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 theory:sensor_technology:st12_impedance_spectroscopy [2019/03/19 07:25]glangereis [Downloads] — (current) Both sides previous revision Previous revision 2019/03/19 07:25 glangereis [Downloads] 2019/03/19 07:22 glangereis [How to measure impedance?] 2019/03/19 07:21 glangereis [Downloads] 2018/11/01 19:46 glangereis [How to measure impedance?] 2018/11/01 19:45 glangereis [How to measure impedance?] 2018/11/01 19:42 glangereis [How to measure impedance?] 2018/11/01 19:39 glangereis [How to measure impedance?] 2018/10/09 20:03 ↷ Links adapted because of a move operation2018/10/08 22:58 ↷ Links adapted because of a move operation2018/10/08 18:26 glangereis [Sensor Technology TOC] 2018/10/08 18:25 glangereis ↷ Links adapted because of a move operation2017/10/10 18:40 glangereis [Sensor Technology TOC] 2017/10/10 17:57 glangereis [Sensor Technology TOC] 2017/10/10 17:47 glangereis ↷ Links adapted because of a move operation2017/10/10 17:46 glangereis ↷ Page name changed from theory:sensor_technology:st10_impedance_spectroscopy to theory:sensor_technology:st12_impedance_spectroscopy2017/10/10 17:36 glangereis ↷ Links adapted because of a move operation2017/10/10 17:30 glangereis ↷ Links adapted because of a move operation2017/10/10 17:25 glangereis ↷ Page name changed from theory:sensor_technology:stbs6_impedance_spectroscopy to theory:sensor_technology:st10_impedance_spectroscopy2017/10/10 17:20 glangereis ↷ Links adapted because of a move operation2017/10/10 17:00 glangereis ↷ Links adapted because of a move operation2017/10/10 16:55 glangereis ↷ Links adapted because of a move operation2017/10/10 16:55 glangereis ↷ Page moved from theory:stbs:stbs6_impedance_spectroscopy to theory:sensor_technology:stbs6_impedance_spectroscopy2016/12/05 17:46 geert [Introduction] 2016/12/05 17:45 geert [How to measure impedance?] 2016/12/05 17:43 geert [How to measure impedance?] 2016/12/05 15:56 geert 2019/03/19 07:25 glangereis [Downloads] 2019/03/19 07:22 glangereis [How to measure impedance?] 2019/03/19 07:21 glangereis [Downloads] 2018/11/01 19:46 glangereis [How to measure impedance?] 2018/11/01 19:45 glangereis [How to measure impedance?] 2018/11/01 19:42 glangereis [How to measure impedance?] 2018/11/01 19:39 glangereis [How to measure impedance?] 2018/10/09 20:03 ↷ Links adapted because of a move operation2018/10/08 22:58 ↷ Links adapted because of a move operation2018/10/08 18:26 glangereis [Sensor Technology TOC] 2018/10/08 18:25 glangereis ↷ Links adapted because of a move operation2017/10/10 18:40 glangereis [Sensor Technology TOC] 2017/10/10 17:57 glangereis [Sensor Technology TOC] 2017/10/10 17:47 glangereis ↷ Links adapted because of a move operation2017/10/10 17:46 glangereis ↷ Page name changed from theory:sensor_technology:st10_impedance_spectroscopy to theory:sensor_technology:st12_impedance_spectroscopy2017/10/10 17:36 glangereis ↷ Links adapted because of a move operation2017/10/10 17:30 glangereis ↷ Links adapted because of a move operation2017/10/10 17:25 glangereis ↷ Page name changed from theory:sensor_technology:stbs6_impedance_spectroscopy to theory:sensor_technology:st10_impedance_spectroscopy2017/10/10 17:20 glangereis ↷ Links adapted because of a move operation2017/10/10 17:00 glangereis ↷ Links adapted because of a move operation2017/10/10 16:55 glangereis ↷ Links adapted because of a move operation2017/10/10 16:55 glangereis ↷ Page moved from theory:stbs:stbs6_impedance_spectroscopy to theory:sensor_technology:stbs6_impedance_spectroscopy2016/12/05 17:46 geert [Introduction] 2016/12/05 17:45 geert [How to measure impedance?] 2016/12/05 17:43 geert [How to measure impedance?] 