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

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: Z(f-FLW)(omega) = Z(0).tanh (omega tau(0))(n) . (omega tau(0)) (n). tau(0) 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 delta-functions that appear in the T-domain at positions given by T-k = T-0/[pi(2)(k - 1/2)(2)] with k = 1, 2, 3, . . . infinity. The mathematical surface areas of these d-functions are proportional to T-k. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (RkC0)-circuits, with R-k = C-0 x T-k(1) and T-k as defined above. R-k = 2T(k)xZ(0) and C-0 = 0.5 x Z(0)(1). Z(0) 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 T-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. (C) 2017 Elsevier Ltd. All rights reserved.