Understanding quartz crystals and oscillators 9781608071180, 1608071189

Table of contents :
Introduction to Quartz Crystal Resonators
Quartz Crystal Characteristics
Advanced Crystal Resonator Topics
MEMS Resonators and Oscillators
Choosing the Correct Crystal for the Application
Basic Oscillator Theory
Jitter and Phase Noise
Specifying Crystal Oscillators
Pierce-Gate Crystal Oscillator
Colpitts Crystal Oscillator Design
Butler Gate Oscillator Design
Characterization of High Performance Crystal Oscillators
Techniques of High Frequency Oscillator Designs
Crystal Oscillators Requirements in Telecommunications
Testing Crystal Oscillators.

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Understanding Quartz Crystals and Oscillators

For a complete listing of titles in the Artech House Microwave Library, turn to the back of this book.

Understanding Quartz Crystals and Oscillators Ramón M. Cerda

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by Igor Valdman

ISBN 13: 978-1-60807-118-0

© 2014 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book 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 permission in writing from the publisher.   All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

10 9 8 7 6 5 4 3 2 1

To my kids, Cynthia, Ricardo, and Christian.

Contents

Preface

xxi



Acknowledgments

1

Quartz Crystals

1

1.1

Introduction

1

1.2

Mother Nature Used Quartz First

2

1.3

The Curie Brothers

3

1.4

Piezoelectricity

3

1.5

Quartz

4

1.6

Left-Handed and Right-Handed Quartz

5

1.7

Quartz Is Anisotropic

5

1.8

A Timeline of Quartz Crystals and Oscillators

6

1.9 1.9.1 1.9.2 1.9.3 1.9.4 1.9.5

Important Definitions Time Second Frequency Nominal Frequency Clock

7 7 7 8 8 9

vii

xxiiI

viii

Understanding Quartz Crystals and Oscillators

1.9.6 1.9.7 1.9.8 1.9.9 1.9.10 1.9.11 1.9.12 1.9.13 1.9.14 1.9.15 1.9.16 1.9.17 1.9.18 1.9.19 1.9.20 1.9.21 1.9.22 1.9.23 1.9.24 1.9.25 1.9.26 1.9.27 1.9.28 1.9.29 1.9.30 1.9.31 1.9.32 1.9.33 1.9.34

Room Frequency or 25°C Frequency Fractional Frequency Allan Deviation Accuracy, Precision, and Stability Accuracy Precision Stability Frequency Stability Short-Term Frequency Stability Medium-Term Frequency Stability Long-Term Frequency Stability Aging and Drift Ambient Temperature Frequency-Temperature Stability (Frequency Versus Temperature Stability) Tolerance Calibration Jitter Oscillator Phase Shift Phase Noise Resonance Resonance Frequency Quality Factor, Q Resonator Quartz Crystal Quartz Crystal Oscillator Random Deviations Systematic/Deterministic Deviations Uncertainty

13 13 13 13 14 14 14 14 14 14 14 15 15 15 15 15

1.10

Frequency Stability in Perspective

15

1.11

Growing Quartz

15

1.12

Swept Quartz

17

1.13

A Crystal Is Born

18

1.14

Inside the Crystal Unit

19

1.14.1 The Glass Seals 1.14.2 The Conductive Epoxy Cement

20 20

1.15

21

Sealing the Crystal Unit

9 9 10 10 10 12 12 12 12 12 12 12 13



Contents

ix

1.15.1 1.15.2 1.15.3 1.15.4 1.15.5

Solder Seal Resistance Weld Cold Weld Seam Weld Epoxy Seal

21 21 21 21 22

1.16

Testing for Moisture

22

1.17

Crystal Resonator Mechanical Equivalent Model

22

1.18

Crystal Resonator Electrical Equivalent Circuit

22

1.19

Derivation of Equivalent Circuit Equations

24

1.20 Series-Resonant and Parallel-Resonant Oscillators 1.20.1 Definitions of Series and Parallel Crystals

26 26

1.21

Load Capacitance

27

1.22

Fundamental Mode Crystals

28

1.23

Overtone Mode Crystals

28

1.24

Spurious Modes

29

1.25

Expanded Quartz Resonator Equivalent Circuit Model

29

1.26

The Ideal Phase Angle of the Quartz Crystal Resonator

30

1.27

Pulling the Crystal Frequency by Changing the Load Capacitance

31

1.28

Zero-to-Pole Spacing

33

1.29

Trim Sensitivity

34

1.30

Important Unitless Quantities

37

1.31

Resistance of the Crystal Above Series Resonance (ESR) 38 References

39

2

Quartz Crystal Characteristics

41

2.1

Introduction

41

2.2

Defining the Frequency Versus Temperature Curve

41

x

Understanding Quartz Crystals and Oscillators

2.3 2.3.1

Quartz Crystal Cuts The AT, BT, and SC Cuts

42 42

2.4 2.4.1 2.4.2 2.4.3

Temperature Characteristics of AT Cut, BT Cut, and SC Cut AT-Cut Frequency-Temperature Curves BT-Cut Frequency-Temperature Curves SC-Cut Frequency-Temperature Curves

43 43 43 43

2.5

Thickness Versus Frequency of Quartz Wafers (Blanks)

45

2.6

Bechmann Frequency-Temperature Curves

47

2.7 2.7.1

AT Cut Versus SC Cut AT-Cut Versus SC-Cut Pros and Cons

49 49

2.8

The SC-Cut B-Mode Temperature Characteristic

50

2.9

Vibrational Displacements of AT Versus SC Cuts

51

2.10 2.10.1 2.10.2 2.10.3 2.10.4

Drive Level High Drive Level Low Drive Level Correlation Drive Level Maximum Drive Level

52 52 52 53 53

2.11

Drive Level Dependence (DLD) or Drive Level Sensitivity (DLS)

54

2.12

Aging

54

2.13

How Drive Level Affects Aging

55

2.14

Activity Dips

56

2.15

Sleepy Crystals Phenomenon

57

2.16 2.16.1 2.16.2 2.16.3 2.16.4 2.16.5 2.16.6

Specifying Crystals Specifying the Nominal Frequency Specifying the “Mode” of Operation Specifying a Fundamental Crystal Specifying an Overtone Crystal Specifying a Parallel or Series-Resonant Crystal and Load Capacitance Specifying the Crystal’s Resistance

58 59 59 60 60 60 60



Contents 2.16.7 2.16.8 2.16.9 2.16.10 2.16.11 2.16.12 2.16.13 2.16.14 2.16.15 2.16.16 2.16.17 2.16.18 2.16.19 2.16.20 2.16.21 2.16.22 2.16.23 2.16.24 2.16.25 2.16.26

Specifying an AT-Cut Crystal Specifying a BT-Cut Crystal Specifying an SC-Cut Crystal Specifying the Frequency Calibration (Tolerance) Specifying the Frequency-Temperature Stability Specifying the Operating Temperature Range Specifying the Shunt Capacitance, C0 Specifying the Aging Rate Specifying an Overall Accuracy Specifying the Trim Sensitivity Specifying a Pullable Crystal (“Pullability”) Specifying the Drive Level Specifying Drive Level Dependence (DLD) Specifying Spurious Responses Specifying the Quality Factor Q Specifying the Motional Inductance, L1 Specifying Inverted Mesa Crystals Specifying a Tuning Fork Crystal Specifying Strip Crystals Specifying Mechanical Shock Resistance

xi

60 61 61 61 61 62 62 63 63 63 64 64 65 65 66 67 67 68 70 71

2.17 Crystal Unit Handling Precautions 2.17.1 High-Temperature Storage Precautions 2.17.2 Electrostatic Discharge (ESD) Precautions

71 71 71

2.18

72

Crystal Specification Template References

73

3

Advanced Quartz Crystal Resonator Topics

75

3.1

Introduction

75

3.2

Flicker Noise

75

3.3

Introduction to Fluctuation Equations

76

3.4

Quartz Resonator Flicker Noise Model

77

3.5

Quartz Resonator Drive Level Sensitivity

80

3.6

Resonator Q and 1/f  Noise Versus Drive Level

82

3.6.1 3.6.2 3.6.3

1/f  Flicker Noise Versus Drive Level Resonator Q Versus Drive Level Resonator Q Versus Manufacturing Defects

82 83 83

xii

Understanding Quartz Crystals and Oscillators

3.7 3.7.1 3.7.2 3.7.3

The Effect of Acceleration on Quartz Resonators Gravitational Acceleration Sinusoidal Acceleration/Vibration Random Acceleration/Vibration

84 85 85 86

3.8

Drive Level Dependency Testing

89

References

89

4

MEMS Resonators and Oscillators

91

4.1

Introduction

91

4.2 4.2.1

Some MEMS Terminology Electromechanical Systems

92 92

4.3 4.3.1 4.3.2 4.3.3 4.3.4

MEMS Resonators Quartz MEMS (QMEMS) MEMS Resonator Equivalent Circuit Model Frequency-Temperature Performance of MEMS Phase Noise and Jitter Performance of MEMS Oscillators

93 94 94 95

4.4 4.4.1 4.4.2

MEMS Oscillators Versus Quartz Oscillators Performance Claims/Reports from MEMS Vendors Present Challenges Facing Commodity MEMS Oscillators References

97 98

96

99 100

5

Choosing the Correct Crystal for the Application

103

5.1

Introduction

103

5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7

Choosing the Correct Crystal for a Low-Cost CLOCK Selecting the Nominal Frequency of Operation Selecting the Type of Cut (AT, BT, or SC Cut) Selecting the Load Capacitance Determining the Maximum ESR Value Determining If the Feedback Resistor Is Built In Frequency-Temperature Stability Calibration Tolerance

103 105 105 106 108 110 110 110

5.3 5.3.1

Choosing the Correct Crystal for a VCXO Changing/Varying the Load Capacitance

111 112



Contents

xiii

5.3.2

VCXO Design Example

112

5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5

Choosing the Correct Crystal for a TCXO Frequency-Temperature Hysteresis Activity Dips (Perturbations) Aging Rate Frequency-Temperature Characteristics Summary Comments for TCXO Crystals

115 116 116 117 118 118

5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 5.5.6

Specifying a Crystal for an OCXO Application Turnover Temperature Aging Rates Calibration Tolerance G-Sensitivity 2-g Tip-Over Test Holder Options References

119 119 121 121 121 121 121 121

6

Oscillator Theory

123

6.1

Introduction

123

6.2

Feedback Oscillator Model

124

6.3

Negative-Resistance Model

126

6.4

Bode’s Gain Phase Method

128

6.5

Root Locus Method

129

6.6 6.6.1 6.6.2

The LC Tank Circuit Component Quality Factors The Basic Tank Circuit

132 132 133

6.7

Loaded Q

135

6.8

LC Resonators

137

6.9

LC Oscillators

137

6.10

Crystal Oscillator Topologies

139

6.11

Choosing a Topology

143

6.12

Load-Reactance Stability

143

xiv

Understanding Quartz Crystals and Oscillators

6.13 6.13.1 6.13.2 6.13.3

How Crystals Oscillate and Control the Frequency: A Qualitative Discussion How a Crystal Oscillator Starts Up How the LLATOR Generates Negative Resistance How an Oscillator Maintains Frequency Stability References

144 144 146 147 148

7

Phase Noise and Jitter

149

7.1

Introduction

149

7.2 7.2.1 7.2.2 7.2.3

The Concept of Noise Density The Gaussian (Normal) Distribution Central-Limit Theorem Power Spectral Density (PSD) Versus Probability Distribution (Density) Function (PDF)

149 150 150

7.3

The Noise Floor

151

7.4 7.4.1 7.4.2 7.4.3

Oscillator Phase Noise Oscillator Frequency-Stability Physical Meaning of Power Spectral Density (PSD) Single-Sideband (SSB) Noise Spectrum

153 155 157 158

7.5

Power-Law Noise Processes

158

7.6

Deterministic Signals

159

7.7

Measuring Phase Noise on Spectrum Analyzer

159

7.8

Leeson’s Oscillator Noise Model

161

7.9 7.9.1 7.9.2 7.9.3 7.9.4 7.9.5 7.9.6

Oscillator Jitter Random Jitter (RJ) Deterministic Jitter (DJ) Total Jitter (TJ) Cycle-to-Cycle Jitter Period Jitter Phase Jitter

161 162 162 162 163 163 163

7.10

Units of Jitter

163

7.11

Measuring Jitter

164

151



Contents 7.12

Transforming Phase Noise to Phase Jitter References

xv

165 168

8

Specifying Crystal Oscillators

169

8.1

Introduction

169

8.2

Crystal Oscillator Types

170

8.3

Available Oscillator Output Waveforms

170

8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.4.6 8.4.7

Output Structures of PECL, LVDS, CML, CMOS, and Clipped-Sinewave Output Voltage Swings of LVPECL, LVDS, and CML PECL Termination LVDS Termination CML Termination TLL and CMOS Output Voltage Logic Levels CMOS Termination Clipped-Sinewave Termination

171 174 174 175 175 176 176 178

8.5 8.5.1

Specifying the Output Waveform Specifying the Oscillator Output Waveform for the Lowest Noise Floor

178

8.6

Spread-Spectrum CLOCKs

180

8.7 8.7.1 8.7.2 8.7.3 8.7.4 8.7.5 8.7.6 8.7.7 8.7.8

Specifying CLOCK Oscillators Specifying the CLOCK’s Frequency Accuracy Specifying the CLOCK’s Phase Noise Specifying the CLOCK’s Jitter Specifying the CLOCK’s Symmetry (Duty Cycle) Specifying the CLOCK’s Rise/Fall Times Specifying a CLOCK’s Frequency Aging Rate Specifying Harmonics Specifying Subharmonic(s)

181 181 182 182 183 183 184 184 184

8.8 8.8.1 8.8.2 8.8.3 8.8.4 8.8.5

Specifying VCXOs Specifying Frequency Deviation (Pullability) Specifying APR Specifying the VCXO Deviation Linearity VCXO Transfer Gain (Kv ) Input Modulation Bandwidth of VCXO

185 185 185 186 187 188

179

xvi

Understanding Quartz Crystals and Oscillators

8.9 8.9.1

Specifying TCXOs Specifying a TCXO’s Frequency Aging Rate

188 189

8.10 8.10.1 8.10.2 8.10.3 8.10.4 8.10.5

Specifying OCXOs Specifying the OCXO Maximum Operating Temperature Specifying the OCXO Frequency-Temperature Stability Specifying the OCXO Short-Term Stability Specifying the OCXO Frequency Aging Rate Specifying Warm-Up

