Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks

In this paper, a numerical solution is presented for the generalized BLT equation with inhomogeneous transmission lines. In particular, a fully automatized method is developed for the computation of the transmission line propagation matrices and the junction scattering matrices from network structural speciflcation and from transmission line characteristic parame- ters. Two numerical examples are presented, one for a tree-shaped network, and the other for a network involving a circuit loop. The Baum-Liu-Tesche (BLT) equation (1) is widely used for the modeling and analysis of complex transmission line networks, such as wired telecommunication networks and power lines in auto- motive vehicles, railway infrastructures, aircrafts, etc.. The original BLT equation for networks with homogeneous transmission lines has been generalized to the case of inhomogeneous transmis- sion lines in (2). This generalized BLT equation is parameterized by the propagation matrices of inhomogeneous transmission lines and by the scattering matrices at network junctions. However, it is not indicated in (2) how these propagation and scattering matrices can be computed from the structural speciflcation of a network and from the inhomogeneous characteristic parameters of the transmission lines constituting the network. In this paper, a fully automatized method is presented for the computations of the propagation matrices and of the scattering matrices from the speciflcation of the topological structure of a network and from the inhomogeneously distributed resistance, inductance, capacitance and conductance (RLCG) characteristic parameters of all the transmission lines. It is shown that the propagation and scattering matrices are independent of the choice of the directions of the currents in transmission lines, despite the fact that the deflnition of the two opposite waves on each line (as linear combinations of voltage and current) depends on the chosen current direction. It is then possible to compute the propagation and scattering matrices with some local convention for current directions on each line and at each junction, without taking care of the consistency between all the local choices in a network. The automatized computation method is greatly simplifled thanks to these well deflned local conventions. The computation of the scattering matrices has been partially inspired by the results reported in (3). A new conven- tion for the notations involved in the generalized BLT equation is also introduced to facilitate the implementation of the automatized approach to network simulation through the construction and numerical solution of the generalized BLT equation.