Positive‐real property of passive fractional circuits in W‐domain

Fractional circuits have attracted extensive attention of scholars and researchers for their superior performance and potential applications. Fractional circuits constitute a new challenge for the analysis and synthesis methods of traditional circuits theory. Passivity is the fundamental property of traditional circuits (integer order electric circuits). As is known to all, passivity is equivalent to positive realness in traditional linear circuits. However, this equivalence is broken down by introducing fractional elements into electrical networks in s‐domain. To address this issue, on the basis of s‐W transformation, we study the passive criteria of fractional circuits with rational order elements in this paper. Definitions of positive‐real (matrix) function in W‐domain are given, and the equivalence conditions of positive realness are derived. In addition, a conclusion is proposed in which the immittance (matrix) function of passive fractional circuits with rational order elements is positive real in W‐domain. The applications of passive criteria in circuit synthesis are shown.