Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems

277 Pages·1998·7.17 MB·English

by  Alper Demir

ANALYSIS AND SIMULATION
OF NOISE IN NONLINEAR
ELECTRONIC CIRCUITS AND SYSTEMS
ANALYSIS AND SIMULATION
OF NOISE IN NONLINEAR
ELECTRONIC CIRCUITS AND SYSTEMS
by
Alper Demir
Bell Laboratories
and
Alberto Sangiovanni-Vincentelli
University of California
Springer Science+Business Media, LLC
ISBN 978-1-4613-7777-1 ISBN 978-1-4615-6063-0 (eBook)
DOI 10.1007/978-1-4615-6063-0
Library of Congress Cataloging-in-Publication Data
A CLP. Catalogue record for this book is available
from the Library of Congress.
Copyright © 1998 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1st edition 1998
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC
Printed on acid-free paper.
Contents
Acknowledgments IX
1. INTRODUCTION 1
2. MATHEMATICAL BACKGROUND 5
2.1 Probability and Random Variables 6
2.1.1 Events and their probabilities 6
2.1.2 Random variables and their distributions 7
2.1.3 Expectation 9
2.1.4 Convergence of random variables 11
2.1.4.1 Strong law of large numbers 12
2.1.4.2 Central limit theorem 12
2.2 Stochastic Processes 13
2.2.1 Mean and autocorrelation 14
2.2.2 Gaussian processes 15
2.2.3 Markov processes 16
2.2.4 Stationary processes 16
2.2.5 Cyclostationary processes 17
2.2.6 Spectral density 18
2.2.7 Wiener process 20
2.2.8 Poisson process 21
2.2.9 Continuity and differentiability of stochastic processes 22
2.2.10 White noise 24
2.2.11 Ergodicity 26
2.2.12 Numerical simulation of stochastic processes 30
2.3 Filtering of Stochastic Processes with Linear Transform ations 31
2.3.1 Dynam ical system representation 31
2.3.2 Linear dynamical system representation 32
2.3.3 Stochastic processes and linear systems 35
2.3.3.1 WSS processes and LT I systems 35
v
VI NOISE IN NONLINEAR ELECTRONIC CIRCUITS
2.3.3.2 Cyclostationary processes and LPTV systems 37
2.4 Matrix Algebra, Linear Differential Equations and Floquet Theory 38
2.4.1 Eigenvalues and eigenvectors of a matrix and its transpose 38
2.4.2 Sim ilar matrices 39
2.4.3 Function of a square matrix 40
2.4.4 Positive definite/semidefinite matrices 41
2.4.5 Differential equations 42
2.4.6 Linear homogeneous differential equations 43
2.4.7 Linear inhomogeneous differential equations 44
2.4.8 Linear differential equations with constant coefficients 44
2.4.9 Linear differential equations with periodic coefficients 45
2.5 Stochastic Differential Equations and Systems 48
2.5.1 Overview 49
2.5.2 An example 53
2.5.3 Stochastic integrals 55
2.5.4 Stochastic differential equations 58
2.5.5 Ito vs. Stratonovich 60
2.5.6 Fokker-Planck equation 62
2.5.7 Numerical solution of stochastic differential equations 63
3. NOISE MODELS 67
3.1 Physical Origins of Electrical Noise 67
3.1.1 Nyquist's theorem on thermal noise 70
3.1.2 Shot noise 72
3.1.3 Flicker or 1/ f noise 73
3.2 Model for Shot Noise as a Stochastic Process 74
3.3 Model for Thermal Noise as a Stochastic Process 81
3.4 Models for Correlated or non-White Noise 84
3.4.1 1/ f noise model for time-invariant bias 86
3.4.2 Models for correlated noise associated with time-varying signals 89
3.4.2.1 Probabilistic characterization of the models 91
3.4.2.2 Comparison of the models 93
3.5 Summary 96
4. OVERVIEW OF NOISE SIMULATION FOR NONLINEAR ELECTRONIC CIRCUITS 99
4.1 Overview 99
4.2 Noise Sim ulation with LT I Transformations 102
4.3 Noise Sim ulation with LPTV Transform ations 106
4.4 Monte Carlo Noise Simulation with Direct Numerical Integration 109
