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 fFLW (v) = Z 0 tanh (vt 0 ) n (vt 0 ) n . t 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 d-functions that appear in the t-domain at positions given by t k = t 0 /[p 2 (k 1 / 2 ) 2 ] with k = 1, 2, 3, . . . 1. 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 (R k C 0 )-circuits, with R k = C 0 t k 1 and t k as defined above. R k = 2t k Z 0 and C 0 = 0.5Z 0