Passively realizable approximations of non-realizable fractional order impedance functions

Abstract

Approximating the fractional order differentiation and integration operators is a common approach in implementation of fractional order dynamics. This paper aims to investigate how the procedure of approximating the fractional order operators influences on the realizability of a fractional order impedance function by passive networks. To this aim, conditions for the possibility of passive realization of the approximations of the fractional order impedance functions by using RLC components are obtained. More precisely, considering two general forms for the filters approximating the fractional order operators, the open mapping theorem in complex analysis is applied to obtain the realizability conditions on the polar plots of the approximating filters. It is found that the approximated impedance function may be realizable by a passive RLC network, whereas the original fractional order impedance function cannot be realized by passive networks composed of resistors and fractional inductors and capacitors. Furthermore, for a class of impedance functions, the realizability condition is simplified as a condition on the phase of the filter approximating the fractional order differentiation operator. Some examples are presented to verify the usefulness of the obtained conditions.