Depressed semicircle

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. 

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. 

Assuming that the Nyquist plot of the impedance does not display an ideal semicircle (e.g., it shows a depressed semicircle or a wide arc), it might be described using two or more discrete time constants or a continuous distribution of time constants. In the former case, the equivalent circuit may involve two or more parallel RCs in series. In the latter case, it may involve one or more parallel CPEs and Rs in series. As mentioned, one solution could be to use several CNLS fittings however, a more direct method would be the deconvolution of the imaginary part of the impedance data. 

The Nyquist plot in a complex-plane diagram of the Randles cell with capacitor replaced by a CPE is a depressed semicircle, as shown in Figure 4.136. If co —> the intercept of the depressed semicircle at the real axis equals Reh and rf m — 0, the intercept equals the value of Rei + Rct. The deviation of the n value describes the... 

A result such as shown in Fig. 16L indicates clearly that the system cannot be described correctly by a simple equivalent circuit of the types discussed so far. Sometimes, if the depressed angle a is small (say, less than 10°) the problem may perhaps be ignored, and one may obtain R either as the distance from /I to B or from A to B , which will differ in this case by a few 10%. In any event, changes in the angle of depression or in the radius of the depressed semicircle still can be taken as an indication of variation in the properties of the... 

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

Yovtng impedance, true CPE behavior is not fovmd. It should be noted that the CPE model corresponds to a specific distribution of time constants that may or may not correspond to a given physical situation. Local impedance measurements can give information about the nature of this distribution, whether 2-D, 3-D, or both. This example shows that not all depressed semicircles correspond to a CPE behavior. 

Figure 1 shows the dielectric relaxation properties of the Tween microemulsions plotted on the complex permittivity plane (from Foster et al ( 1). The mean relaxation frequency (corresponding to the peak of each semicircle) decreases gradually from 20 GHz for pure water at 25°C to ca. 2 GHz for a concentrated microemulsion containing 20% water. Since the permittivity of the suspended oil/ emulsifier is 6 or less at frequencies above 1 GHz, this relaxation principally arises from the dipolar relaxation of the water in the system. Therefore, the data shown in Figure 1 clearly show that the dielectric relaxation times of the water in the microemulsions are slower on the average than those of the pure liquid. The depressed semicircles indicate a distribution of relaxation times (9), and were analyzed assuming the presence of two water components (free and hydration) in our previous studies. 

Figure 1. Plots of the complex permittivity of the 0/W microemulsions prepared with Tween 60 on the complex dielectric plane ("Cole-Cole" plots), showing the depressed semicircles that indicate a distribution of relaxation times. Figure lb is an expanded portion of Figure la. A few frequencies are indicated for reference. Reproduced with permission from Reference 1. Copyright 1982 Academic Press.

Samples with depressed semicircles have been adjusted with a CPE. The CPE is a nonintuitive circuit element. In case of a resistance R and a CPE in parallel, we observe the arc of a circle with the center some distance below the x-axis in the Argand plot. The corresponding circuit equivalent is called a ZARC-element.44... 

The influence of the nonlinearity of diffusion on the observed complex plane plots is shown in Fig. 13. Spherical mass transfer causes the formation of a depressed semicircle at low frequencies instead of the linear behavior observed for linear semi-infinite diffusion. For very small electrodes (ultramicroelectrodes) or low frequencies, the mass-transfer impedances become negligible and the dc current becomes stationary. On the Bode phase-angle graph, a maximum is observed at low frequencies. 

Figure 12.40 shows the Cole-Cole plot of specimen (a), (b), and (c) respectively. In all three cases, the nature of the Cole-Cole plot is found to be a depressed semicircle over that of pure base matrix gum Arabica [17]. The Cole-Cole plot depicted indicates the nature of electroactivity in the complexes. The inclusion of micro/meso-sized chromatophore enhanced the electroactivity in the complexes to increase the electronic conductivity in them. Analysis of the Cole-Cole plot shows that it is a depressed semicircle showing overall EABP characteristics. The obtained semicircular impedance variation is more depressed compared to that of pure gum Arabica [17]. The depressed nature... 

