Finite transmission-line equivalent circuit

When discussing the ionic conductivity of catalyst layers, one must mention the finite transmission-line equivalent circuit, which is widely used to model porous electrodes and was shown as Figure 4.33 in Chapter 4. For ease of discussion, the circuit is re-plotted here as Figure 6.23. 

Figure 6.23. Finite transmission-line equivalent circuit describing the impedance behaviour of a PEMFC electrode [24], (Reprinted from Electrochimica Acta, 50(12), Easton EB, Pickup PG. An electrochemical impedance spectroscopy study of fuel cell electrodes. Electrochim Acta, 2469-74, 2005, with permission from Elsevier and the authors.)...

Figure 11.17. Illustration of the finite transmission-line equivalent circuit for a porous electrode...

Figure 23.4. Finite transmission-line equivalent circuit describing the impedance behavior of a PEMFC electrode [29]. (Reproduced by permission of ECS— The Electrochemical Society, from Lefebvre M, Martin RB, Pickup PG. Characterization of ionic conductivity profiles within proton exchange membrane fuel cell gas diffusion electrodes by impedance spectroscopy.)...

For porous electrodes, an additional frequency dispersion appears. First, it can be induced by a non-local effect when a dimension of a system (for example, pore length) is shorter than a characteristic length (for example, diffusion length), i.e. for diffusion in finite space. Second, the distribution characteristic may refer to various heterogeneities such as roughness, distribution of pores, surface disorder and anisotropic surface structures. De Levie used a transmission-line-equivalent circuit to simulate the frequency response in a pore where cylindrical pore shape, equal radius and length for all pores were assumed [14]. 

Greszczuk et al. [252] employed the a.c. impedance measurements to study the ionic transport during PAn oxidation. Equivalent circuits of the conducting polymer-electrolyte interfaces are made of resistance R, capacitance C, and various distributed circuit elements. The latter consist of a constant phase element Q, a finite transmission line T, and a Warburg element W. The general expression for the admittance response of the CPE, Tcpr, is [253]... 

A very important issue to consider when working with porous electrodes is that the capacitance is only accessible through a distribution of ohmic resistances, due to the finite resistance of flie supporting electrolyte inside the pores. These situations can be roughly represented by an equivalent circuit, as shown in Fig. 11, where the porous electrode is described by a truncated RC transmission line of R and C elements representing the double Ityer capadtance and the electrolyte resistance in a particular pore size. 

The equivalent circuit analog of this situation is a finite-length transmission line terminated with an open circuit. A constant activity or concentration is also a common condition for the interface removed from x = 0. In this case the finite-length transmission line would be terminated in a resistance, and the impedance is given by the expression... 

Diffusion-Related Elements. Although we usually employ ideal resistors, capacitors, and inductances in an equivalent circuit, actual real elements only approximate ideality over a limited frequency range. Thus an actual resistor always exhibits some capacitance and inductance as well and, in fact, acts somewhat like a transmission line, so that its response to an electrical stimulus (output) is always delayed compared to its input. All real elements are actually distributed because they extend over a finite region of space rather than being localized at a point. Nevertheless, for equivalent circuits which are not applied at very high frequencies (say over 10 or 10 Hz), it will usually be an adequate approximation to incorporate some ideal, lumped-constant resistors, capacitors, and possibly inductances. 

But an electrolytic cell or dielectric test sample is always finite in extent, and its electrical response often exhibits two generic types of distributed response, requiring the appearance of distributed elements in the equivalent circuit used to fit IS data. The first type, that discussed above, appears just because of the finite extent of the system, even when all system properties are homogeneous and space-invariant. Diffusion can lead to a distributed circuit element (the analog of a finite-length transmission line) of this type. When a circuit element is distributed, it is found that its impedance cannot be exactly expressed as the combination of a finite number of ideal circuit elements, except possibly in certain limiting cases. 

Mathematical models based on probability have been developed to analyze the system as a whole on the basis of the statistical behavior of the EMI [100]. These models are useiul in developing computer aided design and analysis procedures to solve EMI problems. Computational models of EMI and EMC problems from circuit level to a more complex system levels such as aircraft EMC have also been extensively studied. Techniques such as FEM, method of moments (MoM), Transmission Line method (TLM), Finite Difference Time Domain method (FDTD), Finite Difference Frequency Domain method (FDFD), Partial Element Equivalent Circuit model (PEEC) and a number of such methods have been used for this purpose for various applications. Interested readers can see reference 100 for a nice review of such approaches. 

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