Interconnects in VLSI Design

233 Pages·2000·19.374 MB·English

by  J. M. Wang, E. S. Kuh (auth.), Hartmut Grabinski (eds.)

INTERCONNECTS IN VLSI DESIGN
Interconnects in
VLSI Design
Edited by
Hartmut Grabinski
Hannover, Germany
Universităt
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-6954-7 ISBN 978-1-4615-4349-7 (eBook)
DOI 10.1007/978-1-4615-4349-7
Printed on acid-free paper
AlI Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
OriginalIy published by Kluwer Academic Publishers in 2000
Softcover reprint ofthe hardcover lst edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner.
CONTENTS
Foreword ......................................... H. Grabinski vii
Recent Development in Interconnect Modeling ...... 1.M. Wang, E.S. Kuh
Study of Parallel Plane Mode Excitation at a Double-Layer Via Interconnect
Using the FDTD Method . . . . . . . . . . . . . . . . . . .. C. Schuster, W. Fichtner 25
A Testchip for Analysis of Signal Integrity and Ground-Bounce Effects in
Deep-Submicron Technology .......................... T. Steinecke 37
Measurement of Signal Integrity within Deep-Submicron Interconnect ....
· .......................... F. Caignet, S. Delmas-Ben-dhia, E. Sicard 49
Considering Magnetic Interference in Board-Level Interconnect Design ...
· ............................................. G. Muller, K. ReifJ 61
Input Shape Influence over Interconnect Performances ............... .
· ... L.B. Kenmei, F. Huret, E. Paleczny, P. Kennis, G. Servel, D. Deschacht 71
Comparison of RL and RLC Interconnect Models in the Simultaneous Switch-
ing Noise Simulations .............. S. Hasan, 1. Prince, A. Cangellaris 79
Black-Box Modeling of Digital Devices ........................... .
· ............................ I.S. Stievano, I.A. Maio, F.G. Canavero 89
Advanced Modeling of Nonuniform Interconnects ................... .
· ................................. S. Grivet-Talocia, F.G. Canavero 101
Modeling of Passive Components for Radio Frequency Applications .....
· ........................ 1. Lescot, 1. Haidar, A. Giry, F. Ndagijimana 119
Electrical Performance of Capacitors Integrated in Multi-Layered Printed
Circuit Boards .............................. A. Madou, L. Martens 133
Characteristic Impedance Measurement on Silicon ................... .
· ............................. " U. Arz, D.F. Williams, H. Grabinski 147
VI Interconnects in VLSI Design
Efficient Computation of the Parameters of Parallel Transmission Lines in IC
Interconnects ............................... M. Grimm, H.K. Dirks 155
Modeling of Optical Interconnections for Data Transmission within High-
Speed Electronic Systems ...................................... .
· ..... A. Wallrabenstein, T. Bierhoff, A. Himmler, E. Griese, G. Mrozynski 181
Quantifying the Impact of Optical Interconnect Latency on the Performance of
Optoelectronic FPGA's ... J. Dambre, H. Van Marek, J. Van Campenhout 195
PIN CMOS Receivers for Optical Interconnects ..................... .
· .................. H. Zimmermann, T. Heide, A. Ghazi, P. Seegebrecht 203
BICMOS Receiver OEIC for Optical Interconnect ................... .
· ............... K. Kieschnick, H. Zimmermann, H. Plefi, P. Seegebrecht 213
Electrical/Optical Circuit Boards: Technology -Design -Modeling ...... .
· .................................... E. Griese, D. Krabe, E. Strake 221
FOREWORD
This book presents an updated selection of the most representative contributions to
the 2nd and 3rd IEEE Workshops on Signal Propagation on Interconnects (SPI) which
were held in Travemtinde (Baltic See Side), Germany, May 13-15, 1998, and in
Titisee-Neustadt (Black Forest), Germany, May 19-21, 1999.
This publication addresses the need of developers and researchers in the field of
VLSI chip and package design. It offers a survey of current problems regarding the
influence of interconnect effects on the electrical performance of electronic circuits
and suggests innovative solutions. In this sense the present book represents a continua
tion and a supplement to the first book "Signal Propagation on Interconnects", Kluwer
Academic Publishers, 1998.
The papers in this book cover a wide area of research directions: Beneath the des
cription of general trends they deal with the solution of signal integrity problems, the
modeling of interconnects, parameter extraction using calculations and measurements
and last but not least actual problems in the field of optical interconnects.
