Order selection in passive transmission line macromodels based on the Lie decomposition

In a passive macromodel for lossy, dispersive multiconductor transmission lines (MTL's) has been proposed. The macromodel uses a multiplicative approximation of the matrix exponential known as the Lie product. The circuit implementation of the macromodel is a cascade of elementary cells, each cell being the combination of a pure delay element and a lumped circuit representing the transmission line losses. Compared with passive rational macromodeling, the Lie product macromodel is capable of efficiently simulating long, low-loss MTL's while preserving passivity. In this paper, we build on the results of and use transmission line theory to derive a new time-domain error criterion for the Lie product macromodel. We also show how this criterion can be used to determine the minimum number of cells needed in the macromodel to guarantee that the magnitude of the time-domain error is below a given engineering tolerance.