Parallel RC-Circuit

In this section, the physical modeling of impedance of a CCL is discussed. The model is based on a macrohomogeneous model for the CCL performance discussed in Chapter 4. To demonstrate the idea of EIS, the impedance of a simple parallel RC-circuit is considered first. 

To calculate the total current between a and b, it is noted that the current in the resistor R is simply 4 /R, while the AC current in the capacitance C is dq/dt = Cd(p/dt, where q is the instant charge of the capacitor. Summing these contributions gives the total current between a and b  

The potential (p (t) is harmonic in time. Since oiu circuit is linear, the total current induced in the system is also harmonic. Thus, (p and I can be represented as 

Substituting these Fourier fransforms into Equation 5.75, one obtains 

This shows that R and l/(/ C) are parallel impedances. Solving Equation 5.78 for Z, results in 

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According to this model, the SEI is made of ordered or disordered crystals that are thermodynamically stable with respect to lithium. The grain boundaries (parallel to the current lines) of these crystals make a significant contribution to the conduction of ions in the SEI [1, 2], It was suggested that the equivalent circuit for the SEI consists of three parallel RC circuits in series combination (Fig. 12). Later, Thevenin and Muller [29] suggested several modifications to the SEI model ... 

Fig. 12. Parallel RC circuit for the automatic correction of thermograms. Reprinted from (40) with permission.

Figure 8.9 Parallel RC circuit (a) circuit (b) current-voltage relationships (c) frequency dependence of impedance Z and phase angle (j>.

Fig. 2.55 (a) Series-parallel RC circuit which could, depending on component values, produce the frequency response shown in (b). 

Therefore, the real and imaginary components, Zre and Zim, in the AC impedance of the parallel RC circuit are given by... 

Figure 2.18. Graphical representation of the AC impedance of a parallel RC circuit...

If a resistor is added in series with the parallel RC circuit, the overall circuit becomes the well-known Randles cell, as shown in Figure 4.11a. This is a model representing a polarizable electrode (or an irreversible electrode process), based on the assumptions that a diffusion limitation does not exist, and that a simple single-step electrochemical reaction takes place on the electrode surface. Thus, the Faradaic impedance can be simplified to a resistance, called the charge-transfer resistance. The single-step electrochemical reaction is described as... 

Figure 11.10 plots the imaginary -Z" vs. the real part Z" of the complex impedance (Argand plot) exemplary for undoped CoTi03/La at 400°C under synthetic air. All plots showed semicircles and could be described with the impedance function of a parallel RC circuit equivalent. 

An automated data fitting software was developed allowing to describe the impedance spectra as the impedance function of a circuit equivalent, e.g., consisting of a resistor and a capacitor in parallel. Also other elements, such as inductances or constant phase elements (CPE), could be implemented. The admittance of the parallel RC circuit is just the sum of the admittances of the two elements which give the impedance44 ... 

For this circuit, the representation of versus the so-called Nyquist plot, is shown in Figure 1.9c. For a parallel RC circuit, the total impedance and phase angle become ... 

The effect of DC bias on a contaminated sample at 100% RH is shown in Figure 5. At bias levels corresponding to threshold and super-threshold levels for electrochemical reactions, the impedance spectrum shows the capacitive loop that intersects the real axis at low frequency (.1 Hz). Zero-DC-bias data, which are not shown, form a similar arc that is large compared to the scale of this plot. This behavior is modelled by a parallel RC circuit, whose resistance decreases from 1 x 10 to 1.6 x 10 and whose capacitance remains constant at approximately 30000 pF, as DC bias is raised from 0 to 3.0 V. The resistances agree with those measured in DC leakage current experiments. The capacitances are 100 times larger than those measured on the clean sample at 100 % RH. 

The variation of the impedance with frequency is often of interest and can be displayed in different ways. In a Bode plot, log Z and are both plotted against log cu. An alternative representation, a Nyquist plot, displays Zi vs. Zrc for different values of cu. Plots for the series RC circuit are shown in Figures 10.1.8 and 10.1.9. Similar plots for a parallel RC circuit are shown in Figures 10.1.10 and 10.1.11. 

Figure 10.1.10 Bode plots for a parallel RC circuit with R= 10011 and C = ...

