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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.

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