In this note, we investigate the realization problem of a special class of positive-real admittances, which is common in vehicle suspension designs. The number of inerters and dampers is restricted to one in each case and the number of the springs is arbitrary. To solve the problem, we first convert a previous result by to a more direct form. A necessary and sufficient condition for realizability is then derived and explicit circuit arrangements are provided by assuming that the three-port network consisting of only springs after extracting the damper and the inerter has a well-defined impedance. To remove the assumption on the existence of a well-defined impedance, a condition is established on the topological property of the n -port network without a well-defined impedance to obtain an equivalent class of such networks so that the realizability condition is derived with realization. By combining the conditions with and without a well-defined impedance, the final realization result is obtained.

This paper investigates the simplified realization problem of a special class of positive-real admittances similar to biquadratic functions but with an extra pole at the origin, which is widely used in the analysis of suspension systems. The results in this paper are motivated by passive mechanical control with the inerter. The concept of strictly lower complexity is first defined, whose indices in this paper are the total number of elements, the number of resistors (dampers), and the number of capacitors (inerters). We then derive a necessary and sufficient condition for this class of admittance to be realized by the networks that are of strictly lower complexity than the canonical realization by the Foster Preamble method. To solve this problem, it is shown that it suffices to consider the following: 1) networks with at most four elements, 2) irreducible five-element resistor-inductor (RL) networks, and 3) irreducible five-element resisitor-inductor-capacitor (RLC) networks. Other cases are shown to be impossible. By finding their corresponding network configurations through a series of constraints and deriving the corresponding realizability conditions, the final condition can be obtained. Finally, the U-V plane and numerical examples are provided to illustrate the theoretical results.

This note is concerned with the passive network synthesis problem of one-port networks consisting of one inerter, one damper, and at most three springs. To solve the problem, a necessary and sufficient condition is first derived for the realization of a three-port resistive network containing at most three elements, utilizing graph theory and several existing results of n-port resistive networks. By extracting the damper and the inerter, a necessary and sufficient condition is obtained for the realization of one-port networks containing one damper, one inerter, and at most three springs under an assumption that the admittance of three-port networks containing only springs is well-defined. The covering networks are also presented. Based on properties of circuit topology, a realizability condition is derived for the special case when the earlier assumption does not hold. Combining the two conditions when the assumption holds or not, the final realizability condition is obtained.

This brief is concerned with the realization problem of biquadratic impedances, motivated by the recent development in passive mechanical control. This brief generalizes a realization procedure of a special class of biquadratic impedances to a more general form, and the resulting series-parallel RLC networks whose elements are no more than those in Bott-Duffin's networks are obtained without minimization. The realizability condition is proven by providing a constructive synthesis procedure. The series-parallel network obtained contains elements no more than those of Bott-Duffin's network when the impedance is nonregular.

This paper is concerned with the minimal realization problem of a third-order real symmetric matrix as the admittance of three-port resistive networks. First, a necessary and sufficient condition is derived for a real symmetric matrix to be realizable as the admittance of three-port resistive networks with four terminals and at most k elements, where k ∈ {1,2,...,5}. Since it is well-known that the matrix must be paramount, necessary and sufficient conditions are obtained for any paramount matrix to be realizable as the admittance of three-port resistive networks with at most k elements, where k ∈ {1,2,3,4}. Moreover, a necessary and sufficient condition is derived for a paramount matrix that cannot be realized with less than five elements to be realizable as the admittance of three-port resistive networks with five elements. Finally, some numerical examples are presented to illustrate the results. The results of this paper can contribute to solving minimal realization problems of one-port and multi-port transformerless networks with more than one kind of elements.