PC5271 wiki - User contributions [en]

Magnetic field sensing using a fluxgate magnetometer

2025-04-29T03:15:13Z

Xueqi: /* Modeling and measurement of a permanent magnet */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday's Law of Induction and Lenz's Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.9 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by [7]:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer's functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor's performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 150 pT/√Hz, which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Kominis, I. K., Kornack, T. W., Allred, J. C., & Romalis, M. V. (2003). A subfemtotesla multichannel atomic magnetometer. Nature, 422(6932), 596-599.

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T06:17:12Z

Xueqi: /* How does a fluxgate magnetometer work? */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday's Law of Induction and Lenz's Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by [7]:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer's functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor's performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 150 pT/√Hz, which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Kominis, I. K., Kornack, T. W., Allred, J. C., & Romalis, M. V. (2003). A subfemtotesla multichannel atomic magnetometer. Nature, 422(6932), 596-599.

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T06:16:29Z

Xueqi: /* Possible Improvements to Measurement Accuracy */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by [7]:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer's functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor's performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 150 pT/√Hz, which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Kominis, I. K., Kornack, T. W., Allred, J. C., & Romalis, M. V. (2003). A subfemtotesla multichannel atomic magnetometer. Nature, 422(6932), 596-599.

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Main Page

2025-04-28T06:15:42Z

Xueqi: /* Magnetic field sensing using a fluxgate magnetometer */

'''Welcome to the wiki page for the course PC5271: Physics of Sensors!'''

This is the repository where projects are documented. Creation of new accounts have now been blocked,and editing/creating pages is enabled. If you need an account, please contact Christian.

'''Deadline for report'''

Deadline for finishing your report on this wiki will be '''29 April 2025 23:59:59''' ;) Please be ensured you are happy with your project page at this point, as this will be the basis for our assessment.
Thanks!! Ramanathan Mahendrian and Christian Kurtsiefer

==Projects==
===[[Project 1 (Example)]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join. Once the project has stabilized, keep stuff in the project page linked by the headline.

===[[Laser Gyroscope]]===
Team members: Darren Koh, Chiew Wen Xin

Build a laser interferometer to detect rotation.

===[[Laser Distance Measurer]]===
Team members: Arya Chowdhury, Liu Sijin, Jonathan Wong

This project aims to build a laser interferometer to measure distances.

(CK: We should have fast laser diodes and fast photodiodes, mounted in optics bench kits)

===[[Non-contact Alcohol Concentration Measurement Device At NIR Spectrum]]===
Team members: Lim Gin Joe,Sun Weijia, Yan Chengrui, Zhu Junyi
This project aims to build a sensor to measure the concentration of alcohol by optical method
(CK: you can check Optics Letters 47, 5076-5079 (2022) https://doi.org/10.1364/OL.472890 for some info)

===[[Ultrasonic Acoustic Remote Sensing]]===
Team member(s): Chua Rui Ming

How well can we use sound waves to survey the environment?

(CK: we have some ultrasonic transducers around 40kHz, see datasheets below)

===[[Blood Oxygen Sensor]]===
Team members: He Lingzi, Zhao Lubo, Zhang Ruoxi, Xu Yintong

This project aims to build a sensor to detect the oxygen concentration in the blood.

(CK: We have LEDs at 940nm and 660nm peak wavelenth emission, plus some Si photodiodes)

===[[Terahertz Electromagnetic Wave Detection]]===
Team members: Shizhuo Luo, Bohan Zhang

This project aims to detect Terahertz waves, especially terahertz pulses (This is because they are intense and controllable). We may try different ways like electro-optical sampling and thermopile detectors.

===[[Optical measurement of atmospheric carbon dioxide]]===
Team member(s): Ta Na, Cao Yuan, Qi Kaiyi, Gao Yihan, Chen Yiming

This project aims to make use of the optical properties of carbon dioxide gas to create a portable and accurate measurement device of carbon dioxide.

===[[Photodetector with wavelength @ 780nm and 1560nm]]===
Team members: Sunke Lan

To design photodetector as power monitor with power within 10mW.

(CK: Standard problem, we have already the respective photodiodes)

===[[LED based avalanched photodetector]]===

Team members: Cai Shijie, Nie Huanxin, Yang Runzhi

1.Build a single photon detector using LED. The possible LED is gallium compounds based, emitting wavelength around 800nm(red light).

2. Other possible detector: photomultiplier or avalanche photon detector(do we have that?).

3.Do single double-slit interference experiment.

Other devices needed: use LED as single photon source (wavelength shorter than the emitting wavelength 800nm)

prove the detection is single photon: need optical fibre, counting module

===[[Motor Rotation Speed Measurement via the Hall Effect Sensor]]===

Team members: Mi Tianshuo

This project implements a Hall effect sensor to measure the rotation speed of a circuit board-controlled rotary plate.

===[[STM32-Based IMU Attitude Estimation]]===

Team members: Li Ding, Fan Xuting

This project utilizes an STM32 microcontroller and an MPU6050 IMU sensor to measure angular velocity and acceleration, enabling real-time attitude angle computation for motion tracking.

===[[Magnetic field sensing using a fluxgate magnetometer]]===
Team members: Ni Xueqi

This project investigates magnetic field sensing using a fluxgate magnetometer (FLC100). A 5 V supply drives the sensor, and the output is monitored with an oscilloscope. A permanent magnet is modeled in COMSOL and experimentally measured, showing expected distance-dependent field decay. A toroidal solenoid is also studied; due to imperfections, a magnetic field decaying as <math> 1/r^3 </math> outside the toroid is observed. Measurements confirm dipole-like behavior and linear current dependence, demonstrating the fluxgate magnetometer's sensitivity and validating magnetic field modeling.

===[[CO2 Concentration Detector]]===
Team members: Xie Zihan，Zhao Yun，Zhang Wenbo

Infrared absorption-based CO₂ gas sensors are developed based on the principle that different substances exhibit different absorption spectra. Because the chemical structures of different gas molecules vary, their degrees of absorption of infrared radiation at various wavelengths also differ. Consequently, when infrared radiation of different wavelengths is directed at the sample in turn, certain wavelengths are selectively absorbed and thus weakened by the sample, generating an infrared absorption spectrum.

