Transformerless synthesis of passive electrical networks for effective mechanical control

Abstract

This thesis is concerned with the transformerless synthesis of passive electrical networks, which is motivated by mechanical control based on inerters. Passive network synthesis is a significant topic of circuits and systems theory, which is to physically design a circuit containing passive components according to a given function describing the network, such as an impedance, admittance, transfer function, etc. The most general one-port transformerless realization approach, Bott-Duffin procedure, always generates many redundant elements, and the transformerless realization problem for multi-port passive networks is still an open question. After the 1970s, research interest of passive network synthesis has declined due to the rapid development of integrated circuits.

More than one decade ago, a new kind of passive mechanical elements named the inerter was invented, which completes the analogy between passive mechanical and electrical systems. As a result, passive network synthesis can be applied to the design of passive mechanical networks, which has been successfully utilized in a series of mechanical control systems, such as vehicle suspensions, train suspensions, machine vibration systems, etc. Such a passive control method can have lower cost and higher reliability compared with the active control approach, and it has been shown that the introduction of the inerter can certainly enhance the system performances. Since in practice mechanical systems usually require low complexity due to the limitation of weight and space, research interest in transformerless synthesis of passive networks become significant and urgent for more effective mechanical control.

In this thesis, the minimal realizability problem of a biquadratic impedance with double poles and zeros as a seven-element series-parallel network is investigated, which is the series or parallel connection of a three-element seriesparallel network and a four-element series-parallel network. Such a class of biquadratic impedances can provide fast transitions, which is expected to have advantages in some mechanical systems. A necessary and sufficient condition for such a realizability is presented.

This thesis also solves the minimal realization problem for a third-order real symmetric matrix as the admittance of a three-port resistive network. A necessary and sufficient condition for the realizability is derived and the configurations to cover the condition are also presented. The investigation can provide guidance on minimal realization problems of more general n-port resistive networks and can be a first critical step toward solving the realizability of n-port RLC networks. Moreover, the results can be utilized to solve minimal realizations of one-port or two-port passive mechanical or electrical networks based on the elements extraction approach.

In addition, this thesis is concerned with the realization problem for a class of transfer functions as a two-port RC ladder network with a specified gain. The cascade realization methodology and the continuous gain adjustment technique based on the congruent transformation are utilized. A systematic procedure is established for such a realizability.