A Direct Approximation of Fractional Cole–Cole Systems by Ordinary First-order Processes

Cole-Cole systems are widely used in electrochemistry to represent impedances of galvanic elements like fuel cells. For system analysis of Cole-Cole systems in the time domain fractional calculus has to be applied. A plain representation of fractional differential equations of Cole-Cole systems by means of conventional ordinary differential equations is addressed in this contribution. Usually in literature, the operator for the fractional derivation is appproximated to ensure that the fractional system can be represented by conventional differential equations of integer order. This article presents a new approach which results in a direct approximation of the Cole-Cole system itself by conventional linear time invariant systems. The method considered is based on the distribution density of relaxation times of conventional first-order processes. This distribution density is an alternative representation of the transfer behavior of such systems. Several approximation methods, based on an analysis of the distribution density, are presented in this work. The feasibility of these methods will be demonstrated by a comparison of the approximation model to a reference model for a solid oxide fuel cell (SOFC), respectively.