Once the infrared absorption spectrum of a particular substance is known, its infrared absorption peaks can be identified. For the same substance, when the concentration changes, the absorption intensity at a given absorption peak also changes, and this intensity is directly proportional to the concentration. Therefore, by detecting how the gas alters the wavelength and intensity of the light, one can determine the gas concentration.

===[[Light Sensing System Based on the Photoelectric Effect]]===
Team members: Xu Ruizhe, Wei Heyi, Li Zerui, Ma Shunyu

This project utilizes the principle of the photoelectric effect to design a smart light sensing system. The system can detect ambient light intensity and process the data using Arduino or Raspberry Pi. When the light intensity changes beyond a predefined threshold, the system can trigger responses such as lighting up an LED, activating a buzzer, or automatically adjusting curtains.

===[[Temperature and humidity sensors]]===
Team members: Chen Andi, Chen Miaoge, Chen Yingnan, Fang Ye

This project aims to design and evaluate a real-time temperature and humidity monitoring system based on Arduino and the DHT11 sensor. The system is low-cost, easy to implement, and suitable for applications such as smart homes, agriculture, and storage environments. In addition to system development, the project compares the performance of the DHT11 and SHT31 sensors in various environments—indoor, outdoor, and rainy conditions—to assess their accuracy, stability, and response time. The results help guide practical sensor selection, especially in scenarios where cost and simplicity are prioritized over high precision.

===[[Ultrasonic Doppler Speedometer]]===
Team members: Yang Yuzhen, Liu Xueyi, Shao Shuai

Design and build an ultrasonic Doppler speedometer to measure the velocity of a moving object.

==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]]
* 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
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]
* Ultrasonic detectors:
** plastic detctor, 40 kHz, -74dB: [[Media:MCUSD16P40B12RO.pdf|MCUSD16P40B12RO]]
** metal casing/waterproof, 48 kHz, -90dB, [[Media:MCUSD14A48S09RS-30C.pdf|MCUSD14A48S09RS-30C]]
** metal casing, 40 kHz, sensitivity unknown, [[Media:MCUST16A40S12RO.pdf|MCUST16A40S12RO]]
** metal casing/waterproof, 300kHz, may need high voltage: [[Media:MCUSD13A300B09RS.pdf|MCUSD13A300B09RS]]
* Magnetic field sensor
** Fluxgate magnetometer [[media:Data-sheet FLC-100.pdf|FCL100]]
* 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]]
** Transimpedance amplifiers for photodetectors: [[media:AD8015.pdf|AD8015]]

===Some code snippets===
* For the [[media:Generic FPGA board version 3 - Quantum Optics Wiki.pdf|pattern generator]], you need to send the following text file to it to generate ultrasonic pulses:

# This pattern is to generate a burst of 10..20 oscillations at 40 kHz
# every 100ms for a sonar test. Pulses are TTL level on the AUX output,
# I/O lane 0 bit 7 is a sync pulse (10ns long), I/O lane 0 bit 0 copies the
# aux line, bit 1 indicates the pause periode between bursts.
# Internal counter 0 is for burst counting, int counter 1 for pause cycles

# Set device to programming mode: reset table, reset RAM, program params
config 13
writew 0, 60571; # basic address is 0, input thres -0.5V (not used)
writew 0,0,0,0; # external counter preload (not used)
writew 9,999,0,0; # internal cnt preload only first one is relevant
# and determines the number of pulses (minus 1) and
# number minus 1 of multiples of 100us for pause
writew 0,0,0,0,0,0,0,0; # DAC preload - not used

config 4; # switch to RAM write

# This is the RAM sequence- starting with 40kHz burst
writew 0x80,0,0,0,0,0, 0,0x1010; # ad 0: sync pulse 10nsec, load cnt 0
writew 0x01 0,0,1,0,0,1248,0xc004; # ad 1: pulse on (12.49us), if done go ad4
writew 0x01 0,0,1,0,0, 0,0x1100; # ad 2: pulse on for 10ns, decr int cnt 0
writew 0x00,0,0,0,0,0,1249,0x0001; # ad 3: pulse off for 12.5us, then go 1

# Waiting time / pause
writew 0x02,0,0,0,0,0, 0,0x1020; # ad 4: preload internal cntr 1 (10ns)
writew 0x02,0,0,0,0,0,9998,0x1200; # ad 5: decr cnt1 (10ns)
writew 0x02,0,0,0,0,0, 0,0xd008; # ad 6: if count is down goto ad 8(10ns)
writew 0x02,0,0,0,0,0, 0,0x0005; # ad 7: goto ad 5(10ns)

writew 0x02,0,0,0,0,0, 0,0x0000; # ad 8: restart (goto ad 0; 10ns)

# start pattern and keep output level on AUX line to TTL level
config 0x400;

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

== Old Wiki ==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2]

Main Page

2025-04-28T06:15:21Z

Xueqi: /* Magnetic field sensing using a fluxgate magnetometer */

'''Welcome to the wiki page for the course PC5271: Physics of Sensors!'''

This is the repository where projects are documented. Creation of new accounts have now been blocked,and editing/creating pages is enabled. If you need an account, please contact Christian.

'''Deadline for report'''

Deadline for finishing your report on this wiki will be '''29 April 2025 23:59:59''' ;) Please be ensured you are happy with your project page at this point, as this will be the basis for our assessment.
Thanks!! Ramanathan Mahendrian and Christian Kurtsiefer

==Projects==
===[[Project 1 (Example)]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join. Once the project has stabilized, keep stuff in the project page linked by the headline.

===[[Laser Gyroscope]]===
Team members: Darren Koh, Chiew Wen Xin

Build a laser interferometer to detect rotation.

===[[Laser Distance Measurer]]===
Team members: Arya Chowdhury, Liu Sijin, Jonathan Wong

This project aims to build a laser interferometer to measure distances.