The editor would like to thank the authors as well as the reviewers for their con
tributions to this book. I am very grateful to Uwe Arz for his valuable help concerning
the classification of the papers as well as a lot of helpful hints. In addition lowe a
special tribute to my daughter Daniela for helping to prepare the final version of this
book. Last but not least, a word of appreciation goes to Kluwer's staff in general,
especially to Mr. James Finlay and Mrs. Cindy Lufting, for all the cooperation and
help providing during the production of this book.
I am convinced that this publication will be a valuable source of information for
developers and researchers, offering new and innovative solutions to various problems
in the field of interconnects while simultaneously reflecting the state of the art.
Hartmut Grabinski
Laboratorium flir Informationstechnologie
Universitat Hannover
Hannover, July 2000
RECENT DEVELOPMENT IN
INTERCONNECT MODELING
J. M. Wangl, E. S. Kuh1
Abstract
The Spice simulator which was first introduced 30 years ago is still the dominating
circuit simulator used throughout. It is slow and clearly not suitable to handle large
deep submicron circuits. During the past decade many more efficient circuit simula
tors have been developed. AWE[l] and SWEC[2,3j are based on moment matching
and Pade approximation. Although they work well by and large, both have inherent
problems of stability. The next generation of simulators based on the Lanczos pro
cess[4] and Arnoldi algorithm[5] resolved the numerical stability problem. However,
it is the work on congruent transformation[6] which can preserve passivity that guar
antees stability. Efficient programs based on these are now available[7,8]. However,
these are strictly formulated from equations of lumped circuits. Thus when handling
a transmission line, discretalization must be used. It is clear that at high frequen
cies, discretalization causes inaccuracy. In references[9,1O], interconnect is treated as
transmission lines characterized by Telegraph equations. State equation is desired and
an integrated congruent transformation is introduced. It is shown that passivity is pre
served and a lumped model can be derived which match the moments of the original
distributed circuit at arbitrary frequencies. An efficient algorithm based on the Hilbert
space theory is developed. Furthermore, the method can handle mixed lumped and
distributed circuits with coupled transmission lines. Another approach[ll] similar to
the above is to use Chebyshev polynomial in both the z-domain and s-domain. Be
cause of the orthogonal property of Chebyshev polynomials, the resulting algorithm
is more efficient. In addition, it is possible to determine in advance the order of the
approximation needed for a prescribed error tolerance. Experimental results indicate
that the method is very attractive.
1 EEeS Department
University of California at Berkeley
Berkeley, USA
H. Grabinski (ed.), Interconnects in VLSI Design, 1-23.
© 2000 Kluwer Academic Publishers.
2 Interconnects in VLSI Design
1. Introduction
In recent years, with the rapid increase of signal frequency and decrease of feature
sizes of high speed electronic circuits, interconnect has become a dominating factor
in determining circuit performance and reliability in deep submicron designs. At high
frequencies, people usually use distributed model in order to do accurate interconnect
timing analysis and optimization. Previously, moment matching techniques [4,8,9] are
used to generate passive reduced order models for interconnects. Because the mo
ment matching approaches lack optimality property, they require multipoint and/or
high order for practical circuit simulation. The situation is worsen when the mod
els need to capture skin effect[12], where too high order of moments are generated.
Moverover, for most practical circuits, it is relative difficult to know the expansion
points beforehand. Model reduction using balanced realization[13] is developed to
minimize the difference of the reduced order model with the original model over all
frequencies. However, in this method we are guaranteed to obtain a stable but not a
passive reduced order system given the original system is a passive one. In this paper,
we explore using the Chebyshev expansion to generate an optimal passive reduced
order model. This new method does not require the knowledge of expansion points
and it includes an automatic order selection scheme. Our algorithm consists of two
main steps. In the first step, each line's voltage and current in the frequency domain
are modeled by a set of finite order Chebyshev polynomials with respect to the spa
tial variable or the frequency variable. The passivity is preserved and the Chebyshev
expansion coefficients of the input admittance matrix are also matched. In the second
step, an L2 Hilbert space theory based integrated congruence transform is applied to
the network to form its reduced order model. Because of the orthogonality of Cheby
shev polynomials, we can simplify the Modified Gram-Schmidt Algorithm. The main
contribution of this paper is that we provide an optimal passive reduced order model
for distributed lines via the Chebyshev expansion without the knowledge of expansion
points. We simplify the construction of integrated congruence transform by using the
orthogonality property of Chebyshev polynomials. Theoretically, we can extend the
integrated congruence transform concept to other orthogonal polynomials. In order to
control the approximation order, we also develop an automatic order selection scheme.
