A theory of nonlinear networks. I

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

$$L\left( i \right)\frac{{di}}{{dt}} = \frac{{\partial P\left( {i,v... ...ht)\frac{{dv}}{{dt}} = - \frac{{\partial P\left( {i,v} \right)}}{{\partial v}}.$$

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.