(CK: We should have fast laser diodes and fast photodiodes, mounted in optics bench kits)

===[[Non-contact Alcohol Concentration Measurement Device At NIR Spectrum]]===
Team members: Lim Gin Joe,Sun Weijia, Yan Chengrui, Zhu Junyi
This project aims to build a sensor to measure the concentration of alcohol by optical method
(CK: you can check Optics Letters 47, 5076-5079 (2022) https://doi.org/10.1364/OL.472890 for some info)

===[[Ultrasonic Acoustic Remote Sensing]]===
Team member(s): Chua Rui Ming

How well can we use sound waves to survey the environment?

(CK: we have some ultrasonic transducers around 40kHz, see datasheets below)

===[[Blood Oxygen Sensor]]===
Team members: He Lingzi, Zhao Lubo, Zhang Ruoxi, Xu Yintong

This project aims to build a sensor to detect the oxygen concentration in the blood.

(CK: We have LEDs at 940nm and 660nm peak wavelenth emission, plus some Si photodiodes)

===[[Terahertz Electromagnetic Wave Detection]]===
Team members: Shizhuo Luo, Bohan Zhang

This project aims to detect Terahertz waves, especially terahertz pulses (This is because they are intense and controllable). We may try different ways like electro-optical sampling and thermopile detectors.

===[[Optical measurement of atmospheric carbon dioxide]]===
Team member(s): Ta Na, Cao Yuan, Qi Kaiyi, Gao Yihan, Chen Yiming

This project aims to make use of the optical properties of carbon dioxide gas to create a portable and accurate measurement device of carbon dioxide.

===[[Photodetector with wavelength @ 780nm and 1560nm]]===
Team members: Sunke Lan

To design photodetector as power monitor with power within 10mW.

(CK: Standard problem, we have already the respective photodiodes)

===[[LED based avalanched photodetector]]===

Team members: Cai Shijie, Nie Huanxin, Yang Runzhi

1.Build a single photon detector using LED. The possible LED is gallium compounds based, emitting wavelength around 800nm(red light).

2. Other possible detector: photomultiplier or avalanche photon detector(do we have that?).

3.Do single double-slit interference experiment.

Other devices needed: use LED as single photon source (wavelength shorter than the emitting wavelength 800nm)

prove the detection is single photon: need optical fibre, counting module

===[[Motor Rotation Speed Measurement via the Hall Effect Sensor]]===

Team members: Mi Tianshuo

This project implements a Hall effect sensor to measure the rotation speed of a circuit board-controlled rotary plate.

===[[STM32-Based IMU Attitude Estimation]]===

Team members: Li Ding, Fan Xuting

This project utilizes an STM32 microcontroller and an MPU6050 IMU sensor to measure angular velocity and acceleration, enabling real-time attitude angle computation for motion tracking.

===[[Magnetic field sensing using a fluxgate magnetometer]]===
Team members: Ni Xueqi

This project investigates magnetic field sensing using a fluxgate magnetometer (FLC100).

A 5 V supply drives the sensor, and the output is monitored with an oscilloscope.

A permanent magnet is modeled in COMSOL and experimentally measured, showing expected distance-dependent field decay.

A toroidal solenoid is also studied; due to imperfections, a magnetic field decaying as <math> 1/r^3 </math> outside the toroid is observed. Measurements confirm dipole-like behavior and linear current dependence, demonstrating the fluxgate magnetometer's sensitivity and validating magnetic field modeling.

===[[CO2 Concentration Detector]]===
Team members: Xie Zihan，Zhao Yun，Zhang Wenbo

Infrared absorption-based CO₂ gas sensors are developed based on the principle that different substances exhibit different absorption spectra. Because the chemical structures of different gas molecules vary, their degrees of absorption of infrared radiation at various wavelengths also differ. Consequently, when infrared radiation of different wavelengths is directed at the sample in turn, certain wavelengths are selectively absorbed and thus weakened by the sample, generating an infrared absorption spectrum.

Once the infrared absorption spectrum of a particular substance is known, its infrared absorption peaks can be identified. For the same substance, when the concentration changes, the absorption intensity at a given absorption peak also changes, and this intensity is directly proportional to the concentration. Therefore, by detecting how the gas alters the wavelength and intensity of the light, one can determine the gas concentration.

===[[Light Sensing System Based on the Photoelectric Effect]]===
Team members: Xu Ruizhe, Wei Heyi, Li Zerui, Ma Shunyu

This project utilizes the principle of the photoelectric effect to design a smart light sensing system. The system can detect ambient light intensity and process the data using Arduino or Raspberry Pi. When the light intensity changes beyond a predefined threshold, the system can trigger responses such as lighting up an LED, activating a buzzer, or automatically adjusting curtains.

===[[Temperature and humidity sensors]]===
Team members: Chen Andi, Chen Miaoge, Chen Yingnan, Fang Ye

This project aims to design and evaluate a real-time temperature and humidity monitoring system based on Arduino and the DHT11 sensor. The system is low-cost, easy to implement, and suitable for applications such as smart homes, agriculture, and storage environments. In addition to system development, the project compares the performance of the DHT11 and SHT31 sensors in various environments—indoor, outdoor, and rainy conditions—to assess their accuracy, stability, and response time. The results help guide practical sensor selection, especially in scenarios where cost and simplicity are prioritized over high precision.

===[[Ultrasonic Doppler Speedometer]]===
Team members: Yang Yuzhen, Liu Xueyi, Shao Shuai

Design and build an ultrasonic Doppler speedometer to measure the velocity of a moving object.

