EE3901/EE5901 Sensor Technologies

Chapter 4 Notes

Capacitive sensors and signal conditioning circuits for reactive sensors

Capacitance

Firstly, let’s review the physics of capacitors. A capacitor consists of two electrodes separated by an insulating material. An accumulation of charge on the electrodes creates an electric field between the plates. The electric field causes the effects we know as capacitance.

Capacitance is measured in farads (F), which is equivalent to coulombs/volt. In other words the capacitance tells you how much electric charge will accumulate on each electrode per volt.

Suppose we have rectangular electrodes, as shown in Figure 1.

These are called “plane parallel” electrodes. These serve as a prototypical example of a capacitor, and serve to explain most of the important physics.

The capacitance of plane parallel electrodes is given by

where is the permittivity of the material between the electrodes, is the surface area of each electrode, and is the distance between the electrodes. The permittivity (also called the dielectric constant) is a physical property of a material that characterises how strongly it can polarise in the presence of an electric field. It is often expressed as

where is the relative permittivity and pF/m is the permittivity of free space.

Fringing effects

The geometric capacitance () is only valid when the lateral dimensions of the electrodes are much larger than the separation distance. In real capacitors, there are slight differences due to edge effects at the corners of the electrodes (Figures 2-4).

As shown in Figure 4, the electric field lines curve outwards near the edge of the capacitor. If this system were a sensor, and our goal was to measure changes in capacitance, these fringing effects may cause interference. For sensing purposes, we can minimise the edge effects by using guard plates (Figure 5).

Using these guard electrodes, the capacitance between the middle electrodes is closer to the ideal geometric value, .

Mechanisms of capacitance sensing

Using the equation we can identify three ways that capacitance can vary:

1. Changes in area (e.g. by moving the electrodes sideways so that the area of overlap varies); or
2. Changes in separation distance (e.g. by moving the electrodes closer together or further apart); or
3. Changes in permittivity (e.g. by changing the properties of the material between the plates).

When we consider also the equivalent circuit of multiple capacitors, we can identify another mechanism:

4. Introducing or removing nearby objects that are capacitively coupled to the sensor, thereby adding or removing a capacitors from the equivalent circuit of the sensor.

Some mechanisms are shown in Figure 6.

Capacitive humidity sensors

Capacitance is a useful mechanism to detect water because water has a high permittivity. (This is because water molecules are polar, and can easily orient themselves in response to an applied electric field.) The relative permittivity of air is while for water it varies from at 0 °C to at 100 °C. Hence, adding water into the dielectric medium produces noticeable changes in permittivity.

A typical mechanism is to use some absorbent material that takes up moisture from the air. A properly chosen absorbent polymer film can be used to create an almost linear relationship between capacitance and relative humidity. However, the permittivity varies with temperature, so the temperature must also be measured and corrections applied.

Capacitive level sensors

Capacitance can be used to measure the level of liquid, as shown in Figure 7.

The electrodes may have various geometries, for example, they may be cylindrical with an inner electrode and an outer electrode where the liquid travels up the middle like a pipe. Alternatively they may have flat electrodes arranged as parallel plates or coplanar plates. If the electrodes are inside the liquid, then they must be covered with an insulating material. Alternatively, the electrodes may be placed on the outside of the container walls.

Since the air and liquid have different permittivity values, the capacitance per unit length in the air and liquid region are different. As the water level rises, the proportion of the capacitor with a high permittivity increases, resulting in a close to linear increase in capacitance.

Again there is a temperature dependence that must be accounted for. One strategy to do this is to include two more electrode pairs to measure the capacitance of the liquid and air. This can be used to correct for changes in temperature.

Other applications

• Capacitance can be used to measure small displacements by moving the electrodes with respect to each other. For example some accelerometers measure the deflection of a proof mass suspended on a miniature spring.
• Studio condenser microphones measure the deflection of a diaphragm through change in capacitance.
• An electrically conducting object can be used as one electrode in a single ended capacitance probe.

