# Sensor Systems

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theory:sensor_technology:st12_impedance_spectroscopy

# Transducer Characterization by Impedance Spectroscopy

Transducers (both sensors and actuators) can be modelled as a network of discrete equivalent components by a method called Impedance Spectroscopy. The method uses the fact that a two-port passive linear network is completely defined by its response in case we know the phase and magnitude of its impedance over a sufficiently broad bandwidth. As a result, we can find an equivalent network for a transducer by measuring the electrical impedance at several frequencies. The result is a network, both the structure and the values of discrete passive components (R, L and C), that describe the transducer sufficiently.

## Introduction

This method is commonly used in the field of electrochemistry, and good tutorails can be found when searching for Electrochemical Impedance Spectroscopy. An example of such a tutorial is published by Gamry Instruments on line1). The original method was developed mainly by J.Ross McDonald 2)3) in 1987 and also published by Bernard Boukamp in 19954).

Consider a two-port black box circuit as drawn in figure 1. The only thing we know about the box is that it contains time-invariant, linear, and passive components. In practice, a network of resistors, inductors and capacitors (RLC) satisfies this criterion.

Fig. 1: The time response on a step function applied to a two-port black box

Now imagine we apply a known voltage to the two-port, for example a step

$$\begin{cases} U(t)=0 & \text{ if } t< 0 \\ U(t)=U_{0} & \text{ if } t\geqslant 0 \end{cases}$$

and we measure the current as a function of time as

$$I(t)=G\left ( U(t) \right ) \label {eq:TransferAdmittanceForm}$$

with $G$ the transfer function for $U(t)\rightarrow I(t)$. Note that in a similar way, we can apply a current $$\begin{cases} I(t)=0 & \text{ if } t< 0 \\ I(t)=I_{0} & \text{ if } t\geqslant 0 \end{cases}$$

and measure the potential as a function of time

$$U(t)=H\left ( I(t) \right ). \label {eq:TransferImpedanceForm}$$

Now $H$ is a transfer function for $I(t)\rightarrow U(t)$. We call $G$ the admittance function and $H$ the impedance function5). What is most important, is that based on observing the transfer function, we have a strong clue what is inside the box. For example, the response of figure 1 strongly reminds us of an RC-series network. Based on the time constant $\tau$, we can even determine the RC-product.

The assumed RC-network is an equivalent circuit: it is not necessarily the real network inside the box. Later on we will see that some responses can be implemented by multiple different equivalent circuits: in most cases there is not a unique topology. However, the shape of the network has to be determined to resemble logical physical phenomena. For example, when the response of figure 1 is observed with a electrolytic capacitor, the series network of a resistor and a capacitor makes sense: we have modelled it as an ideal capacitor with a series resistor representing the resistance of the connection wires.

Because a step function contains all frequencies, the response of a network on a step function gives a broad-spectrum fingerprint. Therefore, the transfer function of the black box is completely characterised by applying a step. Both for mathematical and practical reasons it is more convenient to apply a series of harmonic signals (sine-waves) to characterise the black box. This needs some knowledge on the analysis by Fourier series.

## Example 1: Fourier analysis of RC networks

Assume the black box is filled with a series network of a resistor and a capacitor like figure 2. The differential equation is $$I(t)=C\frac{\mathrm{d} }{\mathrm{d} t}\left [ U(t)-RI(t) \right ]$$

and for the boundary condition

$$\begin{cases} U(t)=0 & \text{ if } t< 0 \\ U(t)=U_{0} & \text{ if } t\geqslant 0 \end{cases}$$

we find the solution $$I(t)=\frac{U_{0}}{R}e^{-\frac{t}{RC}}. \label{eq:TransferFunctionTimeRCseries}$$

Fig. 2: Example for a two-port that is a series network of a resistor and a capcitor

Equation \eqref{eq:TransferFunctionTimeRCseries} is the transfer function $G$ in the admittance form \eqref{eq:TransferAdmittanceForm}. Although this response characterises the circuit completely, it is easier to approach the identification in the frequency domain than in the time domain. For the frequency domain we have to apply the theory of Fourier transforms.

Any time dependent function can be written as an infinite series of harmonic terms. This is the Fourier transform where the time signal is represented by the sum of the harmonics

$$U(t)=\frac{1}{2 \pi}\int_{-\infty }^{\infty } \! \overline{U}(\omega)e^{j \omega t} \ \mathrm{d} \omega$$

with the coefficients $\overline{U}(\omega)$ given by

$$\overline{U}(\omega)=\int_{-\infty }^{\infty } \! U(t)e^{-j \omega t} \ \mathrm{d} t$$

with all $\overline{U}(\omega)$ orthogonal. Note that this makes use of Euler's identity

$$e^{jx}=\cos (x) + j \sin (x).$$

We can directly describe the impedance of the circuit of figure 2 as

$$\overline{Z}_{RCs}(j \omega)=R+\frac{1}{j \omega C}$$

and the admittance as $$\overline{Y}_{RCs}(j \omega)=\frac{R^{-1}\cdot j \omega C}{R^{-1} + j \omega C}=\frac{j \omega C}{1+j \omega RC}$$

where the convention is to write $H(t) \rightarrow \overline{Z}(j \omega)$ for the impedance and $G(t) \rightarrow \overline{Y}(j \omega)$ for the admittance expressions of the transfer function in the time- and frequency domain respectively.

In figure 3 the Bode diagram of the series circuit figure 2 is drawn. In a Bode diagram, the magnitude $\left | \overline{Z} \right |= \sqrt { Re^{2} + Im^{2} }$ is plotted as a function of the frequency on a log-log scale. The second part of the Bode diagram is the phase of $\overline{Z}$ as a function of frequency on a log-lin scale. As a reference, the Bode diagram for a parallel circuit of a resistor and a capacitor is drawn as well, that will result into an impedance of

$$\overline{Z}_{RCp}(j \omega)= \frac{R\cdot \frac{1}{j \omega C}}{R + \frac{1}{j \omega C}}=\frac{R}{1+j \omega RC }$$

$$\overline{Y}_{RCp}(j \omega)=R^{-1} + \frac{1}{j \omega C}$$

Fig. 3: The bode plot for the impedance of a series and parallel RC network

For electrical engineerins, the Bode diagram is a common representation of the transfer function. However, there are other options to plot the same transfer functions. For recognising patterns of networks, the Polar plot is more convenient because it represents both phase and magnitude in a single graph. In a Polar plot, the horizontal axes is the real part and the vertical axes the imaginary part. Here the convention is used to plot the imaginary part with a minus sign. A vector from the origin to a certain frequency has a length $|\overline{Z}|$ and an angle $arg(\overline{Z})$.

Fig. 4: The polar plot for the impedance of a series and parallel RC network

In the series case, we can see a pure resistive element of $10 k\Omega$ on the horizontal real axes for $\omega \rightarrow \infty$. In the same figure, the phase goes to $\pi /2$ for DC, meaning $\omega \rightarrow 0$. An upward line in the upper-right quadrant represents pure capacitive behaviour. The right-hand picture represents the case where the resistor and capacitor are in parallel. In the impledance plot, this combination becomes a clear semi-circle on the upper-right quadrant. Using polar plots, we can find combinations of resistors and capacitors by pattern recognition.

## Example 2: Fourier analysis of RL networks

In case the black-box circuit of figure 2 is not filled with a resistor and a capacitor, but with a resistor and an inductor, the impedance equation becomes for a series circuit

$$\overline{Z}_{RLs}(j \omega)=R+j \omega L$$

$$\overline{Y}_{RLs}(j \omega)=\frac{R^{-1}\cdot \left ( j \omega L \right )^{-1}}{R^{-1} + \left ( j \omega L \right )^{-1}}=\frac{1}{R+j \omega L}$$

The Bode diagram and the polar plots can be seen in figure 5 and figure 6 respectively. It can be seen that in an resistor-inductor configuration, the polar plot goes from infinite in the lower-right quadrant to the real axes for DC for a series circuit. A semi-circle in the lower-right quadrant can be seen for the parallel configuration.