2016/12/05 15:56 geert 2016/09/30 12:53 geert [Example 1: Fourier analysis of RC networks] 2016/09/30 12:51 geert 2016/09/30 12:50 geert [Systematic data fitting: elimination method] 2016/09/30 12:50 geert [Systematic data fitting: elimination method] 2016/09/30 12:48 geert [Systematic data fitting: elimination method] 2016/09/30 12:46 geert [Manual data fitting of RC circuits] 2016/09/27 06:39 geert [ST TOC] 2016/09/26 19:25 geert [ST TOC] 2016/09/24 21:29 glangereis [ST TOC] 2016/09/24 13:33 glangereis 2016/09/23 09:54 geert [Example 1: Fourier analysis of RC networks] 2016/09/23 09:53 geert [Example 1: Fourier analysis of RC networks] 2016/09/23 09:51 geert 2016/09/22 12:55 glangereis ↷ Links adapted because of a move operation2016/09/22 12:54 glangereis ↷ Links adapted because of a move operation2016/09/22 12:52 glangereis ↷ Links adapted because of a move operation2016/09/22 12:51 glangereis ↷ Links adapted because of a move operation2016/09/22 12:51 glangereis ↷ Links adapted because of a move operation2016/09/22 12:50 glangereis ↷ Links adapted because of a move operation2016/09/22 12:50 glangereis ↷ Links adapted because of a move operation2016/09/22 12:49 glangereis ↷ Page moved from background:modelling:stbs6_impedance_spectroscopy to theory:stbs:stbs6_impedance_spectroscopy2016/09/22 12:49 glangereis ↷ Links adapted because of a move operation2016/09/22 12:48 glangereis ↷ Links adapted because of a move operation2016/09/22 12:45 glangereis ↷ Links adapted because of a move operation2016/09/22 12:44 glangereis ↷ Links adapted because of a move operation Line 1: Line 1: - ====== 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 line((Gamry Instruments,​ Basics of Electrochemical Impedance Spectroscopy [[ - http://​www.gamry.com/​application-notes/​EIS/​basics-of-electrochemical-impedance-spectroscopy/​]])). The original method was developed mainly by J.Ross McDonald ((Mcdonald, J. Ross. "​Impedance spectroscopy:​ emphasizing solid materials and systems."​ Impedance Spectroscopy Emphasizing Solid Materials and Systems (1987).))((Macdonald,​ J. R., & Barsoukov, E. (2005). Impedance spectroscopy:​ theory, experiment, and applications. History, 1(8).)) in 1987 and also published by Bernard Boukamp in 1995((B.A. Boukamp, J. Electrochem. Soc, 142, 1885 (1995). )). - - Consider a two-port black box circuit as drawn in ​. 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. - - <​imgcaption timeresponse|The time response on a step function applied to a two-port black box>​{{theory:​sensor_technology:​modelling:​impedancespectroscopy_fig1.png?​500}}​ - - 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//​ function((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]])). 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 ​ 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 ​ 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 ​. 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} - - - <​imgcaption rctimeresponse|Example for a two-port that is a series network of a resistor and a capcitor>​{{theory:​sensor_technology:​modelling:​impedancespectroscopy_fig2.png?​300}}​ - - 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 ​ 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 ​ the Bode diagram of the series circuit ​ 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 } - - - and the admittance of - - - \overline{Y}_{RCp}(j \omega)=R^{-1} + \frac{1}{j \omega C} - - - <​imgcaption bodeplotRC|The bode plot for the impedance of a series and parallel RC network>​{{theory:​sensor_technology:​modelling:​bodeplotrc.png?​400}}​ - - 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 $arg(\overline{Z})$. - - <​imgcaption polarplotRC|The polar plot for the impedance of a series and parallel RC network>​{{theory:​sensor_technology:​modelling:​polarplotrc.png?​500}}​ - - 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 ​ 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 - - - and the admittance is - - - \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 ​ and ​ 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. - - <​imgcaption bodeplotRL|The bode plot for the impedance of a series and parallel RL network>​{{theory:​sensor_technology:​modelling:​bodeplotrl.png?