Capacitive interface circuits

Interface circuits for capacitive sensors often use a sinusoidal voltage or current. In this case, the capacitive sensor can be represented by a complex impedance where is a (typically undesirable) series resistance and is the reactance of the sensor. Note where is the frequency of the AC power source.

For some intuition, consider a capacitance sensor with pF. An AC excitation frequency of 10 kHz would give kΩ, whereas an excitation frequency of 100 MHz would give Ω. These reactances could be measured using 2-wire or 4-wire measurement circuits, just as for resistance sensors. Simply use a sinusoidal source and measure the RMS voltage and current, and find the impedance with Ohm’s law.

However, there are also various circuit designs that are more specialised to capacitive sensor interfacing.

The auto-balancing bridge

The auto-balancing bridge (Figure 8) is a simple circuit that is common in LCR meters. To understand this circuit, notice that the op-amp creates a virtual ground at . Therefore is completely determined by the voltage source and capacitor. However, all the current must flow through the resistor, creating a voltage at the output . Therefore by measuring , it is simple to calculate the capacitance.

Op-amp circuits with linear responses

When impedance is linearly proportional to the measurand

Consider a capacitive sensor where the separation distance is varied. Specifically, consider that the distance is given by

where is a typical distance and represents the influence of the measurand. In this case the sensor response is of the form:

where is the capacitance at .

Now consider the impedance of this capacitive sensor:

Notice that impedance is linearly proportional to the measurand. A suitable interface circuit is shown in Figure 9.

The resistor is necessary to provide feedback at DC. The AC behaviour of this circuit is that of an inverting amplifier, i.e.

where is the impedance of the sensor and is the impedance of the fixed capacitor . If we choose then and the output voltage becomes

Notice that the output is linear with respect to .

When impedance is inversely proportional to the measurand

Other types of capacitive sensors may have variations in permittivity or surface area, i.e. a response of the form:

In this case the capacitor’s impedance is

i.e. impedance is inversely proportional to the measurand. In this case, swap the placement of the variable and fixed capacitors:

The resulting output voltage is

Again notice that the output voltage is linearly proportional to .

AC bridges

Bridge circuits can also be used for capacitive sensors, as shown in Figure 11.

The analysis is similar to the Wheatstone bridge except that impedances are used instead of resistances.

A variation of this concept is the Blumlein bridge, where a centre-tapped transformer is used on one side (Figure 12).

The advantage of this design is that the sensing side has galvanic isolation from the excitation source, so that it can be separately grounded. Notice that is a single ended voltage with respect to the indicated ground.

To analyse this circuit, notice that the action of the transformer is to produce a voltage source with magnitude where is the turns ratio. Using a voltage divider formula and recognising that the halfway point of the centre-tapped transformer has half the voltage, we can write the output voltage as:

Circuits based on charging & discharging capacitors

Transformers cannot be miniaturised, hence other interfacing circuits have been developed for integrated circuits. Instead of using sinusoidal excitation, capacitance can also be measured based on its charging and discharging behaviour.

Charge transfer

Charge can be transferred from an unknown capacitor to a known capacitor (Figure 13).

This circuit operates in three steps:

1. The unknown capacitor is charged to a fixed voltage .
2. The energy stored in is shared with by opening S1 and closing S2. The output voltage can then be measured.
3. is discharged by closing S3, and the process repeats.

The output voltage can be derived by conservation of charge. You will show in tutorial questions that

can then be calculated from this equation.

Variable frequency oscillators

An oscillator is a circuit that produces a periodic waveform. A common strategy to measure capacitance is to design an oscillator circuit whose output frequency depends upon the capacitance of the sensor. An example is the relaxation oscillator that uses a comparator (Figure 14):

The sensor can be either or . The oscillation period is proportional to the time constant of the circuit, so changes in either or will change the oscillation frequency. In this design, the output voltage is a square wave, the frequency of which can be monitored using a microcontroller.

