Complex Plane and Bode Plots

Complex plane (Nyquist) plots are the most often used in the electrochemical literature because they allow an easy prediction of the circuit 

The Bode phase-angle graph is shown in Fig. 4(e). The phase angle is described by 

The dc transient response of electrochemical systems is usually measured using potentiostats. In the case of EIS, an additional perturbation is added to the dc signal to obtain the frequency response of the system. The system impedance may be measured using various techniques  

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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 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. 

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... 

Fig. 2.39 Complex plane and Bode plots for a nested circuit in Fig. 2.37, i(f 2C 2))), using...

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. 

Fig. 4.2 Complex plane and Bode plots fin- redox system with diffusion = 10 fi, Cdi = 25 pF,...

Fig. 4.14 Complex plane and Bode plots for semi-infinite external spherical diffusion Tq. (a) qo semi-infinite linear diffusion, (b) 0.005, (c) 0.01, (d) 0.02, (e) 0.05 cm other parameters as in Fig. 4.12...

Exercise 4.1 Write a program in Maple or Mathematica for the Randles model and create the corresponding complex plane and Bode plots. 

Exercise 4.4 Write a program in Maple or Mathematica to simulate semi-infinite spherical fusion (external) diffusion and create the corresponding complex plane and Bode plots. The parameters are as in Exercise 4.3, except Kq = 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 ... 

Fig. 8.3 Complex plane and Bode plots for the R-CPE circuit m series, R = 100 Q,...

Fig. 8.4 Complex plane and Bode plots for circuit consisting of solution resistance in series with parallel connection of CPE and resistance R. Parameters = 10 D, T = 20 pF cm s, ...

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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