Asymptotic Waveform Evaluation, 1994 And Moment Matching for Interconnect Analysis The Springer International Series in Engineering and Computer Science Series, Vol. 252

The intense drive for signal integrity has been at the forefront ofrapid and new developments in CAD algorithms. Thousands ofengineers, intent on achieving the best design possible, use SPICE on a daily basis for analog simulation and general circuit analysis. But the strained demand for high data speeds, coupled with miniaturizationon an unprecedented scale, has highlighted the previously negligible effects of interconnects; effects which are not always handled appro­ priately by the present levels of SPICE. Signals at these higher speeds may be degraded by long interconnect lengths compared to the increasingly shorter sig­ nal rise times. Interconnect structures can be diverse (pins, connectors, leads, microstrips, striplines, etc. ) and present at any of the hierarchical packaging levels: integrated circuits, printed circuit boards, multi-chip modules or sys­ tem backplanes. Analysis of these effects in any CAD package has become a necessity. Asymptotic waveform evaluation (AWE) and other moment matching tech­ niques have recently proven useful in the analysis of interconnect structures and various networks containing large linear structures with nonlinear termi­ nations. Previously, all that was available to the designer was a full SPICE simulation or a quick but uncertain timing estimation. Moment matching, used in linear systems analysis as a method of model reduction, describes a method to extract a small set of dominant poles from a large network. The information is obtained from the Taylor series coefficients (moments) of that system.

1 Introduction.- 1.1 Motivation.- 1.2 Overview.- 1.3 Notation.- 2 Asymptotic Waveform Evaluation.- 2.1 State space formulation.- 2.2 Generating an approximate response.- 2.3 MNA formulation.- 2.4 Examples.- 3 Transmission Lines.- 3.1 Linear subnetwork formulation.- 3.2 Moments of distributed networks.- 3.3 Uniform lossy coupled transmission lines.- 3.4 Nonuniform transmission lines.- 3.5 Miscellaneous combinations.- 4 Padé Approximations.- 4.1 The rational approximation.- 4.2 Accuracy.- 4.3 Laurent expansion about o0.- 5 Accuracy Improvements.- 5.1 Higher precision programming.- 5.2 Moment scaling.- 5.3 Padæ table analysis.- 5.4 Optimal pole selection.- 5.5 Partitioning.- 5.6 Method of characteristics.- 5.7 Frequency shifting.- 6 Complex Frequency Hopping.- 6.1 Complex expansions.- 6.2 Pole convergence.- 6.3 Pole selection.- 6.4 Search strategy.- 6.5 Extra CPU requirements.- 7 Nonlinear Analysis.- 7.1 Notation.- 7.2 Differential equations (SPICE) approach.- 7.3 Recursive convolution approach.- 8 Sensitivity Analysis.- 8.1 Pole and zero sensitivity.- 8.2 Coefficient sensitivity.- 8.3 Moment sensitivity: lumped elements.- 8.4 Moment sensitivity: distributed elements.- 8.5 Sensitivity of multi-conductor transmission line stamp.- 9 Other Applications.- 9.1 PEEC simulation.- 9.2 3-D RC mesh analysis.- 9.3 Symbolic analysis.- A Appendix.- A.1 Transmission line equations.