189

8.11

The Bare Essentials Needed to Specify a Crystal CLOCK Oscillator

191

8.12

The Bare Essentials Needed to Specify a VCXO

191

8.13

The Bare Essentials Needed to Specify a TCXO

191

8.14

The Bare Essentials Needed to Specify a OCXO

192

References

189 190 190 190 190

192

9

Pierce-Gate Oscillator

193

9.1

Introduction

193

9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5

The Basic Pierce-Gate Oscillator Feedback Resistor Rf Resistor Rs Inverter U1 Measuring the Inverter Transconductance and Output Conductance How the PI Network Consisting of X1, C1, and C2 Provides the Necessary Phase Shift

194 194 195 196

9.3

Pierce-Gate Open-Loop Gain Phase Analysis

199

9.4

Determining the Sufficient Gain Margin of the Pierce-Gate Oscillator

201

9.5

Negative Resistance Versus the Crystal Shunt Capacitance

204

9.6 9.6.1

Pierce Gate for Third Overtone Mode Crystals Pierce-Gate Inductorless Third Overtone Design

205 207

197 198



Contents

xvii

9.7

Pierce-Gate Start-Up Characteristics

208

9.8

Optimizing the Pierce Gate for High Loaded Q

209

9.9

Pierce-Gate VCXO Configuration

211

9.10

Measuring the Crystal Drive Level

213

9.11

Pierce-Gate CLOCK Design Example

213

References

217

10

Colpitts Oscillator

219

10.1

Introduction

219

10.2

Derivation of Gain Equation

219

10.3 10.3.1 10.3.2 10.3.3 10.3.4

Colpitts CC Quick Design Procedure Using a Fundamental-Mode, Parallel-Resonant Crystal Selection of the Split Caps Resistor Bias Setting the Crystal to Frequency Selecting a Transistor

222 222 222 223 224

10.4 10.4.1 10.4.2 10.4.3

Colpitts CC Quick Design Procedure Using a Third Overtone Parallel- or Series-Resonant Crystal Design of the Third Overtone Mode Selection Setting a Parallel-Resonant Third Overtone Crystal to Frequency Setting a Series-Resonant Third Overtone Crystal to Frequency

10.5

Transient Analysis of Colpitts CC

231

10.6

Colpitts VCXO Design

233

References

225 226 226 227

236

11

Butler Crystal Oscillator Design

237

11.1

Introduction

237

11.2

Butler’s Emitter Follower Oscillator Operation

237

11.3

Butler’s Emitter Follower Design Procedure

239

xviii

Understanding Quartz Crystals and Oscillators

11.4

Butler’s Emitter Follower VCXO Design Example

240

11.5

Butler Gate Oscillator

247

References

248

12

Characterization of High-Performance Crystal Oscillators

249

12.1

Introduction

249

12.2

Why the Allan Variance?

249

12.3 Defining the Allan Variance 12.3.1 Defining the Allan Deviation 12.3.2 Tau-Sigma Plots

250 252 252

12.4

Modified Allan Variance

252

12.5

Overlapped Allan Variance

254

12.6 Defining Time Variance 12.6.1 Time Deviation Defined

255 255

12.7

Time-Interval Error

257

12.8 12.8.1 12.8.2 12.8.3

Frequency and Phase Data Collection Methods Time Interval Counter Method of Data Capturing Heterodyne Method of Data Capturing Dual-Mixer Time-Difference Method of Data Capturing

257 258 258

12.9

Allan Variance Estimate from Frequency-Domain Measures References

259 259 260

13

Frequency Multiplication Techniques

261

13.1

Introduction

261

13.2

The Effect on the Signal by Multiplying

261

13.3

PLL Multiplication

262

13.4

Step Recovery Diode (SRD) Multiplication

263



Contents

xix

13.5

Nonlinear Transmission Line Multiplication

263

13.6

Direct Multiplication

264

13.7

Mixer Multiplication

265

13.8

Low Noise Schottky Diode Odd-Order Multiplier

265

13.9

Times Three Frequency Multiplier Design Example

266

References

267

14

Crystal Oscillator Requirements in Telecommunications

269

14.1

Introduction

269

14.2

Some Telecommunication Definitions

270

14.3

PLL Design Criteria in Telecommunications Networks 272 References

272

15

Testing Crystal Oscillators

273

15.1

Introduction

273

15.2 15.2.1 15.2.2 15.2.3 15.2.4 15.2.5

Military Standards MIL-PRF-55310 MIL-PRF-38534 MIL-STD-810 “Environmental Engineering Considerations and Laboratory Tests” MIL-STD-883 “Test Method Standard Microcircuits” MIL-STD-202 “Test Methods for Electronic and Electrical Component Parts”

273 274 274

15.3

Organizations That Write Standards Other Than Military

274 274 275 275

15.4 Testing Crystal Oscillators 275 15.4.1 Equipment Required for the Testing Crystal Oscillators 275 15.4.2 Testing Frequency-Temperature Stability 277 15.5

Frequency Counters

277

15.6

Test Fixtures

279

References

281

xx

Understanding Quartz Crystals and Oscillators



Glossary

283



About the Author

295



Index

297

Preface Practicing and new engineers faced with the task of specifying a quartz crystal or even designing a simple crystal oscillator may be in the dark on how to accomplish the task. Crystal oscillators are considered by some to be black magic, like RF. If you were one of those fortunate engineers who took a course in college on crystal oscillators, it was either so theoretical or so cookbook that it was useless to the practicing engineer. Frustrated with this situation, you try to find textbooks with concise and reliable design information, but can’t find any. I also could not find many understandable texts as a practicing engineer. Sure, there are some very good textbooks for the hardcore oscillator design engineer (i.e., Parzen and Bottom), but in contrast to these advanced texts, this book offers a complete introduction to the subject matter. The goal of the author is to present the practicing and new engineer with comprehensible material about quartz crystals and oscillators to demystify the field. Although this book is an introduction to the frequency control field, it does include advance subjects on the crystal resonator (Chapter 3) and quartz crystal oscillator design (Chapters 9, 10, and 11). Chapter 6 introduces the newcomer to oscillator theory using three different analysis techniques. Specifying the crystal unit in great detail is covered in Chapters 2 and 5. The specifying crystal oscillator is covered in Chapter 8. Phase noise and jitter are introduced in Chapter 7.

xxi

Acknowledgments This humble author would like to personally thank: • Agilent Technologies for lending me the entire GENESYS software suite (all the schematics and simulations in this book were done using GENESYS. • PTC for lending me MathCad Prime 2 and MathCad 15. • John R. Vig for all his work done over the years in the field of frequency control and for his drawings that were used directly or with minor modifications to illustrate many of the subjects covered in this book. • Wally Galla and Ralph Peduto, who both passed away a few years ago. Wally and Ralph were former colleagues who were great people and excellent crystal oscillator design engineers. • My present employer, Crystek Crystals Corporation, for the opportunity to create and design my true passion, crystal oscillators. On the lighter side, I would like to thank all my friends at the Boulevard Tavern and the World Famous Cigar Bar in Fort Myers, Florida, for all their encouragement in this four-year endeavor. Last, but not least, I would like to give a special thank you to Jack Battaglia for all his life advice. Thank you all!

xxiii

1 Quartz Crystals 1.1  Introduction Today, quartz, unique in its chemical, electrical, mechanical, and thermal properties is unparalleled in its use as a frequency control element in applications where stability of frequency is an absolute necessity. There is no doubt that quartz crystals have been a major contributor to the electronic revolution that has taken place for the last 50 years. This is especially true in the area of wireless communications that, without crystal controlled transmission, radio and TV would not be possible in their present form. The quartz crystals allowed the individual channels in communication system to be spaced closer together to make better use of one of the most precious resources, wireless bandwidth. In World War II, quartz crystal oscillators were recognized as an important part of winning the war effort. They were an essential and valuable element in radio telephone communications. Before the war, quartz crystal oscillators were made in very small numbers. The quartz crystal unit is one of the most remarkable electronic devices developed by man. Today, the low cost for the frequency precision that is derived from it is unparalleled by any other device. Recently, however, it is beginning to get some competition from MEMS resonators. Quartz is a compound/mineral comprised of silicon and oxygen (SiO2) as shown in Figure 1.1 [1]. Silicon and oxygen are two of the most abundant elements on earth. Quartz is all around us in large quantities (about 10% to 14% of the Earth’s crust is SiO2) and is easily seen. The next time you visit the

1

2

Understanding Quartz Crystals and Oscillators

Figure 1.1  Atomic lattice of silicone dioxide, quartz. The structure as a whole has no center of symmetry. The absence of a center of symmetry is a necessary condition for the piezoelectric effect [1].

beach or a desert, take a scoop of sand in the palm of your hand and search for glass-looking pebbles. The glass-looking pebbles are quartz.

1.2  Mother Nature Used Quartz First Corolla spiders have discovered that quartz pebbles, when used in a certain way, can form an insect direction detection trap. It is an ingenious trap and deserving of the first Nobel Prize for Physics. The trap is constructed in the following way. The spider makes a hole in the sand in which it can hide so not to be seen from the surface. It then collects quartz pebbles, which it places all around the perimeter of the hole on the surface. The spider then spins a web string from each quartz pebble to itself inside the hole and just below the surface so as not to be seen. The trap is set. As an insect approaches the trap from any direction and is none the wiser, it touches one of the quartz pebbles. The delicate touch is amplified by the properties of quartz on that particular web string to the spider hiding below ground. In a fraction of a second, the spider then jumps up and out of the hole in the direction of the vibrating string to get its prize. To this day, corolla spiders continue building this wonderful and ingenious trap.



Quartz Crystals

3

Shortly we will recount the history of quartz as man discovered its properties in 1880. However, we now know who really made the first discovery and invented the first mechanical transducer using quartz: the spider. Therefore, are we simply newcomers to quartz, as nature has been using its properties for thousands of years?

1.3  The Curie Brothers In 1880, the brothers Jacques and Pierre Curie discovered the direct piezoelectric effect on quartz. They were able to measure a charge on the surface of natural quartz whose magnitude was proportional to the pressure applied on it. In 1881, the French physicist Gabriel Lippmann predicted the converse piezoelectric effect (a crystal is strained when a voltage is applied to it) using the principle of conservation of electricity. Later that same year, it was verified by the Curie brothers. These two phenomena, direct and converse piezoelectric effects, were first called “piezoelectricity” in 1881 by Hankel [2]. The prefix “piezo” is derived from the Greek word piezein meaning “to press.” Hence, piezoelectricity means “pressure electricity.” Pierre Curie was born in Paris on May 15, 1859. His older brother Jacques was born four years earlier on October 29, 1856. This important discovery of piezoelectricity in quartz by the two brothers was not by chance. Pierre had previously studied crystal symmetry and its pyroelectric phenomena, which helped the brothers to predict and look for piezoelectricity in quartz. The brothers were both highly educated at a young age and were the sons of a medical doctor, Eugene Curie. Pierre developed a passion for mathematics at 14, matriculating at 16, and at 18 received his license ès sciences. Jacques, at 20, was the preparatory of chemistry courses in the School of Pharmacy at the Sorbonne. The Curie brothers are related to famous Madame Curie. Pierre met Marie Sklodowska in the spring of 1884 and married her on July 25, 1895. Marie and Pierre together discovered two radioactive elements in 1898, radium and polonium, for which they received the Nobel Prize for Physics in 1903.

1.4  Piezoelectricity Professor Walter Guyton Cady [3] defined piezoelectricity as follows: “Piezoelectricity is electric polarization produced by mechanical strain in crystals belonging to certain classes, the polarization being proportional to the strain and changing direction with it.” The key words are “changing direction with it,”

4

Understanding Quartz Crystals and Oscillators

which means that if the applied pressure is replaced by a stretch (i.e., a reversal in the sign of pressure), then the sign of the electric polarity also reverses. Cady also stated that: “… a piezoelectric crystal must have certain one-wayness in its internal structure; in other words, it must have a structural ‘bias’ that determines whether a given region on the surface shall show a positive or negative charge on compression.” It is the reversal of a sign of strain with a sign of field that distinguishes piezoelectric materials from nonpiezoelectric materials. Figure 1.2 shows an underformed and strained lattice of quartz.

1.5  Quartz The crystalline form of SiO2 at temperatures below 573°C is called alpha quartz, or simply quartz. The name quartz was derived from the German word “quaz.” The melting point of quartz is above 1,700°C. Above 573°C, the Curie temperature of quartz, the form of SiO2 is called the beta quartz. Most of the piezoelectric properties are lost with the transformation to beta quartz. Quartz occurs naturally (Brazil originally supplied virtually all of the world’s natural quartz) in the form of crystals of various sizes and shapes. With the rapid acceptance and use, the natural supply has almost been exhausted. Fortunately, the diminishing supply led to the discovery of growing synthetic quartz. This is accomplished by dissolving SiO2 in an alkaline water solution at approximately 400°C and at a pressure of 10,000 n/cm2 inside a steel autoclave.

Figure 1.2  Lord Kelvin’s model of the charge distribution of quartz. The deformation of the lattice moves the center of gravity of the negative and positive charges, which then produces surface charges [1].



Quartz Crystals

5

Growing synthetic quartz is a slow process (about 1 mm/day), but the slow growth rate ensures a more homogeneous crystal with high purity and with a quality factor of the same order as natural quartz. Additionally, synthetic quartz can be grown in many different dimensions and orientations to increase production yields. Both natural and synthetic quartz generally have a hexagonal cross section, and are usually capped by hexagonal pyramids with six cap faces at both or either ends as depicted in Figure 1.3. The various directions within the crystal are identified by three axes. The Z, or optical, axis runs longitudinally through the center of the quartz. The direction of greatest electrical sensitivity is called the electrical, or X, axis. This axis joins two points at opposite corners of the hexagon. The Y, or mechanical, axis joins two opposite faces of the hexagon. As there are six faces, there are three X and Y axes.

1.6  Left-Handed and Right-Handed Quartz Quartz is enantiomorphous, that is, both right-handed and left-handed crystal versions exist as depicted in Figure 1.4. A difference in optical rotation (mirror image) creates the left-handed and right-handed quartz, but their physical properties are identical. Generally right-handed quartz is used in the manufacture of quartz crystal resonators.