4.5 Summary 111
5. TIME-DOMAIN NON-MONTE CARLO NOISE SIMULATION 113
5.1 Form ulation of Circuit Equations with Noise 114
Contents Vll
5.2 Probabilistic Characterization of the Circuit with Noise 117
5.3 Sm all Noise Expansion 120
5.4 Derivation of a Linear Time Varying SDE Model for Noise Analysis 124
5.5 Derivation of Linear Time Varying ODEs for the Autocorrelation Matrix 128
5.6 Solution of the Linear Time Varying ODEs for the Autocorrelation Matrix 133
5.7 Numerical Computation of the Autocorrelation Matrix 133
5.7.1 Computation of the coefficient matrices 133
5.7.2 Numerical solution of the differential Lyapunov matrix equation 135
5.7.3 Numerical solution of the algebraic Lyapunov matrix equation 139
5.8 Alternative ODEs for the Autocorrelation Matrix 141
5.8.1 Alternative ODE for the variance-covariance matrix 141
5.8.2 Alternative ODE for the correlation matrix 142
5.8.3 Numerical computation of the autocorrelation matrix 143
5.9 Time-Invariant and Periodic Steady-State 144
5.9.1 Time-invariant steady-state 144
5.9.2 Periodic steady-state 146
5.10 Examples 148
5.10.1 Parallel RLC circuit 148
5.10.2 Switching noise of an inverter 150
5.10.3 Mixer noise figure 150
5.10.4 Negative resistance oscillator 155
5.11 Summary 159
6. NOISE IN FREE RUNNING OSCILLATORS 163
6.1 Phase Noise and Tim ing Jitter Concepts 165
6.2 Phase Noise Characterization with Time Domain Noise Simulation 169
6.2.1 Definition of timing jitter and phase noise 170
6.2.2 Probabilistic characterization of phase noise 172
6.2.3 Examples 176
6.2.3.1 Ring-oscillator 176
6.2.3.2 Relaxation oscillator 179
6.2.3.3 Harmonic oscillator 180
6.2.3.4 Conclusions 183
6.2.4 Phase noise "spectrum" 183
6.3 Phase Noise: Same at All Nodes 185
6.4 Kaertner's Work on Phase Noise 191
6.5 Alternative Phase Noise Characterization Algorithm 192
6.5.1 Example~ 194
6.6 Non-White Noise Sources and Phase Noise 200
6.7 Phase Noise of Phase-Locked Loops 208
6.8 Summary 210
V III NOISE IN NONLINEAR ELECTRONIC CIRCUITS
7. BEHAVIORAL MODELING AND SIMULATION OF PHASE-LOCKED LOOPS 215
7.1 PLLs for Clock Generators and Frequency Synthesizers 219
7.2 Behavioral Models of PLL Components 224
7.2.1 Reference oscillator and the VCO 224
7.2.2 Frequency dividers 226
7.2.2.1 Characterization of tim ing jitter for frequency dividers 227
7.2.3 Phase-frequency detector 228
7.2.4 Charge pump and the loop filter 230
7.3 Behavioral Sim ulation Algorithm 232
7.3.1 Numerical integration with threshold crossing detection 233
7.3.2 Simulating the timing jitter in numerical integration 236
7.3.3 Acquisition detection and the sim ulation output 236
7.4 Post Processing for Spurious Tones and Timing Jitter/Phase Noise 237
7.4.1 Spurious tones 238
7.4.2 Timing jitter/phase noise 240
7.4.2.1 Variance of the timing jitter of transitions 241
7.4.2.2 Spectral density of phase noise/timing jitter 245
7.5 Examples 246
7.5.1 Acquisition behavior 246
7.5.2 Tim ing jitter characterization 249
7.5.3 Phase noise spectrum 252
7.6 Summary 256
8. CONCLUSIONS AND FUTURE WORK 261
References 265
Index
273
Acknowledgments
The authors would like to acknowledge the financial support of the Semicon
ductor Research Corporation (SRC) , the California MICRO program (Corpo
rate sponsors: Harris, Philips, Hewlett-Packard, Rockwell, Motorola, Texas
Instruments), Bell Laboratories (Lucent Technologies), Cadence Design Sys
tems, Motorola, and CNR (Consiglio Nazionale delle Ricerche) under a VIP
grant.
They would like to thank the following individuals for their input and col
laboration: Prof. Paul Gray and Prof. Dorit Hochbaum (U.C. Berkeley), Peter
Feldmann and Jaijeet Roychowdhury (Bell Laboratories), Ken Kundert (Ca
dence Design systems), Prof. Jacob White (MIT), Marcel van de Wiel (Philips
Research Laboratories), Ed Liu, Iasson Vassiliou, Henry Chang and the other
members of the Analog CADgroup at U.C. Berkeley, and Todd Weigandt (U.C.
Berkeley).
1
INTRODUCTION
In electronic circuit and system design, the word noise is used to refer to any
undesired excitation on the system. In other contexts, noise is also used to
refer to signals or excitations which exhibit chaotic or random behavior. The
source of noise can be either internal or external to the system. For instance,
the thermal and shot noise generated within integrated circuit devices are in
ternal noise sources, and the noise picked up from the environment through
electromagnetic interference is an external one. Electromagnetic interference
can also occur between different components of the same system. In integrated
circuits (Ies), signals in one part of the system can propagate to the other
parts of the same system through electromagnetic coupling, power supply lines
and the Ie substrate. For instance, in a mixed-signal Ie, the switching activity
in the digital parts of the circuit can adversely affect the performance of the
analog section of the circuit by traveling through the power supply lines and
the substrate.
Prediction of the effect of these noise sources on the performance of an
electronic system is called noise analysis or noise simulation. A methodology
for the noise analysis or simulation of an electronic system usually has the
following four components:
A. Demir et al., Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems
© Kluwer Academic Publishers 1998
2 NOISE IN NONLINEAR ELECTRONIC CIRCUITS
• Mathematical representations or models for the noise sources.
• Mathematical model or representation for the system that is under the in
fluence of the noise sources.
• A numerical analysis/simulation algorithm to "analyze" or "simulate" the
effect of the noise sources on the system in some useful sense.
• Post-processing techniques to characterize the effect of the noise sources on
the system by calculating useful performance specifications using the "data"
created by the analysis/simulation of the system.
In this work, we will be concentrating on the type of noise phenomena caused
by the small current and voltage fluctuations, such as thermal, shot and flicker
noise, which are generated within the integrated-circuit devices themselves.
This type of noise is usually referred to as electrical or electronic noise, be
cause it originates from the fact that electrical charge is not continuous but
is carried in discrete amounts equal to the electron charge. Electrical noise is
associated with fundamental processes in integrated-circuit devices. In prac
tical electronic circuits and systems, the effect of external noise sources, such
as digital switching noise coupled through the power supply lines and the IC
substrate, can be overwhelming compared with the effect of the electrical noise
sources on the performance. The effect of such external noise sources can and
should be minimized by using techniques such as differential circuit architec
tures, separate power supply lines for the analog and digital portions of the
circuit, and isolation of the sensitive analog portion from the rest of the sys
tem. However, the effect of electrical noise sources can not be eliminated, since
it is generated within the electronic devices that make up the system. Thus,
electrical noise represents a fundamental limit on the performance of electronic
circuits [1]. Even though the noise analysis and simulation methodology we
will be presenting was developed to analyze the effects of electrical noise, it can
also be quite useful in analyzing the effects of other types of noise on electronic
circuits. The effects of electrical noise on the performance is most relevant for
analog and mixed-signal electronic circuits, which will be the types of circuits
and systems this work is concerned with.
Noise analysis based on wide-sense stationary noise source models and the
theory of linear time-invariant systems has been used by analog circuit and
system designers for quite some time. This type of noise analysis, usually re
ferred to as A C noise analysis, is implemented in almost every circuit simulator
such as SPICE, and it has been quite an invaluable tool in linear analog IC
design. For instance, in most cases, analog amplifier circuits operate in small
signal conditions, that is, the "operating-point" of the circuit does not change.