EE3901/EE5901 Sensor Technologies
Week 7 Tutorial

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College of Science and Engineering, James Cook University
Last updated: 07 April 2022

Question 1

Two rectangular metal plates of dimensions 10 cm x 5 cm are separated by 1 mm of air, which has ϵr=1.0006\epsilon_{r}=1.0006. Calculate the capacitance of this structure.

Question 2

The device from Question 1 is being used as a displacement sensor. The distance between the plates is varied, and the capacitance is measured. Given a measured capacitance of 5.56 pF, what is the separation distance between the plates?

Question 3

Consider the capacitive strain gauge shown in Figure Q3.

Figure Q3: A capacitive strain gauge formed by micropatterned metal foil electrodes on a flexible polymer substrate. (a) Top down view of part of the electrode structure. (b) The overall device consists of a long string of interdigitated electrodes. (c) Perspective view of the device. Zoom:

For a real example of such a device, see the scanning electron microscope image in Figure 2 of this paper by Jiseok Kim et al.

The working principle of this strain gauge is that deformation of the sensor changes the distance between the electrodes. Each pair of electrodes forms a capacitor. The dimensions are as follows: overlap distance v=400 μmv=400\ \text{μm}, electrode width w=40 μmw=40\ \text{μm}, nominal separation distance d=20 μmd=20\ \text{μm}, total number of capacitors n=200n=200. Each capacitor is connected in parallel.

(a) Given a measured capacitance of 1.5 pF, calculate the thickness (hh) of the metal electrodes above the polymer substrate. You may assume the relative permittivity is ϵr=1\epsilon_{r}=1.

(b) Suppose that the gauge experiences a uniform strain of 0.02 in tension. Neglecting changes in electrode thickness, what will be the new capacitance?

(c) Suggest an interface circuit for this sensor using a single op-amp that will produce an output voltage whose RMS value is linearly proportional to the applied strain.

Question 4

A liquid level sensor is shown below.

Figure Q4: (a) A water level sensor based on plane-parallel electrodes. (b) The equivalent circuit. Zoom:

The electrodes are covered with a thin layer of insulating material to avoid shorting them when a conductive liquid is being stored in the container. The dimensions are as follows: the plate height is hmax=100 mmh_{max}=100\ \text{mm}, the gap is d=1 mmd=1\ \text{mm}, the width of the plates (in the direction into the page) is w=15 mmw=15\ \text{mm}, and the liquid level is h=65 mmh=65\ \text{mm}.

(a) Given that the total capacitance is measured to be 247.5 pF, find the relative permittivity of the liquid.

(b) Suggest an interface circuit for this sensor using a single op-amp that will produce an output voltage whose RMS value is linearly proportional to the level of the liquid.

Question 5

Capacitance is being measuring using the method of charge redistribution (Figure Q5).

Figure Q5: The circuit for Question 5. The unknown capacitor is CxC_x, while C1C_1 is a fixed capacitor whose value is already known. Zoom:

Recall that the charge stored in a capacitor is given by Q=CVQ=CV.

(a) For times t<0t<0, S1 is closed, S2 is open, and S3 is closed. Hence CxC_{x} charges to the reference voltage VrV_{r}, while C1C_{1} completely discharges. Write down an expression for the stored charge in CxC_{x}.

(b) At the moment t=0t=0, S1 is opened, S2 is closed, and S3 is opened. Hence CxC_{x} and C1C_{1} must come to the same voltage by exchanging charge. Using the principle of conservation of charge, calculate the voltage V0V_{0}.

(c) Extension question, which is not examinable in this subject but is surprising if you haven’t seen it before: Choose Cx=C1=CC_{x}=C_{1}=C, then calculate the energy in the system for t<0t<0 and t>0t>0. Recall the energy in a capacitor is W=12CV2W=\frac{1}{2}CV^{2}. Does this circuit violate conservation of energy??