Transmission line finite

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.)...

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]... 

The linear can readily be extended to consider the case of a finite transmission line. In this case the initial and boundary conditions are... 

The shape of the chronoamperometric current response depends to a large extent on the value of the uncompensated solution resistance R . We note from Fig. 1.54 that is in series with the finite transmission line element. The presence of uncompensated solution resistance effects can be clearly identified by examining the current/time data when the latter is plotted in i(t) versus format. In many cases (see Fig. 1.56) such plots are nonlinear, so we observe deviation from the expected linear response at both short and long time periods. At short time... 

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.)...

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

T. Tischler, and W. Heinrich, The perfectly matched layer as lateral boundary in finite difference transmission line analysis, Int. Microwave Symposium Digest 1, 121-124 (2000). 

In all wave methods the transmission line is ideally matched except the sample holder. If the value of Z (go) is differ from Zq, one can observe the reflection from a mismatch of the finite magnitude. A similar type of wave analysis also applies in free space and in any other wave systems, taking into account that in free space Zq k, 377 il for plane waves. 

The finite thickness of a film on the resonator surface makes the calculation of the mechanical impedance at the surface analogous to that of an appropriately terminated transmission line [41]. Noting the correspondence between stress and voltage and between particle velocity and current, the stress-firee upper film surface is analogous to a short-circuited electrical transmission line. From this analogy, the input impedance seen at the resonator/film interface is [40]... 

T. Kokkinos, C. Sarris, and G. Eleftheriades, Periodic finite-difference time-domain analysis of loaded transmission-line negative-refractive-index metamaterials, IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1488-1495, Apr. 2005. doi 10.1109/TMTT.2005.845197... 

The slip-type joint (Figure 7-112) is particularly useful in steam and hot-water transmission lines where expansions and contractions are frequent and of large magnitude. Slip joints contain no highly-stressed flexing element subject to failure after a finite number of cycles. 

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. 

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]. 

In addition to the interaction between the electromagnetic wave and the materials, the heating effect of applicators is also an important issue in microwave sintering. This is simply because applicators are necessary to build the cavity resonators. Various methods, such as finite-difference time domain (FDTD) [61-63], finite element method (FEM) [64], transmission line matrix (TLM) [65, 66], method of moments (MOM), have been used for such purposes. 

A modified approach in [23] is to apply a transmission line model with Finite-Difference Time-Domain method (FDTD). But this method has some drawbacks. 

Transmission line models can be used for inert electrodes and it is a modification of the Randles model (Fig. 6.3). Since the Randles-circuit can be used to describe a nondistributed system, the transmission line models invokes a finite diffusional Warburg impedance, Z, in place of concentration hindered impedance (Fig. 6.4). Randles model is concerned with Qi (the double layer capacitance), [the resistance to charge transfer) and Z by describing the processes occurring in the film. The expression of total impedance, Ztot, is given by following equation ... 

Because of the assumption of semiinfinite diffusion made by Warburg for the derivation of the diffusion impedance, it predicts that the impedance diverges from the real axis at low frequencies, that is, according to the above analysis, the dc-impedance of the electrochemical cell would be infinitely large. It can be shown that the Warburg impedance is analogous to a semi-infinite transmission line composed of capacitors and resistors (Fig. 8) [3]. However, in many practical cases, a finite diffusion layer thickness has to be taken into consideration. The first case to be considered is that of enforced or natural convection in an... 

Historically, the Warburg impedance, which models semi-infinite diffusion of electroactive species, was the first distributed circuit element introduced to describe the behavior of an electrochemical cell. As described above (see Sect. 2.6.3.1), the Warburg impedance (Eq. 38) is also analogous to a uniform, semi-infinite transmission line. In order to take account of the finite character of a real electrochemical cell, which causes deviations from the Warburg impedance at low frequencies. 

Each type of transmission line has different characteristic impedance and propagation constant which are functions of the complex permittivity of the substrate and superstate. The complex permittivity of methanol is extracted in this method for the frequency up to 40 GHz using a hybrid method that combines experimental data with finite element analysis. 

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... 

It is therefore apparent that in passing from high to low freqnency in a system of this kind, there is an additional impedance dne to concentration polarization. Macdonald and HnU [1984] considered this effect on the electrical response of this type of system. Under many drcnmstances, the presence of concentration polarization might be confnsed with an interface impedance. At different ratios of mobiU-ties of anions and cations, either diffnsion-Iike response (finite-length transmission line behavior) or parallel capacitative-resistive behavior may appear. Ac impedance methods have been nsed to determine ionic transference nnmbers in polymeric electrolytes using this principle (Sorensen and Jacobsen [1982]). 

---