Fig. 5: The bode plot for the impedance of a series and parallel RL network

Fig. 6: The polar plot for the impedance of a series and parallel RL network

## Equivalent topologies

When it is claimed that combinations of capacitors, resistors and inductors result into characteristic patterns in the polar plot, we may think all more complex circuits can be recognised as well. This is not the case. The two circuits in figure 7 result in the same polar plot and Bode diagrams.

Fig. 7: These two basic structures can not be distinguished form the polar plots or Bode diagrams

The shared polar plot is represented in figure 8 and is equal if \begin{aligned} R_{1} & = R_{a}\left ( 1+\frac{R_{a}}{R_{c}} \right )\\ C_{2} & = \left ( \frac{R_{c}}{R_{a}+R_{c}} \right )^{2}C_{b}\\ R_{3} & = R_{a}+R_{c}. \end{aligned}

In this specific plot, the values $R_{a} = 1k\Omega$, $R_{c} = 10k\Omega$ and $C_{b} = 1\mu F$ are chosen.

Fig. 8: The polar plot for the two equivalent circuits

Fletcher6) descibes the equivalences in a more standardized way, and gives many more equivalent circuit topologies.

## Manual data fitting of RC circuits

We have seen that in the polar plot

• a series capacitor results into a vertical line in the upper-right quadrant
• a series resistor results into a shift along the real axis
• an RC parallel combination results into a semi-circle in the upper-right quadrant.

In fact, a series resistor is an RC parallel combination with an infinitesimally small capacitor, and a series capacitor is an RC parallel combination with an infinitely large resistor. Knowing this, we can assume that any complex circuit of resistors and capacitors consists of a series circuit of parallel combinations as represented in the circuit of figure 9.

Fig. 9: A complex RC circuit can be modelled as a series of R-C parallel combinations

The circuit of figure 9 is described by

$$\overline{Z}_{fitted}(\omega)=\sum_{i=1}^{N}\frac{R_{i}}{1+j \omega R_{i}C_{i}} \label{eq:TransferFunctionFrequencyRCseries}$$

with $R_{i}$ and $C_{i}$ the $N$ combinations of parallel resistor-capacitor combinations.

The number $N$ of the series is equal to the number of semi-circles observed in the polar plot, plus one for the optional series resistor and one for the optional series capacitor. Note that this is an empirical model: to change the topology to a model structure that makes physically sense, the strcutuce can be transformed into an equivalent circuit that matches physical phenomena by the method of Fletcher7) can be used.

As an example, consider the polar plot of figure 10. In figure 11 the same polar plot is used to fit two circles. Note that the circles must touch because when $\omega$ becomes infinite, the sum of all resistances is equal to the real part.

Fig. 10: An example of a polar plot for an unknown circuit Fig. 11: The same polar plot of an unknown circuit with fitted circles

Now we can derive:

• This circuit has at least two parallel resistor-capacitor combinations;
• There is an offset along the real axis: for high frequencies, the plot becomes completely real at $1k \Omega$. This means there is a series resistor of $R_{0} = 1k \Omega$;
• For low frequencies, there is a completely imaginary vertical line, indicating a series capacitor. There is one measurement point taken along this line (not indicated in the graph): at $\omega = 1 rad/s$ the imaginary part is $10.105 k \Omega$. This means that the series capacitor is $C_{3}=99.0 \mu F$ because $\overline{Z}=1/j\omega C$.

