Passive Network Synthesis In Bi-quadratic Function Implementation

The main purpose of this thesis is to investigate the realization problem of biquadratic impedances, which can provide guidance to the development of more general functions.First, the new research background of passive network synthesis is reviewed as well as its fundamental contents. The invention of a new mechanical element named "inerter" completes the analogy between passive mechanical systems and electrical ones. This makes it possible to design the passive mechanical systems using the theory of passive network synthesis. It has been shown that using the inerter the performances of practical systems such as vehicle suspensions can be enhanced. Since mechanical systems require as fewer elements as possible, interest in passive network synthesis has revived.Furthermore, the realization problem of biquadratic impedances with at most four elements is investigated. Based on structure constraints on possible realizations, a series of necessary conditions for realizations are established. Applying these conditions to all the possible network graphs, a necessary and sufficient condition is obtained for a biquadratic impedance to be realizable with at most three elements, and a group of configurations to cover the condition is provided. Then, irreducible four-element networks are presented and classified as three quartets. Based on the principle of duality and a transformation of s<->s-1, the final result is obtained by deriving the realizability condition of one network in each quartet. The result is illustrated by U-V planes and numerical examples.Finally, the realization procedure based on biquadratic impedances is also studied. Tirtoprodjo has established a procedure for the class of biquadratic impedances with negative real poles or zeros of multiplicity two. This thesis generalizes the procedure to more general biquadratic impedances. It is derived that using the method of decomposition or that after multiplying a common factor, any positive-real biquadratic impedance is realizable by a series-parallel network with at most seven or nine elements. If the function is not regular, then the number of elements is no more than Bott-Duffin’s networks. The results are also illustrated by a U-V plane and two numerical examples.