5 years ago

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

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(ω)= Z 0 ·tanh(ωτ 0) n ·(ωτ 0)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 τ k =τ0/[π2(k − ½)2] with k =1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τ k . It is found that the FLW impedance can be simulated by an infinite series combination of parallel (R k C0)-circuits, with Rk = C 0×τ k −1 and τ k as defined above. Rk = k ×Z 0 and C 0 =0.5×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 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.

Publisher URL: www.sciencedirect.com/science

DOI: S0013468617318145

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