Fast Time-Domain Simulation of 200+ Port S-Parameter Package Models

S-Parameters are quickly becoming the standard method in PCB SI analysis to describe packages, channels, and connectors. This paper will describe techniques for significantly improving the generation and transient simulation of typical s-parameter models and large (200+ port) s-parameter package models used for power distribution and simultaneous switching noise analysis. We show that direct representation of sparameters by poles/residues is more efficient than creating equivalent circuits or performing direct convolution. This paper will also describe why it is important to simulate channels in transient analysis, the caveats of alternate frequency-domain channel analysis techniques, and how AMS models and these new transient s-parameter simulation techniques makes it feasible to perform transient simulation of hundreds of thousands of data cycles. Author(s) Biography Vadim Heyfitch is an MTS engineer with the Altera High-Speed IO Applications Group in San Jose, CA. Lately he has been working on modeling SSN in FPGA packages. Vadim has worked in the field of signal integrity both as an employee and as a consultant at system, board and package level at Intel, Cadence, MEMS, and telecom startups. He has a M.S. in Physics from MIPT (Moscow Institute of Physics and Technology), Russia, and has done some post-graduate work at Michigan State University and the University of Washington. Vladimir Dmitriev-Zdorov is an engineer with Mentor Graphics Corp.'s System Design division. He graduated with honors at the Taganrog State University of Radio Engineering (Russia). Later, Vladimir received Ph.D. and D.Sc. degrees based on his work on methods for circuit and system simulation. For years he was a university professor, and twice stayed at the German National Research Center of CS and IT (GMD) as an invited researcher. The results of his work have been published in numerous papers, conference proceedings and a monograph. Gary Pratt is the manager of high speed partnerships for Mentor Graphics Corp.'s System Design division. He is a graduate of the University of Wisconsin at Madison, a member of IEEE, and a licensed professional engineer with 23 years experience in power electronics; control systems; digital image and signal processing; analog, digital and software design; and engineering management. Gary has been an evangelist for emerging EDA technologies throughout his career. Sherri Azgomi is a Senior Applications Engineer at Altera Corporation, and has been with the company for the last nine years. She has specialized in signal integrity and has written numerous white papers, application notes, and other literature on design features of FPGAs. She has a BSEE from San Jose State University. Introduction At operational frequencies now reaching several GHz, the conventional lumped circuit component models become inadequate. By developing new high-speed models such as W-element (coupled transmission line) we can solve a number of relatively simple problems with special geometry, such as a set of parallel traces along ideal ground plane. However, any realistic design would contain a variety of more complicated geometrical constructions whose detailed description is available only with electrodynamics’ methods (ED). Still, pure ED models are quite expensive and so developers have to combine them with traditional circuit components. A natural bridge between ED and the circuit world is a frequency-domain description (Y, Z and S-parameters) of multi-port blocks. Normally, frequency-domain data has a form of touchstone files or other formats generated from ED simulation or measurement. Recently, S-parameters became a new type of library primitive which modern simulators need to support for PCB SI applications. Due to its distributed-parameter nature, size and complexity, S-parameters present many challenges. These include simulation accuracy, stability, causality and performance, as well as the ability to simulate IC packages and connectors with hundreds of ports. A number of techniques were developed over time dealing with convolution, rational approximation, enforcing model passivity & causality and generating equivalent circuits. Interested reader can find hundreds of related publications. However, none of the existing implementations seems perfect; each one has its own bias and makes compromises between accuracy and performance. Let us briefly consider them in more details. Direct Convolution and Pole-Residue Approximation Methods As the original model description exists in frequency domain, we need to use some form of a convolution integral ∫ ∫ − = − = n n t t d x t d d dx t a t y 0 0 ) ( ) ( ) ( ) ( τ τ τ β τ τ τ (1) to compute time-domain (transient) solution. Here, x(t) and y(t) are respectively the input and response of the linear block, while a(t) and β(t) are unit step and unit pulse test response of the block. Mathematically, the existing methods can be grouped by the way they compute convolution in (1). In the first group, they find system step or pulse response numerically by inverse fast Fourier transform and then use direct convolution to compute (1) as a discrete sum. This method is implemented in a number of simulators, including HSPICE, ELDO (with DSP method), Apache Spice and others. This approach is straightforward however it suffers from the following: Increasingly slow computations Model evaluation time grows as the simulation progress. Each time we estimate (1), we integrate from time zero to the current moment. Since the step/pulse response is a sampled dependence, there is no way to reuse the portion of the integral/sum computed at the previous step. Because of above, they truncate the step/pulse response in order to limit the number of summations. However, such truncation changes the property of the model that results in loss of accuracy or even instability if passivity is violated. Limited dynamic range Since the inverse Fourier transform can only accept equally distant points and the number of sampling points cannot exceed a few thousands, the effective frequency range of the model cannot be made wider than approximately 3 decades. Normally, the model has poorer accuracy at low frequencies, where the number of samples per decade is the smallest; therefore much of inaccuracy appears near the steady state. Sampled step/pulse response has its own time granularity independent on step selection mechanism used by the simulator. In presence of nonlinear devices, such as transistors, simulators may greatly reduce the step. The lumped models can naturally adjust to the smaller step and produce the correct behavior. However, S-parameter models have a fixed granularity of the step/pulse response determined at the model initialization step (see Fig. 3 below). This may result in discontinuity and cause the step selection mechanism to fail. Possible non-causality of the solution If for some reason the sampled frequency data does not satisfy causality conditions (e.g., Kramers-Kronig relations), its Dirac pulse or unit step response, computed by the inverse Fourier transform, may start at negative time, that is, prior to the input causing this response. Practically, they always remove this forgoing portion of the test response so it does not appear in the convolution integral. However, such simple elimination of unwanted portion creates imbalance between the responses in frequency and time domain. In simpler words, the model we finally get in time domain is not the same as the original dependence. Sometimes this appears as mismatch between AC and transient analysis. Excessive memory is needed to store all the step/pulse responses of the large size model. With N ports, we need to store NxN (or Nx(N+1)/2 if symmetric) sampled dependences, each one consisting of thousands of points. But this is not all: during simulation, we also need to store the past history x(t) that has to be integrated over and over. The methods from the second group are approximate because of the sampled dependence by rational polynomials, which can also be represented as a sum of simple components defined by their respective pole and residue. Such approximation helps leverage efficient simulation in a number of ways. First, rational polynomial approximation is strictly causal therefore we obtain an exact match between timeand frequency domain models. Next, since pulse and step response of each pole/residue pair is an exponential function possibly with complex argument the resulting pulse/step responses in (1) can now be expressed as sums of exponential functions, too. Hence, it becomes possible to reevaluate the current value of the integral sum based on the value it had at the previous point (so-called recursive convolution). Time-consuming integration of past history can be replaced by fast and compact update of state variables that makes the overall algorithm linear in complexity. The other advantages include time-continuous model with no time granulation, unlimited dynamic range (6-8 and more decades of frequency range), and no need in response truncation. Although mathematically equivalent, there exists conceptually different and more popular way of exploiting the same type of rational polynomial approximation, where each poleresidue pair is represented by the element of the lumped circuit. The advantage of this approach is evident: the sampled frequency dependence is replaced by the portable Spicecompatible sub-circuit that can be used directly without adding any special primitive or making any modifications to the simulator code. Due to that reason, there were almost no efforts made to adopt the pole/residue approximation directly in circuit simulators, in contrast to simply using the equivalent circuits. However, with increasing size and complexity of the distributed models, several limitations of equivalent circuit models become evident. The performance of the simulator is greatly affected by the following. With thousands of additi