Simulation of lossy transmission lines

Abstract

Methods for the simulation of nonlinearly terminated, lossy transmission lines are discussed. Filtering of the characteristic line impulse response and incorporation of DC analysis are treated more detailed. Simulation results are presented. 1. DEFINING RELATIONS The waves propagating on the lossy, uniform (N + 1)-conductor line are assumed to be of TEM nature such that the line voltages and currents are related by the telegraphist’s equations in the frequency domain. The properties of the line system are determined by the the per-unit-length transmission line parameters R, L, G, and C which in turn define Z(w) = R(w) + jwL(w) and Y(w) = G(w)+jwC(w). A transmission line equivalent circuit, as given in Figure l., can be derived [l] where the symbol YO = Z-lr represents the (N x N) characteristic admittance matrix and I’ is the generally complex wave propagation matrix calculated from r2 = 2 - Y. The vectors of voltage sources at the near and far end are denoted by Es and ER, respectively. They depend on voltage values of the opposite line end according to the following expressions: The equivalent circuit is characterized by the entries of the matrices exp(-rd) and Yo which can be interpreted as transfer functions of linear, causal systems. Due to the nonlinear character of semiconductor devices contained in the terminating circuits the transients in the transmission lines should be analyzed in the time domain. For this purpose, the equivalent circuit in Figure 1. is transformed into its time domain equivalent. The entries of the matrices W = exp(-rd) and YO become impulse responses denoted by wlk(t) and yo(t), respectively. These impulse responses are calculated using the inverse Fast Fourier Transform (FFT) and discrete-time convolution is used to determine the instantaneous line response. For example, the value of the I-th sending-end voltage source values is es~[k] = win[k] * (2v~~[k] - e~~[lC]) where the brackets indicate time-discrete values. Similarly, the current flowing into the I-th port of the sending-end characteristic admittance is determined from jsl[k] = yoin[&] * (~sn[k] - esn[IC]).