As drawn, this circuit is for a comparator with push-pull outputs, for example the TLV3491. The pullup resistor is necessary to make this circuit work with a single power rail. This circuit is suitable for oscillations up to tens of kHz, which limits the range of capacitances that can be measured.

Other types of oscillators (e.g. Wein bridge oscillators, Hartley oscillators, and Colpitts oscillators) operate at higher frequencies, and hence can be used with sensors whose capacitance is smaller. The principle is the same: changes in capacitance cause changes in oscillation frequency, and the frequency is then measured with a microcontroller or other digital circuit.

Designing capacitive sensors

If a capacitor has plane-parallel geometry, then the capacitance is given by the well-known formula , where is the permittivity of the material in between the electrodes, is the surface area of their overlap, and is the separation distance.

However, what if the system does not have this geometry? There are a few special geometries where exact solutions are known, but in general, there is no formula sheet for the capacitance of any arbitrary shape. If there is no simple formula, then we must turn to computer software to calculate capacitance. This week, we will learn how to calculate capacitance for any geometry that you can draw in a CAD package. We will do this using the finite element solver in Matlab’s PDE toolbox.

The meaning of capacitance

The meaning of capacitance can be understood through the formula

where is the stored charge, is the capacitance, and is the potential difference between the conductors.

Equation (1) will be our recipe for calculating capacitance. We will set up a computer simulation with a given voltage applied to some geometry of conductors. Then, we will use the computer to find the charge . Knowing both of these quantities, we can easily find the capacitance .

Recap of electrostatics

Electricity and magnetism are described by Maxwell’s equations. However, for the purposes of analysing capacitors, we will simplify Maxwell’s equations by applying the electrostatics approximation. We will also assume that there is no free charge outside of any conductor.

With these approximations, the key physics is Gauss’s law,

where is called the electric displacement, and the 0 on the RHS indicates that there is no free charge.

However, for numeric calculations, it is convenient to calculate the electric potential (aka voltage) at every point, instead of the electric displacement. This is because the potential is a scalar value. The relevant equations are

where is the electric field, and is the electric potential. Substituting into Eq. (2), we obtain

Capacitors are created when there are two or more conductors. We will assume that the conductors are ideal. This means that the potential is the same everywhere within the conductor. (In the language of circuit theory, the entire conductor is a single node.)

Therefore, the important and interesting physics is what happens in the region outside the conductors. Our problem formulation will look like Figure 15. This is a 2D slice through a general 3D geometry. In this example figure, there are two rectangular conductors shown in white. These should be viewed as extending into the page to create the overall 3D structure.

We will solve for the potential in the region between and around the conductors. To set up the problem, we must specify:

1. The voltage at every edge, and
2. The permittivity at every point within the domain.

From this, we can numerically solve Eq. (5) to obtain the electric potential.

There is one final piece of electrostatics theory that we will need. Gauss’s law can be converted into the so-called “integral form”, which reads:

where the integration is around a closed contour , and is the charge located within the contour. We will numerically integrate around the contour, for example, using the trapezoidal rule. From this, we obtain the amount of charge on each conductor. Knowing the amount of charge and the voltage, we can then find the capacitance.

The case of more than two conductors

Some capacitive sensor designs will need electrodes to act as shields. Therefore, we need to define a capacitance matrix, as a way to describe the capacitive effects between systems of conductors.

Firstly, let’s consider just one conductor. Even a single, isolated conductor has capacitance. Suppose that you place some electric charge on this isolated conductor; then the charge will induce an electric field in its vicinity. These electric field lines must terminate somewhere. By convention, we define self-capacitance as the capacitance between the object and a conducting sphere at infinite distance.

Now, suppose that there are multiple conductors in the system. As shown in Figure 16, there will be interactions between all pairwise combinations of conductors. These are called mutual capacitances. We can express the mutual capacitances in a matrix as follows:

where is the number of conductors. The diagonal entries are the self-capacitances, and the off-diagonal elements are the mutual capacitances. Notice that mutual capacitances are symmetric: .

