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Phase Lock Loops and Frequency Synthesis Phase Lock Loops and Frequency Synthesis V.F. Kroupa  2003 John Wiley & Sons, Ltd ISBN: 0-470-84866-9

Phase Lock Loops and Frequency Synthesis Vˇenceslav F. Kroupa Academy of Sciences of the Czech Republic, Prague

Copyright  2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com 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, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Ofﬁces John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-84866-9 Typeset in 10.5/13pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

To my wife Magda for her encouragement in starting and ﬁnishing this work

Contents Preface 1 Basic Equations of the PLLs 1.1 Introduction 1.2 Basic Equations of the PLLs 1.3 Solution of the Basic PLL Equation in the Time Domain 1.3.1 Solution in the Closed Form 1.3.2 Linearized Solution 1.4 Solution of Basic PLL Equations in the Frequency Domain 1.5 Order and Type of PLLs 1.5.1 Order of PLLs 1.5.2 Type of PLLs 1.5.3 Steady State Errors 1.6 Block Diagram Algebra References 2 PLLs of the First and Second Order 2.1 PLLs of the First Order 2.2 PLLs of the Second Order 2.2.1 A Simple RC Filter 2.2.2 Phase Lag-lead RRC or RCC Filter 2.3 PLLs of the Second Order of Type 2 2.3.1 PLLs of the Second Order of Type 2 with Voltage Output PD 2.3.2 PLLs of the Second Order of Type 2 with Current Output Phase Detector 2.4 Second-order PLLs with Frequency Dividers in the Feedback Path References 3 PLLs of the Third and Higher Orders 3.1 General Open-loop Transfer Function G(s) 3.1.1 Additional RC Section 3.1.2 Two RC Sections xiii 1 1 1 4 4 5 6 7 8 9 9 10 12 13 13 15 15 17 20 21 23 26 27 29 30 30 30

viii CONTENTS 3.1.3 Active Second-order Low-pass Filter 3.1.4 Twin-T RC Filter 3.1.5 PLLs with a Selective Filter in the Feedback Path 3.1.6 Time Delays in PLLs 3.2 Higher-order Type 2 PLLs 3.2.1 Third-order Loops: Lag-lead Filter with Additional RC Section 3.2.2 Third-order Loop: Second-order Lag Filter Plus RC Section 3.2.3 Fourth-order Loops 3.2.4 Fifth-order Loops 3.3 PLLs with Transmission Blocks in the Feedback Path 3.3.1 Divider in the Feedback Path 3.3.2 IF Filter in the Feedback Path 3.3.3 IF Filter and Divider in the Feedback Path 3.4 Sampled Higher-order Loops 3.4.1 Third-order Loops with the Current Output Phase Detector 3.5 Higher-order Loops of Type 3 3.6 Computer Design of a Higher-order PLL References 4 Stability of the PLL Systems 4.1 Hurwitz Criterion of Stability 4.2 Computation of the Roots of the Polynomial P (s) 4.3 Expansion of the Function 1/[1 + G(s)] into a Sum of Simple Fractions 4.3.1 Polynomial S(s) Contains Simple Roots Only 4.3.2 Polynomial S(s) Contains a Pair of Complex Roots 4.3.3 Polynomial S(s) Contains Multiple-order Roots 4.4 The Root-locus Method 4.5 Frequency Analysis of the Transfer Functions – Bode Plots 4.5.1 Bode Plots 4.5.2 Polar Diagrams 4.6 Nyquist Criterion of Stability 4.7 The Effective Damping Factor 4.8 Appendix References 5 Tracking 5.1 Transients in PLLs 5.1.1 Transients in First-order PLLs 5.1.2 Transients in Second-order PLLs 5.1.3 Transients in Higher-order Loops 5.2 Periodic Changes 5.2.1 Phase Modulation of the Input Signal 5.2.2 Frequency Modulation of the Input Signal 32 33 35 38 41 41 43 46 49 55 55 56 56 56 56 59 60 63 65 66 68 69 69 69 70 70 73 74 81 83 88 91 92 93 93 94 94 101 101 104 104

