Efficient time-domain simulation of lossy multiconductor transmission lines by means of generalized method of characteristics

The paper investigates some crucial aspects in the derivation of time-domain equivalent circuits of lossy multiconductor transmission lines, obtained by using the generalized method of characteristic. The paper highlights how an exact evaluation of the effect of the propagation delays not only provide an accurate time-domain equivalent circuit, but suggests naturally the way to obtain a reduced-order model. The propagation is described correctly by means of damped delayed sources, while the effects of losses are efficiently taken into account by a low-order approximated lumped circuit. In this way it is easy to circumvent the main problems arising when using the generalized method of characteristic that is the correct evaluation of the impulse responses describing the time-domain model and the computational cost of the time-domain convolutions. Introduction In today’s VLSI circuits the high operating frequencies impose to model properly the electrical interconnects present at various hierarchical levels. Therefore the recent literature presents many efforts not only to propose interconnect models that can describe accurately the propagation, but also to derive from them reduced-order models requiring lower computational costs (e.g., [1]). In the quasi-TEM hypothesis of propagation, the interconnect may be modeled as a transmission line, which, with a suitable definition of the per-unit-length parameters may describe the most general case of lossy multiconductor lines with frequency dependent parameters. There are many approaches to derive from this model a macromodel which can be at same time accurate and efficient (e.g., see [2]). Here the generalized method of characteristic is adopted to derive a time-domain equivalent multiport representing the line, based on impedances and delayed sources. The strength of this method is the delay extraction which is particularly important when dealing with long lines [3]. The main weakness of the method, on the contrary, is the costly convolutions which arise in the time-domain model. It is well-known that a model able to accurately describe the line behavior cannot be derived if the delays are not extracted, especially when long lines are considered. Therefore, in literature many approaches are presented to extract approximately these delays (e.g., [3],[5]-[6]). Here a general procedure is reviewed which allows to extract exactly these delays, so guaranteeing the causality of the model. This extraction allows, from one hand, to describe analytically the highly irregular and unbounded terms contained in the line impulse responses, which are then synthesized through damped delayed sources and resistive multiports. From the other hand, the remainders are enough regular to be described by a low-order lumped circuit approximation, so reducing the problem of the computational cost of the convolutions. The delay extraction procedure, based on the theory of the perturbation of the spectrum of symmetric operators [4], can handle the general case of lossy multiconductor lines with frequency dependent parameters, regardless of the way in which the parameters are known (analytically of experimentally). The delay extraction procedure Let us consider a line of length d consisting of n signal conductors and a reference one, and let us define as T n s x V s x V s x )] , ( ),..., , ( [ ) ; ( 1 = V and T n s x I s x I s x )] , ( ),..., , ( [ ) ; ( 1 = I the Laplace transforms of the voltage and current distributions along the line. In the quasi-TEM hypothesis of propagation, the interconnect may be modeled through the Telegrapher’s equations Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 31 178 . 0 ) , ( ) ( ) , ( , 0 ) , ( ) ( ) , ( = + = + s x s Y dx s x d s x s Z dx s x d V I I V (1) With a suitable definition of the per-unit-length impedance ) (s Z and admittance ) (s Y these equations describe the most general case of lossy multiconductor lines with frequency dependent parameters. Having defined the terminal variables as ) , 0 ( ) ( 1 s x s = = I I , ) , ( ) ( 2 s d x s = − = I I , ) , 0 ( ) ( 1 s x s = = V V , ) , ( ) ( 2 s d x s = = V V , the following equivalent multiport representation may be derived (e.g., [4]) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 1 1 1 s s s Z s s s s Z s c c W I V W I V = − = − (2) where the characteristic impedance matrix ) (s Z c is the defined by ) ( )] ( ) ( [ ) ( 1 s Z s Y s Z s Zc − = , (3) and the controlled voltage sources 2 1, W W are given by [ ] [ ] ) ( ) ( 2 ) ( ) ( , ) ( ) ( 2 ) ( ) ( 1 1 2 2 2 1 s s s P s s s s P s W V W W V W − = − = , (4) where the propagation operator ) (s P is defined as ) ) ( ) ( exp( ) ( s Y s Z d s P − = . (5) The time-domain model is obtained by reverse transforming (2) and (4): ) ( ) ( * ) ( ) ( ) ( ) ( * ) ( ) ( 2 2 2 1 1 1 t t t z t t t t z t c c w i v w i v = − = − (6) [ ] [ ], ) ( ) ( 2 * ) ( ) ( , ) ( ) ( 2 * ) ( ) ( 1 1 2 2 2 1 t t t p t t t t p t w v w w v w − = − = (7) where the symbol * indicates the time convolution and the impulse responses ) (t zc and ) (t p are the inverse Laplace transforms of ) (s Z c and ) (s P . This model has the main advantage to provide a circuital representation that fits naturally the propagation phenomenon: for instance, when port 2 is matched, 0 w = ) ( 1 t and so the model exactly impose the characteristic impedance ) (s Z c to be the input impedance at port 1. Conversely, the two main drawbacks are: a) the difficult evaluation of the impulse responses ) (t zc and ) (t p , which affects strongly the accuracy; b) the high computational cost of the time convolutions, which lower the efficiency. Therefore this model, successfully used when lossy two-conductor lines or ideal multiconductor lines are considered, presents many challenges when dealing with the general case of lossy multiconductor lines, which cannot be decoupled unless for very special cases (e.g., [4]). The impulse responses ) (t zc and ) (t p cannot be evaluated analytically even when the p.u.l. parameters are given in analytic form. On the other hand, they cannot be computed numerically because of the presence of delayed Dirac pulses and/or highly irregular terms that are “unbounded”. Many solutions are found in literature to overcome these difficulties. Even if in the past many attempts have been made to derive reduced-order approximations of the whole operators ) (s Z c and ) (s P , and some robust algorithms have been used to this purpose (e.g., the Vector Fitting method [7]), the number of poles needed was too high, essentially due to the presence of the delays. Therefore, it is clear that to describe accurately and efficiently the behavior of long lines one has to extract such terms. For shorter line this extraction could be avoided by using, for instance, the method of rational matrix approximation [2]. One of the most efficient approach [3] is based on the following steps: first the matrix ) ( ) ( ) ( s Y s Z s A = is diagonalized as ) ( ) ( ) ( ) ( 1 s U s s U s A − Ω = , then ) (s P is rewritten as ) ( )) ( exp( ) ( ) ( 1 s U s d s U s P − Ω − = . (8) At this point the delay are extracted by imposing ) exp( ) ( )) ( exp( k k k sT s D s d − ≈ η − , (9) where i T is the delay of the k th mode and ) (s Dk is an approximated factor of the type Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 31 179 . ) ( 1 i ∑ = − = m i ik k s s R s D (10) Other approaches may be also found in literature, such as the Hybrid Phase Pole Macromodel [5] which searches an approximation in the form