Lagrangian and Hamiltonian formulation of transmission line systems with boundary energy flow

The classical Lagrangian and Hamiltonian formulation of an electrical transmission line is reviewed and extended to allow for varying boundary conditions. The method is based on the definition of an infinite-dimensional analogue of the affine Lagrangian and Hamiltonian input-output systems formulation. The boundary energy flow is then captured in an interaction Lagrangian. This leaves the associated Hamiltonian equations of motion symplectic in form, while the internal Hamiltonian still coincides with the total stored energy in the transmission line. The framework is, however, limited to a line that is terminated on both ends by independent voltage sources. This stems from the fact that the classical formulation captures only one wave equation for a lossless transmission line in terms of an integrated charge density. Additionally, the inclusion of the usual line resistance and shunt conductance via a Rayleigh dissipation function(al) is nontrivial. To circumvent these problems, a family of alternative Lagrangian functionals is proposed. The method is inspired by a (not so well-known) concept from network theory called ‘the traditor’.