Graph-theoretic properties of dynamic nonlinear networks

Abstract

Graph-theoretic concepts are used to deduce properties of nonlinear networks and properties of nonlinear resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports. The basic result is that if the ports of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port form no cutsets (resp., loops), then each port voltage (resp., port current) is a linear function of the internal element voltages (resp., currents) only; i.e., no other external port voltage (resp., port current) is involved. This result is very general in the sense that it is independent of the constitutive relations of the internal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port elements. It is also a rather subtle result because it forms the basis of a large number of new network and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port theorems. For example, in examining the closure properties of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, this result is used to show, among other things, that if the ports of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port form neither loops nor cutsets, and if the internal resistors are strictly passive, or strictly increasing, then the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port is also strictly passive, or strictly increasing. Moreover, many of these conclusions remain valid when the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port contains independent voltage and current sources. The main result is extended to a network containing capacitors, inductors, resistors, and sources. Here, graph-theoretic conditions are given such that the voltage and current waveforms of the capacitors and inductors are functions of the resistor and source voltage and current waveforms. Dynamic nonlinear networks containing capacitors, inductors, resistors, and sources such that there are loops of capacitors, or cutsets of inductors are shown to be equivalent to networks without such loops or cutsets. Explicit analytical expressions are given for specifying the constitutive relations of the elements of the equivalent circuit. This result allows the generalization of many previous results in nonlinear networks which exclude capacitor loops and inductor cutsets.