The Maxwell capacitance matrix

Let’s analyse the 3-conductor system of Figure 2 more closely. The charge on conductor 1, given a certain arrangement of voltages, can be obtained by summing up the contributions due to each capacitor shown in the figure:

However, Equation (8) is not very convenient to use because the expressions involve differences between voltages. It would be nicer to treat all voltage terms as being defined with respect to a common ground reference, just like we normally do in circuit theory. We would prefer an equation of the form:

If we expand the brackets in Eq. (8) and equate coefficients to those in Eq. (9), we find

In general we have

This leads us to define the Maxwell capacitance matrix as follows:

Notice that the off-diagonal terms in the Maxwell capacitance matrix are negative. The physical meaning of the negative sign is that a positive voltage on one conductor induces a negative charge on another capacitor.

Given some entries in the Maxwell capacitance matrix, we can find the mutual capacitances by inverting Eqs. (5) and (6) to obtain

Overall, we can see that there are two ways to define a “capacitance matrix” for a multi-conductor system. The matrix of mutual capacitances has the advantage that the network can be analysed via circuit theory to find the parallel combination of capacitors between any two conductors. Conversely, the Maxwell capacitance matrix has the advantage that it is easy to calculate. We will describe the process to calculate the Maxwell capacitance matrix next.

How to calculate capacitance

Consider once more the 3 capacitor system of Figure 2. Suppose that we apply a voltage to conductor 1, and we ground all the other conductors. Therefore, substituting and into Eq. (12), we find

which leads to simple expressions for the first column of the Maxwell capacitance matrix:

Next, we apply a voltage to conductor 2 and ground all the other conductors, to calculate the second column of the Maxwell capacitance matrix. Continuing in this way, we obtain the full description of all capacitive interactions in the system.

Practical example

Below is an example of analysing a real example using Matlab’s PDE Toolbox.

Defining the geometry

Firstly, you must draw the geometry that you wish to calculate. The geometry must cover the region surrounding the conductors with a sufficiently large “air gap” around the outside. Typically the easiest way to do this is to define a large rectangle, then draw features within this region. The features you draw will be either conductors or regions of differing permittivity. An example is shown in Figure 17.

The Matlab PDE toolbox requires geometries specified in the STL file format. For 2D problems, such as this, the STL file must have all its geometry lying with a plane. This means that you cannot extrude your sketch into a 3D shape, because the result will be a 3D geometry that will be treated by Matlab as a 3D problem.

To generate a planar geometry in Fusion 360, select the “Surface” toolbar and click “Patch” (as shown in Figure 18).

Next, click on the air gap part of the sketch to create the patch. There will be holes in the patch where the electrodes and other features are defined. You can then export this patch to an STL file by right clicking on it in the browser and selecting the Save As Mesh option (Figure 19).

You should export the STL in binary format using metres as the dimension (Figure 20), since if you specify geometry in SI units then you avoid having to perform unit conversions later.

You may download this example geometry as:

• Editable Fusion 360 file
• Exported planar STL.

Loading the geometry into Matlab

The STL file is loaded and displayed as follows.

%% Load geometry and show the edges
geometry = importGeometry('geometry.stl');

figure();
pdegplot(geometry,'EdgeLabels','on');
xlabel('X distance (m)');
ylabel('Y distance (m)');

By zooming in on the figure in Matlab, you can identify which edges correspond to which features.

% specify the edge IDs
outside_edges = [2 4 11 13];
dielectric_edges = [7 10 15 14];
electrode1_edges = [4 1 8 6];
electrode2_edges = [9 12 5 16];

This geometry currently has three “holes” in it. There is a hole where we intend to place the dielectric, and a hole for each of the electrodes. The dielectric must be “filled in” so that the electric potential will be calculated in this region. This can be done using the addFace function:

% fill in the dielectric region
geometry = addFace(geometry, dielectric_edges);
% leave the conductors as holes in the geometry

% show the faces 
pdegplot(geometry,'FaceLabels','on');

Notice that the electrodes are left as holes. This is important. The Matlab PDE toolbox can only specify voltage boundary conditions at the edges of the domain. Therefore, every conductor must be treated as a gap within the domain. If you accidentally use addFace to fill in a conductor, then the voltage specified there will be completely ignored.

