A vector-space approach to the theory of linear n-port networks

Abstract

The theory of linear (active and passive) networks is investigated from a vector-space viewpoint. It is shown that any linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}</tex> can be associated with a linear operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> and characterized through the kernel or hyperkernel of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> . More precisely, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}</tex> may be associated with a whole class of linear operators. Some of the corresponding matrices (that will be called "canonical") allow a unified treatment for the usual matrix descriptions of n-ports. This also provides a new derivation of the rules for the change of external variables in the description of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}</tex> , as well as some results about the connectability of two <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port networks.