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) = 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 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_kC₀)-circuits, with R_k = C₀×τ_k ⁻¹ and τ_k as defined above. R_k = 2τ_k×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.