Low-frequency approximations to the finite-length Warburg diffusion impedance: The reflexive case

A study on the influence of the characteristic parameters of the finite-length diffusion processes on the galvanostatic dynamic response of electrochemical systems described by the reflexive Warburg impedance, is presented. The characteristic frequencies of the reflexive diffusion impedance, originally including those of the purely diffusive behaviour, are identified. The galvanostatic charge/discharge cycle is interpreted from the truncated series obtained from the development into partial fractions of the impedance. The low-frequency behaviour is also compared to the impedance obtained from the development into continued fractions with an only characteristic frequency. Interesting results associated to the transition between the capacitive and diffusive behaviours, emerge from the use of a hybrid methodology dealing with frequency and time domains. A simple electric circuit physically based on the series developments of the reflexive Warburg impedance with an only relaxation time, is used to interpret the galvanostatic charge curve of a commercial supercapacitor and to validate the theoretical study.