目录 2. A QUICK OVERVIEW OF CIRCUIT SIMULATION 3. FREQUENCY-DOMAIN ANALYSIS OF LINEAR SYSTEMS 4. TIMING ANALYSIS FOR A COMBINATIONAL STAGE 5. TIMING ANALYSIS FOR COMBINATIONAL CI RCUITS 6. STATISTICAL STATIC TIMING ANALYSIS 7. TIMING ANALYSIS FOR SEQUENTIAL CIRCUITS 8. TRANSISTOR-LEVEL COMBINATIONAL TIMING OPTIMIZATION 9. CLOCKING AND CLOCK SKEW OPTIMIZATION 10. RETIMING
This page intentionally left blank TIMIING Sachin Sapatnekar niversity of Minnesota, U.S.A KLUWER ACADEMIC PUBLISHERS NEW YORK. BOSTON. DORDRECHT. LONDON. MOSCOW e Book sbn 14020-8022-0 Print Isbn 140207671-1 @2004 Kluwer Academic Publishers New york Boston dordrecht London moscow Print c2004 Kluwer Academic Publishers Boston All rights reserved No part of this e Book may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the united states of america Visit Kluwer online at http://kluweronline.com and Kluwer's e Bookstore at http:/lebooks.kluweronline.com To ofelia from whom 've learnt that timing isnt everything This page intentionally left blank ontents 1. PREDUCTION/INTROFACE 2. A QUICK OVERVIEW OF CIRCUIT SIMULATION 2.1 Introduction 2.2 Formulation of circuit equations 2. 3 Examples of equation formulation by inspection 2. 4 Solution of nonlinear equations 2.5 Solution of differential equations 2.6 Putting it all togethe 55634845 2.7 A primer on solving systems of linear equations 2.8 Summary 3. FREQUENCY-DOMAIN ANALYSIS OF LINEAR SYSTEMS 3.1 ntroduction 3.2 Interconnect modeling 36 3.3 Typical interconnect structures 38 3. 4 The Elmore delay metric 3.5 Asymptotic waveform evaluation 44 3.6 Krylov subspace-based methods 3.7 Fast delay metrics 3. 8 Realizable circuit reduction 3.9 Summary 70 4. TIMING ANALYSIS FOR A COMBINATIONAL STAGE 4.1 Introduction 4.2 Identifying a logic stage 4. 3 Delay calculation under purely capacitive loads 4.4 Effective capacitance: delays under rc loads 4.5 Capacitive coupling effects 23756 4.6 Summary Ⅴ iii TIMINg 5. TIMING ANALYSIS FOR COMBINATIONAL CIRCUITS 5.1 Introduction 97 5.2 Representation of combinational and sequential circuits 97 5.3 False paths 107 5.4 Finding the k most critical paths 5.5 Summary 6. STATISTICAL STATIC TIMING ANALYSIS 113 6.1 ntroduction 113 6.2 Modeling parameter variations 116 6.3 Early work on statistical Sta 121 6. 4 Statistical Sta in the absence of spatial correlation 6.5 Statistical Sta under spatial correlations 126 6.6 Summary 132 7. TIMING ANALYSIS FOR SEQUENTIAL CIRCUITS 7.1 Introduction 133 7.2 Clocking disciplines: Edge-triggered circuits 7. 3 Clocking disciplines Level-clocked circuits 7. 4 Clock schedule optimization for level-clocked circuits 144 7.5 Timing analysis of domino logic 7.6 Summary 149 8. TRANSISTOR-LEVEL COMBINATIONAL TIMING OPTIMIZATION 151 8.1 Introduction 151 8.2 Transistor sIzIng 152 8.3 The Tilos algorithm 8.4 Transistor sizing using convex programming 161 8.5 Lagrangian multiplier approaches 162 8.6 Timing budget based optimization 166 8.7 Generalized posynomial delay models for transistor sizing 167 8.8 Dual v optimization 8.9 Resolving short path violations 177 8.10 Summary 9. CLOCKING AND CLOCK SKEW OPTIMIZATION 181 9.1 Accidental and deliberate clock skew 181 9.2 Clock network construction 183 9. 3 Clock skew optimization 9. 4 Clock skew optimization with transistor sizing 196 9.5 Timing analysis of sequential circuits for skew scheduling 9.6 Wave pipelining issues 203 9.7 Deliberate skews for peak current reduction 204 9. 8 Summary 205 Contents IX 10. RETIMING 10.1 Introduction to retimil 10.2 A broad overview of research on retiming 211 10.3 Modeling and assumptions for retiming 214 10.4 Minimum period optimization of edge-triggered circuits 216 10.5 Minimum area retiming of edge-triggered circuits 227 10.6 Minimum period retiming of level-clocked circuits 240 10.7 Minimum area retiming of level-clocked circuits 247 10.8 Summary 247 11. CONCLUSION 249 Appendices 251 A-Closed-form formulae for the roots of cubic and quartic equations 251 A 1 Cubic equations A 2 Quartic equations 252 B-a Gaussian approximation of the pdf of the maximum of two 253 Gaussians C-On the convexity of the generalized posynomial formulation for 255 transistor sIzing C1 Proof of convexity 255 C2 Relation of a generalized posynomial program to a posynomial program 257 References 259 Index 291