1.7  Quartz Is Anisotropic Many of the quartz crystal properties (mechanical, electrical, and optical) are dependent upon direction in the quartz bar. Hence, quartz is anisotropic, which is one of the main characteristics of piezoelectric materials. This leads to quartz being direction-sensitive (one-wayness) as stated by Cady. Because of this direction dependency, it will matter which specific angle of cut is used to saw the

Figure 1.3  Quartz bar depicting faces and axes where X is the electrical axis, Y is the mechanical axis, and Z is the optical axis. The major cap faces are called the r-faces and the prism faces are called the m-faces.

6

Understanding Quartz Crystals and Oscillators

Figure 1.4  Left-handed and right-handed quartz exist caused by optical rotation. The x- and s-faces indicate right- or left-handedness [3].

bar to make individual crystal blanks. Every time the angle of cut is changed, it gives rise to a different electrical performance as covered in Chapter 2. Glass, unlike quartz, is isotropic (direction does not matter). The following comparison between glass and quartz will illustrate how striking this direction sensitivity of quartz is. Suppose you cut out a 1-inch cube of quartz and a 1-inch cube of regular glass. Then the amount of pressure necessary to crush the glass and quartz is determined by using a press. One notes a consistent pressure amount no matter which face is used to crush the glass. In other words, glass is not directiondependent. With the quartz cube, however, one measures different pressure numbers depending on which face is used. Surprisingly, one face will be many times stronger than the other. Another example is the electrical conductivity of quarts, which is about 1,000 times greater in one direction then another. Table 1.1 lists some more of the physical properties of quartz whose values are direction-dependent [3].

1.8  A Timeline of Quartz Crystals and Oscillators Nothing came from the profound observation and work by the Curies for nearly 40 years, with the exception of the German crystallographer Voigt, who included the phenomenon in his crystal physics book Lehrbuch der Kristallphysik, published in 1910. In France during World War I, Professor Paul Langevin



Quartz Crystals

7

Table 1.1 Some Anisotropic (Direction-Dependent) Properties of Quartz Coefficient of thermal expansion Dielectric constant Electrical conductivity Hardness Optical index of refraction Piezoelectric constant Solubility

tried to use plates cut from quartz crystals to generate and receive sound waves in water for detecting submarines and other submerged objects. His device was not perfected until after the war. This was the origin of modern sonar. In 1918, Cady recognized that a quartz crystal can be used as a resonator to control and stabilize the frequency of an oscillator and for making filters. In the same year, Nicolson at Bell Telephone Laboratories built the first functional quartz stabilized oscillator. Nicholson also experimented with Rochelle salt, another piezoelectric material which has a larger piezoelectric effect than quartz. He constructed and demonstrated phonograph pickups, microphones, and loudspeakers using Rochelle salt [4]. Table 1.2 is a summary of some of the notable accomplishments made with quartz.

1.9  Important Definitions In defining these terms, the author chose the definitions that are most appropriate for this book. These definitions represent the language [6] of this subject and the reader is encouraged to learn them to more fully understand the material presented going forward. Please note that terms are not in alphabetical but in a learning order. 1.9.1  Time

The first of all definitions relevant to this book is defining time. So what is time? The answer is very complicated but in this book the following definition will suffice: Time is what clocks measure in basic units of seconds. 1.9.2  Second

The International System (SI) defines the second as: “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between

8

Understanding Quartz Crystals and Oscillators

Table 1.2 Some Notable Accomplishments Made with Quartz 1880: The Curie brothers discovered the direct piezoelectric effect on quartz. 1893: Lord Kelvin proposed a model for explaining the piezoelectricity of quartz and was able to calculate approximately the value of the piezoelectric constant. 1918: Nicholson of Bell Telephone Laboratories patents an oscillator employing Rochelle salt. 1920: Professor Cady patents an oscillator whose frequency was definitely quartz crystal-controlled. 1923: Professor G. W. Pierce of Harvard published a circuit with a crystal between grid and plate, the predecessor to today’s Pierce oscillator topology. 1925: Van Dyke derives the electrical equivalent circuit of a quartz resonator. 1926: WEAF in New York became the first crystal-controlled radio broadcasting station. 1927: Warren Marrison develops the first quartz clock standard. 1934: A paper published in the Bell System Technical Journal describes the AT and BT cuts by their inventors, G. W. Willard, I. E. Fair, and Messrs. F. R. Lack from the Bell Telephone Laboratories [5]. 1956: Commercially grown culture quartz becomes available. 1972: Miniature tuning fork quartz crystal units are developed. 1974: The SC cut is first theorized by Dr. R. Holland and is verified in 1976. 2006 to 2007: SiTime and Discera introduced production MEMS oscillators.

the two hyperfine levels of the ground state of the cesium atom 133.” This definition was adapted in 1967 by the 13th General Conference of Weights and Measures, Geneva, Switzerland [1]. 1.9.3  Frequency

Frequency is the number of repetitions per unit time of a complete waveform. One cycle per second is defined as 1 hertz, or 1 Hz. Time and frequency are reciprocal of each other, that is,



Frequency =

1 (Hz ) Time (S )

(1.1)

Hence, if the precise frequency can be generated, then time can be derived accurately and vice versa. 1.9.4  Nominal Frequency

An ideal frequency with zero uncertainty, nominal frequency is the frequency assigned to the crystal or oscillator, as specified in the data sheet. For example, in a 122.88-MHz oscillator, the nominal frequency is 122.88 MHz.



Quartz Crystals

9

1.9.5  Clock

A device that generates accurately spaced, periodic signals that can be used for timing applications, a clock is built with three essential components: a frequency source (i.e., an oscillator), a counting mechanism, and a means of displaying or recording the results. If the oscillator is quartz controlled or stabilized, then the clock is called a quartz clock. 1.9.6  Room Frequency or 25°C Frequency

Room frequency and 25°C frequency mean the same thing in this book and it is the frequency of the crystal or oscillator at room temperature or 25°C. When taking frequency measurements over the temperature of an oscillator or crystal, one can reference these measurements to the room or 25°C frequency or the nominal frequency. Pay attention to this fact when reading frequencytemperature data from a manufacturer of crystals or oscillators. 1.9.7  Fractional Frequency

When frequency is divided by frequency as in ∆f /f, it is called fractional frequency. Fractional frequency can be expressed as a percentage. It turns out that in the context of this book, it will be a very small percentage and that is more appropriate to use parts per million (ppm), when referring to all unitless small fractions; 1 ppm = 1 × 10−6. It is much easier to say “the oscillator frequency is off by 1 ppm” than to say “the oscillator is off by 0.0001%.” You will not need to repeat it in a conversation. One can calculate parts per million with the following equation: ppm =



Frequencymeasured − Frequencynominal × 106 ( ppm ) Frequencynominal

(1.2)

A short-cut version of (1.2) is ppm =



Delta Frequency in Hz (ppm ) Nominal Frequency in MHz

(1.3)

or

ppm =

∆f Fn ( MHz )

(ppm )

(1.4)

10

Understanding Quartz Crystals and Oscillators

Notice that 1 ppm is one part in a million, while 1% is one part in a hundred. This is the same as saying that 1 ppm is one part in 106, pronounced as “one part in ten to the six,” but is written as 1 × 10−6 or one part in 106. All mean the same thing. For the same reason that ppm is used instead of %, units in ppm can also get too small and we change over to parts per billion (ppb); 1 ppb = 1 × 10−9= 0.0000001% (one part in 109). One can calculate parts per billion with the following equation:



ppb =

Frequencymeasured − Frequencynominal × 109 ( ppb ) Frequencynominal

(1.5)

1.9.8  Allan Deviation

A nonclassical statistics used to estimate frequency stability is sometimes called the Allan variance, but its proper name is Allan deviation as it is the square root of the variance. The classical (standard) variance may not converge for a long averaging sample time, but the Allan does and hence its importance. The Allan deviation is a key metric for specifying short-term frequency stability in crystal oscillators. 1.9.9  Accuracy, Precision, and Stability

We will define accuracy, precision, and stability together as the terms are often misused when they relate to crystals and oscillators. The illustration of Figure 1.5(a) will aid in understanding the definitions for accuracy and stability. 1.9.10  Accuracy

Accuracy is the extent to which a given measurement (in our case frequency) agrees with the definition of the quantity being measured; the degree of conformity of a clock’s rate with that of a time standard (MIL-PRF-55310). Accuracy is used to refer to the time offset or frequency offset of a device. If the nominal frequency is 20 MHz, then accuracy is the ability of the measurement to match this actual quantity. Mathematically, frequency accuracy is



Accuracy =

Frequencymeasured − Frequencynominal Frequencynominal

(1.6)



Quartz Crystals

11

Figure 1.5  (a) A view of frequency versus time of accuracy and stability for an oscillator. (b) Accuracy and precision for a marksman.

12

Understanding Quartz Crystals and Oscillators

1.9.11  Precision

Precision is the extent to which a given set of measurements of one sample agrees with the mean of the set. Precision is analogous to the standard deviation. Figure 1.5(b) gives us an understanding of accuracy and precision as it relates to a marksman [1]. 1.9.12  Stability

Stability describes the amount something changes as a function of parameters such as time, temperature, and shock. 1.9.13  Frequency Stability

Frequency stability is the spontaneous and/or environmentally caused frequency change within a given time interval [6]. Frequency stability versus time can be further subcharacterized depending on the observation interval: short-term, medium-term, and long-term frequency stability. 1.9.14  Short-Term Frequency Stability

Short-term frequency stability describes the amount frequency changes in a time interval of 100 seconds or less. It is an interaction between the desired signal and an unwanted signal or noise. In oscillators, short-term frequency stability in the time domain is described by the Allan deviation and by the single-sided power spectral density (phase noise) in the frequency domain. 1.9.15  Medium-Term Frequency Stability

Medium-term frequency stability describes the amount frequency changes in a time interval of minutes to several hours. Frequency versus temperature is a medium-term frequency stability parameter. 1.9.16  Long-Term Frequency Stability

Long-term frequency stability describes the amount that frequency changes in a time interval of 1 day to years. Frequency aging and drift are long-term frequency stability parameters. 1.9.17  Aging and Drift

The Consultative Committee on International Radio (CCIR) states that aging and drift can be defined in the following way: “Aging is the systematic change



Quartz Crystals

13

in frequency with time due to internal changes in the oscillator, that is, the frequency change with time when factors external to the oscillator (environment, power supply, etc.) are kept constant. Drift is defined as the systematic change in frequency with time of an oscillator, that is, drift is due to a combination of factors, i.e., it due to aging plus changes in the environment and other factors external to the oscillator. Aging is what one specifies and what one measures during oscillator evaluation. Drift is what one observes in an application.” Therefore, the term drift should not be used to specify oscillator performance. Other parameters like frequency-temperature stability and frequency-supply voltage tolerance should be used to separately specify a performance as a function of a single variable, which lends itself to ease in measurement. Aging should be expressed as either a maximum rate after a specified time period (e.g., 1 × 10−10/day after 30 days) or maximum total frequency change over a specified time period (e.g., 1 × 10−8/month), or both. 1.9.18  Ambient Temperature

Ambient temperature is the range of temperatures to which the oscillator or crystal will be exposed during operation. 1.9.19  Frequency-Temperature Stability (Frequency Versus Temperature Stability)

Frequency-temperature stability is the maximum permissible deviation of the oscillator frequency due to operation over the specified temperature range at nominal supply and load conditions, with other conditions constant. 1.9.20  Tolerance

Tolerance is the permissible amount of deviation from the nominal frequency at room temperature (+25ºC). 1.9.21  Calibration

Calibration is the specified tolerance in the data sheet. It is the guaranteed tolerance at room temperature. Calibration and tolerance are often interchanged in manufacturer’s data sheets. These two terms are often listed together as calibration tolerance. 1.9.22  Jitter

Jitter can be described as abrupt and unwanted variations of a signal’s successive cycles.

14

Understanding Quartz Crystals and Oscillators

1.9.23  Oscillator

Oscillator is an electronic circuit that generates an alternating waveform from a dc power supply. An oscillator is a source of energy at a specific or range of frequencies. 1.9.24  Phase Shift

The change in position that is introduced into the phase of periodic signal as it passes through any electrical network. 1.9.25  Phase Noise

Phase noise is rapid, short-term, random fluctuations in the phase of a waveform. 1.9.26  Resonance

Resonance is a condition existing when an oscillatory circuit responds with maximum amplitude to an applied source of a given frequency. 1.9.27  Resonance Frequency

Resonance frequency is the natural frequency of an oscillator. This frequency can be divided or multiplied to produce an output frequency and/or converted to time. 1.9.28  Quality Factor, Q

The quality factor is defined as

Q = 2π

Energy stored during a cycle Energy dissipated per cycle

(1.7)

Quality factor is a figure of merit; the higher the quality factor, the sharper the selectivity of the resonant circuit. 1.9.29  Resonator

Resonator is the highest Q component(s) in an oscillator circuit that is intended to control or stabilize the frequency of oscillation. In oscillator circuits, the following can be used as resonators: ceramic resonators, coaxial resonators, inductors, quartz crystals, transmission lines, and surface acoustic wave (SAW) devices, to name a few.



Quartz Crystals

15

1.9.30  Quartz Crystal

Quartz crystal is a two-terminal component made very similar in construction like a capacitor but in which piezoelectric quartz is used as the substrate and housed in a hermetically sealed package. Quartz crystal, piezoelectric crystal, quartz crystal resonator, crystal unit, or simply crystal, all have the same meaning in this book and are used interchangeably. 1.9.31  Quartz Crystal Oscillator

Quartz crystal oscillator is an oscillator circuit that utilizes a quartz crystal as the resonator to control or stabilize its frequency. 1.9.32  Random Deviations

Random deviations are unpredictable except in a statistical sense. 1.9.33  Systematic/Deterministic Deviations

Systematic or deterministic deviations are completely determined as a function of initial conditions and an independent variable. 1.9.34  Uncertainty

Uncertainty is the limits of the confidence interval of a measured or calculated quantity [6].

1.10  Frequency Stability in Perspective Frequency stability can be described in different ways. However, the author wants the reader to have an appreciation and perspective for frequency stability versus time. What are you paying for when you specify this parameter? The following example will convey this: An oscillator has a frequency stability of one part in 1010 (that is, 1 × −10 10 aging). How many seconds that this represent in a human lifetime of 80 years? The answer is ~1/4 second. This is an amazing low number which can be achieved with some of today’s oven controlled crystal oscillators (OCXOs).

