Equivalent circuit Nyquist diagram

Idealised Nyquist diagram corresponding to the equivalent circuit of Ershler-Randles (Fig.2.6). 

Figure 2.37 shows an example impedance spectrum of an electrochemical system with two time constants. Figure 2.37a, b, and c are the equivalent circuit, simulated Nyquist diagram, and Bode plot, respectively. 

By taking into account the double-layer capacity, Q, and the electrolyte resistance, Re, one obtains the Randles equivalent circuit [150] (Fig. 10), where the faradaic impedance Zp is represented by the transfer resistance Rt in series with the Warburg impedance W. It can be shown that the high-frequency part of the impedance diagram plotted in the complex plane (Nyquist plane) is a semicircle representing Rt in parallel with Cd and the low-frequency part is a Warburg impedance. 

Figure 5.26 Nyquist diagram of impedance for a simple equivalent circuit of the electrodeelectrolyte interface, calculated for the electrical circuit shown.

In Nyquist plots (imaginary part vs. real part of the impedance) for each separated physical process, a semicircle is obtained. Mostly, the imaginary part is plotted as an ordinate in positive direction because the inductivities do not play an important role in sohd electrolyte cells. Ideally, the middle point of the semicircle is located on the real axis. This equates an/ C combination. In reahty, due to the inhomogeneity of the electrode, depressed semicircles are obtained with middle points below the real axis. The intercepts with the real axis and maxima are related to a resistance and capacitor. The complex impedance diagram is not unambiguous. It is useful to interpret this by means of an equivalent circuit in terms of values of physical processes [6]. In Fig. 2... 

Figure 6.7 (a] representation of the electrical equivalent circuit in the case of the film formation (b) simulation of the corresponding impedance diagram in the Nyquist plan. 

Figure 8.7 (a) Equivalent electric circuit corresponding to an oxide ion conductor and the corresponding electrochemical Nyquist diagram, (b) Example of a polycrystalline sample. 

Several additional physical processes potentially exist that can create somewhat more complicated equivalent circuit diagrams, as is evident for example from discussions in Chapter 10 on conductive polymer films. For instance, additional macrodefect corrosion and diffusion effects may develop between the coating and the surface, with an additional low frequency relaxation becoming visible in the Nyquist plot. The interface between a pocket of solution and the bare metal is modeled as a double-layer capacitance in parallel with a kinetically controlled charge-transfer resistance R, which can also often include the diffusion element associated with corrosion products in series with R. If the diffusion element represents a finite diffusion, an additional Rpiipp I element appears and a third relaxation at low frequencies,... 

The equivalent circuit for a complete PEM fuel cell can be applied initially to a "symmetrical" gas supply while varying current densities. At low current densities (< 400 mA/cm ) the mass-transport limitations are very minor, and a simple Nyquist diagram with two relaxation times emerges. The overall impedance is dominated primarily by the cathodic process, with resulting higher cathodic overpotentials. The equivalent circuit is represented by two parallel combinations of charge-transfer resistances R and doublelayer capacitors C 20 pF/cm for the anode and cathode processes. The... 

Plotted in a Nyquist diagram (Im Z vs Re Z), the measurements of complex impedance at various frequencies of this simple fuel cell equivalent circuit result in a semicircle (Figure 8-8). At very low frequencies, the resulting impedance is Z = Rr -(- Ract, whereas at very high frequencies, the resulting impedance is Z = Rr. Sometimes, a single measurement at high frequency is used to measure the cell resistance Rr. However, the entire frequency... 

For the fuel cell equivalent circuit shown in Figure 8-7, given with Rr = 0.2mOhm, R ct = 1 mOhm, and C = 5 jJ.F, calculate real and imaginary parts of the impedance, ReZ and ImZ, and plot the Nyquist diagram. 

Fig. 6.16 Nyquist diagrams of (a) uncoated LiMn204 as a function of the state-of-charge after 50 cycles in the potential range 4.0. 3 V and (b) after Zr02 coating. Cells were cycled at 1C rate and maintained at L= 55 °C. The equivalent circuit used for the analysis is shown as insert (see text for symbol meaning)...

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