Synthesis of essential-regular bicubic impedances

This paper provides a complete realisation of a special class of positive-real bicubic impedances. The problem is motivated by the concept of the inerter, which is the mechanical dual of a capacitor. This device allows mechanical network synthesis, by completing the electrical mechanical analogy. With mechanical synthesis, the emphasis is to minimise the number of elements required to allow feasible implementation. The definitions of simple-series-parallel networks and essential-regular positive-real functions are introduced. The simple-series-parallel minimum-reactive networks that can realise all essential-regular bicubics are identified and grouped into six network quartets. One of the advantages of these networks is that they contain the minimal number of reactive elements. The necessary and sufficient realisability conditions for all these networks, as well as corresponding element values, are then derived. Finally, numerical examples are provided to illustrate the validity of the theoretical results. In the course of the argument, interesting conclusions regarding essential-regular bilinear and biquadratic functions have also been presented.