Figure 6.1 Nyquist plots for (a] a capacitor, (b] a capacitor in series with a resistor, (c] a capacitor in parallel with a resistor, and (d] a resistor in series with a parallel RC-circuit.

The analysis of the impedance data can be carried out by a complex plane method using the Nyquist plot vs. The equation for a parallel RC circuit gives rise to a semicircle in the Z (co) plane, as that shown in Figure 2.1a, which has intercepts on the Zjgji axis at oo) and Rq( 0), where (Rq - RJ) is the resistance... 

The impedance plots for the composite HR95 membrane are shown in Figure 2.3, where two different contributions associated with the membrane (m) and the electrolyte solution between the electrodes and the membrane surfaces (e) can clearly be observed. To check this assumption, the impedance data obtained with the electrolyte alone, without any membrane in the measuring cell, are also plotted in Figure 2.3. As can be observed, a parallel RC circuit with only a relaxation process and a maximum frequency of around 10 Hz (similar to that in Figure 2.1) was obtained for the electrolyte solution measured alone (R C. The circuit associated with the composite polyamide/polysulfone HR95 membrane shows two subcircuits. 

GRAPH 11.23 Formal Graph of the parallel RC circuit in the general case with elementary properties (a) and when the conductance is proportional to the capacitance (homothetic conditions), leading to the existence of a scalar kinetic constant representing their composition (b). 

FIGURE 11.9 Cole-Cole (Nyquist) plots of the impedance (a) and of the admittance (b) of a parallel RC circuit in the linear case. The apex of the semicircle occurs when the angular frequency m equates the kinetic constant Kq (=1/Tc), which corresponds to the point intersecting the bisector in the admittance plot... 

The general relationships can be read in the Formal Graph in the case study abstract. From there, the development of the model is identical to what has been done with global state variables in case study H4 Parallel RC Circuit. ... 

The relaxation time corresponds here to the inverse of the kinetic constant. The plots of this impedance and of the corresponding admittance were given in case study H4 Parallel RC Circuit (Figure 11.9) and are reproduced in the case study abstract. 

FIG U RE 11.13 Plots of the complex impedance of a parallel RC circuit according to the Cole-Cole (Nyquist) representation (a) and to the dispersion representation (b, d). A 45° mirror copy (c) of the dispersion curve of the real part is placed under the Cole-Cole plot (a) for evidencing the construction of the semicircle through its projections (b, c) in the dispersion representation. 

Impedance of the Series and Parallel RC Circuits. In the simplest analysis, the circuits I and II represent the electrical configurations encountered in measurement and interpretation of experimental impedance spectroscopic behavior of the double-layer at electrode interfaces. They provide the basis for analysis of impedance spectra of more complex RC networks that arise in representation of supercapacitor behavior. 

In qualitatively considering the impedance behavior of series or parallel RC circuits, it is always useful to deduce the limiting cases that arise for (u 0 or (u —> oo. Then, respectively, and Zc —> 0. The behavior for these cases is usually... 

Figure 4.5.24. Complex-plane or Nyquist plot for the impedance spectrum of a simple parallel RC circuit, showing top-point characteristic frequency (0= 1/RC.

In the complex-plane representation of the impedance behavior of a parallel RC circuit, it is convenient to identify the maximum (so-called top point ) in the semicircular plot which is at a critical frequency O) = l/RC, the reciprocal of the time constant for the response of the circnit. A similar sitnation arises for a series RC... 

The impedance of a parallel RC circuit is always less than die resistance R or capacitive reactance Xc of the individnal branches. The relative valnes of Xc and R determine how capacitive or resistive the circnit line current is. The one that is the smallest and therefore allows more branch current to flow is the determining factor. Thus if Xc is smaller than R , the cnricnt in the capacitive branch is larger than the cnrrcnt in the resistive branch, and the line ciurent tends to be more capacitive (Fignre 16.8) (Jainwen et al., 2006). 

Figure 5.4 Simple parallel RC circuit model of the electrochemical interface.

Figure 5.5 (a) CV of the parallel RC circuit showing two scans. Inset emphasizes the charging current in the initial scan, (b) Increase in capacitive current as scan rate is increased. Parallel RC circuit constructed with a 2-Mf2 resistor, a 150-S2 resistor, and a IO-/4F capacitor. The arrows show the increase in current with increasing scan rate. 

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