New Framework for Passive Macromodeling of High-Speed Distributed Transmission Line Subnetworks

A new framework for passive macromodeling of multiport distributed interconnects is presented in this paper. The proposed framework provides for efficient treatment of various categories of distributed interconnects, such as lossless transmission lines, on-chip RC distributed transmission lines, transmission lines with constant RLCG parameters, transmission lines (TLs) with frequency-dependent RLCG parameters etc. The proposed methodology is based on closed-form matrix-rational approximation of exponential functions describing Telegrapher’s equations and enables the development of the transmission line macromodel to be formulated analytically in terms of known (stored) constants and given perunit-length parameters. The proposed framework can be easily incorporated with conventional circuit simulators such as SPICE and also with the recently developed passive modelreduction techniques. I. INTRODUCTION The rapid increase in operating speeds, density and complexity of modern integrated circuits has made interconnect analysis a requirement for all state-of-the-art circuit simulators. Interconnect effects such as ringing, signal delay, distortion, attenuation and crosstalk can severely degrade signal integrity. Interconnections can be from various levels of design hierarchy, such as on-chip, packaging structures, MCMs, PCBs and backplanes. As the frequency of operation increases, the interconnect lengths become a significant fraction of the operating wavelength, and conventional lumped models become inadequate in describing the interconnect performance and transmission line models become necessary. Skin and proximity effects also become prominent at high frequencies and distributed models with frequency-dependent parameters may be needed. The major difficulty usually encountered while linking the distributed transmission line models and nonlinear simulators is the problem of mixed frequency/time [1], [2]. This is because distributed elements are usually characterized in the frequency-domain whereas nonlinear components such as drivers and receivers are represented only in time-domain. Several publications can be found in the literature, which address this issue. Approaches based on conventional lumped segmentation of transmission lines provide a brute force solution to the problem of mixed frequency/time simulation. However, these methods lead to large circuit matrices, rendering the simulation inefficient. In this paper, a new framework for passive macromodeling of multiport distributed interconnects is presented. The proposed framework provides for efficient treatment of various categories of distributed interconnects, such as lossless transmission lines, on-chip RC distributed transmission lines, transmission lines with constant RLCG parameters, transmission lines with frequency-dependent RLCG parameters etc. The proposed methodology is based on closed-form matrixrational approximation of exponential functions describing Telegrapher’s equations [4]. The method uses pre-determined (stored) coefficients given by the closed-form matrix-rational approximation and the per-unit-length parameters to obtain analytically a macromodel in the form of ordinary differential equations. The proposed model can be easily incorporated with conventional circuit simulators such as SPICE and also with the recent passive model-reduction techniques. II. REVIEW OF DISTRIBUTED TRANSMISSION LINE EQUATIONS Distributed interconnects are described by a set of partial differential equations known as Telegrapher’s equations: