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A fast method for the simulation of lossy interconnects withfrequency dependent parameters
Abstract
Simulation of large interconnect structures has become a necessity for the design of current systems. The efficiency of moment-matching techniques has made the simulation of high-speed circuits containing large numbers of interconnect networks practically possible. In this paper a new method is proposed to extend moment-matching techniques to the analysis of interconnect networks with frequency dependent parameters. The frequency dependent interconnect parameters are extracted using the integral form of the Helmholtz equation
... Simulations methods such as SPICE give the most accurate insight into arbitrary interconnect structures, but are computationally expensive in total IC design stages. Faster methods based on moment matching techniques are proposed [8][9] but are still too expensive to be used during layout optimization. Thus, Elmore delay [2], a first order approximation of delay under step input, is still the most widely used delay model in the performance driven synthesis of clock distribution and Steiner global routing topologies. ...
Recent years have seen significant research in finding closed form expressions for the delay of the RLC interconnect which improves upon the Elmore delay model. However, several of these formulae assume a step excitation. But in practice, the input waveform does have a non zero time of flight. There are few works reported so far which do consider the ramp inputs but lacks in the explicit nature which could work for a wide range of possible input slews. Elmore delay has been widely used as an analytical estimate of interconnect delays in the performance-driven synthesis and layout of VLSI routing topologies. However, for typical RLC interconnections with ramp input, Elmore delay can deviate by up to 100% or more than SPICE computed delay since it is independent of rise time of the input ramp signal. We develop a novel analytical delay model based on the first and second moments of the interconnect transfer function when the input is a ramp signal with finite rise/fall time. Delay estimate using our first moment based analytical model is within 4% of SPICE-computed delay, and model based on first two moments is within 2.3% of SPICE, across a wide range of interconnects parameter values. Evaluation of our analytical model is several orders of magnitude faster than simulation using SPICE. We also discuss the possible extensions of our approach for estimation of source-sink delays for an arbitrary interconnects trees.
Analysis of frequency-dependent lossy transmission lines is very important for designing the high-speed VLSI, MCM and PCB. The frequency-dependent parameters are always obtained as tabulated data. In this paper, a new curve fitting technique of the tabulated data for the moment matching technique in the interconnect analysis is presented. This method based on Chebyshev interpolation enhances the efficiency of the moment matching technique.
The trend of increasing operating frequency and decreasing device size is pushing the complexity of circuit designs to the next level. At high frequency end, interconnect effects give rise to a variety of signal integrity issues. Any overlooked signal integrity issues could be vital and lead to design failures. Signifcant efforts were devoted to developing interconnect modeling techniques. Sophisticated modeling techniques typically result in large macromodels. In order to lower iterative design cost and deliver inspired designs to the market in time, model order reduction (MOR) techniques were widely adopted by advanced design automation tools. In this thesis, a novel hierarchical model order reduction technique is proposed. Using this technique, the overall circuit matrices are very sparse after each level of reduction. The technique also introduces an important transformation mechanism for the preservation of moments and passivity. The proposed technique significantly speeds up the order reduction process and is a major contribution to multi-level interconnect circuit simulation. La tendance de l'augmentation de la frequence d'operation et la diminution de la taille des appareils cause les circuits a ^etre de plus en plus complexe. Les effets des interconnexions exposent des problemes varies au sujet de l'integrite des signaux avec les signaux haute frequence. Les problemes d'integrites negligees peuvent ^etre importants et peuvent resulter en une mauvaise conception. Des techniques de modele sophistiquees existent mais produisent des grandes macromodeles. Pour reduire le co^ut de conception iterative et produire de bonnes conceptions au marche rapidement, des modeles de reduction d'ordre des modeles sont utilise dans beaucoup des outils avances de l'automation de conception. Dans cette these, une nouvelle modele de reduction d'ordre hierarchique est proposee. Avec cette technique, les matrices de circuit deviennent tres clairsemees apres chaque niveau de reduction. Cette technique introduit aussi un mecanisme de transformation important pour la preservation des moments et passivite. La technique proposee accelere le proces de reduction d'ordre et c'est une contribution importante a la simulation des circuits a plusieurs niveaux d'interconnexion.
