(n+2)-node resistive n-port realizability of Y-matrices

Abstract

Sufficient conditions for the realization of a real symmetric matrix as the short-circuit conductance matrix of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 2)</tex> -node resistive n-port network are established. The properties of networks of departure and padding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port networks are used to arrive at these results. It is shown that any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n x n)</tex> real symmetric matrix realizable by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> -node <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port network can also be realized by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 2)</tex> -node <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port network containing at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(n + 1)/2</tex> conductances if all the zero conductances of the corresponding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> - node realization <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> , form part of a fundamental cut-set in N with respect to the port tree of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> .