In the low frequency range, both contact and interfacial polarizations were observed in all samples. These polarizations have to be carefully considered it is important to take into account only the so-called depressed semicircle that does not include contact and interfacial polarization effects. 

Impedance spectra of real electrochemical cells will usually not show a perfect semicircle as the one presented in Fig. 4(c) but depressed semicircles as shown in Fig. 11(a). 

The Nyquist plots of many real electrolyte systems deviate from the ideal Debye response, they tend to show distorted or depressed semicircles, tilted or curved spikes. The deviations can be fitted if a CPE replaces the capacitor in the equivalent circuit. A CPE is viewed as a leaky capacitor with impedance given by ... 

A pure resistor and a pure capacitor in parallel connection gives a perfect semicircle in a Nyquist plot. A resistor and a CPE in parallel connection, on the other hand, gives a depressed semicircle with its centre below the horizontal axis by an angle an/2 as shown in Figure 14. 

Figure 20 shows the experimental results (solid circles) for hexanoyl chitosan (M of 2 X 10 g mol )-based electrolyte system. The Nyquist plot shows a depressed semicircle with its centre below the horizontal axis by an angle 20°. The equivalent circuit may take the form of a resistor connected in parallel with a CPE.The value of R obtained by graphical means was 226 kfl. Open squares semicircle was calculated withi = 226 kO and = 5.3 x 10 s after Equations (19) and (20). The intercept of the open squares semicircle on the Z axis gives = 220 kQ. Thus the equivalent circuit that has been used to simulate the Nyquist plot gives a hue representation of the material. 

It is often found that the double-layer capacitance or a coating capacitance does not behave like an ideal capacitor, experimentally manifested in the complex plane plot by a depressed semicircle whose center lies below the real axis. This behavior is usually attributed to some distribution (or dispersion) in some physical property of the system (e.g., the porous surface of the metal or the varying thickness or composition of a coating) and is modeled by the use of a constant phase element (CPE) [30]. 

Even if the depletion layer contribution can be successfully separated from the other contributiOTis in the system, one may find that it does not behave as an ideal capacitor. A clear indication of this is the observation of frequency dispersion in the Mott-Schottky plot, or a somewhat depressed semicircle instead of a perfectly round one in a Nyquist plot of the impedance spectrum. In such cases, it is often... 

In many cases, complex systems present a distribution of relaxation times and the resulting plot is a depressed semicircle, which is associated with a nonideal capacitor or a constant phase element (CPE), and its impedance is expressed by Q(a) = To(7(o)" , where Tq represents the admittance and n is an experimental parameter (0 < < 1)... 

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). 

Figure 2.7 shows the impedance plots for the different samples studied, as well as for the electrolyte solution alone. As can be observed, both the RC70PP membrane and the ETFE film show a unique depressed semicircle, a distribution of the relaxation times due to the electrolyte and the solid structure. However, two separated... 

Rp 3.37 X 10 ). This indicates the formation of protective and compact rust on WS. Randle model was used to get the electrical elements and equivalent. The depressed semicircle diameter of WS was found higher than MS. This behaviour indicates the presence of diffusion resistant layer at the electrolyte/steel interface. In case of MS, the real component of resistance was found to reduce with decrease in frequency, a phenomenon attributed to inductive behaviour of the electrolyte/ steel interface. The availability of SO2 reduced dissolution in WS. Presence of alloying elements in WS may be attributed as the reason. 

Impedance plots are modified at spherical electrodes. Examples of such plots, using an equivalent circuit in Eig. 4.1, are shown in Eig. 4.14. The high-frequency semicircle is related to the coupling of Ret and C[Pg.111]

ZeARC Depressed semicircle in complex dielectric constant plane (see Fig. 2.2.2a) (20), (21)... 

Experimental data and fits by the above functions (Barsoukov et al. [2003]) is shown in Figure 4.5.10. The spectrum of Li-intercalation cathodes has a depressed semicircle in the high frequency region, whose size and frequency dependence is determined by Ra and Cdi. Combined alectronic and ionic resistance of the porous material, Rm contributes to its stretching along the X-axis. In the lower frequency... 

---