1.11  Growing Quartz Today, all commercial quartz used for the manufacture of crystals is man-made (cultured/synthetic quartz). The natural supply of usable quartz was depleted

16

Understanding Quartz Crystals and Oscillators

many decades ago. No distinction will be required between natural and cultured quartz in the material that follows in this book. Natural quartz is still used in some circumstances but will not be covered here. A photograph of two man-made quartz bars is shown in Figure 1.6. Synthetic quartz is grown in steel autoclaves as depicted in Figure 1.7 [1]. The quartz bars take about 30 to 260 days to grow. In general, the slower the growth, the higher the quality.

Figure 1.6  Photograph cultured (man-made) quartz in deliberate made sizes/shapes to optimize production yields.

Figure 1.7  Illustration of an Autoclave where synthetic quartz bars are grown.



Quartz Crystals

17

The typical temperature of the Autoclave is 350°C with inside pressures of around 2,000 atmospheres. Before growing a bar of quartz, a seedling of quartz is needed. The seeds are hung from wires inside the Autoclave like sausages. Dissolved quartz then evaporates onto the seeds to grow atomic layer after layer to form the bar. The seeds are placed in the above chamber of the Autoclave, which is about 5° cooler than the bottom to promote condensation of the dissolved nutrient onto the seeds. The seeds force the additional growth in the same crystalline structure as its own; hence, a silicon dioxide bar is born. The quartz bars are grown in such shapes and sizes that labor costs and material are minimized.

1.12  Swept Quartz All quartz, whether natural or man-made, has crystalline imperfections and impurities. The quality of the quartz can be graded from its infrared absorption for specific applications [7]. The crystalline imperfections cannot be corrected, but the impurities can be removed by a technique called “sweeping of the quartz,” as illustrated in Figure 1.8. Sweeping is accomplished by placing the quartz bar at a high temperature while at the same time applying a high voltage across it. Under these conditions, the impurities migrate to one side. The side containing the impurities is then cut away. Quartz that undergoes this process of impurity removal is called swept quartz. Swept quartz has improved radiation hardness

Figure 1.8  Technique for removing impurities from a quartz bar, sweeping the quartz.

18

Understanding Quartz Crystals and Oscillators

because the alkaline metals that affect frequency change in a radiation environment have been reduced to an extremely small amount [7]. Hence, using swept quartz is a must for any space-based application of a crystal oscillator. The etching properties of swept quartz are also improved, which is why it is used in the production of high-frequency fundamental inverted mesa crystals.

1.13  A Crystal Is Born It is very useful to understand some of the processes that need to take place in the manufacturing of a modern quartz crystal resonator as illustrated in Figure 1.9. The following will be general in nature with more details forthcoming about the process in this and other chapters. While the crystal unit holders keep evolving and shrinking over time, the crystal resonator remains nothing more than a slab of quartz sandwiched between two metal plates and placed in a hermetically sealed package where the electrical connections are made to each of the plates. Modern resonators have replaced the plates/contacts by forming electrodes by depositing a thin film of metal. This deposition process of adding the electrodes is called base plating in the crystal industry. Silver and gold are common precious metals used for base plating crystal units. Due to its higher price, gold is primarily used in highfrequency and low-aging crystals. The sweep step is usually omitted for low-cost crystals. After the quartz bar is grown and after sweeping of the quartz bars (if required), the next step is to roughly cut the blanks or wafers from the bar. The

Figure 1.9  Quartz resonator major fabrication steps [1].



Quartz Crystals

19

cutting is usually performed with a slurry saw. The saw contains about 100 thin stretched metal bands moving back and forth across multiple quartz bars in a flood of abrasive slurry. Because the angle of the cut must be precisely done, xray diffraction is used on the first cut samples to correct the angle of the blades if necessary. After the rough cut from the slurry saw is performed, lapping, etching, polishing, and cleaning are done to ready the blank for deposition of the metal electrodes. The blanks are ground to the desired initial frequency using lapping techniques with graduated sizes of abrasives to obtain a smooth finish. High-frequency and high-quality crystal units are given optically polished surfaces to increase the quality factor Q. Any irregularities left on the surfaces will degrade the crystal electrical performance and increase the flicker noise of the finished oscillator. After assembly/mounting of the blank is completed, the next step is very critical: final frequency adjustment. The final frequency adjustment step is necessary because up to this point, the frequency of the blank cannot be held precisely (within the calibration requirement, for example, within ±50 ppm, even ±200 ppm is very difficult to achieve). So the nominal frequency is not targeted but instead above or below frequency is targeted. For example, the target below the desired frequency might be −5,000 ppm to +0 ppm. The target above the desired frequency might be +0 ppm to +5,000 ppm. The final frequency adjustment is then performed to set the frequency within specification, for example within ±10 ppm at 25°C. Different final frequency calibration machines are used depending on which method was selected to roughly set the frequency (below or above the target). If the blank is high in frequency, then it will be necessary to deposit a precious metal like gold or silver on one of the electrodes to force the frequency down. If the frequency is below the target, then it will be necessary to remove material from one of the electrodes to force the frequency up. Adding a precious metal is the old method while removing material from the electrodes is the most modern and the cleaner process that affords lower aging rates on the finished crystal unit.

1.14  Inside the Crystal Unit The typical internal construction for a crystal unit in an HC-49/U hermetic holder is shown in Figure 1.10. Let us take a tour at the construction of this crystal. This holder is designed to hold a round quartz crystal blank instead of rectangular. The blank is cemented to the internal posts with a conductive epoxy at the flags of each of the electrodes. The leads are isolated from the metal base and cover with two glass seals. The cover is attached to the base in a special sealing chamber to evacuate the normal atmosphere. If one replaces the quartz

20

Understanding Quartz Crystals and Oscillators

Figure 1.10  Internal pictorial view of an HC-49 crystal unit.

dielectric with a nonpiezoelectric one, this construction technique would simply produce a capacitor. In fact, a crystal will not pass direct current just like a normal capacitor. 1.14.1  The Glass Seals

The glass seals shown in Figure 1.10 can be damaged when the leads are bent or cut. If the glass seals fracture, the hermetic properties of the holder are compromised. Within months to a year, the frequency of the crystal will age downward by permitting the normal atmosphere to slowly enter the holder. The rate downward will depend on the size of the damage to seals. The big problem here is that the crystal may go out of frequency specification when the crystal unit is in the customer’s hands. Bending the leads to make this crystal holder a surface mountable (SMT) is a common practice. However, when the crystal unit manufacturer finishes performing the SMT transformation, a gross-leak test is performed. The grossleak test will confirm that the glass seals have not been damaged during the bending of the leads. Hence, if an SMT package is needed, then it should be ordered that way from the crystal unit manufacturer. 1.14.2  The Conductive Epoxy Cement

In the attachment of the crystal blank to the posts, a conductive epoxy cement is used. Solder is never used in modern crystals. These epoxy compounds have been specifically formulated for the cementing of crystals blanks. They have been designed to minimize any “outgassing” that can deposit onto to crystal



Quartz Crystals

21

blank or change their holding properties versus time, temperature, and mechanical stress. A good epoxy, after it is cured, will not expand or contract over time, because any physical change, like mechanical relaxation, will result in a frequency shift or jump. Such a mechanical relaxation can take place when the crystal experiences a shock force. In some cases, the shock induces an abrupt and permanent frequency change.

1.15  Sealing the Crystal Unit All modern crystals are housed in a hermetically sealed package to reduce the fast aging rate that would occur if exposed to the normal atmosphere. There are several sealing methods discussed next with their own advantages and disadvantages. Some, but not all, of the sealing methods follow. 1.15.1  Solder Seal

Solder seal holders have lower lead-to-case capacitance, giving them an advantage for some filter designs. Solder sealed units can be easily reopened for rework purposes. Solder sealing is seldom used in modern crystal units due to the large amount of contaminants that are always trapped inside the package resulting in poor frequency aging. 1.15.2  Resistance Weld

Resistance welding is commonly used for through-hole crystal holders. This method of sealing introduces much less contaminants than solder sealing. This sealing process in conjunction with normal atmosphere replacement, discussed later, results in low-frequency aging units. 1.15.3  Cold Weld

Sealing of very high Q crystals is performed in high vacuum. This sealing process is called cold weld. This method of sealing costs more than resistance weld but introduces virtually no contaminants, which results in superior electrical (in particular high Q) and frequency stability performance. The minimization of contaminants and the absence of high temperatures during the cold weld sealing process results in the lowest-frequency aging rates. 1.15.4  Seam Weld

Seam weld sealing is the most commonly used method for low-profile SMD crystals with ceramic base/metal cover packages. The performance is similar to

22

Understanding Quartz Crystals and Oscillators

resistance weld but the sealing cost is lower due to the larger amount of packages that can be sealed at a time. 1.15.5  Epoxy Seal

Epoxy sealing is widely used in high volume commodity SMD crystals. The frequency aging of this sealing method is not as good as seam weld.

1.16  Testing for Moisture Whichever of these sealing methods are used, the process must produce an internal dew point lower than the intended lower operating temperature. Sealing must take place at a very low dew point of −40°C or lower to prevent trapping moisture, which can create excessive aging of the crystal frequency, and possible high resistance and hysteresis problems at low temperatures. To meet MIL-PRF-1090, the sealing dew point must be −55°C or lower. Performing a frequency-temperature hysteresis on the crystal unit will reveal any moisture problems inside it.

1.17  Crystal Resonator Mechanical Equivalent Model A quartz crystal resonator is a mechanical vibrating system that is linked to the electrical word via the piezoelectric effect. Figure 1.11 shows this link between the mechanical and electrical world. In Figure 1.11, the mechanical vibrating system and electrical circuit are equivalent as each can be represented by the same differential equations. The capacitor, inductor, and resistor correspond to the spring, mass, and dashpot (i.e., the damping element). The capacitor C represents mechanical elasticity, the inductor L represents mechanical inertia, and the resistor R represents mechanical losses. This link adds insight to the analysis of the electrical equivalent circuit comprised of these three components. For example, moving the mass a greater distance corresponds to an increase in the value of the capacitor C. It is evident that an increase in losses means a larger value for R.

1.18  Crystal Resonator Electrical Equivalent Circuit The derivation of the equivalent circuit is beyond the scope of this book, and it requires the solution of a second-order differential equation describing the motion of the vibrating particles in the quartz crystal including the effects of damping. The steps involve solving the differential equation and then applying



Quartz Crystals

23

Figure 1.11  Equivalent electrical circuit and mechanical vibrating system.

boundary conditions to arrive at the general equation for the particle displacement. Following differentiating the equation for displacement to obtain the strain, polarization is calculated by multiplying the strain by the piezoelectric constant. The resulting equation can then be expressed in terms of the charge density, which is then used to calculate the potential at any point in the crystal. Applying boundary conditions, the equation for potential is then evaluated at the surface of the crystal, from which the field intensity can be determined and subsequently the current density. The ratio of voltage to current is then determined in terms of the physical parameter, constants, and excitation frequency of the crystal resulting in the topology and electrical parameters of the equivalent circuit [3]. In 1925, Van Dyke used a similar procedure to the one outlined above and derived the equivalent circuit that follows. In Figure 1.12, C1, L1, and R1 form the motional arm of the crystal resonator while C0 is the static arm. These are the same three electrical components from Figure 1.11. C0 is the shunt or static capacitance, which is primarily formed from the electrodes of the crystal plus the strays of the holder. The shunt capacitance C0 is the only physical component in the equivalent circuit and is present if the crystal is oscillating or not. This capacitor can actually be measured with a simple capacitance meter. However, the motional arm elements (C1, L1, and R1) are equivalent and therefore are not real. The motional capacitance C1 represents the stiffness of the crystal transformed into electrical terms through the piezoelectric effect. The motional inductance L1 represents the effective mass of the crystal also transformed into electrical terms, while R1 represents the frictional loses. These motional components are valid over a narrow frequency

24

Understanding Quartz Crystals and Oscillators

Figure 1.12  Schematic symbol and the single-mode, one-port, electrical equivalent circuit of the crystal resonator. This electrical model of a quartz resonator is also known as the Van Dyke model.

range and will change with drive level, temperature, time, shock, and vibration, to name a few. Also, due to the piezoelectric effect, the motional arm values are only valid in the range of mechanical resonance. This is a critical fact when using the crystal equivalent electrical model for simulation purposes. The impedance of the two reactive elements of the motional arm, C1 and L1 is extremely high. The inductance value of L1 can be greater than 1 Henry and the capacitance C1 can be less than 0.1 fento-farad depending on the type of cut and overtone involved. The motional resistance R1 can range from a few ohms to tens of thousands of ohms. The enormous L1/C1 ratio cannot be physically built with discrete components, and this is the major reason why quartz resonators are used instead. In summary, 1. C1 is the motional capacitance and represents mechanical elasticity. Mechanical elasticity will be transformed to frequency deviation by the piezoelectric effect. 2. L1 is the motional inductance and represents mechanical inertia. 3. C0 is the shunt or static capacitance and it is formed from the electrodes, holder, and leads. 4. R1 is the motional resistance and represents mechanical losses.

1.19  Derivation of Equivalent Circuit Equations The complex impedance equation of the crystal equivalent circuit of Figure 1.12 is the parallel combination of the motional arm and the static arm impedance. The motional arm impedance is (R1 +1/JwC1 + JwL1) while the static arm impedance is (1/JwC0). Hence, the crystal impedance is



Quartz Crystals



Z ( jw ) =

(R1 + 1

(R1 + 1

jwC1 + jwL1 )(1 jwC 0 )

jwC1 + jwL1 ) + (1 jwC 0 )

25



(1.8)

Equation (1.8) is a complex impedance but our interest is in the imaginary part of it. The imaginary portion of a complex impedance is called the reactance. In Figure 1.13 the crystal’s reactance curve has been plotted without going through the algebra of separation of the impedance equation. There are several frequency points of interest in Figure 1.13. The first is fs and it is the frequency at which the motional capacitance C1 cancels the motional inductance L1 (mechanical resonance). The frequency fs is called the series resonance or series-resonant frequency of the crystal resonator and is given by

fs =

1 2 π L1C1

(1.9)

The anti-resonant or parallel-resonant (parallel resonance) frequency fa is where the motional inductance L1 resonates with the parallel combination of C1 and C0. Therefore, fa is given by



1

fa = 2 π L1

C 0C1 C 0 + C1

(1.10)

The series and anti-resonant frequencies are related by the formula

f a = f s 1 + C1 C 0

Figure 1.13  Reactance curve of the crystal resonator equivalent circuit.

