Note on the dynamics of LC circuits with elements in excess

In this note we propose the formulation of the dynamics of LC-circuits containing elements in excess and with interaction ports in terms of an implicit Hamiltonian system. The Hamiltonian function is the total electromagnetic energy of the circuit and the state space is defined by the inductors `fluxes and capacitors' charges. But the state space is endowed with a geometric structure which generalizes the Poisson bracket and is called Dirac structure. This Dirac structure is the geometric representation of the interconnection structure of the LC-circuit. It is shown to depend solely of the network graph and the partition of its edges according to the places of the inductors and the capacitors. Then we also propose a change of coordinates, based on the topology of the circuit, in which the implicit Hamiltonian system is decomposed into an explicit Hamiltonian system and a set of constraint equations. The proposed formulation is illustrated by means of an example.

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