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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 impedancez_c(s) = 1/(c_αs^α) with constant parameters (c_α, α) instead of the ideal capacitor of impedancez(s) = 1/(cs) 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 $z_{\alpha }\left(s_n\right)=s_n^{-\alpha /2}\coth \left(s_n^{\alpha /2}\right)$, wheres_n = jω_nis 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.