The primary implications of adaptive finite element analysis (FEA)
on simulating the electromagnetic performance of microelectronic system
interconnection (MSI) structures are investigated. The potential
advantages and related costs of using advanced strategy adaptive FEA in
MSI electromagnetic modeling, for computing sufficiently accurate
solutions more efficiently than non-adaptive methods, are examined.
Practical applications are illustrated for the principal adaptive FEA
refinement models
Analysis of frequency-dependent lossy transmission lines is very
important for designing high-speed VLSI, MCMs and PCBs. The
frequency-dependent parameters are always obtained as tabulated data. In
this paper, a new curve fitting technique of the tabulated data for the
moment matching technique in interconnect analysis is presented. This
method is based on Chebyshev interpolation and enhances the efficiency
of the moment matching technique
Recently there has been intense discussion related to the role of
interconnects in designing high performance systems and the methodology
offered for interconnect analysis. This paper offers an inside view on
interconnect CAE tools and their capabilities. Only those CAE tools
which, according to the author's knowledge, are used for interconnect
design rule generation and for final analysis and/or verification of
interconnect are discussed in this paper
In this paper, we present some numerical techniques for the
time-domain characterization of on-chip interconnects near semiconductor
substrate in high-speed very large scale integrated circuits. A
time-domain full-wave method for the extraction of frequency-dependent
equivalent circuit parameters of these lines is described first. Then
the method, based on numerical inversion of Laplace transform, is used
for the transient simulation of interconnection lines with
frequency-dependent parameters. The frequency-dependent effects over a
very broad range of frequencies are included in the analysis. A
numerical example for realistic geometry and material parameters
demonstrates the effects of lossy substrate on the signal propagation,
including delay, waveform distortion and crosstalk
The nodal method has been widely used for formulating circuit equations in computer-aided network analysis and design programs. However, several limitations exist in this method including the inability to process voltage sources and current-dependent circuit elements in a simple and efficient manner. A modified nodal analysis (MNA) method is proposed here which retains the simplicity and other advantages of nodal analysis while removing its limitations. A simple and effective pivoting scheme is also given. Numerical examples are used to compare the MNA method with the tableau method. Favorable results are observed for the MNA method in terms of the dimension, number of nonzeros, and fill-ins for comparable circuit matrices.
The asymptotic waveform evaluation (AWE) technique provides a generalized approach to lumped RLC circuit response approximations. Two results are described: (1) generalization of the AWE method to handle interconnect models which contain distributed elements; and (2) application of the generalized AWE technique to the case where the distributed elements can be modeled as lossy coupled transmission lines. The generalized AWE technique is useful for delay and crosstalk estimation and can be used to evaluate transient responses of high-speed interconnect circuits with negligible error compared with conventional circuit simulators, while being two to three orders of magnitude faster
A method of calculation of eddy currents in thin plates is presented. The stationary eddy-current problem is solved by using a scalar potential. The reaction of the eddy currents on the magnetic field is taken into account by application of an integral equation. Further, a combination of finite elements with the integral equation is presented to calculate an approximated solution for optional configuration of the plate. The paper is a continuation of [2].Es wird eine Methode zur Berechnung der Wirbelstrme in dnnen Platten dargestellt. Zur Lsung der stationren Wirbelstromproblem wird ein skalares Potential verwendet. Der Einflu der Wirbelstrme auf das Magnetfeld wird mit Hilfe einer Integralgleichung erfat. Auerdem wird eine Kombination der Methode der finiten Elemente mit der Integralgleichung beschrieben, die eine Nherungslsung fr eine beliebige Konfiguration der Platte gestattet. Die Arbeit ist als Erweiterung der Untersuchungen von [2] gedacht.
