RADIOENGINEERING, VOL. 20, NO. 1, APRIL 2011 221 Computer Simulation of Nonuniform MTLs via Implicit Wendroff and State-Variable Methods Lubomír BRANČÍK, Břetislav ŠEVČÍK Dept. of Radio Electronics, Brno University of Technology, Purkyňova 118, 612 00 Brno, Czech Republic brancik@feec.vutbr.cz, xsevci34@stud.feec.vutbr.cz Abstract. The paper deals with techniques for a computer simulation of nonuniform multiconductor transmission lines (MTLs) based on the implicit Wendroff and the state- variable methods. The techniques fall into a class of finite- difference time-domain (FDTD) methods useful to solve various electromagnetic systems. Their basic variants are extended and modified to enable solving both voltage and current distributions along nonuniform MTL’s wires and their sensitivities with respect to lumped and distributed parameters. An experimental error analysis is performed based on the Thomson cable whose analytical solutions are known, and some examples of simulation of both uniform and nonuniform MTLs are presented. Based on the Matlab language programme, CPU times are analyzed to compare efficiency of the methods. Some results for nonlinear MTLs simulation are presented as well. Keywords Multiconductor transmission line, Wendroff method, state-variable method, sensitivity analysis, Matlab. 1. Introduction The simulation of multiconductor transmission lines (MTL) plays an important role in a design of today’s high- speed electronic systems, especially due to continuously increasing clock frequencies resulting in signal integrity problems at the transmission structures [1]. Here not only possibilities to evaluate waveforms of voltage or current signals propagated, but also to determine their sensitivities with respect to various parameters are needed to be able to optimize the design. Besides more frequent uniform MTLs, the nonuniform ones should sometimes be considered to model more general transmission structures. The paper focuses its attention to two principles of the computer simulation of the nonuniform MTLs. First, the implicit Wendroff method [2], [3], originally used to solve transient phenomena on single and three-phase TLs in field of a power engineering, is discussed and further extended to enable such the solution. The uniform MTLs considered in [4] have served as a starting point for generalizing the method towards nonuniform MTLs [5]. To show further potential of the method, first experiments with a simulation of nonlinear MTLs are also shortly discussed. Second, the state-variable methods [6], [7], [8] are presented and ex- tended for nonuniform MTLs simulation, here both in the time and the Laplace domain. The latter is connected with a proper technique of the numerical inversion of Laplace transforms to get the required time-domain solution. To evaluate computational efficiencies of the methods the CPU times have been assessed depending on the number of MTL’s wires and points of discretization. Both approaches are in relation to a broad class of similar finite-difference time-domain (FDTD) techniques being elaborated for solving various electromagnetic systems in the time do- main, [9], [10], including the MTLs, [11], [12]. Let us consider a simple MTL system containing an (n+1)-conductor transmission line, terminated by lumped- parameter circuits, left (L), right (R), as shown in Fig. 1. Fig. 1. MTL system containing (n+1)-conductor transmission line. Let us first consider a linear MTL defined by its length l and per-unit-length (p.-u.-l.) n × n matrices R 0 (x), L 0 (x), G 0 (x) and C 0 (x), i.e. nonuniform in general. The MTL telegraphic equations are [13] 0 0 0 0 (, ) (,) ()(,) () , (, ) (,) ()(,) () , tx tx x tx x x t tx tx x tx x x t               v i R i L i v G v C (1) where v(t,x) and i(t,x) are n × 1 column vectors of voltages and currents of n active wires at the distance x from the MTL’s left end, respectively. Equation (1) is supplemented by boundary conditions reflecting terminating lumped- parameter circuits, by using their generalized Thévenin or Norton equivalents, for example, as will be shown later. LUMPED PARAMETER CIRCUIT (L) LUMPED PARAMETER CIRCUIT (R) (n+1) - conductor transmission line i L i R v L v R 0 l x