Minimal series-parallel network realizations of bicubic impedances

TL;DR

This paper addresses the challenge of designing minimal series-parallel passive networks that realize arbitrary bicubic impedance functions, extending known solutions to networks with up to three energy storage elements.

Contribution

It introduces a novel continuity-based method to eliminate redundant elements, solving the minimal realization problem for networks with up to three energy storage components.

Findings

Successfully characterizes minimal series-parallel networks with up to three energy storage elements.

Develops a new approach to eliminate redundant elements from network realizations.

Extends the class of impedance functions for which minimal passive network realizations are known.

Abstract

An important open problem in the synthesis of passive controllers is to obtain a passive network that realizes an arbitrary given impedance function and contains the least possible number of elements. This problem has its origins in electric circuit theory, and is directly applicable to the cost-effective design of mechanical systems containing the inerter. Despite a rich history, the problem can only be considered solved for networks that contain at most two energy storage elements, and in a small number of other special cases. In this paper, we solve the minimal network realization problem for the class of impedances realized by series-parallel networks containing at most three energy storage elements. To accomplish this, we develop a novel continuity-based approach to eliminate redundant elements from a network.