Synthesis of N + 1 Node Resistor N-Ports

Abstract

Determination of the necessary and sufficient conditions that a matrix be the short-circuit admittance matrix of a resistor n-port without transformers is a classic problem in network theory. This paper presents a new procedure for determining the realizability of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n + 1</tex> terminals. This technique has the advantage of being considerably simpler than any previous technique; in fact, it proceeds almost by inspection. Furthermore, it completely eliminates the Gould algorithm used by Cederbaum, and it avoids the "tree growing" process of Guillemin, Biorci, and Civalleri. It is entirely algebraic in nature. The only restriction is that the given matrix have no zero elements. The method is applied to a tenth-order matrix to illustrate its simplicity. The method is based on a new equation relating the short-circuit admittance matrix to the inverse of the connection matrix of port voltages. A simple standard form for the short-circuit admittance matrix which leads to a unique solution of the equation for the connection matrix is given. The realizability is then completed by checking the result of a congruence transformation for hyperdimonance. A new method of obtaining tip ports is also included. This is applied to the realization of a linear tree, and it is shown that the usual permutation into linearly tapered form may be completely avoided.