Complex-plane impedance

Figure 15 shows a set of complex plane impedance plots for polypyr-rolein NaC104(aq).170 These data sets are all relatively simple because the electronic resistance of the film and the charge-transfer resistance are both negligible relative to the uncompensated solution resistance (Rs) and the film s ionic resistance (Rj). They can be approximated quite well by the transmission line circuit shown in Fig. 16, which can represent a variety of physical/chemical/morphological cases from redox polymers171 to porous electrodes.172... 

Figure 15. Complex plane impedance plots for polypyrrole at (A) 0.1, (B) -0.1, (C) -0.2, (D) -0.3, and (E) -0.4 V vs. Ag/AgCl in NaCl04(aq). The circled points are for a bare Pt electrode. Frequencies of selected points are marked in hertz. (Reprinted from X. Ren and P. O. Pickup, Impedance measurements of ionic conductivity as a probe of structure in electrochemi-cally deposited polypyrrole films, / Electmanal Chem. 396, 359-364, 1995, with kind permission from Elsevier Sciences S.A.)...

The interface impedance for a case such as Ag/Ag4Rbl5 will consist of a capacitance (derived from the Helmholtz formula) in parallel with i et so that in the complex plane impedance a semi-circle will be found from which Qi and can be evaluated. Rq will cause this semicircle to be offset from the origin by a high frequency semicircle due to the bulk impedance between the interface and the reference electrode (Fig. 10.12). There can be no Warburg impedance (a line at 45° to the real axis generally due to diffusion effects) in this case. 

Fig. 10.12 Expected complex plane impedance diagram for an electrolyte with one mobile species which is contacted by two non-blocking metal electrodes, e.g. Ag/Ag Rblj/Ag. is the bulk resistance of the electrolyte and R is the charge transfer resistance for the Ag/Ag Rbls interface.

Fig. 7 Complex plane impedance plot for p-type Si in 1% HF solution, in the dark, at bias potential 0 V vs. SCE (after Vanmaekelberg et al. [9]).

FIGURE 1.19 Complex-plane impedance plot (Nyquist plane) for an electrochemical reaction under kinetic control. 

FIGURE 1.20 Complex-plane impedance plot (Nyquist plane) for an electrochemical system, with the mass transfer and kinetics (charge transfer) control regions, for an infinite diffusion layer thickness. 

FIGURE 1.21 An example of a complex-plane impedance plot (Nyquist plane) for an electrochemical system under mixed kinetic/diffusion control, with the mass transfer and kinetics (charge transfer) control regions, for a finite thickness 8N of the diffusion layer. Assumption was made that Kf Kh at the bias potential of the measurement, and D0I = Dmd = D, leading to RB = RCT (krb8N/ >). 

FIGURE 1.22 (a) Capacitor C with a series resistance, (b) Complex-plane impedance plot for the equiva-... 

FIGURE 1.23 Complex-plane impedance plot for a 2.3V/10F supercapacitor laboratory cell using acetonitrile-based electrolyte. Bias voltage = 0 V room temperature. 

Fig. A2.2. Resistance and capacitance in series (a) Electrical circuit (b) Complex plane impedance plot.

This is a vertical line in the complex plane impedance plot, since Z is constant but Z" varies with frequency, as shown in Fig. A2.2b. 

At low frequencies (m — 0). it tends to a finite value. The complex-plane impedance of the BW will be shown in Section 4.2. 

In the complex-plane impedance diagram, the Nyquist plot of resistance and capacitance in parallel is an ideal semicircle, as depicted in Figure 4.3b. The diameter equals the value of the resistance, R. The imaginary part of the impedance reaches a maximum at frequency [Pg.146]

Figure 4.4b shows the simulated Nyquist plot of resistance and a CPE in series connection, in a complex-plane impedance diagram. More examples of the effect of parameters on the spectra can be found in Appendix D (Model D3). 

The complex-plane impedance diagram is given in Figure 4.126. At both high and medium frequencies the complex-plane impedance is characterized by a well pronounced semicircle, while at the low-frequency range a tail appears. This tail s shape is strongly dependent on the value of the CPE exponent, as can be seen in AppendixD (Model Dll). 

