On the Distributed Resistor-Constant Phase Element Transmission Line in a Reflective Bounded Domain

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 _α s ^α) with constant parameters (c _α, α) 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}_α({s}_n)={s}_n^−α/2coth({s}_n^α/2), where s _n = jω _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.