Complex plane plot

The CPE appears to arise solely from roughening of the surface by the corrosion process. This was verified with IS experiments on iron and several steels in 15% HC1 at 25°C. The electrodes were polished with alumina and maintained at 150 mV cathodic of the rest potential. Complex plane plots of the impedance responses were nearperfect semi-circles centered on the V axis. Analyses via EQIVCT using the Rq+P/R circuit, gave rise to n-values of the CPE in excess of 0.93 in all cases and remained constant throughout the tests. 

Fig. 5. Complex-plane plot of impedance spectrum for a polycrystalline diamond film between two ohmic contacts. Frequency/kHz shown on the figure. Solid circles data obtained with ac bridge. Open circles data obtained with phase-sensitive analyzer. Top equivalent circuit [30].

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. 

Recall that the faradaic resistance can be determined as a low-frequency cut-off at the complex-plane plot of impedance spectrum (compare the equivalent circuit in Fig. 10b). Such plots measured in the Fc(CN)63 /4 solutions of different concentrations are given in Fig. 23a [104] (similar results were obtained in [111]). The plots are (somewhat depressed) semicircles, whose radii decreased with increasing redox couple concentration. Figure 23b shows the line plotted by using the data in Fig. 23a, in accord with Eq. (6). We notice that all three methods yielded similar results. 

These components are represented as a complex plane plot in Fig. 11.6... 

In general, the form of complex plane plots of the type shown in Fig. 11.6 alters on changing the concentration of the electroactive species16. It has been shown for the various kinetic regimes that the variation of Z" with concentration has the form in Fig. 11.8. The equation of this curve... 

Fig. 16.7. Impedance of AISI316 stainless steel in 3 per cent (wt) NaOH at 80°C (a) Complex plane plot (b) Bode plot (from Ref. 10 with permission).

Figure 8.8 illustrates the semicircular complex plane plot predicted by equation (8.31). 

The complex plane plot of VpEis exhibits two semicircles, and it can be fitted using the series/parallel circuit shown in Fig. 8.19. 

Figure 21 Common graphical representations of EIS data in corrosion studies, (a) Complex plane plot, (b) Bode magnitude and Bode phase angle plots. (From Gamry, EIS Manual, pp. 2-3, 2-5.)...

On a complex plane plot, a CPE exhibits a straight line whose angle is n/2 a (1 < a < -1) with respect to the real axis. In a Bode magnitude plot a straight line response like that of a capacitor is obtained. The slope deviates from an ideal value of -1 as a decreases below 1. 

EIS can also detect defects arising from lack of adhesion at adhesively bonded surfaces (111). The presence of such defects produces pronounced changes in the character of the data presented either in complex plane plots or in Bode plots. Figure 38 illustrates the measurement configuration and provides examples of EIS data for defective and defect-free samples. Studies have shown that the presence of defects is readily revealed and that the geometry of the defects and their spatial extent can be inferred from a detailed analysis of EIS spectra. 

Complex plane plot — The complex number Z = Z + iZ", where i = v/-i, can be represented by a point in the Cartesian plane whose abscissa is the real part of Z and ordinate the imaginary part of Z. In this representation the abscissa is called the real axis (or the axis of reals) and the ordinate the imaginary axis (the axis of imaginaries), the plane OZ Z" itself being referred to as the complex plane [i]. The representing point of a complex number Z is referred to as the point Z. 

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

The - complex plane plot of the -> admittance of a solitary CPE is a straight line which makes an angle of (1 - af) 7t/2 with the Y axis. 

The complex plane plot of the impedance of the resistor R in parallel with a CPE is a depressed semicircle with the center below the Z axis. 

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

An impedance response can be interpreted graphically as a vector on the complex plane. The imaginary axis is the out-of-phase response (Z"), and the real axis is the in-phase response (Z ). The magnitude of the impedance response Z is the length of the vector, and the phase angle (]) describes its direction (Fig. 3). Each point on the plane defines an impedance response at a particular frequency. Such representations are commonly referred to as complex plane plots, Nyquist diagrams, or Cole-Cole plots. However, the Cole-Cole plot is actually the complex plane representation of the dielectric response of a material. 

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 ... 

Consider now a more realistic situation, in which both the series and the parallel resistance must be taken into account. The equivalent circuit and the corresponding complex-plane plot are shown in Fig. lOG. 

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. 

A further characteristic IMPS signature for nanocrystalline films is a spiral in the high-frequency regime as exemplified by the complex plane plot in Figure 33 [349]. This spiral is typical of a constant time lag (i.e., frequency-dependent phase shift), and it arises simply from the transit time required for the majority carriers to move... 

Here = Jra — Jua, where and are, respectively, the unrelaxed and relaxed compliance functions in the a relaxation process. The values of these quantities could in principle be obtained by extrapolation methods from complex plane plots of /"( ) versus / (co). If (j(t) = Gq Im exp(/o)r), then Eq. (12.1) can be written as... 

The impedance behavior of electrode reactions is often complex but can be conveniently simulated by computer calculations, especially in the case of the method based on kinetic equations (108, 113). The forms of the frequency response represented in terms of the Z versus Z" complex-plane plots and by relations of Z or phase angle to frequency ai or log (o (Bode plots) are often characteristic of the reaction mechanism and involvement of one or more adsorbed intermediates, and they thus provide diagnostic bases for mechanism determination complementary to those based on dc, steady-state rate versus potential responses. The variations of Z versus Z" plots with dc -level potential, in controlled-potential experiments, also give rise to useful diagnostic information related to the dc Tafel behavior. 

Figure 3.42 Complex plane plots of the ER response of the Hm GC electrode in the solution specified in Figure 3.42 in the high frequency (A) and wide frequency (B) regions for Ejc= —0.325 V, AEac = 10.25 mV and A, = 433 nm. Open circles are experimental values and solid circles in (A) and full lines in (A) and (B) represent best fits to the model.

Figure 4. Complex plane plot for CdTe - DMF (5% H2O) Interface containing O.IM TBAP. Conditions as In Fig. 2.

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