​400}}​ - - <​imgcaption polarplotRL|The polar plot for the impedance of a series and parallel RL network>​{{theory:​sensor_technology:​modelling:​polarplotrl.png?​500}}​ - - ===== 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 ​ result in the same polar plot and Bode diagrams. - - <​imgcaption equivalent_circuits|These two basic structures can not be distinguished form the polar plots or Bode diagrams>​{{theory:​sensor_technology:​modelling:​equivalent_circuits.png?​450}}​ - - The shared polar plot is represented in ​ 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. - - <​imgcaption equivalent_polarplot|The polar plot for the two equivalent circuits>​{{theory:​sensor_technology:​modelling:​equivalent_polarplot.png?​300}}​ - - Fletcher((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.)) 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 ​. - - <​imgcaption series_rc_parallel|A complex RC circuit can be modelled as a series of R-C parallel combinations>​{{theory:​sensor_technology:​modelling:​series_rc_parallel.png?​200}}​ - - The circuit of ​ 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 Fletcher((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.)) can be used. - - As an example, consider the polar plot of ​. In ​ 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. - - <​imgcaption polar_plot_fit0|An example of a polar plot for an unknown circuit>​{{theory:​sensor_technology:​modelling:​polar_plot_fit0.png?​300}}​ - <​imgcaption polar_plot_fit1|The same polar plot of an unknown circuit with fitted circles>​{{theory:​sensor_technology:​modelling:​polar_plot_fit1.png?​300}}​ - - 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 ,​ 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 $\omega_{\pi /4} = RC$. - - <​imgcaption semi_circle|How to derive the R and C from a polar plot semi-circle>​{{theory:​sensor_technology:​modelling:​semi_circle.png?​300}}​ - - To go back to ,​ 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 ​. - - <​imgcaption fitted_circuit|The structure of the emperically fitted circuit>​{{theory:​sensor_technology:​modelling:​fitted_circuit.png?​400}}​ - ===== Systematic data fitting: elimination method ===== - - Equation \eqref{eq:​TransferFunctionFrequencyRCseries} which describes the universal structure of ​ 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 ​. 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 ​ 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 ​ 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 ​. - - <​imgcaption eliminated_circuit|Our dataset is reduced in four steps by subtracting the calculated responses of the found RC combinations>​{{theory:​sensor_technology:​modelling:​eliminate.png?​300}}​ - - What has happened in fact is illustrated in :​ we model the equivalent circtui by equation \eqref{eq:​TransferFunctionFrequencyRCseries} which describes the universal structure of ​. - - <​imgcaption polarplotfit|The overall procedure of recognizing several RC combinations from a polar plot>​{{theory:​sensor_technology:​modelling:​polarplotfit.png?​400}}​ - ===== Automated model fitting ===== - - Now have a model (equation \eqref{eq:​TransferFunctionFrequencyRCseries} and ​),​ 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 ​. 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 Boukamp((B.A. Boukamp, J. Electrochem. Soc, 142, 1885 (1995). )), is that the universal model of equation \eqref{eq:​TransferFunctionFrequencyRCseries} and ​ 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 (HP4194, HP4294, E4990, E4991), or the 2-port option of a network analyzer((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]])). ​ - * A configuration around the Analog Devices AD5933((Analog Devices, 1 MSPS, 12 Bit Impedance Converter Network Analyzer, [[http://​www.analog.com/​en/​products/​rf-microwave/​direct-digital-synthesis-modulators/​ad5933.html]])),​ like the evaluation kit of Analog Devices((Analog Devices, AD5933 Evaluation Board, [[http://​www.analog.com/​en/​design-center/​evaluation-hardware-and-software/​evaluation-boards-kits/​EVAL-AD5933.html]])) or the PmodIA module of Digilent((Digilent PmodIA, [[http://​store.digilentinc.com/​pmodia-impedance-analyzer/​]])). See my [[theory:​background:​digilent_pmodia|explorations]] of using this board and the LabVIEW code below. - * A high-end LCR meter like the [[https://​www.keysight.com/​en/​pd-2279116-pn-E4980AL/​precision-lcr-meter-20-hz-to-300-khz-500-khz?