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 \[ L ( i ) d i d t = ∂ P ( i , v ) ∂ i , C ( v ) d v d t = − ∂ P ( i , v ) ∂ v . L ( i ) {{di}}{{dt}} = {{∂ P ( {i,v} )}}{{∂ i}},C ( v ) {{dv}}{{dt}} = - {{∂ P ( {i,v} )}}{{∂ v}}. \] The function, P ( i , v ) P ( {i,v} ) , 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.