Impedance: Bode plot complex plane

Fig. 2.33 Complex plane impedance. Bode, and complex plane admittance plots for a connection of R and C in parallel (RQ, R = 100 i2, C = 20 pF...

Fig. 2.35 Complex plane impedance, Bode, and complex plane admittance plots for circuit...

Different kinds of plots based on impedance Z, admittance Z 1, modulus icoZ, or complex capacitance (z coZ) 1 can be used to display impedance data. In solid state ionics, particularly plots in the complex impedance plane (real versus imaginary part of Z) and impedance Bode-plots (log(Z) log(co)) are common. A RC element (resistor in parallel with a capacitor) has, for example, an impedance according to... 

The most common graphical representation of experimental impedance is a Nyquist plot (complex-plane diagram), which is more illustrative than a Bode plot. However, a Bode plot sometimes can provide additional information. 

When a range of frequencies is applied to the DUT, both El and ECI techniques are called spectroscopies, i.e., electrical impedance spectroscopy and electrochemical impedance spectroscopy. Electrochemical impedance spectroscopy (EIS) profiles, measured as a function of the interrogating frequency, can be presented by two popular plots complex plane impedance diagrams, sometimes called Nyquist or Cole-Cole plots, and Bode (I Z I and 6) plots (Fig. 2). As the impedance, Z, is composed of a real and an imaginary part, the Nyquist plot shows the relationship of the imaginary component of impedance, Z" (on the Y-axis), to the real component of the impedance, Z (on the X-axis), at each frequency. A diagonal line with a slope of 45° on a Nyquist plot represents the Warburg... 

Equations 2.37-2.40 result in the commonly used presentation of the impedance, e.g. the Nyquist and the Bode plots. The first one shows the total impedance vector point for different values of co. The plane of this figure is a complex plane, as shown in the previous section. Electrochemical-related processes and effects result in resistive and capacitive behaviour, so it is common to present the impedance as ... 

Fig. 16.7. Impedance of AISI316 stainless steel in 3 per cent (wt) NaOH at 80°C (a) Complex plane plot (b) Bode plot (from Ref. 10 with permission).

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

Fig. 13L Comparison of a) complex-plane impedance, (b) complex-plane admittance, (c) complex-plane capacitance and (d) Bode magnitude and Bode angle plots for the same equivalent circuit. C =20 uF R = 10 kFl R = I kO.. Values of li) (Rad/s) at which some of the points were calculated are shown.

The impedance behavior of electrode reactions is often complex but can be conveniently simulated by computer calculations, especially in the case of the method based on kinetic equations (108, 113). The forms of the frequency response represented in terms of the Z versus Z" complex-plane plots and by relations of Z or phase angle to frequency ai or log (o (Bode plots) are often characteristic of the reaction mechanism and involvement of one or more adsorbed intermediates, and they thus provide diagnostic bases for mechanism determination complementary to those based on dc, steady-state rate versus potential responses. The variations of Z versus Z" plots with dc -level potential, in controlled-potential experiments, also give rise to useful diagnostic information related to the dc Tafel behavior. 

There are two limits of the impedance (O = 0, Z = / and ro —> o°, Z = 0. The corresponding complex plane and Bode plots for the same values of R and C elements as those used in the series R-C model above, are shown in Fig. 3. The Nyquist plot shows a semicircle of radius RH with the center on the real axis and the frequency at the semicircle maximum equal to (0= RC. The circuit s characteristic breakpoint frequency (the inverse of the characteristic time constant), as observed in the impedance Bode graph, is the same as for the series and the parallel R-C circuits. The complex plane admittance plot represents a straight line parallel to the imaginary axis [Fig. 3(c)], which is similar to the impedance complex plane plot for the series R-C connection. 

Figure ll(b-d) also shows complex plane and Bode plots for the total electrode impedance in the presence of slow charge-transfer kinetics. It should be stressed that the Warburg impedance caimot be represented by... 

The influence of the nonlinearity of diffusion on the observed complex plane plots is shown in Fig. 13. Spherical mass transfer causes the formation of a depressed semicircle at low frequencies instead of the linear behavior observed for linear semi-infinite diffusion. For very small electrodes (ultramicroelectrodes) or low frequencies, the mass-transfer impedances become negligible and the dc current becomes stationary. On the Bode phase-angle graph, a maximum is observed at low frequencies. 

The impedance technique is often applied to electrochanical systems that have not been studied before. The complex plane and Bode plots obtained often displayed shapes that had never been encountered previously. Before starting the analysis and modeling of the experimental results, one should be certain that the impedances are valid. There is a general mathematical procedure that allows verification of the impedance data. It was introduced by Kramers and Kronig, further developed by Bode, and later applied to During the impedance measurements, a small ac... 

In addition to comparing the sum of squares, the experimental and simulated data should be compared by using complex plane and Bode plots. The phase-angle Bode plot is particularly sensitive in detecting time constants. Boukamp proposed to study the residual sum of squares after subtracting the assumed model values from the total impedance data. If the model is valid, the residuals should behave randomly. If they display regular tendencies, it may mean that the model is not correct and further elements should be added. However, the variations of the residuals should be statistically important. 

Fig. 2 Representative (a) complex plane diagrams (Nyquist or Cole-Cole plots) and (b) Bode plots from electrochemical impedance spectroscopy measurements...

