Complex-plane capacitance

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.

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. 

On the complex-plane capacitance plot the semicircle results from... 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

Another way of presenting the data is in the complex-plane capacitance form ImZ 1... 

On the complex-plane capacitance plot, the semicircle results from the series combination of R and Cdi while the intercept with the real axis yields the value of Cdi. The vertical line is due to the parallel combination of Rp and Cdi- While the physical meanings of complex impedance and complex admittance are clear, that of the complex capacitance is questionable. Nevertheless, presenting the data as in Figure 16.5d has some merit, because the numerical value of Cdi is given direcdy as the diameter of the semicircle. 

It would seem then that the complex-plane impedance plot is the best way of presenting the data, if one is mainly interested in the value Rp and its variation with time or potential. The complex-plane capacitance plot, on the other hand, brings out more directly the value of the capacitance and its variation with the different parameters of the experiment. The Bode plots are also informative, but it could be argued that they provide a lower resolution, because they are presented on a logarithmic scale. 

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. 

Table I shows the details of surface treatment and coating along with the calcxilated values of total resistance R and effective capacitance C. For specimens with Initial mechanical surface preparation, the Nyqulst Impedance plot shows the characteristic semicircular behavior with a resistance of the order of 1800 n cm and a capacitance of about 40 yF cm. As different surface treatments are Incorporated on a sequential basis, the complex plane diagram shows a gradual evolution.

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 ... 

The above-described situation is but an exception rather than the rule. Generally, the diamond electrode capacitance is frequency-dependent. In Fig. 12 we show a typical complex-plane plot of impedance for a single-crystal diamond electrode [69], At lower frequencies, the plot turns curved (Fig. 12a), due to a finite faradaic resistance Rp in the electrode s equivalent circuit (Fig. 10). And at an anodic or cathodic polarization, where Rf falls down, the curvature is still enhanced. At higher frequencies (1 to 100 kHz), the plot is a non-vertical line not crossing the origin (Fig. 12b). Complex-plane plots of this shape were often obtained with diamond electrodes [70-73],... 

It goes without saying that the frequency dependence of capacitance, which follows from the complex-plane plots of the type shown in Fig. 12, manifests itself in a frequency-dependent slope of Mott Schottky plots (Fig. 13) [78], The complications in calculating Na thus involved will be discussed at length in Section 5.3 below. 

A2.4 Representation in the complex plane A2.5 Resistance and capacitance in series A2.6 Resistance and capacitance in parallel A2.7 Impedances in series and in parallel A2.8 Admittance... 

Fig. A2.1. Representation in the complex plane of an impedance containing resistive and capacitive components.

We exemplify the use of vectors in the complex plane with a resistance and capacitance in series (Fig. A2.2a). The total potential difference is the sum of the potential differences across the two elements. From Kirchhoff s law the currents have to be equal, that is... 

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

Complex plane plot — Figure. Theoretical - impedance of a parallel connection of a resistance and a capacitance... 

The simulated Nyquist plot of resistance and capacitance in series is a vertical line in the complex-plane impedance diagram, as shown in Figure 4.2(b). The effect of the parameter R on the position of the line is presented in Appendix D (Model Dl). 

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]

The complex-plane impedance diagram of the modified bounded Randles cell is given in Figure 4.206. If distortions in the double-layer capacitance are assumed, a CPE can be used to replace Cdl. More examples of this modified bounded Randles cell can be found in Appendix D (Model D19). 

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. 

Conversely, a purely capacitive response is completely out of phase with the perturbation wave. The capacitive impedance response varies continuously and inversely with frequency and has no real component. In the complex plane, an ideal capacitance (C) appears as a vertical line that does not intercept the real axis. 

The impedance response of the R-RC circuit in Figure 4a is illustrated on a complex plane plot in Figure 4b. At low frequencies, the data approach the real axis at R + Rp (pathway 2), and the capacitive response is illustrated by the arc in the data. The frequency at the apex of the arc (to ) corresponds to the characteristic relaxation time (f ) of the circuit ... 

In many cases, the use of ideal equivalent circuits is convenient but not always appropriate. Nonideal behavior might arise from interactions of species, resulting in frequency-dependent capacitances [C((D)]. Under these conditions, the physical process is more accurately described by a range of relaxation time constants instead of a unique value. Such distributed relaxation events are usually manifested as semicircles depressed below the real axis in the complex plane, and the angle of depression is related to the degree of nonideality. Various distribution functions and constant phase elements have been employed to describe such events. These nonidealities are especially evident in biological systems. 

The complex plane plots in Fig. 18 illustrate the characteristic components of the impedance response for p-type silicon and heavily doped n-type silicon in the absence of illumination. In the region of pore formation where dt//dlog(/) = 60 mV, the impedance response is characterized by an inductive loop at low frequencies and a capacitive loop at higher frequencies, as shown in Fig. 18 a. In the transition region, a second capacitive loop is observed related to oxide formation at the surface (Fig. 18 b). At more positive potentials in the electropolishing domain (Fig. 18 c) only the two capacitive loops are seen. 

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]. 

---