Basic Equivalent Circuits

Impedance models are constructed according to the electrochemical phenomena. The total impedance of an electrochemical system can be expressed by different combinations of the electrical elements. This section covers the features of basic equivalent circuits commonly used in electrochemical systems. In Appendix D, the effect of an element parameter change on a spectrum related to a given equivalent circuit is described in detail. 

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Figure 1.23 shows the Nyquist plot of a 2.3V/10F supercapacitor laboratory cell. When compared to Figure 1.22b, it appears that the basic equivalent circuit proposed is unable to describe the real behavior of an EDLC. 

In summary, simple combinations of elements and basic equivalent circuits for electrochemical systems have been introduced in this section. Although these models are relatively simple, they are commonly employed in the investigation of electrochemical systems, including fuel cells. A real electrochemical system may be much more complicated. However, complicated electrochemical systems can still be constructed from these basic equivalent circuits. 

Figure G.2 Basic equivalent circuit for calculating the volt-drop of a 500 kW HV motor.

Figure 11.13 illustrates a basic equivalent circuit to represent a general electrochemical reaction. Rs represents the electric resistance, which consists of the ionic, electronic, and contact resistances. Since the electronic resistance is typically much lower than the ionic resistances for a typical fuel cell MEA, the contribution of the electronic resistance to Rs is often negligible. Cj is the double-layer capacitance associated with the electrode-electrolyte interfaees. Since a fuel cell electrode is three-dimensional, the interfaces include not only Arose between Are surfaces of the electrodes and the membrane but also those between the catalysts and the ionomer within the electrodes. Ret is the resistanee associated with the charge transfer process and is called charge transfer resistanee. Z is called the Warburg impedance it deseribes the resistance arising from the mass transport processes. 

Figure 11.13. Illustration of a basic equivalent circuit for an electrochemical reaction. -Ohmic resistance R., - charge transfer resistance Cd - double-layer capacitance Z -Warburg impedance.

Figure 7.7. Basic equivalent circuit for metal microelectrode.

Figure 9.2 Basic equivalent circuit used to model electrode in Figure 9.1.

Fig. 19.36 Basic circuit for a poiemiostat. (a) Basic circuit for a potentiostat and electrochemical cell, (b) Equivalent circuit, (c) Circuit of a basic potentiostat. A.E. is the auxiliary electrode, R.E. the reference electrode and W.E. the working electrode (6 and c are from Polen-tiostat and its Applications by J. A. von Fraunhofer and C. H. Banks, Butlerworths (1972))...

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

In an analysis of an electrode process, it is useful to obtain the impedance spectrum —the dependence of the impedance on the frequency in the complex plane, or the dependence of Z" on Z, and to analyse it by using suitable equivalent circuits for the given electrode system and electrode process. Figure 5.21 depicts four basic types of impedance spectra and the corresponding equivalent circuits for the capacity of the electrical double layer alone (A), for the capacity of the electrical double layer when the electrolytic cell has an ohmic resistance RB (B), for an electrode with a double-layer capacity CD and simultaneous electrode reaction with polarization resistance Rp(C) and for the same case as C where the ohmic resistance of the cell RB is also included (D). It is obvious from the diagram that the impedance for case A is... 

Fig. 5.21 Basic types of impedance spectra and of corresponding equivalent circuits (Eqs 5.5.16 to 5.5.19). (According to R. D. Armstrong et al)...

At the heart of impedance analysis is the concept of an equivalent circuit. We assume that any cell (and its constituent phases, planes and layers) can be approximated to an array of electrical components. This array is termed the equivalent circuit , with a knowledge of its make-up being an extremely powetfitl simulation technique. Basically, we mentally dissect the cell or sample into resistors and capacitors, and then arrange them in such a way that the impedance behaviour in the Nyquist plot is reproduced exactly (see Section 10.2 below on electrochemical simulation). 

Modeling and optimization of chemical sensors can be assisted by creating equivalent electrical circuits in which an ordinary electrical element, such as a resistor, capacitor, diode, and so on, can represent an equivalent nonelectrical physical parameter. The analysis of the electrical circuit then greatly facilitates understanding of the complex behavior of the physical system that it represents. This is a particularly valuable approach in the analysis and interpretation of mass and electrochemical sensors, as shown in subsequent chapters. The basic rules of equivalent circuit analysis are summarized in Appendix D. Table 3.1 shows the equivalency of electrical and thermal parameters that can be used in such equivalent circuit modeling of chemical thermal sensors. 

This expression is the basic description for the use of the pyroelectric effect in a host of sensor applications including the well known optical detection devices (82,83). A particularly useful way of describing this type of system is with an equivalent circuit where the pyroelectric current generator drives the pyroelectric impedance and the measuring amplifier circuit as shown in Figure 11. 

A basic electric equivalent circuit to describe an EDLC is presented in Figures 1.22a and b, which shows the Nyquist (Figure 1.22b) plot of an ideal capacitor C, in series with a resistance... 