(1.11)

26

Understanding Quartz Crystals and Oscillators

If we call the series frequency a zero and the anti-resonant a pole, then the zero-to-pole spacing of the equivalent circuit is

fa − fs C ≈ 1 × 106 (in ppm ) fs 2C 0

(1.12)

1.20  Series-Resonant and Parallel-Resonant Oscillators The region between fs and fa is called the area of usual parallel resonance. This entire region is inductive and many oscillator topologies operate in this region. Ignoring C0, very important to note is that the series resonance point fs is resistive and is equal to R1. Many oscillators are also designed to operate the crystal at fs. Consequently we can separate most crystal oscillator circuits or topologies in two major categories: series-resonant oscillators and parallel-resonant oscillators. In this book, the following oscillator definitions will apply. A series-resonant oscillator is a circuit topology that forces the crystal resonator to oscillate at series resonance point fs of the crystal’s reactance curve, its resistive point. A series-resonant oscillator will contain no reactive components in its feedback loop. A parallel-resonant oscillator is a circuit topology that forces the crystal resonator to oscillate in the area of usual parallel resonance on the crystal’s resonator reactance curve; its inductive region. A parallel-resonant oscillator will contain reactive components (capacitors) in its feedback loop. In these two definitions, I have taken some liberty for this book. Seriesresonant and parallel-resonant oscillators have been described differently (and correctly) in other books. The key to remember is that once the oscillator topology and values have been chosen, the crystal will be forced to operate at either series resonance or its inductive region. Figure 1.14 presents examples of parallel-resonant and series-resonant topologies as defined earlier. Oscillators that operate outside these two conditions are not covered in this book. 1.20.1  Definitions of Series and Parallel Crystals

These definitions for series-resonant and parallel-resonant oscillators require a crystal unit that has been specifically calibrated to the nominal frequency for that particular oscillator topology. If specified incorrectly, the frequency of the oscillator will be off the tolerance of the crystal. The series-resonant topology will use a series-resonant crystal while the parallel topology will use a parallelresonant crystal.



Quartz Crystals

27

Figure 1.14  (a) Parallel-resonant and (b) series-resonant topologies.

The following definitions for series-resonant and parallel-resonant crystals will be used in this book. A series-resonant crystal (or series crystal ) is calibrated such that its nominal frequency is at the series resonance point fs, and it is intended to be used in series-resonant oscillators. A parallel-resonant crystal (or parallel crystal ) is calibrated such that its nominal frequency is within the region of usual parallel resonance and it is intended to be used in parallel-resonant oscillators. Note that there is no physical construction difference between a series and parallel crystal. In fact, all crystals can be operated at either series resonance or parallel resonance.

1.21  Load Capacitance It was stated that a parallel crystal is calibrated such that its nominal frequency is in the inductive region of the crystal’s reactance curve. However, in specifying it, one must narrow down the region to an exact point on the reactance curve. The exact point in the inductive region is specified by the load capacitance

28

Understanding Quartz Crystals and Oscillators

value. A parallel-resonant circuit will present a certain capacitive load. The crystal manufacturer needs to know what that capacitive load value is to calibrate the crystal so that when placed in circuit, the frequency of the oscillator will be within the calibration tolerance of the crystal. For example, a popular load capacitance value is 20 pF and therefore the equivalent in-circuit series capacitance value should be designed to match it.

1.22  Fundamental Mode Crystals The crystal’s reactance response shown in Figure 1.13 is only one of many as depicted in Figure 1.15. A crystal unit that is calibrated to frequency on the lowest response is called a fundamental crystal or a fundamental mode crystal. The fundamental mode response is the most active and has the lowest resistance. A fundamental crystal can be calibrated at series resonance or in its inductive region. Hence, there are series and parallel calibrated fundamental crystals.

1.23  Overtone Mode Crystals An overtone crystal is a crystal resonator that is calibrated to frequency at a major response other than the fundamental (see Figure 1.15). Technically the fundamental response is the first overtone. The next major response above the fundamental is called the third overtone. Above the third overtone are the fifth and then the seventh. There are only odds and no evens. Note that overtones are not harmonics of the fundamental response. The major responses are not exact multiples of each other.

Figure 1.15  Crystal resonator reactance curve depicting fundamental, third, and fifth overtone responses (modes). Also shown are the spurious responses, which are always to the right of the major responses.



Quartz Crystals

29

1.24  Spurious Modes In describing the spurious modes, Dr. Virgil Bottom [7] said it best, “The inharmonic overtone modes, or spurs, are not an expression of the perversity of nature or of demon possession. On the contrary they are natural phenomena just as are the overtones of an organ pipe or piano string. They are more closely analogous to the overtones of a bell which, in general, are all higher in pitch than the fundamental and have frequencies which are not harmonically related to the fundamental or to each other.” The spurs are unwanted modes but are there by design and cannot be eliminated. The frequency location and amplitude can be calculated given the specific geometry design of the crystal resonator. These calculations are not presented in this book, but the following generalities will be stated. The spurs amplitude and frequency separation from the main mode can be controlled by varying the shape and size of the electrodes. Increasing the curvature of the quartz blank will, in general, increase the frequency difference between the main mode and a particular spur. The size of the electrodes and curvature of the quartz blank will influence the amplitude of the spurs. Spurs at −6 dB or lower with respect to the desired mode (fundamental or overtone) will not create problems for the oscillators described in this book. For those designing crystal filters, having very low spur responses is a major design requirement. Please note that spurs only occur to the right of the fundamental or overtone responses. This is why in crystal filters the out-of-band attenuation below the passband can be much better than the out-of-band attenuation above the passband.

1.25  Expanded Quartz Resonator Equivalent Circuit Model The crystal resonator equivalent circuit of Figure 1.12 will now be expanded to include the first two overtone motional arms. Because the shunt capacitance C0 is real, it stays in the circuit unchanged. The overtone arms motional values are related to the fundamental arm by



CN ≅

C1 N2

LN ≅ L1

(1.13)

(1.14)

30



Understanding Quartz Crystals and Oscillators

R N ≅ N 2R1

(1.15)

where N = 3, 5, 7, … is the overtone number. Figure 1.16 shows the expanded equivalent circuit with the third and fifth overtone arms added. The important outcome of the above relationships between the fundamental motional arm compared to the overtone arms is that L1 is virtually unchanged, while C1 is reduced by a factor of N 2. This results in larger numerical separation between CN and LN and then C1 and L1, thus giving the overtone arms a higher-quality factor of Q. Hence, overtone crystals can be designed with higher Q than fundamental ones. However, this higher Q makes pulling the frequency of an overtone crystal more difficult to accomplish.

1.26  The Ideal Phase Angle of the Quartz Crystal Resonator The phase plot of the crystal resonator is easy to deduce. Below the frequency fs in Figure 1.13, the crystal’s reactance is capacitive, between fs and fa it is inductive, and finally above fa it is back to being capacitive. Applying our elementary knowledge that a capacitor has a phase angle of −90° and an inductor +90°, we obtain the phase plot shown in Figure 1.17.

Figure 1.16  Multimode, one-port, crystal resonator equivalent circuit with fundamental, third and fifth overtone arms. Additional arms can be added to represent spurious responses as needed.



Quartz Crystals

31

Figure 1.17  Ideal phase plot of a quartz crystal resonator.

1.27  Pulling the Crystal Frequency by Changing the Load Capacitance Many applications require the frequency of the crystal to be changed (pulled or deviated). This may be necessary to trim out the frequency tolerance or to design a voltage controlled crystal oscillator (VCXO). Adding a load capacitance in series with the crystal will move the frequency of the crystal from the point fs towards the fa as its value is decreased. How far up the reactance curve with respect to fs will depend on the value of CL and is given by

∆f C1 = × 106 (in ppm ) fs 2 (C L + C 0 )

(1.16)

∆f = f L − f s

(1.17)

where

and fL is the frequency of the crystal with load capacitance CL. A graphical representation of (1.16) is the pulling curve shown in Figure 1.18. Combining (1.16) and (1.17) and then solving for fL give another insightful version of the formula. That is,



  C1 f L = f S 1 +   2 (C 0 + C L ) 

(1.18)

32

Understanding Quartz Crystals and Oscillators

Figure 1.18  Plot of (1.16). Crystal frequency pulling curve versus load capacitance where the motional capacitance C1 = 10 fF and the shunt capacitance C0 = 5 pF. From the curve one can read that a 20-pF load capacitance in series with crystal will move the frequency +200 ppm above the series resonance frequency.

Equation (1.18) as a function of CL is called the crystal-frequency equation and it is very useful in understanding the operation of crystals in oscillators. Example 1.1

In an ideal 20-MHz parallel-resonant crystal oscillator, the crystal unit that should be used is a parallel crystal with a load capacitance of 20 pF. However, a series-calibrated 20-MHz crystal was installed with C1 = 10 fF and C0 = 5 pF. How far off in frequency will the oscillator be in ppm and hertz? Solution:

One can make the following observation and say that the oscillator will be higher and not lower than 20 MHz. This can be deduced from the reactance curve of Figure 1.13 as the oscillator will force the crystal to operate in the inductive region, which is higher than its 20-MHz series calibrated frequency (at fs  ). This ideal topology presents an equivalent 20-pF series load to the crystal as intended. Hence, the problem involves finding the shift in frequency from series calibrated nominal frequency to a new frequency with 20 pF physically in series with the crystal. Let us be watchful with all the “series” words being used in this example. I understand that this is one of the main confusion with



Quartz Crystals

33

crystals. The load capacitance that a crystal “sees” in circuit should be effectively in series with the crystal’s two terminals. That is, to operate the crystal at parallel resonance, use the appropriate load capacitance in series with it. However, to operate the crystal at series resonance, use an infinite load capacitance in series with it. Infinite load means no reactive component across the crystal’s two terminals. Using (1.16), we have

∆f 10 × 10 −15 = × 106 = +200 ppm −12 −12 fs 2 5 × 10 + 20 × 10

(

)

Hence, the frequency is high by +200 ppm, which is

20 × 200 = +4000 Hz above 20 MHz or 20,004, 000 Hz

1.28  Zero-to-Pole Spacing We can derive (1.12) in a different manner. That is,



fL = fS 1+

  C1 C1 = f S 1 +  C0 + CL  2 (C 0 + C L ) 

(1.19)

Letting CL approach infinity results in



  C1 lim   =0 C L →∞  2 (C L + C 0 ) 

(1.20)

and using (1.20) in (1.19) gives

f L = f s (1 + 0) = f s

(1.21)

34

Understanding Quartz Crystals and Oscillators

The infinite load capacitance (such as coupling capacitance) in series with the crystal does not change or move the frequency of the crystal (i.e., it remains at fs ). Reducing the value of the load capacitance will increase the frequency of the crystal and eventually the frequency of fa will be reached, which will be avoided in the crystal oscillators covered in this book. This leads to

  C1 C1 lim   = 2C C L →0  2 (C L + C 0 )  0



(1.22)

The result of (1.22) is the fractional frequency distance from fs to fa. The result of (1.22) reconfirms the zero-to-pole spacing of the crystal resonator. It represents the maximum usable bandwidth of the crystal without resonating out C0.

1.29  Trim Sensitivity An essential formula that can be derived from the crystal’s reactance curve is the slope at any load capacitance CL. Taking the first derivative of (1.16) with respect of CL, we obtain



  C1 −6 TS = −   × 10 (in ppm pF)  2 (C + C )2  0 L

(1.23)

Crystal unit manufacturers call this equation the trim sensitivity of the crystal at a given load capacitance. It gives the incremental frequency change, for an increment change in load capacitance. This equation can be used to determine how the oscillator component tolerances affect the frequency of the crystal. A plot of this equation is shown in Figure 1.19. Do not use (1.23) to determine the frequency deviation of the crystal over a certain load capacitance range; instead, use the following equation,



C1 (C L 2 − C L1 ) ∆f f − f L1 = L2 = × 106 (in ppm ) (1.24) f f 2 (C 0 + C L1 )(C 0 + C L 2 )

where CL1 and CL2 are two different load capacitances.



Quartz Crystals

35

Figure 1.19  Plot of equation (1.23), the crystal’s trim sensitivity (TS) versus load capacitance, with motional capacitance C1 = 10 fF and the shunt capacitance C0 = 5 pF.

Example 1.2

Consider a 15-MHz crystal described by C1 = 20 fF and C0 = 4 pF. A load capacitance is placed in series with the crystal and varied from 22 pF to 11 pF. Calculate the fractional frequency change in parts per million and hertz. Solution:

Because the capacitance is being changed from a high to a lower value, we know that the frequency will rise and hence a plus fractional frequency change. Using (1.24), we get

(

)

20 × 10 −15 22 × 10 −12 − 11 × 10 −12 ∆f = × 106 = +282 ppm −12 −12 −12 −12 fs 2 4 × 10 + 11 × 10 4 × 10 + 22 × 10

(

)(

)

In hertz, the frequency change is 15 × 282 = +1,230 Hz. This MathCad file calculates the load capacitance range values for a desired pull and, given crystal’s shunt, motional, and load capacitance. C1 := 15 · 10–15 ⇐ Enter here motional capacitance

36

Understanding Quartz Crystals and Oscillators

C0 := 3 · 10–12 ⇐ Enter here shunt capacitance CL := 20 · 10–12 ⇐ Enter Load capacitance value of crystal Plus_Pull := 140 ⇐ Enter here plus pull desired in ppm Minus_Pull := 140 ⇐ Enter here minus pull desired in ppm Solution:

To solve this problem, we will set up two equations, two unknown with the following: ∆F :=



C1 2 (C 0 + CL )

Guesses CL1:= 10 ⋅ 10 −12



CL2 := 10 ⋅ 10 −12

Given ∆F + Plus_Pull ⋅ 10 −6 =



C1 2 (C0+CL1)

∆F − Minus_Pull ⋅ 10 −6 =

C1 2 (C0+CL2 )



 CapMax   CapMin  := Find (CL2, CL1) CapMax = 3.73 × 10 −11

CapMin = 1.309 × 10 −11

Let’s check the results



Total_Pull :=

C1(CapMax-CapMin ) ⋅ 106 2 (C0+CapMin )(C0+CapMax )

Total_Pull = 280



Quartz Crystals

37

1.30  Important Unitless Quantities There are three unitless quantities that are very useful in characterizing crystal resonators. The first of these is the capacitance ratio between the shunt capacitance C0 to the motional capacitance C1, which is an important ingredient of the equivalent circuit. That is, r=



C0 C1

(1.25)