​cc=NL&​lc=dut|Keysight E4980AL]]. This LCR meter can measure impedances at a range of frequencies. By controlling it with MATLAB or LabVIEW (see code below), complete Bodeplots ans Polar plots can be made. - - ===== Downloads ===== - - ^ File ^ Program ^ Version ^ Description ^ - | [[http://​www.geertlangereis.nl/​data_extra/​Keysight_Impedance_Spectroscopy_LabVIEW_v1_0.zip| Keysight_Impedance_Spectroscopy_LabVIEW_v1_0.zip]] | LabVIEW 2018 |  1.0  | LabVIEW vi for an impedance sweep (polar or Bode plot) on a Keysight E4980AL | - | [[http://​www.geertlangereis.nl/​data_extra/​Impedance_Spectroscopy_AD5933_LabVIEW_2018.zip| Impedance_Spectroscopy_AD5933_LabVIEW_2018.zip]] | LabVIEW 2018 |  1.0  | LabVIEW vi for an impedance sweep (polar or Bode plot) on a AD5933 using a BusPirate for I2C | - - ====== Sensor Technology TOC ====== - - These are the chapters for the Sensor Technology course: - * Chapter 1: [[theory:​sensor_technology:​st1_measurement_theory|Measurement Theory]] - * Chapter 2: [[theory:​sensor_technology:​st2_measurement_errors|Measurement Errors]] - * Chapter 3: [[theory:​sensor_technology:​st3_measurement_technology|Measurement Technology]] - * Chapter 4: [[theory:​sensor_technology:​stb_conventions_for_good_graphs_and_tables|Circuits,​ Graphs, Tables, Pictures and Code]] - * Chapter 5: [[theory:​sensor_technology:​st4_sensor_theory|Basic Sensor Theory]] - * Chapter 6: [[theory:​sensor_technology:​st6_sensoractuatorsystems|Sensor-Actuator Systems]] ​ - * Chapter 7: [[theory:​sensor_technology:​st7_modelling_main|Modelling]] - * Chapter 8: [[theory:​sensor_technology:​st8_accelerometer_model|Modelling:​ The Accelerometer]] - example of a second order system - * Chapter 9: [[theory:​sensor_technology:​st9_scaling|Modelling:​ Scaling]] - why small things appear to be stiffer - * Chapter 10: [[theory:​sensor_technology:​st10_lumped_element_models|Modelling:​ Lumped Element Models]] - * Chapter 11: [[theory:​sensor_technology:​st11_finite_element_models|Modelling:​ Finite Element Models]] - * Chapter 12: Modelling: Transducer Characterization by Impedance Spectroscopy - * Chapter 13: [[theory:​sensor_technology:​st13_lumped_element_models_advanced|Modelling:​ Systems Theory]] <- Next - * Chapter 14: [[theory:​sensor_technology:​st14_differential_equation_numerical_models|Modelling:​ Numerical Integration]] - * Chapter 15: [[theory:​sensor_technology:​st15_signal_conditioning_and_sensor_read-out|Signal Conditioning and Sensor Read-out]] - * Chapter 16: [[theory:​sensor_technology:​st16_resistive_sensors|Resistive Sensors]] - * Chapter 17: [[theory:​sensor_technology:​st17_capacitive_sensors|Capacitive Sensors]] - * Chapter 18: [[theory:​sensor_technology:​st18_magnetic_sensors|Magnetic Sensors]] - * Chapter 19: [[theory:​sensor_technology:​st19_optical_sensors|Optical Sensors]] - * Chapter 20: [[theory:​sensor_technology:​st20_actuators|Actuators]] - an example of an electrodynamic motor - * Chapter 21: [[theory:​sensor_technology:​st21_actuator_models|Actuator principles for small speakers]] - * Chapter 22: [[theory:​sensor_technology:​st22_adc_and_dac|ADC and DAC]] - * Chapter 23: [[theory:​sensor_technology:​st23_bus_interfaces|Bus Interfaces]] - SPI, I<​sup>​2​C,​ IO-Link, Ethernet based - * Appendix A: [[theory:​sensor_technology:​sta_easyunitconversion|Systematic unit conversion]] - * Appendix B: [[theory:​sensor_technology:​stc_common_mode_rejection_ratio_cmrr|Common Mode Rejection Ratio (CMRR)]] - * Appendix C: [[theory:​sensor_technology:​std_schmitt_trigger|A Schmitt Trigger for sensor level detection]] -