A Topological Condition for a Linear Time-Varying Electric Circuit to be Represented by a Canonical Equation

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As is well known, electric circuits consisting of a finite number of resistors, capacitors, and inductors including mutual inductances can be described by the Lagrange-Maxwell equations, which are equivalent to loop equations due to Kirchhoff. However, if we further intend state-space approach to the representation of electric circuits, there arises a problem of finding conditions under which such an electric circuit may be represented by a canonical equation, i.e., by a state equation. It is decisive that electric circuits are distinctive from the Newtonian mechanical systems in the point that the quadratic form which represents magnetic energy stored in all the inductors, which corresponds to the kinetic energy of a mechanical system, is not necessarily positive definite. This paper presents, in the necessary and sufficient manner, a topological condition for an electric circuit prescribed by a Lagrange-Maxwell equation to be represented by a state equation, without any modification such as insertion of any excess inductors.
1999-12-20