==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]]
* 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
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]
* Ultrasonic detectors:
** plastic detctor, 40 kHz, -74dB: [[Media:MCUSD16P40B12RO.pdf|MCUSD16P40B12RO]]
** metal casing/waterproof, 48 kHz, -90dB, [[Media:MCUSD14A48S09RS-30C.pdf|MCUSD14A48S09RS-30C]]
** metal casing, 40 kHz, sensitivity unknown, [[Media:MCUST16A40S12RO.pdf|MCUST16A40S12RO]]
** metal casing/waterproof, 300kHz, may need high voltage: [[Media:MCUSD13A300B09RS.pdf|MCUSD13A300B09RS]]
* Magnetic field sensor
** Fluxgate magnetometer [[media:Data-sheet FLC-100.pdf|FCL100]]
* 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]]
** Transimpedance amplifiers for photodetectors: [[media:AD8015.pdf|AD8015]]

===Some code snippets===
* For the [[media:Generic FPGA board version 3 - Quantum Optics Wiki.pdf|pattern generator]], you need to send the following text file to it to generate ultrasonic pulses:

# This pattern is to generate a burst of 10..20 oscillations at 40 kHz
# every 100ms for a sonar test. Pulses are TTL level on the AUX output,
# I/O lane 0 bit 7 is a sync pulse (10ns long), I/O lane 0 bit 0 copies the
# aux line, bit 1 indicates the pause periode between bursts.
# Internal counter 0 is for burst counting, int counter 1 for pause cycles

# Set device to programming mode: reset table, reset RAM, program params
config 13
writew 0, 60571; # basic address is 0, input thres -0.5V (not used)
writew 0,0,0,0; # external counter preload (not used)
writew 9,999,0,0; # internal cnt preload only first one is relevant
# and determines the number of pulses (minus 1) and
# number minus 1 of multiples of 100us for pause
writew 0,0,0,0,0,0,0,0; # DAC preload - not used

config 4; # switch to RAM write

# This is the RAM sequence- starting with 40kHz burst
writew 0x80,0,0,0,0,0, 0,0x1010; # ad 0: sync pulse 10nsec, load cnt 0
writew 0x01 0,0,1,0,0,1248,0xc004; # ad 1: pulse on (12.49us), if done go ad4
writew 0x01 0,0,1,0,0, 0,0x1100; # ad 2: pulse on for 10ns, decr int cnt 0
writew 0x00,0,0,0,0,0,1249,0x0001; # ad 3: pulse off for 12.5us, then go 1

# Waiting time / pause
writew 0x02,0,0,0,0,0, 0,0x1020; # ad 4: preload internal cntr 1 (10ns)
writew 0x02,0,0,0,0,0,9998,0x1200; # ad 5: decr cnt1 (10ns)
writew 0x02,0,0,0,0,0, 0,0xd008; # ad 6: if count is down goto ad 8(10ns)
writew 0x02,0,0,0,0,0, 0,0x0005; # ad 7: goto ad 5(10ns)

writew 0x02,0,0,0,0,0, 0,0x0000; # ad 8: restart (goto ad 0; 10ns)

# start pattern and keep output level on AUX line to TTL level
config 0x400;

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

== Old Wiki ==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2]

Main Page

2025-04-28T06:14:35Z

Xueqi: /* Magnetic field sensing using a fluxgate magnetometer */

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T06:07:42Z

Xueqi: /* Analysis of noise */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by [7]:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 150 pT/√Hz, which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Kominis, I. K., Kornack, T. W., Allred, J. C., & Romalis, M. V. (2003). A subfemtotesla multichannel atomic magnetometer. Nature, 422(6932), 596-599.

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:55:58Z

Xueqi: /* References */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by [7]:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Kominis, I. K., Kornack, T. W., Allred, J. C., & Romalis, M. V. (2003). A subfemtotesla multichannel atomic magnetometer. Nature, 422(6932), 596-599.

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:53:59Z

Xueqi: /* Modeling and measurement of a toroidal solenoid */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by [7]:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3]

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:53:17Z

Xueqi: /* References */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3]

[4] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[5] Caruso, M. J. (1997). Applications of magnetoresistive sensors in navigation systems (No. 970602). SAE Technical Paper.

[6] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[7] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:52:36Z

Xueqi: /* Applications */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants [4].
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable [5].
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart [6].
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[4] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[5] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:51:08Z

Xueqi: /* Different types of magnetometers */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium [2]. These sensors can achieve extreme sensitivity while operating at room temperature. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[4] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[5] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:50:38Z

Xueqi: /* Different types of magnetometers */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability [1]. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature [2]. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz [3]. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[4] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[5] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:49:33Z

Xueqi: /* References */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature [1]. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
[1] Primdahl, F. (1979). The fluxgate magnetometer. Journal of Physics E: Scientific Instruments, 12(4), 241.

[2] Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007).

[3] Bass, S. D., & Doser, M. (2024). Quantum sensing for particle physics. Nature Reviews Physics, 6(5), 329-339.

[4] Aslam, N., Zhou, H., Urbach, E. K., Turner, M. J., Walsworth, R. L., Lukin, M. D., & Park, H. (2023). Quantum sensors for biomedical applications. Nature Reviews Physics, 5(3), 157-169.

[5] Griffiths, D. J. (2023). Introduction to electrodynamics. Cambridge University Press.

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:44:16Z

Xueqi: /* Different types of magnetometers */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature [1]. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, which are useful in automotive and robotics applications.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-28T05:39:59Z

Xueqi: /* Different types of magnetometers */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>E. Miller, ''The Sun'', (New York: Academic Press, 2005), 23–25.</ref> . The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:41:46Z

Xueqi: /* Analysis of noise */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Environmental Magnetic Noise

The experimental setup exhibited small variations in the measurements, which can be attributed to environmental magnetic noise. Magnetic fields from nearby electronic devices, power lines, or even the Earth's geomagnetic field could interfere with the sensor’s readings. A potential solution to minimize these effects is to conduct the experiments in a magnetically shielded environment. Additionally, enhancing the sensor's ability to distinguish between external noise and the magnetic signal of interest would improve measurement accuracy. This could be achieved through software-based filtering techniques or hardware modifications to the sensor design, which would help isolate the desired signal more effectively.

