Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg.

•An exact Distribution Function of Relaxation Times (DFRT) has been derived for the fractal Finite Length Warburg (f-FLW).•The DFRT for a true FLW consists of an infinite series of δ-functions.•The impedance of a FLW can be presented by an infinite series of (RC) circuits

An analytic Distribution Function of Relaxation Times (DFRT) is derived for the fractal Finite Length Warburg (f-FLW, also called 'Generalized FLW') with impedance expression: Zf·FLW(ω) = Z0 · tanh(ωτ0)n · (ωτ0)−n. τ0 is the characteristic time constant of the f-FLW. Analysis shows that for n → 0.5 (i.e. the ideal FLW) the DFRT transforms into an infinite series of δ-functions that appear in the τ-domain at positions given by τk = τ0/[π2(k − ½)2] with k = 1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τk. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (RkC0)-circuits, with Rk = C0×τk−1 and τk as defined above. Rk = 2τk×Z0 and C0 = 0.5×Z0−1. Z0 is the dc-resistance value of the FLW.A full analysis of these DFRT expressions is presented and compared with impedance inversion techniques based on Tikhonov regularization and multi-(RQ) CNLS-fits (m(RQ)fit). Transformation of simple m(RQ)fits provide a reasonably close presentation in τ-space of the f-FLW, clearly showing the first two major peaks. Impedance reconstructions from both the Tikhonov and m(RQ)fit derived DFRT's show a close match to the original data.

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