Series-Parallel Six-Element Synthesis of Biquadratic Impedances

The object of this paper is to give a complete treatment of the realizability of positive-real biquadratic impedance functions by six-element series-parallel networks comprising resistors, capacitors, and inductors. This question was studied but not fully resolved in the classical electrical circuit literature. Renewed interest in this question arises in the synthesis of passive mechanical impedances. Recent work by the authors has introduced the concept of a regular positive-real functions. It was shown that five-element networks are capable of realizing all regular and some (but not all) nonregular biquadratic positive-real functions. Accordingly, the focus of this paper is on the realizability of nonregular biquadratics. It will be shown that the only six-element series-parallel networks which are capable of realizing nonregular biquadratic impedances are those with three reactive elements or four reactive elements. We identify a set of networks that can realize all the nonregular biquadratic functions for each of the two cases. The realizability conditions for the networks are expressed in terms of a canonical form for biquadratics. The nonregular realizable region for each of the networks is explicitly characterized.