2. Intrinsic Sensor Noise

Intrinsic noise from the fluxgate sensor, such as thermal noise or electronic noise in the circuitry, may also contribute to measurement uncertainty. The FLC 100 sensor has a specified noise level of 0.5 nT RMS (0.1–10 Hz), which is significantly lower than the measured magnetic field and therefore does not serve as the primary source of noise. Nevertheless, operating the sensor in a low-temperature environment could help suppress thermal noise from the electronics, potentially further reducing the overall noise contribution.

3. Vibrations of the Experimental Setup

Vibrations in the experimental setup, whether originating from the laboratory environment or the sensor itself, could lead to fluctuations in the measured signal. These vibrations might alter the position or orientation of the magnet or sensor, introducing noise in the output signal. To mitigate this, vibration isolation techniques, such as placing the sensor on an optical table with active vibration damping, could be employed. Furthermore, improving the mechanical stability of the mounting system would ensure that the sensor remains stationary during measurements, minimizing noise caused by positional shifts.

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:36:09Z

Xueqi: /* Possible Improvements to Measurement Accuracy */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup

As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources

While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise

Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Due to environmental magnetic nose

2. Intrinsic sensor noise

3. Vibration of the setup

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:35:56Z

Xueqi: /* Possible Improvements to Measurement Accuracy */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. Improved Alignment of Experimental Setup
As observed in the experiment, slight misalignments between the magnet and the fluxgate sensor could contribute to discrepancies between the experimental measurements and the simulation results, particularly when measuring the magnetic field distribution around the disk magnet. To improve measurement accuracy, better alignment methods, such as automated positioning systems could be utilized.

2. Use of Alternative Controllable Magnetic Sources
While a permanent magnet and a toroidal solenoid were used to demonstrate the fluxgate magnetometer’s functionality, exploring other controllable magnetic sources could offer further advantages in calibration and measurement accuracy. For example, Helmholtz coils could generate highly uniform and adjustable magnetic fields, enabling more precise control over the field strengths.

3. Suppression of Magnetic Noise
Environmental magnetic noise and sensor-induced noise are significant contributors to the variability in the measurements. To reduce the effects of magnetic noise, incorporating magnetic shielding could provide a more stable magnetic environment. In addition, implementing active noise cancellation techniques or low-pass filters could help suppress high-frequency noise that may affect the sensor’s performance. These improvements would help achieve more reliable data by minimizing unwanted interference from external magnetic fields.

=== Analysis of noise ===

1. Due to environmental magnetic nose

2. Intrinsic sensor noise

3. Vibration of the setup

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:31:29Z

Xueqi: /* Discussions */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

=== Possible Improvements to Measurement Accuracy ===

1. With better alignment of

2. Use other controllable magnetic sources

3. suppress of magnetic noise

=== Analysis of noise ===

1. Due to environmental magnetic nose

2. Intrinsic sensor noise

3. Vibration of the setup

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:20:13Z

Xueqi: /* Modeling and measurement of a toroidal solenoid */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

In conclusion, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:17:41Z

Xueqi: /* Modeling and measurement of a permanent magnet */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a toroidal solenoid ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

Thus, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:14:53Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in this experiment has an inner radius of approximately <math> a = 0.3\text{cm} </math>, an outer radius of approximately <math> b = 0.7\text{cm} </math>, and a total number of turns <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, it ideally vanishes.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) Illustration of a toroidal solenoid. (b) Simulated magnetic field distribution showing field confinement inside the toroid.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Although the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite external field is observed. This residual magnetic field arises primarily from the finite size of the coil, imperfections in the winding (such as turns that are not perfectly azimuthal or uniformly spaced), and edge effects at the inner and outer boundaries of the toroid.

At distances much larger than the toroid's dimensions, the external magnetic field behaves similarly to that of a magnetic dipole and decays approximately as <math> 1/r^3 </math>. In our measurements, by varying both the current amplitude and the distance from the toroid, the magnetic field was found to follow this <math> 1/r^3 </math> trend, as shown in Figure 7.

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid as a function of distance for different currents. A residual external field, decaying approximately as <math> 1/r^3 </math>, is observed due to non-ideal winding and finite coil size.]]

To quantitatively describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant that depends on the current, and <math> r_0 </math> accounts for geometric offsets such as probe size or alignment errors. The fitting results show good agreement with the expected dipole behavior, and the linear dependence of <math> A </math> on current further supports the consistency with the Biot–Savart law.

Thus, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current, consistent with dipole field behavior due to practical imperfections.

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:11:03Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in the experiment has an inner radius around <math> a = 0.3 \text{cm} </math>, an outer radius around <math> b = 0.7 \text{cm} </math>, and a total winding number <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, the field is nearly zero.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

However, even though the ideal toroidal solenoid predicts no magnetic field outside, in practice, a finite magnetic field is observed. This residual field is primarily due to the finite size and imperfect winding of the coil, where the turns are not perfectly azimuthal and uniform. Outside the toroid, we can consider the coil as a magnetic dipole, where the magnetic field decays as <math> 1/r^3 </math> in general. In our measurement results, varying the amplitude of the current and the distance from the toroid, the corresponding magnetic field follows the <math> 1/r^3 </math> trend as shown in Figure 7:

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid with varying distances and currents.]]

To better describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant depending on the current, and <math> r_0 </math> accounts for potential geometric offsets such as probe size or positioning errors. The fitting results show good agreement with the expected model, and the linear dependence of <math> A </math> on current amplitude further verifies the consistency with the Biot–Savart law.

Thus, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current.

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:07:55Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound into a circular loop, forming a torus-shaped structure. The toroid used in the experiment has an inner radius around <math> a = 0.3,\text{cm} </math>, an outer radius around <math> b = 0.7,\text{cm} </math>, and a total winding number <math> n = 50 </math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside under ideal conditions. The magnetic field inside the toroid is theoretically given by:

<math display="block"> B(r) = \frac{\mu_0 nI}{2\pi r} </math>
where <math> I </math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric, while outside the toroid, the field is nearly zero.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution.]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Outside the toroid, we can consider the coil as a magnetic dipole, where the magnetic field decays as <math> 1/r^3 </math> in general. This behavior arises because the current forms a closed loop, and at distances much larger than the size of the loop, the field resembles that of a magnetic dipole. In our measurement results, varying the amplitude of the current and the distance from the toroid, the corresponding magnetic field follows the <math> 1/r^3 </math> trend as shown in Figure 7:

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid with varying distances and currents.]]

