# EE3901/EE5901 Sensor TechnologiesWeek 9 NotesInductive and magnetic sensors

**Dr. Bronson Philippa**

Inductive sensors are those where the variation of inductance is used to detect changes in some physical quantity. Meanwhile, another type of sensing mechanism is to utilise magnetic fields in the sensing process. This week, we will consider both of these types.

## Inductors

Any current-carrying wire will create a magnetic field. However, for
sensing purposes, we usually consider devices designed to have a higher
inductance, i.e. those that will create a larger magnetic field in
response to a current. A common such design is called a solenoid,
which is a wire wrapped into a coil (**Figure 1**).

The inductance of a solenoid is given by

where $\mu$ is the magnetic permeability of the material inside the coil, $N$ is the number of turns of wire, $A$ is the cross-sectional area of the solenoid, and $l$ is the length of the solenoid. This equation is an approximation that is valid when the solenoid is long compared with its cross-sectional dimensions and when the magnetic field lines wrapping around the solenoid experience a homogeneous environment free of conductors.

Just like permittivity, the permeability is often split into two parts using the formula

where $\mu_{r}$ is the dimensionless relative permeability and $\mu_{0}\approx4\pi\times10^{-7}$ H/m is a physical constant called the permeability of free space. Historically $\mu_{0}$ was defined to be exactly $4\pi\times10^{-7}$ H/m, but since 2019 it has been redefined such that it is now an experimentally measured quantity. Nevertheless the value $4\pi\times10^{-7}$ H/m remains a highly accurate and compact representation.

Studying Eq. (1), the parameter that is most likely to be used for sensing purposes is the permeability. It is difficult to change the number of turns or the dimensions of the solenoid in a precise, repeatable manner. Hence the primary way to use inductance for sensing will be to insert or remove a material with different permeability inside the coil, or by disturbing the magnetic environment such as by placing other conductors near the solenoid.

### Interface circuits for inductive sensors

There are various options for interface circuits. The simplest approach is to measure the impedance (e.g. using a two-wire or four-wire measurement circuit), just like capacitive sensors. Of course the power supply must be sinusoidal.

Another strategy is to use an AC bridge (**Figure 2**).

Finally, an LC circuit or LR circuit can be used to define the frequency of an oscillator. Changes in inductance will change the oscillation frequency. This is, again, very similar to the situation for capacitance sensors.

Each of these interface circuits is essentially the same as those used for capacitors, since both capacitive and inductive sensors provide reactive impedance in a circuit. Refer back to the capacitive sensor notes for more details on these circuit designs.

## Eddy currents

The inductance of a solenoid (as stated above) is sometimes called its “self-inductance”. On the other hand, if there are two coils in proximity, then the magnetic field from one will interact with the other. This is an effect called “mutual inductance”, which should be familiar to you from circuit theory. Recall that mutual inductance is the operational mechanism behind transformers.

It is also possible to have mutual inductance where the conductor(s) are not deliberately made into coils. For example, a time-varying magnetic field will induce a circular motion of charges even in a rectangular slab of metal. In a sense, the magnetic field is “making its own coil” by inducing currents to flow in a particular way. This is called an eddy current. Eddy currents are an undesired loss mechanism in power systems and transformers, but they can also be used for sensing purposes.

The simplest eddy current sensor is a solenoid placed at right angles to a conductive sheet:

The eddy currents in some sense behave like another solenoid. They create their own magnetic field, which in turn induces a voltage back in the original solenoid. The polarity of the induced voltage is such that it counteracts the original excitation voltage. In other words, the electrical load of the original solenoid is increased because some energy is transferred to the eddy currents and dissipated there as heat.

In electric circuit terminology, this arrangement is akin to a transformer where the solenoid is the primary winding and the eddy currents act as a secondary winding. Consequently some impedance from the conducting slab is reflected back into the excitation circuit, much like a transformer can reflect an impedance from one winding to another. The increase in resistance will affect the current-voltage characteristic of the solenoid.

In addition to the changes in resistive load, the presence of the eddy currents will also change the inductance of the primary winding. The eddy current creates a magnetic field that opposes the magnetic field of the solenoid, and since inductance is proportional to magnetic flux, the solenoid inductance decreases.

