A geometric Birkhoffian formalism for nonlinear RLC networks

The aim of this paper is to give a formulation of the dynamics of nonlinear RLC circuits as a geometric Birkhoffian system and to discuss in this context the concepts of regularity, conservativeness, dissipativeness. An RLC circuit, with no assumptions placed on its topology, will be described by a family of Birkhoffian systems, parameterized by a finite number of real constants which correspond to initial values of certain state variables of the circuit. The configuration space and a special Pfaffian form, called the Birkhoffian, are obtained from the constitutive relations of the resistors, inductors and capacitors involved and from Kirchhoff’s laws. Under certain assumptions on the voltage–current characteristic for resistors, it is shown that a Birkhoffian system associated with an RLC circuit is dissipative. For RLC networks which contain a number of pure capacitor loops or pure resistor loops the Birkhoffian associated is never regular. A procedure for reducing the original configuration space to a lower dimensional one, thereby regularizing the Birkhoffian, it is also presented. In order to illustrate the results, specific examples are discussed in detail.