The next question is how to deal with the two semi-circles. In figure 12, it can be seen that the width of the semi circle is equal to the resistor value $R$. The capacitor value can be found from the frequency where we have $\pi /4$ phase shift, because there $\omega_{\pi /4} = RC$.

Fig. 12: How to derive the R and C from a polar plot semi-circle

To go back to figure 10, we find: $R_{1}=20k \Omega$ and $C_{1}=0.01 \mu F$ as a pair and $R_{2}=10.1k \Omega$ and $C_{2}=99.0 \mu F$ as the second pair as shown in figure 13.

Fig. 13: The structure of the emperically fitted circuit

## Systematic data fitting: elimination method

Equation \eqref{eq:TransferFunctionFrequencyRCseries} which describes the universal structure of figure 9 shows something important: every parallel RC-combination adds a term to the equation. This means that if we find one RC comination, we can subtract this from the response. Next, another subcircuit RC combination will become visible.

Consider the measured response of figure 10. Assume it is measured as $N$ points with each a real impedance $\overline{Z}_{Re,i}$ and an imaginary impedance $\overline{Z}_{Im,i}$ (that is what impedance analyzers do). For each point, we know the applied radial frequency. So we have three data vectors $\mathbf{\omega}$, $\mathbf{\overline{Z}_{Re}}$, and $\mathbf{\overline{Z}_{Im}}$ with the $N$ elements $\omega_{i}$, $\overline{Z}_{Re,i}$, and $\overline{Z}_{Im,i}$.

First, we observe that the biggest fitted circle (but we can also start with another feature) in figure 11 has a diameter of $20 k \Omega$. This measn that there is one RC-combination in the equivalent circuit that has an $R_{1}=20 k \Omega$. With the method of figure 12 and finding the top of the corresponding circle at $\omega = 5012 rad/s$, we can find $C_{1}=0.01 \mu F$.

So far, this is the same as in the previous paragraph. But now we subtract the spectrum of the found subcircuit from the datasets $\overline{Z}_{Re,i}$ and $\overline{Z}_{Re,i}$:

\begin{aligned} \overline{Z}^{-R_{1}C_{1}}_{Re,i} & = \overline{Z}_{Re,i}-Re\left ( \overline{Z}_{R_{1}C_{1}} \right )\\ & = \overline{Z}_{Re,i}-Re\left ( \frac{R_{1}}{1+j\omega_{i} R_{1}C_{1}} \right )\\ & = \overline{Z}_{Re,i}- \frac{R_{1}}{1+\left (\omega_{i} R_{1}C_{1}\right )^{2}} \label{eq:EliminateReal} \end{aligned}

and

\begin{aligned} \overline{Z}^{-R_{1}C_{1}}_{Im,i} & = \overline{Z}_{Im,i}-Im\left ( \overline{Z}_{R_{1}C_{1}} \right )\\ & = \overline{Z}_{Im,i}-Im\left ( \frac{R_{1}}{1+j\omega_{i} R_{1}C_{1}} \right )\\ & = \overline{Z}_{Im,i}+ \frac{\omega_{i}R_{1}^{2}C_{1}}{1+\left (\omega_{i} R_{1}C_{1}\right )^{2}} . \label{eq:EliminateImag} \end{aligned}

Next repeat this elimination to make $\overline{Z}^{-R_{2}C_{2}}_{Re,i}$ from $\overline{Z}^{-R_{1}C_{1}}_{Re,i}$ and $\overline{Z}^{-R_{2}C_{2}}_{Im,i}$ from $\overline{Z}^{-R_{1}C_{1}}_{Im,i}$ until only one small dot at the origin remains.

This procedure is applied to the dataset in figure 14.

Fig. 14: Our dataset is reduced in four steps by subtracting the calculated responses of the found RC combinations

What has happened in fact is illustrated in figure 15: we model the equivalent circtui by equation \eqref{eq:TransferFunctionFrequencyRCseries} which describes the universal structure of figure 9.