The ratio r is a measure of the electrical energy stored in C0 to the energy stored elastically in C1, due to the lattice strains produced by the piezoelectric effect. Note that (1.25) is the same as the zero-to-pole equation in (1.12) without the 2 factor in the denominator. Hence, this ratio is also used in specifying the maximum frequency deviation possible with the resonator. The ratio r is a measure of the piezoelectric coupling factor, which depends on crystal cut, type, and mode of vibration [8]. The second very important unitless quantity is the quality factor Q, which is defined by the relation Q=



w L 1 = S 1 wSC1R1 R1

(1.26)

Let us rewrite (1.26) by solving for wS in the second part and substituting it in the first part. Hence, we have wS = QR1/L1 and Q is now equal to

Q=

L1 L 1 = ⇒ Q 2 = 21 2 R1 C1 [QR1 L1 ]R1C1 QR1 C1

(1.27)

Finally, taking the square root on both side of the final part of (1.27) yields

Q=

1 L1 R1 C1

(1.28)

Consequently, the high Q in a crystal resonator lies in the fact that the C1 value can be made very small (fento-farads) and the L1 value very large (milli-

38

Understanding Quartz Crystals and Oscillators

henries). Also note that Q is proportional to the zero-to-pole spacing. That is, the smaller the spacing, the higher the Q. The third unitless quantity is the figure of merit M. This quantity is defined as the ratio of impedance of the static arm to the impedance of the motional are at series resonance. The static arm impedance is 1/wSC0, while the impedance of the motional arm at series resonance is simply R1. Therefore, the equation for M is given by

M=

(1/ wSC 0 ) = R1

1 1 = wSC 0R1 2 πf SC 0R1

(1.29)

The figure of merit M must be greater than 2 in order for the resonator to have an inductive region. In general, the greater M is, the more useful the resonator is for oscillator applications. These three unitless quantities are not independent of each other and are related by

Q = rM

(1.30)

1.31  Resistance of the Crystal Above Series Resonance (ESR) It was learned earlier that the resistance of the crystal at series resonance is simply the motional arm resistance R1. However, when the crystal is operated with a load capacitance, the equivalent resistance of the crystal is no longer R1. This important fact is stated incorrectly in many articles about crystal resonators because of the name used in describing the resistance above series resonance. The resistance of the crystal above series resonance is called equivalent series resistance (ESR). The equation for the ESR of a crystal is 2



 C  ESR = R1 1 + 0   CL 

(1.31)

An engineer needs to be vigilant when reading the crystal performance data from the manufacturer that the resistance measurement is listed correctly: ESR for a parallel crystal and R1 for a series crystal. The motional resistance R1 is also called resonance resistance (RR) by some crystal test equipment manufacturers. If, in the crystal performance data, the column for resistance is labeled RR, then it is R1 and not ESR.



Quartz Crystals

39

When calculating the gain margin of series-resonant oscillators, use R1, but use the ESR value for parallel-resonant oscillators.

References [1] Vig, J. R., “Quartz Crystal Resonators and Oscillators,” www.ieee-uffc.org, January 2007. [2] Buchanan, J. P., Handbook of Piezoelectric Crystals for Radio Equipment Designers, WADC Technical Report 56-156, October 1956. [3] Cady, W. G., Piezoelectricity, Vol. 1, New York: Dover Publications, 1964. [4] Mason, W. P., Chapter 1, “Quartz Crystal Applications,” in Heising, D., (ed.), Quartz Crystals for Electrical Circuits, New York: Van Nostrand Company, 1946. [5] Heising, R. A., Quartz Crystals for Electrical Circuits, New York: Van Nostrand Company, 1946. [6] NIST, “Time and Frequency from A to Z for Definitions,” www.nist.gov/general/glossary. htm. [7] Bottom, V. E., Introduction to Quartz Crystal Unit Design, New York: Van Nostrand Reinhold Company, 1982. [8] Parzen, B., Design of Crystal and Other Harmonic Oscillators, New York: John Wiley & Sons, 1983.

2 Quartz Crystal Characteristics 2.1  Introduction This chapter expands on the work started in Chapter 1 by going into greater detail of many of the crystal characteristics. For example, it will cover the electrical behavior and performance under different conditions such as temperature, overtone, drive level, mechanical stress, and type of design. “Stress” is the key word. Stress the quartz in any way such as gravitational, mechanical, and temperature gradient, and the quartz will respond with a frequency shift.

2.2  Defining the Frequency Versus Temperature Curve It is instructive to define some points of interest in a typical frequency versus temperature curve of a crystal resonator. Figure 2.1 shows this curve defining these points of interests. Notice that this curve is cubic in nature and that the inflection point is close to room temperature for this particular curve. Also notice that frequency rises above the upper turning point and falls below the lower turning point. This curve shape is also known as an s-curve in the crystal industry for obvious reasons. The definitions in Figure 2.1 will be referenced throughout the book onward. Also from this point forward, frequency versus temperature can be referred to as frequency-temperature or f-T.

41

42

Understanding Quartz Crystals and Oscillators

Figure 2.1  Frequency versus temperature curve defining: inflection point and lower and upper turnover points.

2.3  Quartz Crystal Cuts Quartz crystal units are first categorized by how the blank is cut from the quartz bar [1]. Recall from Chapter 1 that quartz is anisotropic and, therefore, depending on the angle(s) used to cut blanks out, it will result in different thermal and other important properties. The angle of cut is the primary determining factor for a crystal resonator’s frequency-temperature characteristics. Secondary factors are overtone, drive level, blank geometry, and electrodes size and shapes, to name a few. The scope of this book will only cover the following three important cuts: AT, BT, and SC cuts. 2.3.1  The AT, BT, and SC Cuts

In 1929, it was discovered by groups in America, Germany, and Japan that the temperature coefficient of a Y-cut plate could be improved to become zero at certain temperatures. The first zero temperature coefficient cut discovered was named the AT cut and the second zero temperature coefficient cut discovered was named the BT cut [2]. The BT cut is about 50% thicker than the AT cut for a given frequency. This fact about the BT cut will become an important advantage at high frequencies where the higher the frequency, the thinner the crystal blank. However, the f-T characteristics for the BT-cuts are worse than the AT cuts as it will be illustrated next. The SC in SC cut is an acronym for “stress compensated.” It was first theorized by Dr. R. Holland in 1974 to overcome some of the shortcomings of the AT and BT cuts like thermal shock. In 1975, a paper by E. P. EerNisse from the U.S. Army Signal Corps predicted that a resonator with angle coordinates of φ = 22.5° and θ = −34.3° would produce a resonator with reduced frequen-



Quartz Crystal Characteristics

43

cy shift due to mechanical stress. This new resonator was said to be stress compensated and, hence, the name SC cut.

2.4  Temperature Characteristics of AT Cut, BT Cut, and SC Cut 2.4.1  AT-Cut Frequency-Temperature Curves

The true AT-cut temperature curve is derived by cutting a quartz blank at an angle θ = +35.25° from the Z-axis as shown in Figure 2.2. The frequencytemperature characteristics that this angle produces is shown in Figure 2.3 and it is assigned θ0, which is zero minutes from θ = +35.25°. This is the angle for which the AT was named as mentioned above, the first zero temperature coefficient is around 26°C. It was very important at the time of discovery because the finished oscillator could maintain good frequency stability around room temperature. If the cut angle θ is changed from +35.25°, a different f-T curve will be created. In fact, an entire family can be generated as depicted in Figure 2.4. Selecting and controlling the tolerance of this cut angle are the principle technique that a crystal unit manufacturer uses to meet a certain f-T requirement. Consequently, in a finished crystal unit, its f-T characteristics cannot be changed. 2.4.2  BT-Cut Frequency-Temperature Curves

The BT-cut family is derived by setting the saw angle to 49° from the Z axis as indicated in Figure 2.2. The f-T performance of the BT is a parabola as shown in Figure 2.5. 2.4.3  SC-Cut Frequency-Temperature Curves

The SC cuts are members of doubly rotated cuts, which means that they are cut from the quartz bars with respect to two angles as shown in Figure 2.6. The

Figure 2.2  Illustration of quartz bar showing saw angles for the AT and BT cuts.

44

Understanding Quartz Crystals and Oscillators

Figure 2.3  True AT-cut frequency versus temperature when cut angle θ = +35.25°. This is the curve for which the family was named the AT cut. Note that the zero temperature coefficient is around +26°C.

Figure 2.4  Family of AT-cut curves. Shown is the θ = +35.25° curve, in addition to cut angles of ±5 minutes from it.

f-T curves are essentially the same as the AT cut except the inflection point is around 92°C instead of 26°C as in Figure 2.7.



Quartz Crystal Characteristics

45

Figure 2.5  BT-cut frequency-temperature performance.

Figure 2.6  Depiction of singly and doubly rotated cuts. The AT and BT are singly rotated cuts and the SC is a doubly rotated cut [3].

2.5  Thickness Versus Frequency of Quartz Wafers (Blanks) Usually in a mechanical vibrating system, the thinner the substrate material is, the higher the vibrating frequency will be. This is the case for quartz blanks, where the thinner the quartz blank is made, the higher the frequency of the crystal resonator will be. The frequency constant N relates the thickness of the quartz substrate to the vibrating frequency. The values of N can be expressed in

46

Understanding Quartz Crystals and Oscillators

Figure 2.7  Family of SC-cut curves and similar to the AT-cut curves except that the inflection point is around +92°C for the SC.

megahertz-mils (1 mil is 0.001 inch) or megahertz-microns (1 micron is 0.001 mm). For example, using the value of N for an AT-cut crystal given in Table 2.1, we can calculate the thickness of a 20-MHz blank as follows: 65.5/20 = 3.275 mils. In general, the formula is

Thickness in mils or microns =

N Frequency in MHz

(2.1)

where the values of N are given in Table 2.1 for the different cuts. In the example calculation above, a 20-MHz AT-cut blank is only about 0.003 inch. This is only three times the thickness of an average human hair,

Table 2.1 Thickness Versus Frequency Constants of AT, BT, and SC-Cut Quartz Cut Type MHz-mils MHz-microns AT cut 65.5 1,664 BT cut 100.3 2,549 SC cut 70.9 1,800



Quartz Crystal Characteristics

47

which is about 1 mil thick. So at 20 MHz the crystal is already very thin. Because of this, low-cost crystals are limited to about 30 MHz for a fundamental AT-cut unit. Now if we make the same thickness calculation for 20 MHz but this time we use the N value for a BT-cut blank instead, we obtain 100.3/20 = 5.015 mils. Therefore, the BT cut offers a thicker blank for the same frequency. Due to this, today a low-cost fundamental BT cut can be as high as 45 MHz. Another advantage of the BT cut is a higher capacitance ratio and a larger motional inductance, which is sometimes beneficial for filter applications [2]. Although the BT cut has some advantages over the AT cut, its biggest limitation is its frequency temperature. When good frequency-temperature stability is needed (for example, ±20 ppm over 0 to 70°C) then, it has to be an AT cut because it is not possible with a BT cut. Figure 2.8 clearly shows the superior difference of the AT cut compared with the BT cut.

2.6  Bechmann Frequency-Temperature Curves The frequency-temperature curves of AT, BT, and SC cuts can be generated mathematically with the following power series expansion

∆f 2 3 = a 0 (T − T0 ) + b0 (T − T0 ) + c 0 (T − T0 ) f

(2.2)

Figure 2.8  Typical AT-cut (solid line) versus BT-cut (dotted line) frequency-temperature.

48

Understanding Quartz Crystals and Oscillators

where a0, b0, and c0 are the first-, second-, and third-order coefficients of frequency. These constants depend primarily on the quartz properties and angle of cut. To a lesser extent, they depend on the overtone number and physical design of the electrodes. T0 is the inflection temperature while T is the temperature variable. T0 is approximately 26°C for AT cuts and 92°C for SC cuts. For the AT-cut resonator, these coefficients are:

a 0 = −0.08583 × 10 −6 ( θ − θ0 )

(2.3)



b0 = 0.39 × 10 −9 − 0.07833 × 10 −9 ( θ − θ0 )

(2.4)



c 0 = 109.5 × 10 −12 − 0.033 × 1012 ( θ − θ0 )

(2.5)

where (q - q0) is the difference between the intended angle and the zero temperature coefficient reference angle, in degrees of arc. These coefficients were derived by Dr. Bechmann in 1955. These coefficients change with overtone and other important factors beyond the scope of this book. The reference angles for the AT-cut resonator are listed in Table 2.2. Using (2.2) and the coefficients listed above, I plotted Figures 2.3, 2.4, 2.5, and 2.7. If T0 is taken where the f-T slope is zero, then a0 = 0 and temperature curves can be determined by b0 and c0 coefficients only. Table 2.3 lists the higher-order coefficients for the AT, BT, and SC cuts.

Table 2.2 Reference Angles for AT-Cut Resonators Mode Reference Angle, θ0 AT-cut fundamental 35°15′ AT-cut third overtone

35°20′

AT-cut fifth overtone

35°21′

Table 2.3 Second- and Third-Order Temperature Coefficients of Quartz Cuts Cut b0 (10–9/°C2) c0 (10–12/°C3) AT 0.39 109.5 BT –40 –128 SC –12.3 58.2



Quartz Crystal Characteristics

49

2.7  AT Cut Versus SC Cut The SC-cut crystal resonator is primarily used in ovenized applications. It can be made with much higher Q than the AT cut. Quality factor values of a million are easy to achieve for the SCs while those of hundreds of thousands are difficult for the AT. However, the SC can be problematic for the oscillator designer due to its many modes. In fact, the SC-cut b-mode is stronger than the desired c-mode. Figure 2.9 shows some of the many modes of the SC cut. The manufacturing of the SC cut is much more difficult than the AT cuts. Not only are there two angles of cuts with which to be precise (compared with one for the AT cuts), but the lower C1 values that result in SC cuts means that greater precision is required in all the frequency calibration steps. Recall from Chapter 1 that the smaller the C1 value, the “stiffer” the resonator will be in terms of pulling its frequency. Hence, greater precision in the cutting process and greater precision in all the frequency calibration steps results in a much higher per unit price for an SC-cut resonator. The following are some of the pros and cons for the AT cut and SC cut. 2.7.1  AT-Cut Versus SC-Cut Pros and Cons

The SC-cut pros are: 1. Higher Q, which results in lower close-in phase noise;

Figure 2.9  Spectrograph of SC-cut crystal resonator. The b-modes’ strengths are greater than the c-modes’ requiring mode selection circuitry to operate properly on the desired cmode.

