Transmission line semi infinite length

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 ... 

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). 

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

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