A theory of algebraic n-ports

The foundational aspects of an important subclass of timeinvariant nonlinear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports are dealt with; namely, the class of algebraic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports that includes, among other things, resistors, inductors, capacitors, and memristors as special cases. Sufficient conditions that guarantee an algebraic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port to admit all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^n</tex> hybrid representations are given. Both global and local characterizations are considered in detail. In particular, certain global properties are shown to be invariants relative to the various modes of hybrid representation. The concept of reciprocity is explored in depth and shown to play an important role in determining such global properties as losslessness and passivity. Several generalized potential functions are defined for reciprocal algebraic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports. These functions are then used to derive a number of interesting circuit theoretic properties for nonlinear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports.