer/tl071/datasheet TL071]
** Transimpedance amplifiers for photodetectors: [[media:AD8015.pdf | AD8015]]

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

==Old wikis==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2425S2 AY2024/25 Sem 2] and [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2].