The importance of matching the frequency range for electromagnetic analysis and circuit simulation of VLSI interconnects is discussed in this paper. The electromagnetic analysis utilizes Helmholtz equation to calculate eddy-current loss and dielectric loss. Modeling is based on assembling filters with high cutoff frequencies and small numbers of components. The performance of a gate-array interconnect at different clock frequencies is analyzed. © 1994 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
A generalization of the asymptotic waveform evaluation method to handle interconnect models which contain distributed elements is described. As an example, analysis of lossy coupled transmission lines is considered. The method is useful for both delay and crosstalk estimation
The authors point out that moment-matching techniques that have
been proposed for efficient transient waveform estimation of
interconnect networks used in modeling MCMs can be inaccurate in
high-speed systems by failing to detect some of the dominant high
frequency network poles which lie far from the expansion point but near
the imaginary axis in the frequency plane. Here, an approach for
generating, with an accuracy check, all the dominant poles within the
frequency range of interest using complex frequency hopping (CFH) is
presented. The method, based on a binary search strategy, uses multiport
complex moment-matching in the frequency s plane. CFH allows
for the efficient analysis of large networks which include lossy,
coupled transmission lines and nonlinear terminations, with estimated
waveforms converging to simulation accuracy. Several examples which
demonstrate the accuracy of the proposed technique are presented
The interconnect analysis problem is treated in a general fashion,
exploiting the efficiency and accuracy of asymptotic waveform evaluation
(AWE) for the linear interconnect portion of the circuit while using the
SPICE simulator for the nonlinear portions. An algorithm that allows the
use of partitioned circuit solution techniques in conjunction with AWE
is described. A special technique for determining the moments of a
partition comprising distributed elements is presented.
Moment-computation algorithms for AWE are combined with the method of
characteristics to form another type of macromodel for systems of
coupled lossy lines. The AWESpice algorithm, which combines the linear
multiport macromodels of interconnect with general nonlinearities in a
time domain simulation, is described. Results of using the techniques
are provided
With increasing miniaturization and operating speeds, loss of
signal integrity due to physical interconnects represents a major
performance limiting factor of chip-, board- or system-level design.
Moment-matching techniques using Pade approximations have recently been
applied to simulating modelled interconnect networks that include lossy
coupled transmission lines and nonlinear terminations, giving a marked
increase in efficiency over traditional simulation techniques.
Nevertheless, moment-matching can be inaccurate in high-speed circuits
due to critical properties of Pade approximations. Further,
moment-generation for transmission line networks can be shown to have
increasing numerical truncation error with higher order moments. These
inaccuracies are reflected in both the frequency and transient response
and there is no criterion for determining the limits of the error. In
this paper, a multipoint moment-matching, or complex frequency hopping
(CFH) technique is introduced which extracts accurate dominant poles of
a linear subnetwork up to any predefined maximum frequency. The method
generates a single transfer function for a large linear subnetwork and
provides for a CPU/accuracy tradeoff. A new algorithm is also introduced
for generating higher-order moments for transmission lines without
incurring increasing truncation error. Several interconnect examples are
considered which demonstrate the accuracy and efficiency in both the
time and frequency domains of the new method
Based on numerical results obtained from solutions to integral
equations, other numerical results and theoretical considerations, a set
of closed-form formulas is presented for approximate calculation of
frequency-dependent resistance and inductance per unit length of
microstrip and strip transmission lines. Extensive verifications reveal
that the accuracy obtained by these closed-form formulas is better than
10% for a wide range of parameters
Calculation of capacitances of chipto-chip interconnections on a high density multi-chip package
- D Nayak
- L Hwang
- I Turlik
D. Nayak, L. Hwang and I. Turlik, " Calculation of capacitances of chipto-chip interconnections on a high density multi-chip package, " Proc. 1988 IEPS Conf., pp. 202-212,1988.