### Skin depth

Eddy currents do not penetrate far into the conductor. They are primarily localised close to the surface. There is an exponential drop-off with depth, i.e. current density $J$ as a function of depth $d$ has functional form $J\sim e^{-d/\delta}$, where $\delta$ is called the “skin depth” and is given by

where $\rho$ is the resistivity of the conducting slab, $\omega=2\pi f$ is the natural frequency of the AC current that is driving the solenoid, and $\mu$ is the magnetic permeability of the conducting slab. Some example numbers are as follows. In aluminium ($\mu_{r}=1$, $\rho=2.68\times10^{-8}$ Ω.m), we have $\delta=1.2$ cm at 50 Hz, $\delta=0.23$ mm at 125 kHz, and $\delta=82$ μm at 1 MHz.

### Applications of eddy current sensors

Various designs of eddy current sensor are possible. There are single winding sensors where the same solenoid is used for excitation and sensing. There are also versions with multiple coils, for example, an excitation coil and one or more measuring coils.

An everyday example of eddy current sensors are the induction loops buried under roads near traffic lights to detect when vehicles are waiting. These are wire loops with a sinusoidal power source in the range of 10 - 200 kHz. When a vehicle drives over the loop, eddy currents in the metal parts of the vehicle reduce the inductance of the loop. This is detected (by a change in frequency in an oscillator) and used to trigger the traffic lights.

Other applications for eddy currents include:

- Measuring the distance between the probe and a conductive surface.
- Measuring the thickness of a non-conductive object by placing it between a probe and a metal surface.
- Finding cracks and defects in metal objects, because cracks will interrupt the eddy currents.

In all of these applications the conductor must be thick compared with the skin depth $\delta$. An approximate guide is that the thickness should be at least 3 times the skin depth.

## Linear variable differential transformers (LVDTs)

An LVDT is a robust and reliable sensor for measuring linear displacement (**Figure 4**).
It is a transformer with a split winding, but half of one winding
is wired backwards. The iron core is connected to a rod that slides horizontally. As the
rod slides, the mutual inductances vary. In typical construction each
winding fully encircles the iron core.

This circuit can be analysed by first finding the excitation current $i$ (which is controlled by the impedance of the primary winding and the excitation voltage). Once $i$ is known, the induced voltages on the secondary side can be calculated. Writing KVL around the secondary side and assuming no load (i.e. no current flowing in the secondary), we have:

Notice the negative sign. The output voltage is 0 when $M_{1}=M_{2}$. The mutual inductances are almost linear with respect to displacement up until some maximum range, after which the output voltage drops because of the loss of magnetic coupling between the two windings. Notice that the sign of the displacement can only be detected if the phase of $V_{0}$ is measured and compared to the phase of the excitation current.

From this equation, we can deduce that $V_{0}$ is proportional to the excitation frequency (so long as the frequency remains low enough to avoid magnetic non-linearities due to hysteresis, etc). Typical operating frequencies for LVDTs are in the range of 50 Hz to 20 kHz. In terms of displacement, various LVDTs are available with measurement ranges that go from hundreds of microns to tens of centimeters.

LVDT datasheets will typically specify sensitivity in terms of a transfer function

where $V_{0}$ is output voltage, $S$ is the sensitivity at a given excitation frequency, $V_{i}$ is the input voltage (i.e. the power supply connected to the primary winding), and $x$ is displacement. The sensitivity has units of output voltage per meter of displacement per volt of input. The units of sensitivity are V/m/V, or 1/m, or mV/mm/V, or similar.

The key advantages of LVDTs include:

- No friction between the moving element and the sensor, allowing for very high reliability.
- The sensor contains no electronics and so can be exposed to extreme environments like high/low temperatures and be subjected to vibration and shock.
- In principle arbitrarily high resolution (limited mainly by the voltage sensing electronics).
- Excellent repeatability (especially in returning to the same zero point).

The limitations of LVDTs include:

- The zero position is not perfectly at the centre, due to imperfections in the windings and stray capacitances between windings. However, a given device will repeatedly measure the same point as zero, so the offset can be corrected by a calibration if needed.
- In practice the magnetic interaction is not perfectly linear, so the sinusoidal input is distorted by these non-linearities. The result is higher frequency harmonics that appear in the output signal $V_{0}$. Low pass filtering of the output is sometimes helpful.
- Temperature changes will affect the resistance of the primary winding, and hence the excitation current $i$ if the power supply is a constant voltage. Consequently it is better to use a constant current source to power the primary winding if temperature stability is important.