50

Understanding Quartz Crystals and Oscillators

2. 3. 4. 5. 6.

Lower aging rates; Less sensitivity to mechanical stress; Less sensitivity to vibrations and gravity; Less sensitivity to drive level; Far fewer activity dips.

The SC-cut cons are: 1. Expensive price; 2. Requirement of an oven; 3. Low motional capacitance values that make it difficult to electrically shift its frequency; 4. Sensitivity to electrical fields, which means that it must be packaged in a vacuum to obtain optimum Q; 5. Complicated accompanying oscillator circuitry due to its many modes. AT-cut pros are: 1. Low cost; 2. Ability to be made with high motional capacitance for VCXOs applications; 3. Availability in the most packages. AT-cut cons are: 1. Sensitivity to vibration, stress, gravity, and acceleration; 2. Long warm-up time due to temperature gradients in the quartz during initial power-up; 3. Prone to activity dips, especially in small physical and high-frequency designs; 4. Sensitivity to drive level.

2.8  The SC-Cut B-Mode Temperature Characteristic In Figure 2.10 the SC-cut resonator b-mode f-T characteristic is compared to its c-mode. The b-mode f-T is monotonic with a slope of about −25.5 ppm/°C.



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Figure 2.10  SC-cut f-T for b- and c-modes (roughly scaled to each other).

The b-mode frequency can be used to measure the resonator internal temperature, which, in turn, can be used for temperature compensation. It is possible to design an oscillator to run on both the b-mode and cmode simultaneously. The b-mode frequency can also be used as a temperature sensor.

2.9  Vibrational Displacements of AT Versus SC Cuts The vibrational displacements of the AT and SC cuts are different from each other as illustrated in Figure 2.11. The AT is a simple shear while the SC has a more complex mode of motion. This more complex mode of motion for the SC leads to its many more resonant modes compared to the AT.

Figure 2.11  Modes of motion for singly rotated cuts like the AT cut, and doubly rotated cuts like the SC cut. The shaded areas represent the electrodes of the crystal unit.

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2.10  Drive Level The amplitude of mechanical vibrations of a quartz resonator increases proportionally to the amplitude of the applied power/voltage across it or the current through it. In IEC-444-6 drive level is defined as follows: “The drive level (expressed as power/voltage across or current through the crystal) forces the crystal resonator to produce mechanical oscillation by way of the piezoelectric effect. The parameters/characteristics of quartz crystals are influenced/changed by the drive level. As the drive level increases, the frequency and resistance of the crystal resonator change through non-linear effects.” 2.10.1  High Drive Level

The “high” drive level (e.g., above 1 mW or 1 mA for AT-cut crystals) parameter changes are observed on all crystal units, which may also result in irreversible amplitude and frequency changes. Figure 2.12 shows how excessive drive level affects the frequency of the crystal resonator. Increasing the drive level further may destroy the crystal substrate or the vaporization of the evaporated electrodes. Also, at a high drive level, a spurious mode may replace the main mode as the selective element and cause the oscillator to jump to another frequency (for example, a spur). 2.10.2  Low Drive Level

The “low” drive level (e.g., below 1 µW or 50 µA for AT-cut crystals) frequency and resistance changes are also observed on some crystal units. The frequency

Figure 2.12  Due to nonlinearities of quartz, resonance curves become asymmetric due to high drive level [3].



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change may not be a problem, but the resistance change can rise high enough to make oscillator start-up difficult, or not possible when the circuit’s gain is marginal. Figure 2.13 shows drive level versus resistance for both high and low drive levels. 2.10.3  Correlation Drive Level

Because the crystal resonator parameters do change with the specific drive level applied, a correlation level has been established in the crystal industry. This correlation level is typically 100 µW. Hence, when data is supplied with the crystal, the drive level used to measure is usually 100 µW. Therefore, when comparing parameters on crystals from different manufacturers, verify that the parameters were measured at the same drive level. 2.10.4  Maximum Drive Level

Quartz crystal manufacturers will state a maximum drive level for each specific crystal unit that is based on physical size and/or frequency. There are many oscillator designs that typically exceed the maximum drive level seemingly without any ramification. In fact, the maximum drive level is not an absolute maximum in which, if exceeded, it can destroy the crystal. Per IEC 444-6, the maximum drive level is defined as follows: “The maximum drive level shall be selected so that with a function increase of the drive level by 50%, the resistance does not increase reversibly by more than 10% nor the frequency changes by more than 0.5 ppm.” However, in general, it is not a good idea to exceed the stated maximum drive level in the data sheet of the crystal unit. A crystal will fracture when the

Figure 2.13  Resistance versus drive level current through a crystal resonator [3].

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drive level forces the quartz to exceed its elastic limit. Remember that it is a mechanical vibrating system.

2.11  Drive Level Dependence (DLD) or Drive Level Sensitivity (DLS) Drive level dependence (DLD) is the changing of the crystal resonator parameters (such as frequency and resistance, for example) with the power/voltage across or current through the crystal unit. The drive level can cause reversible or irreversible changes to the crystal. Most reversible DLD effects are due to excessive crystal drive level, but irreversible effects are usually due to production defects. Some of the production defects that cause DLD effects are: 1. Particles (loosely or permanently attached) on the crystal blank surface; 2. Mechanical damage of the quartz blank from excessively coarse lapping and/or scratches on the surface from mishandling; 3. Gas and/or oil contamination from poor vacuum during the electrode evaporation process.

2.12  Aging Any cumulative process that contributes to the deterioration of a crystal unit and that results in a gradual change on its operating characteristics (a drift in frequency especially) is considered aging. The aging behavior can be in a positive or negative direction. A reversal of the aging direction is even possible when a stronger aging mechanism takes over. For a crystal, the first year of aging will exhibit the highest drift in frequency as illustrated in Figure 2.14. The aging rate will accelerate at higher temperatures and with higher drive levels. Low aging oscillators are designed with low drive levels to the crystal to minimize the overall frequency drift of the oscillator. Some high-performance oscillators will even employ an automatic gain circuit (AGC) to keep the drive level constant and low. Temperature is another catalyst that increases the aging rate of a crystal. The crystal unit can be placed at an elevated temperature (i.e., +85°C) unpowered to accelerate the aging. Many aging parameters for crystals can only be met with accelerated aging by the manufacturer. These include baking of the unit before and after sealing for days depending on the requirement. There are many aging mechanisms in a crystal unit that are listed here but are not limited to:



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Figure 2.14  Typical frequency aging behaviors of crystal resonators. Aging can occur in a positive or negative direction. A reversal in direction is even possible [3].

1. 2. 3. 4. 5. 6. 7. 8. 9.

Absorption of moisture; Stress relief from the mounts; Excessive drive level; A fine or gross leak in the crystal unit holder; A mechanical shock which can permanently and abruptly change the frequency; Chemical reactions at the electrode-quartz interface; Outgassing of materials inside the holder; Contamination during the manufacturing process; Small irreversible alterations in the crystal lattice.

2.13  How Drive Level Affects Aging A crystal resonator is a low loss device (i.e. high Q); however, it is not lossless. When a driving electrical force is applied to the resonator, it vibrates in response to it. Due to the frictional losses, heat is generated. The higher the drive level the greater the heat. Intuitively, we know that heat is an aging catalyst, which is the case for a quartz crystal resonator. Hence, the higher the drive level is, the higher the frequency drift is. There are many mechanisms in which drive level affects the resonance frequency drift (aging), including:

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1. Partial vaporization of electrodes; 2. Acceleration of the mounting stress relief; 3. Changes in the background pressure within the enclosure. Because drive level affects the aging rate, a specification for aging will only be meaningful if defined under a certain drive level and temperature. Typical aging rates for a commodity AT-cut crystal unit are less than ±2 ppm per year. Aging rates of less than ±0.5 ppm are possible with some crystal unit designs.

2.14  Activity Dips Anomalies in the f-T and resistance-temperature (r-T) characteristics, as illustrated in Figure 2.15, are called perturbations or activity dips. It is caused when a crystal’s spurious mode (response) overlaps in frequency at certain temperatures the desired mode, called an interfering mode. Before the invention of crystal impedance meters (CI-meters), crystals were specified in terms of the oscillator amplitude under certain conditions, termed the activity. A crystal with an interfering mode would dip the amplitude of the oscillator because of the energy being lost in the main mode, hence the name activity dip. During an activity dip, the resistance of the crystal rises, its frequency lowers, and the motional inductance

Figure 2.15  The curves labeled fR and R1 are the f-T and resistance versus temperature (T) without a load capacitor. The fL and RL curves are the f-T and r-T of the same resonator with a series load capacitor. The load capacitor shifts the frequency to higher values. The curves have been vertically displaced for clarity [3].



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and capacitance deviate from their normal values [2]. Figure 2.15 shows the activity dips of a resonator with and without a load capacitance in series. An activity dip can cause intermittent failures. It affects both the frequency and the resistance of the crystal resonators. If the oscillator gain is insufficient, the resistance increase stops the oscillation. Activity dips occur over narrow temperature ranges, which can cause a strange performance in a system when the activity dip temperature is reached. Depending on the slew rate of the temperature, the oscillator may stop functioning and shift in output frequency and/or output power. Pass the activity dip temperature and everything goes back to normal. This intermittent behavior can be difficult to detect. Activity dips are strongly influenced by the crystal’s load capacitance and drive level. The load capacitance CL can influence the activity dip because the interfering mode usually has a large temperature coefficient and a C1 different than the desired mode. Changing the operating load capacitance value may make the activity dip vanish or reduce it so as not cause any problems. This load capacitance dependence dictates that the crystal be tested over temperature with a physical load capacitor to excite the activity dip(s) if they exist [5]. Activity dips can be caused by some but not limited to some of the following: 1. 2. 3. 4. 5. 6.

Particles introduced during base or final plating; Contaminants such as oil from vacuum pumps; Scratch(s) on the crystal blank; Moisture inside the holder; Poorly adhering plating; The physical geometry of the crystal unit design (especially in square or rectangular blanks), including the thickness of the electrodes.

2.15  Sleepy Crystals Phenomenon A continuing problem with the quartz crystal resonators is the increase in series resistance at a certain drive level, temperature, or an elapse in time. This type of problem is referred to by some authors as “sleeping sickness,” “sleepy crystal,” “low level drive sensitivity,” or “hard starting characteristics,” to name a few. I have chosen “sleepy crystal phenomenon.” It is a very frustrating problem with some crystals with regard to the inability of the oscillator to start up after a long time of inactivity. Turn the power off and then on again, and the oscillator may start. The oscillator will continue to restart. Let the equipment rest for at least a few days and the phenomenon may reoccur. In these sleepy crystals, their re-

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sistance increases after a long storage to such a high level that the oscillator gain margin is not sufficient to start up. To make matters worse, it strongly depends on temperature. The same problem that creates activity dips can be the same for a sleepy crystal. In fact, it could be one and the same problem at times but is described by different terms. Among the most often reported problems for a sleepy crystal are: 1. 2. 3. 4. 5. 6.

Surface scratches; Particles of metal on quartz or electrode surface; Flaking of quartz or electrode surface; Poorly adhered electrodes; Blisters on electrodes; Possibly any manufacturing issue that creates an asymmetry on the crystal unit.

2.16  Specifying Crystals Specifying a crystal correctly can be a daunting task for an engineer who is not familiar with them. Why is a device with such a simple equivalent circuit so difficult to specify at times? Some answers for the crystal requirements will also come directly from the system in which the crystal oscillator is to be used. For example, if it is a wireless system, then some regulatory body, like the Federal Communications Commission (FCC) in the United States, will require specific frequency accuracies to be met. The FCC might state that the wireless system be within ±25ppm for the lifetime of the product. With this number, parameters such as frequency-temperature, frequency tolerance, and the aging rate at a minimum can be derived. In some applications it may be simple to specify a crystal, for example, the crystal being used in a microprocessor, which at most times has no special requirements. The engineer just wants the oscillator to run and he or she may not care about the exact frequency. Even for such simple applications, the crystal can become a problem, not because of the frequency but because the crystal oscillator fails to start reliably in production or, at worst, for the customer. Today, there are hundreds of crystal packages for the designer to choose. Every designer wants the smallest package and the highest performance possible at the lowest cost. The reality is that the crystal design engineer cannot violate the laws of physics and there are many compromises in performance, size, and cost associated with every specific crystal unit design. A system designer should



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build in some margin in the specification but without overspecifying. Only knowledge of the crystal parameters will help you not to overspecify or underspecify it. In this chapter, we learn how to specify each of the most important parameters of the quartz crystal resonator. Learning how to write these parameters will aid in reading data sheets from crystal unit manufacturers. Subsequent chapters will teach design examples where most of these parameters will be derived. 2.16.1  Specifying the Nominal Frequency

Specifying the nominal frequency (the ideal frequency with zero uncertainty) of operation of the crystal unit is the leading thing to consider. It may seem trivial but mistakes get made when specifying this nominal frequency. For example, if not enough significant digits are given as demanded by the frequency tolerance, then the crystal frequency may not meet this requirement. Suppose one needs to specify a crystal’s nominal frequency of 20.123456 MHz with a tolerance of ±15 ppm. However, the nominal frequency was rounded off to 20.123 MHz when specifying its requirement. The difference between these two frequencies in ppm is

20,123,000 − 20,123,456 × 106 = −22.66 ppm 20,123,456

Hence, the requested frequency is outside the frequency tolerance limits of ±15 ppm. Always specify the nominal frequency to the last digit needed all the way to 1 Hz if required. There is no need to go below 1-Hz tolerance for most applications. There is no price penalty in specifying a crystal unit as 20.000000 MHz or one as 20.123456 MHz. The cost driver is the frequency tolerance requirement. To specify the crystal’s nominal frequency, state: “Frequency: 25.123456 MHz.” 2.16.2  Specifying the “Mode” of Operation

After specifying the nominal frequency of the crystal, the second most important parameter is the mode of operation. The mode of operation simply means whether the crystal is going to be operated at the fundamental, third, fifth, or the nth overtone mode response. Hence, the engineer specifying the crystal must determine or design the oscillator for which mode the crystal is to be operated.

