A theory of nonlinear networks. I

Brayton, R. K.; Moser, J. K.

doi:10.1090/qam/169746

Authors: R. K. Brayton and J. K. Moser
Journal: Quart. Appl. Math. 22 (1964), 1-33
DOI: https://doi.org/10.1090/qam/169746
MathSciNet review: 169746
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Abstract | References | Additional Information

Abstract: This report describes a new approach to nonlinear RLC-networks which is based on the fact that the system of differential equations for such networks has the special form \[ L\left ( i \right )\frac {{di}}{{dt}} = \frac {{\partial P\left ( {i,v} \right )}}{{\partial i}},C\left ( v \right )\frac {{dv}}{{dt}} = - \frac {{\partial P\left ( {i,v} \right )}}{{\partial v}}.\] The function, $P\left ( {i,v} \right )$, called the mixed potential function, can be used to construct Liapounov-type functions to prove stability under certain conditions. Several theorems on the stability of circuits are derived and examples are given to illustrate the results. A procedure is given to construct the mixed potential function directly from the circuit. The concepts of a complete set of mixed variables and a complete circuit are defined.

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