Transmission line semi infinite

Fig. 20 Schematic representation of a two-terminal device. The scattering region (enclosed in the dashed-line frame) with transmission probability T(E) is connected to semi-infinite left (L) and right (R) leads which end in electronic reservoirs (not shown) at chemical potentials Eu and r, kept fixed at the same value p for linear transport. By applying a small potential difference electronic transport will occur. The scattering region or molecule may include in general parts of the leads (shaded areas) (adapted from [105] with permission Copyright 2002 by Springer)...

Fig. 11.17. A resistive-capacitive transmission line that describes a semi-infinite...

Fig. IL5.4 Resistive-capacitive semi-infinite transmission line infinite diffusion. R and C are normalized to unit length...

Equation (IL5.36) shows that the Warburg impedance cannot be represented as a series combination of frequency-independent elements in an equivalent circuit. This is possible, however, by a semi-infinite resistive-capacitive transmission line with a series resistance R per unit length and a shunt capacity C per unit length (Fig. IL5.4). 

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... 

Fig. 8 Infinite length transmission line that describes the behavior of semi-infinite diffusion ([3] Chapter 2.1).

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. 

Mathematical analogy versus physical resemblance. In case Study G5 Transmission Line the algebraic model of a semi-infinite electric line has been established, having an equation... 

The porous electrode model described in Eq. (9.7) cannot be represented by a simple connection of R, L, and C elements. However, it can be represented by a semi-infinite series of R-C elements called a transmission fine [410,411], shown in Fig. 9.4. Of course, this representation is equivalent to Eq. (9.7). Some authors tried to use a transmission line to approximate experimental data using a sufficient number of RC elements and verifying whether the number of these parameters was sufficient. This procedure can approximate, then, experimental impedances, but the use of Eq. (9.7) is more appropriate because it allows for the direct estimation of certain parameters and their standard deviations. This model is included in the recent version of the ZView program. 

As the first approximation, impedance of a porous electrode can always be considered as a series combination of two processes—a mass-transport resistance inside the pores and impedance of electrochemical reactions inside the pores. De Levie was the first to develop a transmission line model to describe the frequency dispersion in porous electrodes in the absence of internal diffusion limitations [66]. De Levie s model is based on the assumption that the pores are cylindrical, of uniform diameter 2r and semi-infinite length /, not intercoimected, and homogeneously filled with electrolyte. The electrode material is assumed to have no resistance. Under these conditions, a pore behaves like a imiform RC transmission line. If a sinusoidal excitation is applied, the transmission line behavior causes the amplitude of the signal to decrease with the distance from the opening of the pore, and concentration and potential gradients may develop inside the pore. These assumptions imply that only a fraction of the pore is effectively taking part in the double-layer charging process. The RpQi i- [ohm] resistance to current in a porous electrode structure with number of pores n, filled with solution with resistivity p, is ... 

The impedance of a porous electrode can be simulated with the transmission line model, and the penetration depth can be evaluated [24]. For the non-porous Pt-modified as-deposited surface, the methanol oxidation reaction can be simulated as a simple Randles equivalent circuit comprising a parallel combination of a double layer capacitance and a semi-infinite Warburg impedance in series with a charge transfer resistance. 

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