Continuous approximation of the ZARC element with passive components

Abstract

The ZARC element is a parallel connection between a constant phase element and an ohmic resistor which describes the charge transfer and the double-layer capacitance at an electrode–electrolyte interface. However, this mathematical object has been determined using measurement data and cannot be derived from physical or chemical processes. In order to understand the dynamics of ZARC and its parameters’ influence in frequency and in time domain, we approximate it using fundamental equivalent circuits. Here, we introduce two approaches using RC circuits whose behaviours are well-known. The first method consists of infinitely many serially connected RC circuits which can be uniquely related to ZARC by explicit equations. In contrast, the second uses just three serially connected RC circuits, but adds a minimization problem. Both approaches depend only on three parameters: an ohmic resistance, a capacitance, and a newly defined parameter which is a measure of the modification of the single capacitances. Moreover, we show a decrease of the total capacitance of both impedances for growing deviations from the behavior of an RC circuit. Finally, since the properties of RC circuits are well known in frequency and in time domain, we deduce the behaviours of both methods in the time domain.