The complex-plane impedance diagram of the bounded Randles cell is given in Figure 4.186. In this example, the parameters of the forward and backward reactions and diffusion coefficients are assumed to be equal. Impedance diagrams with variations in the parameters are presented in Appendix D (Model D17). 

The complex-plane impedance diagram of this simple Maxwell structure is presented in Figure 4.22A. More examples of this model are provided in Appendix D (Model D20). 

The complex-plane impedance diagram of the two-RC Voigt structure is depicted in Figure 4.246. It is characterized by two time constants, Tj and t2. The... 

The simulated complex-plane impedance diagram is shown in Figure 4.27b. As can be seen in the figure, this ladder structure is characterized by two semicircles with two time constants, r, = RclCd] and r2 = R3C2, accounting for the two-step reaction. The element C2 symbolizes the adsorption capacitance, and r2 represents the relaxation of the adsorbing process. 

Figure 5.48. Complex plane impedance plots for a 5 cm2 hydrogen fuel cell at a current density of 0.2 A cm2. The cell was run at ambient temperature with humidified H2 and dry air [64], (Reprinted from Electrochimica Acta, 49, Li G, Pickup PG. Measurement of single electrode potentials and impedances in hydrogen and direct methanol PEM fuel cells, 4119— 26, 2004, with permission from Elsevier and the authors.)...

Adding a solution resistance to the circuit shown in Fig. 12L, one obtains the complex-plane impedance plot shown in Fig. 13L(a), which is similar to Fig. 12L, except that the semicircle is displaced to the right on the ReZ axis. 

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.

It would seem then that the complex-plane impedance plot is the better way of pre.senling the data, if one is mainly interested in the... 

In Fig. 15L(a-d) we show four complex-plane-impedance plots calculated for the concentrations of 100, 10, 1 and 0.1 mM, respective-... 

Fig. 17L Complex-plane impedance plots for the corrosion of iron at two different potentials, (a) two well-separated time constants are shown, (h) An inductive loop is observed. Data from Epelboin, Gabrielli, Keddam and Takenoiiti, Electrochini. Acta, 20, 9J3, 1975).

We shall refer to it as the complex-plane impedance plot, recognizing that the same data can also he represented in the complex-plane capacitance or the complex-plane admittance plots. The terms Cole-Cole plot, Nyqidst plot and Argand plot are also found in the literature. 

Fig. ML Complex-plane-impedance plot for an equivalent circuit with...

Fig. I5L Complex-plane impedance plots showing the gradual change from charge-transfer to mass-transport limitation with decreasing concentration. 10 < id < 10 Radis. C =20 uFIcn, R =...

Fig. 16L Complex-plane impedance plot with depressed semicircle. A -B is the diameter of the semicircle, depressed by an angle a.

The existence of a current hump near Tc is confirmed by several additional facts. In the first place, these are deduced from the results of the quantitative treatment of the impedance spectra of the HTSC/solid electrolyte system [147]. This approach consists of calculating from the experimental complex-plane impedance diagrams the parameters characterizing the solid electrolyte, the polarization resistance of the reaction with the participation of silver, and the double-layer capacitance (Cdi) for each rvalue (measured with an accuracy of up to 0.05°). Temperature dependence of the conductance and capacitance of the solid electrolyte (considered as control parameters) were found to be monotonic, while the similar dependences of two other parameters exhibited anomalies near Tc- The existence of a weakly pronounced minimum of Cji near Tc, which is of great interest in itself, was interpreted by the authors as the result of sharp reconstruction of the interface in the course of superconducting transition [145]. 

The detailed interpretation of data on HTSC electrochemistry obtained until now in both liquid and solid electrolytes is essentially complicated by the same problem -the choice of the equivalent circuit adequately describing the impedance for the nonzero overvoltages. Peck et al. [153] carried out the quantitative treatment of the results for low- and high-frequency regions separately. As mentioned previously, a thorough treatment of complex-plane impedance diagrams was performed for data obtained in solid electrolytes, but only for the equilibrium potential. 

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