To better describe the decay behavior, we fit the data to a model of the form:

<math display="block"> B(r) = \frac{A}{(r - r_0)^3} </math>
where <math> A </math> is a proportionality constant depending on the current, and <math> r_0 </math> accounts for potential geometric offsets such as probe size or positioning errors. The fitting results show good agreement with the expected model, and the linear dependence of <math> A </math> on current amplitude further verifies the consistency with the Biot–Savart law.

Thus, the experimental observations confirm that at distances large compared to the toroid’s dimensions, the magnetic field outside the toroidal solenoid decays as <math> 1/r^3 </math> and scales linearly with the current.

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:01:34Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound in a circular shape, forming a doughnut-like structure. The toroid used in the experiment has an inner radius around <math> a = 0.3</math> cm, an outer radius around <math> b = 0.7</math> cm, and a total winding number <math> n = 50</math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside in ideal conditions. The magnetic field inside the toroid is theoretically given by: <math>B(r) = \frac{\mu_0 nI}{2\pi r}</math>, where <math>I</math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric while outside the toroid, the field is zero.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Outside the toroid, we can consider the coil as a magnet dipole, where the magnetic field decays as <math> 1/r^3</math> in general. In our measurement results, we vary the amplitude of the current and the distance, the corresponding magnetic field follows the <math> 1/r^3</math> trend as Figure 7 shows:

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: Measured magnetic field outside a toroidal solenoid with varying distances and currents.]]

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-27T06:00:42Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound in a circular shape, forming a doughnut-like structure. The toroid used in the experiment has an inner radius around <math> a = 0.3</math> cm, an outer radius around <math> b = 0.7</math> cm, and a total winding number <math> n = 50</math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside in ideal conditions. The magnetic field inside the toroid is theoretically given by: <math>B(r) = \frac{\mu_0 nI}{2\pi r}</math>, where <math>I</math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric while outside the toroid, the field is zero.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Outside the toroid, we can consider the coil as a magnet dipole, where the magnetic field decays as <math> 1/r^3</math> in general. In our measurement results, we vary the amplitude of the current and the distance, the corresponding magnetic field follows the <math> 1/r^3</math> trend as Figure 7 shows:

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T16:09:00Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound in a circular shape, forming a doughnut-like structure. The toroid used in the experiment has an inner radius around <math> a = 0.3</math> cm, an outer radius around <math> b = 0.7</math> cm, and a total winding number <math> n = 50</math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside in ideal conditions. The magnetic field inside the toroid is theoretically given by: <math>B(r) = \frac{\mu_0 nI}{2\pi r}</math>, where <math>I</math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric while outside the toroid, the field is zero.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Outside the toroid, we can consider the coil as a magnet dipole, where the magnetic field decays as <math> 1/r^3</math> in general. In our measurement results, we vary the amplitude of the current and the distance, the corresponding magnetic field follows the <math> 1/r^3</math> trend as Figure 7 shows. With increasing current, the magnetic field increases linearly:

[[File:measurement_toroid1.jpeg|400px|thumb|center|Figure 7: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

== Discussions ==

== References ==
<references />

File:Measurement toroid1.jpeg

2025-04-26T16:08:33Z

Xueqi:

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T16:05:14Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound in a circular shape, forming a doughnut-like structure. The toroid used in the experiment has an inner radius around <math> a = 0.3</math> cm, an outer radius around <math> b = 0.7</math> cm, and a total winding number <math> n = 50</math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside in ideal conditions. The magnetic field inside the toroid is theoretically given by: <math>B(r) = \frac{\mu_0 nI}{2\pi r}</math>, where <math>I</math> is the current through the coil. As shown in Figure 6(b), the magnetic field inside the toroid is strong and azimuthally symmetric while outside the toroid, the field is zero.

[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

Due to the finite size of the magnetometer probe, we were only able to measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

Outside the toroid, we can consider the coil as a magnet dipole, where the magnetic field decays as <math> 1/r^3</math> in general. In our measurement results, we vary the amplitude of the current and the distance, the corresponding magnetic field follows the <math> 1/r^3</math> trend as Figure 7 shows. With increasing current, the magnetic field increases linearly:

[[File:measurement_toroid1.jpeg|600px|thumb|center|Figure 7: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:59:29Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:57:50Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:57:12Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:54:54Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:52:29Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:51:26Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:51:14Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:49:45Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound in a circular shape, forming a doughnut-like structure. The toroid used in the experiment has an inner radius around <math> a = 0.3</math>cm, an outer radius around <math> b = 0.7</math>cm, and a total winding number <math> n = 50</math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside in ideal conditions. The magnetic field inside the toroid is expected to be strong and azimuthally symmetric, while the external field is significantly weaker.

To understand the behavior of the magnetic field, we performed numerical simulations to calculate the magnetic field distribution both inside and outside the toroid. The magnetic field inside the toroid is theoretically given by:
<math>B(r) = \frac{\mu_0 nI}{2\pi r}</math>

The simulation results reveal the expected strong field inside the toroid and a rapidly decaying, much weaker field outside.
[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 6: (a) An illustration of a toroidal solenoid. (b) Simulated magnetic field distribution. ]]

Experimentally, we measured the magnetic field using a magnetometer. Due to the finite size and sensitivity range of the magnetometer probe, we were only able to reliably measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

== Discussions ==

== References ==
<references />

File:Simulation toroid1.jpeg

2025-04-26T13:48:29Z

Xueqi:

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:47:54Z

Xueqi: /* Modeling and measurement of a current carrying coil */

== Introduction to magnetic field sensing==

Magnetic field sensing plays a crucial role in both scientific research and practical applications, ranging from searching for fundamental physics, geophysical nagigations, to industrial and biomedical uses. Magnetic sensors are used to detect, map, and measure the intensity and direction of magnetic fields. There are several methods for sensing magnetic fields, and the choice of sensor depends on the application requirements, such as sensitivity, bandwidth, and environmental robustness.

