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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. 

Measurements acquired on batteries in the form of time signals such as voltage-time and capacity-time to assess their cyclability performance can be supplemented by examining their frequency-domain response. This allows one to determine the global characteristics of the signals and the battery, but not the local behavior, which is very important for determining for example battery fading. In this study we examine the short-time Fourier transform for time-frequency deconstruction of galvanostatic charge/discharge signals of lithium-sulfur batteries, taken as an example. The results displayed in terms of spectrograms show how the frequency content of such signals (e.g. charge and voltage time series) evolve with the lifetime of the batteries allowing the detection of critical changes in the response that may lead to fading and eventually default.