On the Structure and Synthesis of a Class of Admittance Matrices

The problem of realizing a matrix as the short circuit admittance matrix of a network containing tubes, transistors, transformers, etc., as well as RLC elements is considered. The form, positions and signs, of an arbitrary off-diagonal entry in an element admittance matrix as it appears in the short circuit admittance matrix is determined for a graph of specified structure, a vertex complete graph and star shaped tree. A table indicating the relationship between the four possible forms thus obtained and the position of the corresponding entry of the element admittance matrix is given. It is proven that any matrix can be decomposed as a sum of matrices, each of which has one of the forms given or the transpose of one of the forms plus a symmetric matrix. The existence of an element admittance matrix corresponding to any given matrix and the specified graph is established. It is also proven that any real matrix can be decomposed into a sum of matrices each of which has one of the allowable forms plus a symmetric matrix such that the diagonal entries of the corresponding element admittance matrix are non-negative. The above theorems are applied to a nonsymmetric matrix for which a network, including element values, containing a tube and transformer as well as R and C elements is determined.