=== Different types of magnetometers ===

* '''Fluxgate magnetometers''' use ferromagnetic cores and detect magnetic fields by sensing nonlinear changes in core permeability. They are valued for their robustness and precision in low-field measurements, making them ideal for space missions and geological studies.

* '''SQUID magnetometers''' (Superconducting Quantum Interference Devices) are among the most sensitive magnetic field detectors. Operating at cryogenic temperatures, SQUIDs exploit quantum interference in superconducting loops to detect fields as small as femtoteslas. They are widely used in biomagnetism (e.g., MEG – magnetoencephalography) and in fundamental physics experiments.

* '''Optically pumped magnetometers''' rely on the interaction between light and spin-polarized atoms, typically alkali metals like rubidium or cesium. These sensors can achieve extreme sensitivity while operating at room temperature <ref>{{Cite journal | author1 = Budker, Dmitry | author2 = Romalis, Michael | title = Optical magnetometry | journal = Nature Physics | volume = 3 | issue = 6 | pages = 493–500 | year = 2007 | doi = 10.1038/nphys634 | url = https://www.nature.com/articles/nphys634}}</ref>. The most sensitive optical magnetometers, such as spin-exchange relaxation-free (SERF) devices, can reach sensitivities below 1 fT/√Hz, rivaling superconducting magnetometers. More compact or portable designs, while less sensitive, still operate effectively in the picotesla to nanotesla range, making them attractive for biomedical, navigation, and remote sensing applications.

* '''Hall effect sensors''' are solid-state devices that measure the voltage generated by a magnetic field perpendicular to an electric current. They are simple, low-cost, and widely used in consumer electronics and industrial systems where high sensitivity is not critical.

* '''Magnetoresistive sensors''' (AMR, GMR, TMR) rely on changes in resistance due to the presence of a magnetic field. They offer high bandwidth and miniaturization potential, useful in automotive and robotics applications.

* '''Proton precession and Overhauser magnetometers''' use nuclear magnetic resonance techniques to measure absolute magnetic field strength with high accuracy, commonly applied in geophysical surveys.

Each of these technologies offers a unique combination of sensitivity, size, cost, and operational constraints.

=== Applications ===

Magnetic sensors find applications across a wide spectrum of fields:

* In '''fundamental physics''', they are used in searches for exotic particles and precision measurements of fundamental constants.
* In '''geophysics''', magnetometers are employed for mineral exploration, archaeology, and mapping Earth's magnetic field.
* In '''navigation''', especially in aerospace and underwater contexts, magnetic sensing provides orientation and heading information where GPS is unavailable.
* In the '''biomedical field''', ultra-sensitive magnetometers (e.g., SQUID and optical) enable non-invasive detection of weak magnetic signals from the brain and heart.
* In '''industry''', they are used for non-destructive testing, monitoring electric motors, and detecting ferrous materials in quality control.

Modern advances in materials science, quantum optics, and electronics continue to expand the reach and capability of magnetic field sensing technologies.

==Operation principles of a fluxgate magnetometer ==

===How does a fluxgate magnetometer work?===

A fluxgate magnetometer works by exploiting the magnetic response of a ferromagnetic core to detect an external magnetic field. It detects changes in magnetic permeability <math>\mu</math> due to the presence of an external field, typically producing a signal at 2nd harmonic of the excitation frequency. In a fluxgate sensor, the core material has very high magnetic permeability, meaning the core material can concentrate magnetic fields very well. Besides, it reacts strongly to changes in external fields. Its permeability changes nonlinearly as the material saturates, which is crucial for generating detectable signals (like the 2nd harmonic) when there's an external field.

Specifically, as shown in Figure 1, the core of a typical fluxgate magnetometer consists of two identical ferromagnetic rods with magnetic permeability <math>\mu</math>, each wrapped with a coil through which an alternating driving current flows in opposite directions. This setup generates magnetic fields <math>B_1 = \mu H_1</math> and <math>B_2 = \mu H_2</math> that are also in opposite directions, where <math>H_1</math> and <math>H_2</math> are auxiliary magnetic field generated by the coils. These coils are driven by a time-varying current (Figure 1b), typically a square wave. A secondary (sensing) coil is placed around both of the two cores to detect changes in net magnetic flux. In the absence of an external magnetic field, the opposing magnetizations <math>B_1</math> and <math>B_2</math> cancel out exactly, resulting in zero net magnetic flux through the sensing coil. Therefore, no voltage is induced in the sensing coil.

[[File:fluxgate_illustration1.jpeg|800px|thumb|center|Figure 1: (a) Illustration of a fluxgate with two parallel ferromagnetic coils, (b) Illustration of an alternating driving wave and (c) the corresponding magnetic responses of the two ferromagnetic magnetic cores in the absence of external magnetic field]]

However, when an external magnetic field <math>B_{\text{ext}}</math> is applied along the axis of the cores, it breaks the symmetry between the two magnetic responses (Figure 2b). This results in the emergence of non-zero total magnetic flux through the sensing coil. According to Faraday’s Law of Induction and Lenz’s Law, a time-varying magnetic flux <math>\Phi(t)</math> through the coil induces an electromotive force (EMF) <math>V_{\text{ind}} = -\frac{d\Phi}{dt}</math>. As Figure 2b shows, the net flux induced voltage typically appears at twice the driving frequency and forms the output signal of the magnetometer, allowing detection of the external magnetic field.

[[File:fluxgate_illustration2.jpeg|800px|thumb|center|Figure 2: (a) Illustration of a fluxgate in an external magnetic field, and (b) the corresponding magnetic responses of the two ferromagnetic magnetic cores, generating non-zero net magnetic flux in the sensing coil.]]