Preparing the mesh

The finite element solver uses triangular meshes to represent the geometry. You need smaller meshes near the electrodes (to capture the rapid variation in voltage in this region). This can be achieved using the HEdge parameter when calling generateMesh.

%% create the model and look at the mesh
emagmodel = createpde('electromagnetic','electrostatic');
emagmodel.Geometry = geometry;

generateMesh(emagmodel, "HEdge", {[electrode1_edges electrode2_edges], 1e-5}); 
figure();
pdeplot(emagmodel);

You must have the mesh covering all the dielectric regions, and there must be a hole in the mesh for each conductor.

Setting material properties and boundary conditions

%% Set material properties
emagmodel.VacuumPermittivity = 8.8541878128E-12;
electromagneticProperties(emagmodel, "Face", 1, "RelativePermittivity", 1);
electromagneticProperties(emagmodel, "Face", 2, "RelativePermittivity", 15);

%% Set conductor 1 to 1V and others to 0, and find the charge on each conductor
electromagneticBC(emagmodel, "Voltage", 1, "Edge", electrode1_edges);
electromagneticBC(emagmodel, "Voltage", 0, "Edge", electrode2_edges);
electromagneticBC(emagmodel, "Voltage", 0, "Edge", outside_edges);

% Solve
R = solve(emagmodel); 

% Plot electric potential
figure()
pdeplot(emagmodel,'XYData',R.ElectricPotential);
xlabel('X distance (m)');
ylabel('Y distance (m)');
c=colorbar(gca());
c.Label.String = 'Electric potential (V)';
axis tight;
axis equal;
hold on;
pdegplot(geometry);

Calculating capacitance

Recall that to calculate capacitance we need to know the charge on each conductor, and for that, we must perform a line integral around the conductor:

The attached code implements a function charge_on_conductor() that will perform this integral.

Given the solution above (where conductor 1 has a voltage of 1V, and all other conductors are grounded), the entries in the Maxwell capacitance are given by the code below.

C11 = charge_on_conductor(R, electrode1_edges)
C21 = charge_on_conductor(R, electrode2_edges)

% Calculate the mutual capacitance
Cm12 = -C12
Cm11 = C11 - Cm12

The results (for the values in the code above) are

Since this is a 2D problem, it represents a slice through a real 3D geometry. Therefore the units are farads per metre, i.e. the amount of capacitance per depth into the page.

We can interpret these capacitances by recalling the equivalent circuit of these self and mutual capacitances (Figure 24).

A capacitance meter connected between the two electrodes would measure the parallel combination of capacitances, i.e.

Complete code sample

• Click here to download the complete Matlab script used in this notes.

Conclusion

Capacitance is commonly used for sensing purposes because it enables relatively long range, contactless measurements. The most famous example of capacitive sensing is the touchscreen found in many portable electronics devices. Capacitance is also common for sensing the presence or absence of various materials, for humidity sensing, and more.

There are two main types of interface circuits for capacitive sensors. In many designs, an AC voltage source is used to treat the capacitance as a complex impedance, and the impedance measured using similar methods to that of resistive sensors. In other designs, changes in capacitance control the timing of an oscillator or the time constant of an RC circuit.

We have developed a method to calculate the capacitance of arbitrary geometries. You can use this to design capacitive sensors. For example, if the underlying mechanism involves changes in permittivity, then you run the calculations with different values of permittivity and examine the resulting changes in capacitance. If your design involves changes in geometry, then you can draw multiple geometries and calculate the capacitance for each one. In this way, you can trial different design concepts and identify the one that will lead to the highest performing sensor.

You will apply this knowledge in Assignment 2 to design your own capacitive sensor.

References

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

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