Regular Positive-Real Functions and Five-Element Network Synthesis for Electrical and Mechanical Networks

This paper classifies the positive-real biquadratic functions which can be realized by five-element networks. The concept of regular positive-real functions is introduced to facilitate this classification. Networks are grouped into quartets which may sometimes reduce to two or one network(s). It is shown that a biquadratic can be realized by a series-parallel network with two reactive elements if and only if it is regular. Moreover, there are two such network quartets which can realize all regular biquadratics. It is shown that the only five-element networks which can realize nonregular biquadratics can be arranged into three network quartets. The quartets comprise: 1) two bridge networks with two reactive elements; 2) four series-parallel networks with three reactive elements; and 3) two bridge networks with three reactive elements. The necessary and sufficient realizability conditions are derived for each of these networks. The results are motivated by an approach to passive mechanical control which makes use of the inerter device.