To understand the impedance of electrochemical objects, it is necessary to understand the behavior of simple electrical circuits, first in steady state, then in transient conditions. Such circuits contain simple linear electrical elements resistance, capacitance, and inductance. Then the cmicept of electrical impedance will be introduced. It demands an understanding of the Laplace and Fourier transforms, which will also be presented. To understand impedance, it is necessary to thoroughly understand the complex plane and Bode plots, which will be presented for a few typical connections of the electrical elements. They can be computed using Excel, Maple, Mathematica, and specialized programs such as ZView. Several examples and exercises will be included. 

Several examples of impedance plots are presented in Exercises 2.8 and 2.10. First, let us look at the complex plane and Bode plots obtained for an R-C connection in series, RC in Boukamp s notation, with R = 150 Q, C = 40 pF, Exercise 2.8. The impedance of such a circuit is described as... 

Until now, only circuits containing resistances and capacitances have been discussed. Inductive effects in electrical circuits appear when alternative electrical current flow creates a magnetic field interacting with the flowing current of course, in a strait wire the inductance is very small, but in looped wires or a coil it becomes larger. The inductive effects always lead to positive imaginary impedances, as will be shown in what follows. Let us first consider the circuit in Fig. 2.40, which contains inductance L in series with resistance Rq and a nested coimection of two (RQ circuits, i.e., LRo(Ci(Ri(R2C2))). The complex plane and Bode plots for this circuit without inductance were presented in Fig. 2.39. 

Impedance of an electrical circuit containing linear electrical elements R, C, and L can be calculated using the impedance of these elements and Ohm s and Kirchhoff s laws. The complex plane and Bode plots can be easily produced using programming in Excel, Zplot, Maple, Mathematica, etc., which are readily available. It should be stressed that these electrical elements are linear, that is, their impedance is independent of the applied ac amplitude. In subsequent chapters, we will see how the impedance of electrochemical systems can be described. 

Exercise 2.14 Simulate impedances for the circuit RCL in series for / = 1 2, C = 0.01 F, and L = 0.01 H. Make simulations using ZView and Excel. Make complex plane and Bode plots. 

Exercise 4.5 Write a program in Maple/Mathematica to simulate transmissive impedance and create the corresponding complex plane and Bode plots. The parameters are as in Exercise 4.3, with / = 0.01 cm. 

To obtain the total impedance, the faradaic impedance, Eq. (5.19), must be inserted into the total impedance (Fig. 4.1b). The complex plane and Bode plots of the total impedance are as in Fig. 2.35. The circuit parameters R i and Cp depend on the potential, as illustrated in Fig. 5.1. The charge transfer resistance displays a minimum at Ep and its logarithm is linear with the potential further from the minimum, while the pseudocapacitance displays a maximum. These values at the potential Ep are... 

In the presence of a redox reaction without diffusion limitations, the system impedance is described by the electrical equivalent circuit / s(Cdi ct) displayed in Fig. 2.34. Replacing the double-layer capacitance with the CPE produces complex plane and Bode plots (Fig. 8.4) corresponding to the equation for the impedance of such a system ... 

Although the CPE and fractal systems give the same impedance in the absence of redox reactions, a comparison of Eq. (8.9) for the CPE model with Eq. (8.17) for a fractal system in the presence of a redox reaction shows that they are structurally different. In fact, they produce different complex plane and Bode plots. This is clearly visible from Fig. (8.9), which can be compared with Fig. 8.4 for the CPE model. With a decrease in the value of , an asymmetry on the complex plane plot occurs that is also visible oti the phase angle Bode plots. This is related to the different topology of the equivalent circuits they are compared in Fig. 8.10. In the CPE model, only the impedance of the double-layer capacitance is taken to the power while in the fractal model the whole electrode impedance is taken to the power (p. The asynunetiy of the complex plane and Bode plots for fractal systems arises from the asymmetric distribution function of time constants in Eq. (8.4) according to the equation [298, 347]... 

It is evident that these two forms are indistinguishable and produce exactly the same impedances and impedance complex plane and Bode plots for these two circuits. All the equations have three adjustable parameters. However, additional information is necessary to decide which circuit has a physical meaning in the given case. For example, when studying a redox process on an electrode in solution, circuit (a) is more probable because Ri and C2 have a meaning of solution resistance in series, with the electrode impedance consisting of a parallel connection of the double-layer capacitance, C2, and the charge transfer resistance, R2 (Sect. 4.1 and Fig. 4.1). 

The purpose of modeling is to find an appropriate model described by an electrical circuit or equation by minimization of the sum of squares. Such model impedances should lie very close to the experimental ones without any systematic deviations. The first test is a visual comparison of the complex plane and Bode plots, which should agree. To assure that the approximating model is correct, several statistical tests might be used. 

The experimental data that were checked by the Kramers-Kronig transforms may be used in modeling. First, usually, fit to an electrical equivalent model is carried out. It is important to use a proper weighting procedure and start with the simplest model. Then additional parameters can be added and their importance verified by the appropriate F- and r-tests. The number of adjustable parameters must be kept to a minimum. Additionally, comparison of the experimental and model impedances on complex plane and Bode plots should be carried out. Furthermore, plots of the residuals indicate the correctness of the model used. Next, on the basis of this fit, a physicochemical model might be constructed. One should check how the obtained parameters depend on the potential, concentration, gas pressiue, hydrodynamic conditions, etc. If a strange or unusual dependence is obtained, one should check whether the assumed model is physically correct in the studied case. This is the most difficult part of modeling. 

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