Nyquist and Bode Plots

Generally, the impedance spectrum of an electrochemical system can be presented in Nyquist and Bode plots, which are representations of the impedance as a function of frequency. A Nyquist plot is displayed for the experimental data set Z(Zrei,Zim.,mi), (/ = 1,2,. ..,n) of n points measured at different frequencies, with each point representing the real and imaginary parts of the impedance (Zrei Zim4) at a particular frequency . 

A Bode plot is an alternative representation of the impedance. There are two types of Bode diagram, log z log (or z log ) and 9 log , describing the frequency dependencies of the modulus and phase, respectively. A Bode plot is normally depicted logarithmically over the measured frequency range because the same number of points is collected at each decade. Both plots usually start at a high frequency and end at a low frequency, which enables the initial resistor to be found more quickly. 

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. 

Semi-circle Two semi-circles (Two time constants) Semi-circle Warburg diffusion 

Depressed semi-circle Two depressed semi-circles Depressed semi-circle Warburg diffusion 

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For the process considered in Prob. 19.12, generate Nyquist and Bode plots by the rigorous method and by the approximate method using several values of n. 

The simplest and most common model of an electrochemical interface is a Randles circuit. The equivalent circuit and Nyquist and Bode plots for a Randles cell are all shown in Figure 2.39. The circuit includes an electrolyte resistance (sometimes solution resistance), a double-layer capacitance, and a charge-transfer resistance. As seen in Figure 2.39a, Rct is the charge-transfer resistance of the electrode process, Cdl is the capacitance of the double layer, and Rd is the resistance of the electrolyte. The double-layer capacitance is in parallel with the charge-transfer resistance. 

In a situation where a charge transfer is also influenced by diffusion to and from the electrode, the Warburg impedance will be seen in the impedance plot. This circuit model presents a cell in which polarization is controlled by the combination of kinetic and diffusion processes. The equivalent circuit and the Nyquist and Bode plots for the system are all shown in Figure 2.40. It can be seen that the Warburg element is easily recognizable by a line at an angle of 45° in the lower frequency region. 

After analyzing the Nyquist and Bode plots (Figure 8a and b), the equivalent circuit model used to fit the impedance spectra is shown in Figure 8c, where the elements Re/ Rr/ and C are assumed to correspond to the electrolyte resistance, the electrochemical reaction resistance, and the total capacitance, respectively. While... 

Using the relationships developed in Problem 7-4, create Nyquist and Bode plots for the complex impedance Z of the circuit shown in Figure 7-1. You will need to use the relationship Y = 1/Z, where Y is the admittance. Finally develop the relationships between Z and s for this circuit. 

Sketch Nyquist and Bode plots of Qw(ku)- Calculate the value of controller gain that gives a phase margin of 45 . 

Figure 12 describes a simple electrochemical cell with a bare metal, in which the corrosion process is controlled by charge transfer. In this circuit, is the ohmic resistance, corresponding to the solution in the cell plus the cables and connections. Ret is the charge transfer resistance and Cji the capacitance of the double layer at the solution-metal interface. The Nyquist and Bode plots for this circuit are also presented. 

FIGURE 1.23 Nyquist and Bode plots of selected top coats with CORPASSIV primer for different times of immersion in 5% NaCl solution, (a) 2-C epoxy topcoat (b) 2-C acrylic and (c) 1 -C acrylic topcoat. (Reprinted from Wessling, B. and Posdorfer, J., Electrochim. Acta., 44, 2139, 1999. With permission. Copyright 1999 Elsevier Science)... 

The time response with the step response and Nyquist and Bode plots are respectively represented in figure (7). 

FIG U RE 2.2 Nyquist and Bode plots for two different electrode/eleetrolyte/membrane/elec-trode systems, (a, b) Dense and symmetric polyamide membrane, (c, d) Dense and symmetric highly hydrophilic regenerated cellulose membrane. 

Impedance plots (Nyquist and Bode plots) for the PS/PA-PEG5 and PS/PA-PEG25 membranes are shown in Figure 2.5b and 2.5c, where the effect of the asymmetric structure on the impedance (Nyquist) plot is indicated, but the differences depending on the PEG concentration are also evident. The equivalent circuit for the total membrane system, (R C ) - (RmQmX is also indicated in Figure 2.5b the depressed semicircle, attributed to the nonconstant phase circuit element (Q, is due to the porous structure of these membranes and the mixture of the relaxation times associated with their electrical response (polymeric matrix and solution). 