Fig. 15: The overall procedure of recognizing several RC combinations from a polar plot

## Automated model fitting

Now have a model (equation \eqref{eq:TransferFunctionFrequencyRCseries} and figure 9), and we derived the interpretation that the response is characterised by identifying semi-circles to find $R_{i}$ and $\omega_{i}$ and subsequently $C_{i}$ as illustrated in figure 15. It should not be difficult to fit the semi-circles automatically. However, because equation \eqref{eq:TransferFunctionFrequencyRCseries} is not linear towards $R_{i}$ and $C_{i}$, this can not be done using a simple LMS algorithm

To be done……

## The Kramers-Kronig test

From the complex functions theory, we know that under certain circumstances, there is a deterministic relation between the imaginary and real data (or magnitude and phase) in a single spectrum. Mathematically, the conditions are that the system should be causal. Practically for us it means that our systems are passive and stationary: constant resistors, capacitors and inductors. This relation is described by the Kramers-Kronig relations, which state that for causal complex plane spectral data there is a dependency between magnitude and phase. The real part of a spectrum can be obtained by an integration of the imaginary part and vice versa as described in the Kramers-Kronig equations:

\begin{aligned} Z_{Re}\left ( \omega \right ) & = Z_{Re}\left ( \infty \right )+\frac{2}{\pi}\int_{0}^{\infty}\frac{xZ_{im}\left ( x \right )-\omega Z_{im}\left ( \omega \right )}{x^{2}-\omega^{2}}dx \\ Z_{im}\left ( \omega \right ) & =\frac{2}{\pi}\int_{0}^{\infty}\frac{Z_{re}\left ( x \right )- Z_{re}\left ( \omega \right )}{x^{2}-\omega^{2}}dx \end{aligned}

This means that Kramers-Kronig relations can be used to evaluate data quality. What is done by Boukamp8), is that the universal model of equation \eqref{eq:TransferFunctionFrequencyRCseries} and figure 9 is applied to the dataset. If the Kramers-Kronig relations can be acknowledged by the fitted model, we may assume the system is passive, stationary and causal (there is for example no drift). The trick of Boukamp is that he fits the model with $N$ compinations of RC circuits when we have $N$ measurements in the dataset. This makes no physical sense, because we want to have RC combinations that represent physical phenomena, but acording to Boukamp this linearises the fitting algorithm just for the test.

## How to measure impedance?

• An impedance analyzer, or the 2-port option of a network analyzer9).
• A configuration around the Analog Devices AD593310), like the evaluation kit of Analog Devices11) or the PmodIA module of Digilent12). See my explorations of using this board.

# Sensor Technology TOC

These are the chapters for the Sensor Technology course:

1)
Gamry Instruments, Basics of Electrochemical Impedance Spectroscopy http://www.gamry.com/application-notes/EIS/basics-of-electrochemical-impedance-spectroscopy/
2)
Mcdonald, J. Ross. “Impedance spectroscopy: emphasizing solid materials and systems.” Impedance Spectroscopy Emphasizing Solid Materials and Systems (1987).
3)
Macdonald, J. R., & Barsoukov, E. (2005). Impedance spectroscopy: theory, experiment, and applications. History, 1(8).
4) , 8)
B.A. Boukamp, J. Electrochem. Soc, 142, 1885 (1995).
5) , 9)
Keysight Technologies, Impedance Measurement Handbook, A guide to measurement technology and techniques, 6th Edition, Application Note, http://literature.cdn.keysight.com/litweb/pdf/5950-3000.pdf
6) , 7)
Fletcher, S. (1994). Tables of Degenerate Electrical Networks for Use in the Equivalent‐Circuit Analysis of Electrochemical Systems. Journal of The Electrochemical Society, 141(7), 1823-1826.
10)
Analog Devices, 1 MSPS, 12 Bit Impedance Converter Network Analyzer, http://www.analog.com/en/products/rf-microwave/direct-digital-synthesis-modulators/ad5933.html