On the distributed resistor-constant phase element transmission line in a reflective bounded domain

Abstract

In this work we derive and study the analytical solution of the voltage and current diffusion equation for the case of a finite-length resistor-constant phase element (CPE) transmission line (TL) network that can represent a model for porous electrodes in the absence of any Faradic processes. The energy storage component is considered to be an elemental CPE per unit length of impedance $z_c(s)={1}/{(c_{\alpha} s^{\alpha})}$ with constant parameters $(c_{\alpha},\alpha)$ instead of the ideal capacitor of impedance $z(s)={1}/{(c\, s)}$ usually assumed in TL modeling. The problem becomes a time-fractional diffusion equation for the voltage that we solve under galvanostatic charging, and derive from it a reduced impedance function of the form $z_{\alpha}(s_n)=s_n^{-\alpha/2}\coth({s_n^{\alpha/2}})$, where $s_n = j\omega_n$ is a normalized frequency. We also derive the system's step response, and the distribution function of relaxation times associated with it. The analysis can be viewed and used as a support for the fractal finite-length Warburg model.