However, differences uniquely related to the membrane material are not always clearly reflected when the solution/membrane systems are studied if high hydrophilic membranes are considered, as was previously stated. For that reason, measurements with dry PEEK membranes were also performed and the Nyquist and Bode plots obtained for each sample are shown in Figure 2.10, where significant differences in the electrical response of both membranes can be observed. 

Figure 4 4 J8. Nyquist and Bode plots of impedance data for lithium polarized at —2.56 V, in a 12 M KOH electrolyte. Experimental data are represented by circles. Points at three frequencies are highUghted in the Nyquist plots. The optimization parameters, used in the computation of the solid lines, are indicated in the hgure. (After Pensado-Rodriguez etal. [2001]).

Figure 19.4 Experimental and simulated Nyquist and Bode plots for copper in a deaerated 0.1 M Nad + 2 X I0 M Na2S-9H20, T = 25 °C as a function of applied potential. The solid lines show the best fit calculation according to the PDM.

The most common approach to describe the physicochemical processes taking place within an MXC entails the use of equivalent circuits as those listed in Table 8.2. We also show in the table the corresponding Nyquist and Bode plots for each circuit. We describe some of these circuits in brief here. For example, circuit 1 is a simple circuit representing the Ohmic resistance in a system. This is a linear resistor, for which the Bode plot shows a straight line at a single impedance value throughout the range of frequencies. 

The frequency characteristics of corrosion systems of bulk A1 and AI2Q3 layer in 0.5M NaCl solution in the form of Nyquist and Bode plots obtained by the measurements and calculations based on adopted equivalent electrical circuits are shown in Figs. 8 and 9, respectively. 

Figure 7-5 shows a simulation of the impedance of this circuit in Nyquist and Bode plot presentations. In the complex plane (Nyquist plot), an ideal capacitive semicircle with Rp as the diameter is displayed. Adopting this simplified model, analysis of corrosion systems is often reduced to the determination of the polarization resistance Rp available from the low frequency limit... 

Impedance Nyquist and Bode plots for different Ca/HEDP and Zn/HEDP inhibitor molar ratios are presented in Figs. 9-33 and 9-34, respectively. The fit parameters obtained with the two-time-constant transfer function and calculated inhibitor efficiencies of the impedance as a function of Ca/ HEDP and Zn/HEDP molar ratios (chedp= 3x10" mol dm ) are presented in Table 9-16. 

Rt represents polarization resistance and Q, is the double layer capacitance. In this case, Nyquist and Bode plots depict only a time constant. However, in most real systems the plots represent deviations from this response leading to a more difficult interpretation of the impedance spectra. 

As was previously indicated, IS measurements can also be used to determined membrane modifications and Figure 9.13 shows Nyquist and Bode plots for PS-Uf and PS-Uf/BSA fouled membranes in contact with a NaCl solution. Here a significant increase in electrical resistance due to membrane fouling can be observed, but the electrolyte contribution hardly differs in both systems. In both cases, the equivalent circuit for the membrane-electrolyte system is given by (R,C,)-(RM that is, a series association of the electrolyte part, formed by a resistance in parallel with a capacitor and the membrane part, which consists of a parallel association of a resistance and a CPE or non-ideal capacitor (RmQm). Fitting the experimental data allows determination of the electrical parameters (resistance, capacitance) for the different NaCl solutions studied and their variation with electrolyte concentration is shown in Figure 9.13c, d, respectively. 

Directly after immersion, case A showed a Nyquist and Bode plot that deviated from the expected plots for intact coatings. Representative measurements after 1 h and 10 days are given by Figure 17a and b. 

Figure 20 Nyquist and Bode plots measured on coating B directly after perforation, (b) Log Z is given on the left axis and alpha on the right axis.

Nyquist and Bode plots for the above circuits are given in Figs 12.34 and 12.35, where T is the time at which the exponential factor is e = 0.37, the time it takes to decrease to 37% of its value, r represents how slow or how fast is a reaction is. The Bode plot for the system is shown in Fig. 12.35. On extrapolation to the log z-axis, the value of Q (coating capacitance) and Cdi (double layer capacitance) are obtained. For the known values of R, Ret and Re, the pore... 

Impedance is usually expressed as a complex number, where the ohmic resistance is the real component and the capacitive reactance is the imaginary one. The most popular formats for evaluating electrochemical impedance data are the Nyquist and Bode plots. In the former format, the imaginary impedance component Z", out of phase) is plotted against the real impedance component Z in phase) at each excitation frequency, whereas in the latter format, both the logarithm of the absolute impedance, Z, and the phase shift, 0, are plotted against the logarithm of the excitation frequency. 

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