Nyquist and Bode Representations

Figure 17.8 Nyquist and Bode representations of impedance data for the blocking circuit presented as Figure 17.1(b) with a as a parameter, a) complex-impedance-plane or Nyquist representation (symbols are used to designate decades of frequency) b) Bode representation of the magnitude of impedance and c) Bode representation of the phase angle. (Taken from Orazem et al. ° and reproduced with permission of The Electrochemical Society.)...

Nyquist and Bode representation of complex impedance data for ideal electrical circuits... 

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 . 

The complex impedance data involves the interplay of three variables, the imaginary component of the impedance, the real component of the impedance, Zreai, and the phase angle, common types of representation for impedance data are, the Nyquist and the Bode representations. Nevertheless, these have become the most widely used graphical representations of impedance data. 

For sensor applications, the data presentation is usually as a complex plane plot, with the real part (Z ) plotted on the x-axis and the imaginary part (Z") plotted on the y-axis, which is also referred to as a Nyquist plot . An alternative representation is of impedance magnitude and phase angle as a function of the logarithm of frequency—this Bode representation is most used in corrosion studies. 

Both the magnitude and the argument are functions of the frequency. The so-named Bode and Nyquist plots are nothing but graphical representations of this functional dependence. 

The variation of the impedance with frequency is often of interest and can be displayed in different ways. In a Bode plot, log Z and are both plotted against log cu. An alternative representation, a Nyquist plot, displays Zi vs. Zrc for different values of cu. Plots for the series RC circuit are shown in Figures 10.1.8 and 10.1.9. Similar plots for a parallel RC circuit are shown in Figures 10.1.10 and 10.1.11. 

Figure 5.10 Representation of the impedance spectrum of the equivalent circuit in Figure 5.8 for when Warburg impedance is much larger than the charge transfer resistance = 1000 Mil, IZ I = 1 Mil s , Cj, = 100 nF, = 10 il. (a) Nyquist plot and (b) Bode plot.

The impedance plot shown in Figure 2.1a vs. or the Nyquist plot) corresponds to an electrochemical cell (electrode/NaCl solution/electrode) and the equivalent circuit consists of a resistance (R) in parallel with a capacitor (C), which is represented as RQ, while Figure 2.1b shows the variation of the phase angle 4) = arc tan(Zi g/Z,e ) with frequency (4) vs./), but other typical impedance representations correspond to the variation of Z, and -Zj g with frequency (Bode plots), as indicated in Figure 2.1c and 2.1d. This latter representation allows the determination of the interval of frequency associated with a given relaxation process, between KT and 10 Hz, with a maximum frequency around 2 x 10 Hz, for the NaCl solution... 

The relationship of the impedance as a complex function is also what leads to the most common data representation in EIS, known as the Nyquist plot (Fig. 8.3). In the Nyquist plot, -Im(Z) is plotted versus Re(Z) over the entire frequency range of the EIS measurement. One of the main shortcomings of the Nyquist plot is that the frequency is not shown there is only an implicit understanding that high frequencies are at the lower Re(Z) values and decrease in the positive X-direction. In the example provided in Figure 8.3, which is the same as the system represented in the Bode plot in Figure 8.2, three separate resistances are apparent as intercepts on the X-axis. Sueh a response is often observed in fuel cells where reactants and products are supplied in excess, and the only resistances governing potential losses are the Ohmic resistance and the activation losses at the anode and the cathode. 

FIGURE 1.71. Schematic representation of a three-dimensional MacDonald plot consisting of a compilation of Bode and Nyquist representations. 

Figure 4. Results of simulation and measurement of the reference impedance. On the left side, the results in Nyquist representation are presented, on the right side, the Bode plot is presented.

The Nyquist plot is a very convenient representation of the R - R circuit process, as it shows an ideal semicircle as an indication of the activation-energy-controlled charge-transfer process. A depressed semicircle in the Nyquist plot is an indication of multiple processes with similar relaxation time constants, or distributed non-ideal kinetics. These ambiguities can be resolved in the "original" and modified Bode plots, as shown in Figures 2-3 and 2-4. The departure of the slope of log vs. log frequency dependency from the unity indicates a distributed process, with a characteristic frequency that may not even correspond to the highest peak value of... 

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