===Manual of the 'Magnetic Field Sensor FLC 100' used in the experiment ===

The FLC 100 magnetic field sensor was used in this experiment. The magnetic sensor has a compact design, high sensitivity, and low noise characteristics, making it well-suited for measuring weak magnetic fields in a laboratory setting. The working parameters of the sensor are listed below:

<center>

{| class="wikitable"
|+ Working parameters of the FLC 100 magnetic field sensor
|-
! Parameter !! Value
|-
| Measurement range || ±100 µT
|-
| Output voltage || ±1 V per 50 µT (max ±2.5 V)
|-
| Reference output || 2.5 V with respect to ground (OUT−)
|-
| Output impedance || <1 Ω
|-
| Load conditions || >1 kΩ resistance, <100 pF capacitance
|-
| Bandwidth || DC to 1 kHz (−3 dB)
|-
| Noise || <0.5 nT RMS (0.1–10 Hz), ~150 pT/√Hz at 1 Hz
|-
| Zero drift || <2 nT/K
|-
| Supply voltage || 5 V ±5%
|-
| Supply current || ~2 mA
|-
| Operating temperature || −40 °C to +85 °C
|-
| Dimensions || 44.5 mm × 14 mm × 5.5 mm
|-
| Detection coil length || 22 mm
|}

</center>

''Source: Stefan Mayer Instruments – FLC 100 Datasheet''

== Experiment ==

=== Setup of the experiment ===
In this experiment, a power supply provides a 5 V driving voltage for the fluxgate magnetometer (FLC100). The output signal from the magnetometer is monitored using an oscilloscope. A schematic of the experimental setup is shown below.

[[File:setup_fluxgate.jpeg|600px|thumb|center|Figure 3: Schematic of the experiment setup]]

To demonstrate the basic operation of the magnetometer, a permanent magnet is used as a test source. As the magnet is moved in proximity to the sensor, changes in the output voltage are clearly observed, indicating a varying magnetic field, as the video below shows:

[[File:Magnetometer_Thumb.jpg|thumb|center|500px|Click to download or view the [[Media:Magnetometer_demo_video.mp4|magnetometer demonstration video]].]]

For calibration purposes, in the following experiments, both a permanent magnet and a current-carrying coil are used. These sources allow characterization of the magnetometer's response under controlled magnetic field conditions.

=== Modeling and measurement of a permanent magnet ===

Firstly, a disk-shaped permanent magnet is used as the source of the magnetic field. The magnetic field distribution around the magnet is simulated using COMSOL Multiphysics, modeling the magnet as a disk with a diameter of 1.6 cm and a thickness of 3 mm. A stream plot of the three magnetic field components is shown in Figure 4(a). It is important to note that the magnetometer measures only the scalar component of the magnetic field, in our case, the <math>B_y</math> is projected along the axis of the magnetometer. The component <math>B_y</math> is evaluated along the blue cutline indicated in Figure 4(a), where it is observed to be very small—on the order of microtesla—because the magnetic field is predominantly out-of-plane near the disk.

[[File:simulation_disk1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

To experimentally validate the simulation results, we position the magnet adjacent to a fluxgate magnetometer and measure the magnetic field strength as a function of distance (Figure 5a). The measurements are repeated three times to ensure reproducibility. The measured voltages and the corresponding calculated magnetic field strengths are presented in Figures 5b and 5c, respectively.

[[File:measurement_disk1.jpeg|600px|thumb|center|Figure 5: (a) Schematic of measurement setup (b) Measured voltage when varying the distance between the disk magnet and the magnetometer, and (a) measured in-plane magnetic field <math>B_y</math>]]

As expected, the magnetic field strength decreases with increasing distance from the magnet, following an approximate inverse relationship. The repeated measurements exhibit small variations, indicating good measurement consistency and reliability of the setup. Discrepancies between the experimental data and the simulation may be attributed to factors such as misalignments in positioning, imperfections in the magnet’s shape or magnetization, and environmental magnetic noise.

=== Modeling and measurement of a current carrying coil ===

In this section, we present the modeling and experimental measurement of the magnetic field generated by a current-carrying coil.

The coil used in the experiment is a toroidal solenoid, as illustrated in Figure 6. A toroidal solenoid consists of a wire wound in a circular shape, forming a doughnut-like structure. The toroid used in the experiment has an inner radius around <math> a = 0.3</math>cm, an outer radius around <math> b = 0.7</math>cm, and a total winding number <math> n = 50</math>.

One important characteristic of a toroidal solenoid is that its magnetic field is largely confined within the core of the toroid, with negligible field outside in ideal conditions. The magnetic field inside the toroid is expected to be strong and azimuthally symmetric, while the external field is significantly weaker.

To understand the behavior of the magnetic field, we performed numerical simulations to calculate the magnetic field distribution both inside and outside the toroid. The magnetic field inside the toroid is theoretically given by:
<math>B(r) = \frac{\mu_0 nI}{2\pi r}</math>

The simulation results reveal the expected strong field inside the toroid and a rapidly decaying, much weaker field outside.
[[File:simulation_toroid1.jpeg|600px|thumb|center|Figure 4: (a) Simulation of magnetic field around a disk magnet (b) <math> B_y </math> along the blue cut line in (a)]]

Experimentally, we measured the magnetic field using a magnetometer. Due to the finite size and sensitivity range of the magnetometer probe, we were only able to reliably measure the magnetic field outside the toroid. Measuring inside the toroid would require a significantly smaller probe to fit within the core without disturbing the field distribution.

== Discussions ==

== References ==
<references />

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:47:01Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:46:04Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:45:23Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:43:09Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:42:24Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T13:41:54Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-26T11:52:43Z

Xueqi: /* Modeling and measurement of a current carrying coil */

Magnetic field sensing using a fluxgate magnetometer

2025-04-25T14:08:09Z

Xueqi: /* Modeling and measurement of a permanent magnet */

Magnetic field sensing using a fluxgate magnetometer

2025-04-25T14:06:41Z

Xueqi: /* Setup of the experiment */

Magnetic field sensing using a fluxgate magnetometer

2025-04-25T13:59:37Z

Xueqi: /* Modeling and measurement of a permanent magnet */