A unified treatment of the Cottrell-type response and the limiting regime in the Warburg diffusion impedances

A study on the unification into an approximate impedance of the Cottrell-type response and the limiting behaviours of the transmissive and reflexive finite-length Warburg diffusion impedances, is presented. The remainder functions arising from the series expansions into partial fractions of the restricted diffusion impedances after successively isolating low-frequency terms, are analysed. This allows us to distinguee between high and low frequency diffusion processes. A reduced-order model where the remainder functions obtained by isolating the limiting terms, are Gerischer-type impedances that naturally embed the Cottrell-type response into the finite-length diffusion framework, is presented for the first time. Characteristic frequencies associated to poles of order ½ are then included to consistently represent restricted Warburg behaviour. The resulting impedance functions are systematically compared, in both frequency and time domains, with the exact results, demonstrating their ability to capture restricted diffusion dynamics. • The semi-infinite Warburg impedance is bounded by a Gerischer-type impedance. • High and low frequency finite-length diffusion processes are identified. • A reduced-order impedance unifying Cottrell and limiting behaviours is presented. • Unified impedances lead to an easy interpretation of dynamic responses.