Equivalent circuit approach

A successful equivalent circuit approach to Wagner s theory was worked out by Hoar and Price. They developed a simple voltage divider circuit which gives quantitative formulas for emf and scaling rate that are very similar to those derived more rigorously by Wagner. A linear lumped version of their proposed circuit is shown in Fig. 3. The subscripts 1, 2 and 3 refer to M cations, X anions and electrons respectively. 

In part I above, c. Wagner s theory of mixed conduction was reviewed in terms of an equivalent circuit approach. The implications of mixed conduction theory for parabolic scaling of metals in high temperature atmospheres were also detailed. It was pointed out, however, that current interest in mixed conduction theory is no longer motivated by corrosion considerations because far too few systems of practical interest conform to the conditions required for pareibolic oxidation. 

Returning to the main thread here, we note that Equations 33 and 34 apply to each mobile species individually however, in all cases of practical interest simultaneous migration of more than one species must be considered. This is especially true in the case of multicomponent electrolytes in which many kinds of ions may be mobile. But if several species are in motion they merely make simultaneous contributions to the current exchanged with the external circuit. The relative contribution of each will depend on the ease of transport in the elctrolyte, i.e., on Gj and on the chemical and electrical potential gradients which act on it. But these matters are better discussed in terms of the extended equivalent circuit approach which will now be developed. 

Some details of the Randles circuit require further study. First, the double layer capacitance is in many cases more properly modeled with a constant phase element. This gives information on the mesoscopic structure of the oxide-electrolyte interface. Also, in some cases, the diffusion impedance contains a power-law behavior. The reason for this is controversial is it due to the back contact or rather an indicator of an unknown kinetic process in WO3 It should also be mentioned that the adsorption process proposed by Fianceschetti and Macdonald [1982] has not been studied in detail, and the systematic equivalent circuit approach of Janmik [2003] has only been rarely used. 

It is possible to interpret the optoelectrical impedance in terms of an electrical equivalent circuit. When doing so, the system resembles a resistor R = AQ at low frequencies and a capacitor C = tIAQ at high frequencies. At intermediate frequencies the system shows a more complex behavior. One should be careful to have a mental picture of a resistor and a capacitor when using equivalent circuit approach for interpreting IMPS data. Maybe it is better to look upon it in a more abstract mathematical way. 

The major shortcoming of an equivalent circuit approach is that an impedance paAem obtained experimentally can be presented by more than one equivalent... 

In another work, Du et al. (2007a) measured the impedance spectra of a half-cell cathode, with and without 0.5M methanol solution on the anode side of the membrane. The resulting spectra show an increase in the charge transfer resistance due to crossover, and a somewhat larger inductive low-frequency loop. Using the equivalent circuit approach, the increase in the loop radius was attributed to the poisoning of the Pt surface by MOR intermediates. 

At this point it should be cautioned that the equivalent circuit approach can yield detailed information of the physicochemical processes of ohmic, mass transfer, and kinetic resistances for a given system, but it is subject to the assumptions of the equivalent circuit used. That is, the use of equivalent circuit analysis offers infinite possibilities and combinations of electrical circuits which all can be rearranged in different ways. The EIS... 

A more fundamental approach to EIS data interpretation is based on first principles and attempts to describe the frequency response data directly from analytical models. Both single- electrode and full-cell models have been developed to describe observed EIS data for fuel cells, with successful explanation of some of the EIS response [e.g., 3-10]. This approach has the ultimate goal of being able to fully predict the EIS response for a given electrode configuration, so that optimal surfaces can be developed. While the end goal is ultimately more fundamental than the equivalent circuit approach, the complexity involved with the porous and partially flooded electrode structures found in fuel cells has precluded its extensive application. 

M. V. ten Kortenaar, C. Tessont, Z. I. Kolar, H. van der Weijde, Anodic oxidation of formaldehyde on gold studied by electrochemical impedance spectroscopy an equivalent circuit approach, J. Electrochem. Soc., 1999,146,6, pp. 2146-2155. 

Fig. 10.1 Equivalent circuits used to represent the semiconductor-electrolyte interface, (a) A more complete approach taking into account the series resistance (Ry), the depletion layer (Csc, Rsc), an oxide surface film...

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. 

The equivalent electrical circuit approach has already been introduced in connection with analysis of mass sensors (Chapter 4). Its application is older and somewhat... 

Commercial impedance analyzers offer equivalent circuit interpretation software that greatly simplifies the interpretation of results. In this Appendix we show two simple steps that were encountered in Chapters 3 and 4 and that illustrate the approach to the solution of equivalent electrical circuits. First is the conversion of parallel to series resistor/capacitor combination (Fig. D.l). This is a very useful procedure that can be used to simplify complex RC networks. Second is the step for separation of real and imaginary parts of the complex equations. 

The complications and sources of error associated with the polarization resistance method are more readily explained and understood after introducing electrical equivalent circuit parameters to represent and simulate the corroding electrochemical interface (1,16-20). The impedance method is a straightforward approach for analyzing such a circuit. The electrochemical impedance method is conducted in the frequency domain. However, insight is provided into complications with time domain methods given the duality of frequency and time domain phenomena. The simplest form of such a model is shown in Fig. 3a. The three parameters (Rp, Rs, and C d,) that approximate a corroding electrochemical inter-... 

When we begin to investigate an electrochemical system, we normally know little about the processes or mechanisms within the system. Electrochemical impedance spectroscopy (EIS) can be a powerful approach to help us establish a hypothesis using equivalent circuit models. A data-fitted equivalent circuit model will suggest valuable chemical processes or mechanisms for the electrochemical system being studied. From Chapter 1, we know that a fuel cell is actually an electrochemical system involving electrode/electrolyte interfaces, electrode reactions, as well as mass transfer processes. Therefore, EIS can also be a powerful tool to diagnose fuel cell properties and performance. 

Alternatively, an equally powerful visualization of impedance data involves Bode analysis. In this case, the magnitude of the impedance and the phase shift are plotted separately as functions of the frequency of the perturbation. This approach was developed to analyze electric circuits in terms of critical resistive and capacitive elements. A similar approach is taken in impedance spectroscopy, and impedance responses of materials are interpreted in terms of equivalent electric circuits. The individual components of the equivalent circuit are further interpreted in terms of phemonenological responses such as ionic conductivity, dielectric behavior, relaxation times, mobility, and diffusion. 

If the interphase is ideally polarizable, the faradaic resistance approaches infinity, and the equivalent circuit shown in Fig. lG(a) can be simplified to that shown in Fig. 2G(b). If it is ideally nonpolari-zable, the faradaic resistance tends to zero, and the equivalent circuit shown in Fig. lG(c) results. Real systems never behave ideally, of course they may approach one extreme behavior or the other, or be... 

This approach, and the Randles equivalent circuit , have been very widely used in the years since 1947. The impact of Randles paper can be gauged by the fact that it continues to be cited, typically attracting 10 citations a year in the period since 1981, and by the breadth of research fields on which it has impacted. Thus in the last few years alone Randles paper has been cited in papers on biosensors, corrosion, physiology, displays, and materials research d as well, of course, as in electrochemistry. 

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