CONTENTS ix 5.3 Discrete Spurious Signals 5.3.1 Small Discrete Spurious Signals at the Input 5.3.2 Small Spurious Signals at the Output of the Phase Detector 5.3.3 Small Spurious Signals at the Output of the PLLs References 6 Working Ranges of PLLs 6.1 Hold-in Range 6.1.1 Phase Detector with the Sine Wave Output 6.1.2 The PD with Triangular Output 6.1.3 The PD with a Sawtooth Wave Output 6.1.4 Sequential Phase Frequency Detectors 6.2 The Pull-in Range 6.2.1 PLLs of the First Order 6.2.2 PLLs of the Second Order 6.3 Lock-in Range 6.3.1 PLLs of the First Order 6.3.2 PLLs of the Second Order 6.4 Pull-out Frequency 6.5 Higher-order PLLs and Difﬁculties with Locking 6.5.1 False Locking 6.5.2 Locking on Sidebands 6.5.3 Locking on Harmonics 6.5.4 Locking on Mirror Frequencies References 7 Acquisition of PLLs 7.1 The Pull-in Time 7.1.1 The Pull-in Time of a PLL with Sine Wave PD 7.1.2 The Pull-in Time of a PLL with Sawtooth Wave PD 7.1.3 The Pull-in Time of a PLL with Triangular Wave PD 7.2 The Lock-in Time 7.3 Aided Acquisition 7.3.1 Pretuning of the VCO 7.3.2 Forced Tuning of the VCO 7.3.3 Assistance of the Frequency Discriminator 7.3.4 Increasing the PLL Bandwidth 7.4 Time to Unlock References 8 Basic Blocks of PLLs 8.1 Filters with Operational Ampliﬁers 105 105 107 109 110 111 111 112 112 112 113 113 117 118 125 127 128 130 132 133 134 134 135 136 137 137 138 140 141 141 142 142 142 143 145 148 148 149 149

x CONTENTS 8.2 Integrators 8.2.1 Active Integrators with Operation Ampliﬁers 8.2.2 Passive Integrators 8.3 Mixers 8.3.1 Multiplicative Mixers 8.3.2 Switching Mixers 8.3.3 Ring Modulators 8.4 Phase Detectors 8.4.1 A Simple Switch 8.4.2 Ring Modulators 8.4.3 Sampling Phase Detectors 8.4.4 Digital Phase Detectors 8.5 Frequency Dividers 8.5.1 Regenerative Frequency Dividers 8.5.2 Digital Frequency Dividers 8.6 Digital Circuits 8.6.1 Gates – the Logic Levels and Symbols 8.6.2 Flip-ﬂops References 9 Noise and Time Jitter 9.1 Introduction 9.2 Types of Noise 9.2.1 White Noise 9.2.2 Flicker or 1/f Noise 9.2.3 Noise 1/f 2 9.2.4 Piece-wise Approximations of Noise Characteristics 9.3 Mathematical Theory of Noise 9.3.1 Frequency Domain 9.3.2 Time Domain 9.3.3 Conversion Between Frequency and Time Domain Measures 9.3.4 Phase and Time Jitter 9.3.5 Small and Band-limited Perturbations of Sinusoidal Signals 9.4 Component Noises 9.4.1 Ampliﬁers 9.4.2 Frequency Dividers 9.4.3 Phase Detectors 9.4.4 Noises Associated with Loop Filters 9.4.5 Oscillators 9.4.6 Oscillator Properties 9.4.7 Phase Noise in PLLs 9.4.8 Application of PLLs for Noise Measurement 152 152 152 152 153 155 156 158 158 158 160 164 172 172 173 180 181 183 187 189 189 189 190 194 195 196 197 197 199 199 200 203 205 205 206 208 208 211 215 220 223

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Phase Lock Loops and Frequency Synthesis.pdf

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