Transfer function models for distributed-parameter systems with impedance boundary conditions

Abstract

A transfer function description is derived for a general class of linear distributed parameter systems dependent on time and one spatial variable. Suitable functional transformations are the Laplace transformation for the time variable and the Sturm– Liouville transformation for the space variable. A practical problem is the determination of the eigenfunctions of the Sturm– Liouville transformation since these depend on the type and the parameters of the boundary conditions. This contribution shows that the design of a transfer function model can be separated from the correct treatment of the boundary conditions. The presented approach exhibits strong parallels to state feedback techniques from control theory. Examples for an electrical transmission line demonstrate how terminations with arbitrary complex impedances can be considered without redesigning the transmission line model.