Synthesis of biquadratic impedances with at most four passive elements

In this paper, we consider the passive network synthesis problem of biquadratic impedances with at most four elements, motivated by the passive mechanical control. In order to solve this problem, a necessary and sufficient realizability condition for no more than three elements is obtained by some topological properties derived previously. Furthermore, the constraints on the possible realizations are used to derive the networks which can cover all the cases, and they are classified as several quartets. Through investigating one of the networks in each quartet, we obtain a necessary and sufficient condition for any biquadratic impedance to be realizable with at most four elements. Finally, the interlocking conditions are illustrated graphically, and numerical examples are given.