Multivariate theory‐based passivity criteria for linear fractional networks

Abstract

Summary In traditional linear network theory, the positive‐real (PR) criteria are widely used to judge the passivity of elements and networks in the light of the fact that there exists an equivalent relationship between the passivity and the PR property of their immittance functions (matrices). However, the equivalence will no longer hold when the fractional elements are introduced into the network, and the PR criteria are not suitable in complex frequency domain anymore. On the other hand, the rapid development of fractional‐order circuits and systems and the corresponding study in fractional circuit analysis and designs put forward an urgent requirement for the passivity criterion, which can tackle linear fractional networks. Hence, in this paper, we propose new passivity criteria for linear fractional networks by aid of generalized Tellegen's theorem and multivariable PR theory. By using the proposed criteria, the passivity of linear fractional networks can be judged, and the steps of the proposed criterion are illustrated by examples.

Extended Darlington Synthesis of Fractional Order Immittance Function With Two Element Orders

“…3) p 1 −type and p 2 −type elements are transformed into fractional order elements by p 1 = s α , p 2 = s β . Immittance matrix of a network composed of fractional order elements with their orders ranging from 0 and 1, and other passive elements are multivariable positive real matrix through appropriate variable substitutions s α = p 1 , s β = p 2 [48].…”

Synthesis of passive fractional‐order LCn‐port with three element orders

“…(i) Set s α = p 1 , s β = p 2 and s γ = p 3 in Z s , the fractional-order immittance matrix Z s is transformed into n × n three-variable reactance matrix Immittance matrix of a multi-port network composed of fractional-order elements with their orders ranging from 0 and 1, and other passive elements are multivariable positive real matrix through appropriate variable substitutions, [61]. The whole process of synthesising fractional-order LC n-port with three element orders is shown in Fig.…”

“…Immittance matrix of a multi‐port network composed of fractional‐order elements with their orders ranging from 0 and 1, and other passive elements are multivariable positive real matrix through appropriate variable substitutions, [61].…”

Passivity Criterions of Networks With General Fractional Order Coupled Inductors

“…On the other hand, passivity is also an important prerequisite for synthesizing fractional order passive networks [22], [23]. But until now, there are just two studies on the passivity of fractional order networks, one in the Wdomain [24] and the other in the multivariate domain (just contains reciprocal FCIs with same element order) [21], the sdomain passivity of fractional order networks has not been discussed. Therefore, it is very important to study the passivity of FCI and its network.…”

“…Then the multivariate passivity of general FCI is proved, this work extends the application-scope of multivariate network synthesis methods. And we also discuss the passivity criterion of fractional order linear network by expanding the multivariate domain criterion in [21]. This paper is organized as follows.…”