# EE3901/EE5901 Sensor TechnologiesWeek 9 Tutorial

**Dr. Bronson Philippa**

## Question 1

An eddy current sensor is used to detect defects on a steel target. Steel has resistivity $\rho=1.18\times10^{-7}$ Ω.m and relative permeability $\mu_{r}=100$. If the driving frequency is 50 Hz, what is the minimum thickness of the steel target to ensure proper sensing with the eddy current sensor?

Hint: the magnetic permeability of free space is $\mu_{0}=4\pi\times10^{-7}$ H/m and the formula for the eddy current skin depth is given by

## Question 2

A thin metal tape is being affixed to a non-conductive target to facilitate distance measurements using an eddy current sensor. The tape is made of aluminium, which has a resistivity of $\rho=2.65\times10^{-8}$ Ω.m and a relative permeability of $\mu_{r}\approx1$. The tape has a thickness of 0.3 mm. What is the required frequency for the eddy current power source?

## Question 3

The inclination of a plane is measured with an LVDT that has a 2 kg
mass attached to its rod (**Figure Q3**). The mass is supported by a spring
which exerts a force

where $k=2000$ N/m and $x$ is the displacement from the zero point of the LVDT. Meanwhile the weight of the the mass in the direction of $x$ is

where $g=9.81\ \text{m}/\text{s}^{2}$. Assume that the friction between the mass and the plane is negligible. The LVDT has a sensitivity of 150 mV/cm/V when powered by a 2.5 kHz, 3 V RMS sine wave.

Derive the transfer function between output voltage and angle $\theta.$

Hint: assume static equilibrium to find the relationship between $\theta$ and $x$, then use the sensitivity of the LVDT to find the relationship between $x$ and the output voltage.

## Question 4

**Figure Q4** shows the equivalent circuit of an LVDT. The device produces
a no-load output voltage of $V_{0}=0.5$ V (RMS) when measuring a
deflection of 10 mm. The power supply is $V_{i}=5$ V (RMS) at 2 kHz.
You would like to increase the output voltage (i.e. increase the sensitivity)
by raising the excitation frequency.

(a) You measure the DC resistance of the primary winding to be $R_{1}=200$ Ω. Next you measure its inductance using an LCR meter and find that the primary windings have an inductance of $L_{1}=10$ mH. Calculate the impedance of the primary winding at 2 kHz and hence find the magnitude of the excitation current $i$.

(b) Calculate the net mutual inductance $M=M_{1}-M_{2}$ at 10mm deflection based upon the measured output voltage $V_0 = 0.5$ V.

Hint: write KCL around the output winding to obtain an expression for $V_0$ as a function of $M$.

(c) Assume that $M$ is constant with respect to frequency (i.e. the magnetic media in the core is not saturated). Calculate the new output voltage when the excitation frequency is raised to 20 kHz.

Hint: you will need to recalculate the excitation current $i$ because the impedance of the primary winding will change.

## Question 5

You are measuring the strength of an electromagnet with the Hall effect.
Your sensor element is a 10 mm $\times$ 10 mm $\times$ 0.1 mm wafer
of p-type silicon with doping density $N_{c}=10^{18}\ \text{cm}^{-3}$.
The geometry is shown in **Figure Q5**.

(a) Is $V_{H}$ positive or negative as drawn?

(b) Suppose instead that the sensor material were n-type silicon. What would be the polarity of $V_{H}$?

(c) Returning to the p-type device, and assuming a geometry factor of $G_{H}=1$, calculate $V_{H}$ for an injected current of $i=50$ mA and a magnetic field strength of 100 mT.

(d) Suggest a type of amplification circuit that could be connected to this sensor to obtain a sensitivity of $-3.1211$ V/T.

(e) The sensor is now rotated about the $y$ axis as shown by the angle $\theta.$ What is the minimum value of $\theta$ such that $V_{H}=0$?

(f) What is the minimum value of $\theta$ such that $V_{H}$ has the opposite sign but the same magnitude?