Low-frequency development approximations to the transmissive Warburg diffusion impedance

Abstract

A study of the behaviour of the low-frequency development approximations to the transmissive Warburg diffusion impedance in both the frequency and time domains is presented. Simple low-frequency electric circuits have been derived from the Weierstrass products expansion and the equivalent ladder networks based on Foster, Maxwell and Cauer series. An original comparative study of the different low-frequency approximated impedances, including the characteristic frequencies, and those of the transmissive Warburg element is reported. The galvanostatic small-signal dynamic responses involving current steps and ramps have been compared by using exact numerical solutions and the approximated solutions at low frequencies derived from the different electric circuits. This study allows the dynamic response of electrochemical energy devices, such as fuel cells or rechargeable batteries, to be physically interpreted on the basis of an only relaxation time by means of the charging/discharging process of a capacitor through one or two resistors.