A theory of nonlinear networks. I

Source 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 $$L ( i ) d i d t = ∂ P ( i , v ) ∂ i , C ( v ) d v d t = − ∂ P ( i , v ) ∂ v . L\left ( i \right )\frac {{di}}{{dt}} = \frac {{\partial P\left ( {i,v} \right )}}{{\partial i}},C\left ( v \right )\frac {{dv}}{{dt}} = - \frac {{\partial P\left ( {i,v} \right )}}{{\partial v}}.$$ The function, P ( i , v ) 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.