On Realizability of Specific Biquadratic Impedances as Three-Reactive Seven-Element Series–Parallel Networks for Inerter-Based Mechanical Control

This article solves the minimal realizability problem of a specific biquadratic impedance with double poles and zeros as a three-reactive-element series-parallel network, where the main conclusion shows that any three-reactive seven-element series-parallel network realizing such a specific biquadratic impedance can always be equivalent to a three-reactive six-element series-parallel network (containing three resistors). First, necessary and sufficient conditions are derived for such biquadratic impedances to be realizable as a series-parallel network containing no more than three reactive elements and no more than three resistors, and the corresponding realizability configurations are presented. Then, the main conclusion is presented, which further implies a corollary about the synthesis of three-reactive series-parallel networks containing an arbitrary number of resistors. The results of this article can contribute to the minimal realization problems of network synthesis especially for the biquadratic impedance case, and can be directly utilized in the optimization and design of inerter-based mechanical control systems.