Experimental examples show that our new model works well in practice.
The rest of the paper is organized as follows. In section 2,we briefly review the prop
erties of the Chebyshev expansion. In section 3, Chebyshev expansion based model
for general lumped RLC circuits is introduced. In section 4 and 5, we derive the
integrated congruence transform based on the Chebyshev expansion with respect to
(w.r.t.) frequency variable and spatial variable, respectively. Then, in section 6, we
introduce the automatic order selection scheme. Finally, in section 7 and 8, we give
examples and conclusions.
Recent Development in Interconnect Modeling 3
2. Background
The Chebyshev polynomials are defined like this:
To(x) = 1;
Tl(X) = x
Tr+1(x) = 2xTr(x) - Tr-1(x) (1)
where x E [-1, 1]. Because of the special range x belongs to, we can also define
Tr(x) = cos (r arccos (x)). The polynomials of even order are obviously even func
tions of x and the polynomials of odd order are odd functions of x. The Chebyshev
polynomial approximation method has been used in a lot of practical applications in
different fields. Suppose we consider Pn(x), a Chebyshev expansion of degree n, as
an approximation to a continuous function f(x). Then
n
L
Pn(x) = crTr(x) (2)
r=O
and
(3)
for r =1= 0,
2 Ln
Cr = - f(Xk)Tr(Xk) (4)
n
k=l
wherexk with k = 1,··· ,n aren zeros of Tn(x).
7r(k - 1)
Xk = cos ( 2 ) (5)
n
We choose Chebyshev expansion method to model the interconnects because it has
three useful properties [1 4] over moment matching methods: 1) It converges exponen
tially; 2) It has the optimal property; It satisfies the discrete least-squares criterion.
This optimal property has been frequently used in filter design where the Chebyshev
expansion is applied as "the equal-ripple function"[15] which provides the minimum
absolute deviation from the ideal-filter curve in passband. 3) It has the orthogonal
property; The Chebyshev polynomials of even order are obviously orthogonal to the
Chebyshev polynomials of odd order. These three properties are best described by the
following three theorems. We consider Pn(x), a Chebyshev polynomial of degree n,
as an approximation to a continuous function f(x).
Theorem 1 Exponential Convergence Theorem: Given a Chebyshev polynomial Pn (x)
of degree n with leading term aoxn,
4 Interconnects in VLSI Design
where k = ao21-n. If ao is bounded, since ITn(x) I < 1, we have exponential conver
gence.
The proof is in [14].
Theorem 2 Weighted Orthogonal Property:
i::j:.j:
i=j::j:.O:
1 Ti(X)Tj(x) dx = ~
/
-1 -JT=X2 2
i=j=O:
This Theorem can be verified by the Chebyshev polynomial definition. Let qn(x) be
any kind of approximation with order n. Let en(x) = f(x) - qn(x) be the error
=
of approximation. We now ask that the average error of our approximation S
tl e; (x )dx should be small. More generally, we may prefer to give different weights
to the required precision in different parts of the range, in which case we ask that
/1
S = w(x)e;(x)dx = minimum
-1
Theorem 3 The Chebyshev expansion based approximation gives us the minimum
average error: minimum = SCheby ::; Sather'
This means that for the order n polynomial approximation, Chebyshev polynomial
approximation satisfies the discrete least-squares criterion. The proof of this theo
rem is by construction. Suppose qn(x) = ao + alx + ... + anxn. Since we need
S = J~1 w(x)(j(x) - (ao + alX + ... + anxn))2dx = minimum, a necessary
cond I·h· on I.S t h at ddaSi = 0 ,~. = 0 , 1 ,"', n. Th'I S I ead s to a set 0 f Im' ear sl. muI tane-
ous equations for the coefficients ai.
(6)
i = 0, 1, ... ,n. The solution of the linear equations is not trivial and would be simpli
fied considerably if the approximation is based on orthogonal polynomials i.e. Cheby
shev polynomial where w(x) = oj I-1 x 2' A rigorous proof in [14] shows that Legendre
and Chebyshev orthogonal polynomials have this kind of optimal property.