There are also rotary devices using the same principle to measure angular displacement. These are called rotary variable differential transformers (RVDTs).

## The Hall effect

A Hall effect sensor measures magnetic field strength.

It is based upon the Lorentz force law, which states that a charged particle (e.g. electron or hole) moving with velocity $\boldsymbol{v}$ in an environment with a magnetic field $\boldsymbol{B}$ experiences a force

where $q=-1.602\times10^{-19}$ C is the charge of an electron. If the electron or hole is moving inside a conductor then its motion results in a voltage perpendicular to the direction of current flow.

**Figure 5** shows a slab of semiconductor with an applied voltage and a magnetic
field that are perpendicular to each other. If the semiconductor is
p-type then holes will drift from high voltage to low voltage, whereas for an n-type material the electrons will travel in the opposite direction.

In the presence of a magnetic field, the Lorentz force causes a deflection of the charge carriers. This creates a net build up of charge on one side of the semiconductor. A build-up of charge can be measured as a voltage, which we call the Hall voltage $V_{H}$. The sign of the Hall voltage is determined from the direction of the Lorentz force as well as the polarity of the majority charge carrier. To find the polarity of the Hall voltage, use the right-hand rule to evaluate the vector cross product in Eq. (6).

The analysis to calculate the Hall voltage is as follows. In this example, we will use the axes as drawn in Figure 5. The Hall voltage creates an electric field along the $z$ direction which is

where $\boldsymbol{\hat{k}}$ is the unit vector in the $z$ direction.

The electric field $\boldsymbol{E}_H$ exerts a force on each carrier:

Meanwhile the magnetic field also exerts a force given by

where $v = |\boldsymbol{v}|$ is the speed of the charge carriers, and $B_y=|\boldsymbol{B}|$ is the strength of the magnetic field along the $y$ axis.

In static equilibrium the forces must balance:

Equation (12) gives us the voltage in terms of the carrier velocity, but we would prefer to express this in terms of the current $i$. Therefore there is one final step in the derivation.

Consider how many charge carriers must leave the device per second to create a current $i$. As shown in Figure 5, the cross-sectional area of the semiconductor in the direction of current is $LW$, and the carriers drift with velocity $v$. Hence all the carriers in a volume $vLW$ will exit the device per unit time. The total number of carriers inside this volume is $N_{c}vLW$ where $N_{c}$ is the doping density of either donors (n-type) or acceptors (p-type). Hence the total amount of charge leaving per unit of time is

Substituting into Eq. (12), the Hall voltage is

In practice the Hall voltage is a little bit less. This simplistic derivation assumed perfectly uniform fields, etc. Hence we introduce a geometry factor $G_{H}$ and write

$G_{H}$ is typically 0.7 to 0.9.

If the magnetic field is not perpendicular to the direction of current flow, then the voltage is reduced by a factor $\sin\theta$ where $\theta$ is the angle between $\boldsymbol{B}$ and $\boldsymbol{v}$.

Hall effect sensors are often used as proximity detectors to find the presence or absence of a permanent magnet. For instance, they are used in wheel encoders to detect shaft rotations. Devices intended for this purpose often include built-in electronics that produces a digital output according to whether the magnetic field is above or below a threshold.

There are also Hall sensors optimised for linear output, to measure the strength of a magnetic field. In this case the datasheet will specify a sensitivity (in units of V/T or mV/mT or similar), assuming a fixed supply current.

## References

Ramon Pallas-Areny and John G. Webster, *Sensors and Signal Conditioning*, 2nd edition, Wiley, 2001.

Clarence W. de Silva, *Sensor Systems: Fundamental and Applications*, CRC Press, 2017.

Winncy Y. Du, *Resistive, Capacitive, Inductive, and Magnetic Sensor Technologies*, CRC Press, 2015.

Jacob Fraden, *Handbook of Modern Sensors: Physics, Designs, and Applications*, 5th edition, Springer, 2016.