On the classification of series-parallel electrical and mechanical networks

The purpose of this paper is to continue to develop the recently introduced concept of a regular positive-real function and its application to the classification of low-complexity two-terminal networks. Previous work showed that series-parallel five-element networks with two reactive elements are always regular and that there are four quartets of series-parallel networks with three reactive elements and three resistive elements which may realise non-regular biquadratic immittances. In this paper we consider six-element networks with four reactive elements. We describe a classification procedure to find an efficient subset of such networks which may realise any non-regular biquadratic that can be synthesised by this class of networks. We also describes a method to derive the necessary and sufficient realisablity conditions for a given network. This paper builds on previous work of the authors as well as the work of Vasiliu. The results are motivated by an approach to passive mechanical control which makes use of the inerter device.