Basically, the impedance behavior of a porous electrode cannot be described by using only one RC circuit, corresponding to a single time constant RC. In fact, a porous electrode can be described as a succession of series/parallel RC components, when starting from the outer interface in contact with the bulk electrolyte solution, toward the inner distribution of pore channels and pore surfaces [4], This series of RC components leads to different time constant RC that can be seen as the electrical response of the double layer charging in the depth of the electrode. Armed with this evidence, De Levie [27] proposed in 1963 a (simplified) schematic model of a porous electrode (Figure 1.24a) and its related equivalent circuit deduced from the model (Figure 1.24b). 

FIGURE 11.9 Basic capacitor electrical equivalent circuit comprising a capacitance, a series inductance, a series resistance, and a parallel resistance. This simple model can fit a DLC behavior in first approximation for a given frequency. 

Finally, the basic equivalence of the two measuring techniques should be appreciated. Although there are many ways to approach such a comparison, the following simplified explanation will, we hope, give a more intuitive feeling for the relationship between EIS and PR measurements. As stated above, both techniques rely on the frequency dependence of the impedance of the double-layer capacitance in order to determine the polarization resistance. EIS uses low frequencies to force the capacitor to act like an open circuit. PR measurements use a slow scan rate to do the same thing. To make comparisons, the idea of equivalent scan rate is useful. Suppose that a particular electrochemical system requires EIS measurements to be made down to 1 mHz in order to force 99% of the current through Rp. What would the equivalent scan rate be for PR measurements A frequency of 1 mHz corresponds to a period of 1000 s. If the sine wave is... 

Transmission line — This term is related to a more general concept of electric -> equivalent circuits used frequently for interpretation of experimental data for complex impedance spectra (-> electrochemical impedance spectroscopy). While the complex -> impedance, Z, at a fixed frequency can always by obtained as a series or parallel combinations of two basic elements, a resistance and a capacitance, it is a much more compli-... 

However, although powerful numerical analysis software, e.g., Zview, is available to fit the spectra and give the best values for equivalent circuit parameters, analysis of the impedance data can still be troublesome, because specialized electrochemical processes such as Warburg diffusion or adsorption also contribute to the impedance, further complicating the situation. To set up a suitable model, one requires a basic knowledge of the cell being studied and a fundamental understanding of the behaviour of cell elements. 

The rapid development of computer technology has yielded powerful tools that make it possible for modem EIS analysis software not only to optimize an equivalent circuit, but also to produce much more reliable system parameters. For most EIS data analysis software, a non-linear least squares fitting method, developed by Marquardt and Levenberg, is commonly used. The NLLS Levenberg-Marquardt algorithm has become the basic engine of several data analysis programs. 

The impedance of the skin has been generally modeled by using a parallel resistance/capacitor equivalent circuit (Fig. 4a). The skin s capacitance is mainly attributed to the dielectric properties of the lipid-protein components of the human epidermis [5,8,9,12]. The resistance is associated primarily with the skin s stratum comeum layer [5,8,9,12]. Several extensions to the basic parallel resistor/capacitor circuit model have appeared in the literature [5,8,9,13]. Most involve two modified parallel resistor/capacitor combinations connected in series [5,8,9]. The interpretation of this series combination is that the first parallel resistor/capacitor circuit represents the stratum comeum and the second resistor/capacitor parallel combination represents the deeper tissues [5,8,9]. The modification generally employed is to add another resistance, either in series and/or in parallel with the original parallel resistor/capacitor combination [8,9]. Realize that because all of these circuits contain a capacitance, they will all exhibit a decrease in impedance as the frequency is increased. This is actually what is observed in all impedance measurements of the skin [5,6,8-15]. In addition, note that the capacitance associated with the skin is 10 times less than that calculated for a biological membrane [12]. This... 

Fig. IG Equivalent circuit for an activation controlled process, showing the three basic circuit elements the double-layer capacitance,...

Furthermore, it is of interest to note that the overall process that leads to intermediates VI and VIII here are basically equivalent in the sense that both are the consequence of the attack by a nucleophilic entity at the imine carbon atom. However, while the introduction of alkoxy at this carbon is the equivalent of a Sn2 substitution, the introduction of hydride requires the more circuitous and conceptually different path of reductive addition followed by nonoxidative elimination. In consequence, the latter implies reduction, the former does not. 

The basic know-how is presented to simulate impedance diagrams of complex equivalent circuits by viewing the electrode surface through fractal patterns. The implications of this model for electrochemical surface technology are also reported in this paper. 

The equivalent circuit of the basic, spatially extended, system as introduced in Ref. 147 is shown in Fig. 38. The electrochemical cell is approximated by a one-dimensional electrode with periodic boundary conditions, while the electrolyte is a two-dimensional, electroneutral... 

Figure 38. Equivalent circuit of the basic spatially extended system, x, direction parallel to the electrode z, direction perpendicular to the electrode.

Figure 43. Equivalent circuit for the basic, spatially one-dimensional system with an external resistor.

The equivalent circuit of a hquid-immersed quartz crystal. Fig. 10, consists of the same basic components. The motional acoustic impedance caused by a (sensing) rigid film and the liquid load can be separated into an inductance, Lc, (coating), an inductance, Tuq, and a resistance, ituq, (liquid). All additional elements are in series to the quartz motional elements. Cuq and Gnq account... 

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