Minimal Realizations of Three-Port Resistive Networks

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.