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2.16.3  Specifying a Fundamental Crystal

Fundamental crystals can be made to about 60 MHz with a flat blank. Higherfrequency fundamentals (to hundreds of megahertz) are made as an inverted mesa types. A fundamental crystal can be specified as series or parallel-resonant. To specify a fundamental mode crystal, simply state: “Mode: Fundamental.” 2.16.4  Specifying an Overtone Crystal

Overtone crystals have higher Q than their fundamental counterparts. In low jitter and low phase noise applications, they are the ideal choice for CLOCK oscillator frequencies above 50 MHz with AT-cut crystals. An overtone crystal can be specified as series or parallel. To specify an overtone mode crystal, simply state: “Mode: Third Overtone” or “Mode: Nth Overtone.” 2.16.5  Specifying a Parallel or Series-Resonant Crystal and Load Capacitance

An important parameter to specify for the crystal is the load capacitance. The load capacitance value is what determines if the crystal is to be series or parallelresonant calibrated. Hence, it determines if the crystal is series or parallel. To specify a parallel-resonant crystal, state as an example: “Load Capacitance: 20 pF.” To specify a series-resonant crystal, state as an example: “Load Capacitance: Series.” 2.16.6  Specifying the Crystal’s Resistance

The fourth most important parameter to specify for a crystal unit is its maximum resistance. This maximum resistance value is very important as it will determine the gain margin of the oscillator. Recall that the resistance of seriesresonant crystal is the motional resistance R1 while for a parallel-resonant crystal is the equivalent series resistance (ESR). The correct way to specify the resistance of a series-resonant crystal is: “Motional Resistance: x ohms maximum.” The correct way to specify the resistance of a parallel-resonant crystal is: “Equivalent Series Resistance (ESR): x ohms maximum.” 2.16.7  Specifying an AT-Cut Crystal

The most common cut by far is the AT cut. It can be made with good f-T characteristics (i.e., ±15 ppm over −20°C to +70°C) without a price premium. The upper frequency limit of an AT-cut flat blank fundamental is about 60 MHz. However, for commodity pricing, it may have to be below about 40 MHz. To specify the AT cut, state the following: “Crystal Cut: AT cut.”



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2.16.8  Specifying a BT-Cut Crystal

The BT cut is usually specified for frequencies above 20 MHz. Its primary advantage over the AT cut is having a thicker blank for the same frequency. Hence, it can be made with higher yields at higher frequencies due to being less fragile. The penalty for using a BT cut is its worse f-T characteristics when compared with the AT cut. The BT cut is only offered with one f-T characteristics while the AT cut can be made in a whole family of f-T curves. To specify the BT cut, simply state: “Crystal Cut: BT cut.” 2.16.9  Specifying an SC-Cut Crystal

The SC cut belongs to a family of doubly rotated cuts. The SC cut is primarily used in ovenized applications. It is by far the most expensive cut covered in this book and also requires a more complex oscillator design due to its many modes. Because the SC is going to be used inside an oven, the upper turnover point of its f-T curve must also be specified. The oven temperature should match this turnover point to maintain excellent frequency stability. To specify the SC cut, state the following as an example: “Crystal Cut: SC-cut” and “Turnover Temperature: 92°C ± 3°.” 2.16.10  Specifying the Frequency Calibration (Tolerance)

The frequency tolerance is the maximum permissible amount of deviation from the nominal frequency at room temperature (+25ºC). Because the crystal manufacturers calibrate the frequency of the crystal unit to meet the tolerance specification, it may be stated as calibration, calibration tolerance, or simply tolerance in the data sheets. Hence, all these terms or words are used interchangeably. Thus, to specify the frequency tolerance/calibration requirement, state any of the following: “Frequency Tolerance: ±25 ppm maximum” “Frequency Calibration: ±25 ppm maximum” “Calibration Tolerance: ±25 ppm maximum” “Tolerance: ±25 ppm maximum” “Calibration: ±25 ppm maximum” 2.16.11  Specifying the Frequency-Temperature Stability

Frequency-temperature stability is the maximum permissible deviation of the crystal frequency to operate over the specified temperature range. “Frequency-

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temperature stability” is the correct term to use. However, frequency stability or frequency versus temperature stability or simply stability is often used. The frequency-temperature stability specification is one of the primary cost drivers for a crystal unit design. The frequency-temperature performance is primarily determined by the angle of cut of the crystal blanks from the quartz bar. The greater the precision of cutting required to meet your requirement (i.e., ±30 seconds versus ±2 minutes of arc angle), the higher the price. Therefore, do not overspecify it. Once a crystal unit is fabricated, its frequency versus temperature stability curve cannot be changed. Any selection or screening process will be limited to the statistical spread. However, screening over temperature is a costly proposition in high volume. To specify frequency-temperature stability, state the following: “Frequency-Temperature Stability: ±30 ppm maximum over −20°C to 70°C.” In some data sheets, the operating temperature range may appear as a separate line. For example, one may encounter the following when reading the data sheet of a certain crystal: “Stability: ±30 ppm maximum” and “Temperature Range: −20°C to 70°C.” 2.16.12  Specifying the Operating Temperature Range

The operating temperature range of the crystal is linked directly (in terms of price) with its frequency-temperature stability. Hence, it is very crucial to determine the crystal’s ambient temperature of operation and not to specify above that range. Its turns out that crystal resonators do not stop working if operated above the high stated temperature. The frequency-temperature stability just keeps getting worse in most cases. However, operating a crystal below its intended low temperature can be problematic if the internal dew point is reached. One can encounter activity dips or, even worse, no operation. To specify the temperature range, simply state: “Operating Temperature Range: −20°C to 70°C.” 2.16.13  Specifying the Shunt Capacitance, C0

The shunt capacitance is the capacitance formed by the quartz substrate being sandwiched between the two plates (the electrodes) plus any holder capacitance. The shunt capacitance is directly linked to the value of the motional capacitance. Some even consider a crystal unit as being a capacitor with resonances at certain frequencies. The modern crystal units discussed in this book all have a maximum shunt capacitance of 7 pF. In later chapters we will learn how the shunt capacitance value affects the oscillator gain margin. Simply specify it as: “Shunt Capacitance = 7 pF maximum.”



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2.16.14  Specifying the Aging Rate

Aging is the systematic change in frequency with time due to internal changes of the crystal unit. These internal changes can be caused and/or accelerated by external factors such as drive level, temperature, and mechanical shock. To be meaningful, aging should be specified as a rate at a certain temperature and drive level. However, it is scarcely written as such. The crystal unit will have a higher aging rate in its first year compared to years thereafter. Hence, it is very common to see the aging rate of a crystal unit written as: “Aging: ±3 ppm maximum first year, ±1 ppm maximum per year thereafter.” This specification is typically written (and will suffice) as such for a commodity crystal. If the aging rate is for an SC-cut crystal, then a more detailed specification should be stated. For example, for an SC-cut crystal, state the aging rate as follows: “Aging: ±0.5 ppb/day maximum after 30 days of operation at +92°C and at a drive level of 100 µW.” Note that this specification has a 30-day power-on period of time before the crystal needs to meet the required aging rate. The main reason is the above SC cut is being used in an oven and that during a period of rest (unpowered), the crystal aging rate retraces back to a higher rate. 2.16.15  Specifying an Overall Accuracy

Instead of specifying the frequency-temperature stability, calibration/tolerance, and aging rate separately, one can choose to specify an overall accuracy inclusive of all three parameters. To specify an overall accuracy, state the following as an example: “Overall Accuracy: ±50 ppm maximum inclusive of frequency-temperature stability, calibration/tolerance and aging for 10 years.” 2.16.16  Specifying the Trim Sensitivity

The trim sensitivity is the slope of pulling curve at every load capacitance as we learned in Chapter 1. The trim sensitivity should not be used to specify the pulling characteristics of a crystal unit but instead to stay within a certain frequency tolerance based on the oscillator component tolerances, for example, if a certain crystal design has a trim sensitivity of −20 ppm/pF at 20 pF. The crystal is placed in an oscillator circuit that has a load capacitance of 18 pF. The combination of the two (the crystal and the oscillating circuit) can be −40 ppm from the nominal frequency right from the start just at room temperature. Hence, the trim sensitivity is useful in calculating the frequency versus the load capacitance tolerance. Again, it is only accurate to within a few pico-farads of the nominal load capacitance. It is not a linear slope.

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To specify the trim sensitivity, state the following: “Trim Sensitivity: −30 ppm/pF typical” or “Trim Sensitivity: −30 ppm/pF maximum −10 ppm/pF minimum.” 2.16.17  Specifying a Pullable Crystal (“Pullability”)

When the crystal unit needs to be pulled or deviated in frequency as in a VCXO application, the crystal must be designed to meet a certain pulling specification. These crystals are termed pullable crystals. Note that all crystals are pullable (especially the fundamental mode), but pullable crystals are designed with large motional capacitance values. Because of large motional capacitances needed, it may not be possible to fit the crystal unit design in certain small packages. To determine how much a crystal can be pulled in frequency, there are only three parameters involved: the motional capacitance C1, the shunt capacitance C0, and the load capacitance CL. These parameters can be specified directly if known or indirectly by specifying the frequency change over load capacitance range. However, to specify pulling knowing the parameters, one must keep in mind that C0 and C1 are linked. To maximize the pull, C1 should be maximized while C0 is minimized. For this reason, one can specify them together as a ratio of C0/C1 maximum. The following are several examples of how to specify pulling. To specify pulling when C0 and C1 are known, use: “Shunt Capacitance C0: 4 pF maximum” and “Motional Capacitance C1: 15 fF minimum.” Alternatively, the following is equivalent but can give the crystal manufacturer more freedom: “C0/C1: 267 max.” To specify pulling without knowing or needing to know C1, probably the easiest way to specify a pullable crystal is in terms of the pulling in parts per million versus the load capacitance. This method leaves the calculation of the motional parameters to the crystal manufacturer, for example, “Pullability: −80 ppm minimum when CL goes from 14 pF to 27 pF and + 80 ppm minimum when CL from 14 pF to 8 pF.” The trim sensitivity can also be used to specify the pullability (which is my least preferred method) of crystal as in the following example: “Pullability: ±15 ppm/pF minimum, ±30 ppm/pF maximum” and “Load Capacitance: 10 pF.” 2.16.18  Specifying the Drive Level

There are actually two drive levels that should be specified. The first is the drive that is to be used in measuring the crystal’s parameters. This drive level is called “correlation drive level” for which the crystal industry usually uses 100 µW. The second and most important drive level is the maximum drive level of operation. The de facto unit for specifying the drive level is watts (microwatts or



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milliwatts). The drive levels are specified as in the following example: “Correlation Drive Level: 100 µW” and “Maximum Drive Level: 500 µW maximum.” 2.16.19  Specifying Drive Level Dependence (DLD)

Drive level dependence (DLD) is the changing of the crystal resonator parameters (such as frequency and resistance for example) with the power/voltage across or current through the crystal unit. The drive level can cause reversible or irreversible changes to the crystal. DLD is performed with a drive level power sweep test. The test is performed from a very low drive level (i.e., 50 nW) to as high as the maximum drive level in discrete steps. To specify a DLD requirement, you must specify a minimum and maximum power level or current as well as the number of points. The number of points is needed because the machine making the measurement will step the power from minimum to maximum power, for example, “Direct Level Dependence (DLD): 1 µW to 500 µW, 20 points minimum.” This statement is not sufficient on its own. You must specify the allowed limits on the ESR and sometimes frequency, as the drive is varied in this range. A typical limit used on the ESR is that it shall not change by more than 30% with the change in drive level. This means 30% from the nominal value, not 30% from the maximum ESR specification limit. Therefore, add a note that also states: “All ESR readings must be within 30% of each other in addition to not a single reading exceed the maximum ESR specification.” 2.16.20  Specifying Spurious Responses

In Chapter 1 we learned that all crystals have spurious responses. The spur amplitude is most commonly specified in two ways: 1. Level of the spurious in decibels below your desired response (the fundamental or an overtone); 2. A minimum ratio of the spur resistance to the desired response resistance. The second way is easy to understand. Your desired response resistance is first measured. Next, the spurs are located and their resistance is also measured. The strongest spur found is then compared in a ratio to your desire response. For example, your spurious specification may be listed as: “Spurious: 2:1 minimum.” For example, if your desired response resistance is measured and found to be 22 ohms, then the strongest spur must be at least 44 ohms or greater to meet your specification requirement.

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The first way to specify the spurious level (which is sometimes called “dB down”) needs a more extensive explanation. To understand how this measurement is performed and calculated, we first need to understand the PI fixture that is used to measure crystals in most modern test equipment. A crystal under test is placed in the PI test fixture whose schematic and values are shown in Figure 2.16. This fixture is designed such that each leg of crystal “sees” 12.5 ohms for a total of 25 ohms. The input and output of the test fixture is designed for 50 ohms to match the instrument making the measurement. Under these conditions, the instrument will calculate the spur level in dB down from your desired response with the following formula:



R + 25  Spur (dB) = 20 log  desired   R spur + 25 

(2.6)

where Rdesired is the crystal resonant resistance of desired response and Rspur is the crystal spurious resistance. Specify spurious or spurs in the following manner: “Spurious: 6 dB down minimum” or “Spurious: -6 dB minimum.” 2.16.21  Specifying the Quality Factor Q

One of the main characteristics of the quartz resonators is the extremely highquality factor Q of the motional arm. Typical values range from 20,000 to several hundred thousand. SC cut can be designed with Q values of several million.

Figure 2.16  The crystal industry has standardized testing crystals with this specific PI fixture.



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The Q of the crystal will dominate the close-in phase of an oscillator, and in general, the higher the loaded Q of the oscillator is, the lower the frequency instability is. Q is a unitless quantity and can simply be specified as: “Quality Factor Q: 50,000 minimum.” 2.16.22  Specifying the Motional Inductance, L1

The motional inductance L1 is rarely specified when C1 is specified. If you know one, you know the other, knowing fs and using the resonance equation of the motional arm. 2.16.23  Specifying Inverted Mesa Crystals

An inverted mesa crystal is a unit that has at least one side of the quartz blank edged to create a thinner region in the middle while keeping the perimeter thicker. The thickness of the perimeter is usually that of a 20-MHz crystal blank. The mesa region thickness is that of the actual nominal frequency needed. It is a clever way of making high-frequency fundamental (HFF) crystals. Figure 2.17 shows an illustration of an inverted mesa crystal. With this inverted mesa technique, a fundamental AT-cut crystal can today be made as high as 650 MHz. Higher frequencies are possible, but the yields are very low. Most of today’s inverted mesa crystal market is from 75 to 350 MHz in the fundamental mode. Inverted mesa crystals are also made in overtone versions for very high frequency CLOCKs. Overtone inverted mesa crystals can be made from 150 to 800 MHz. The aging rate for inverted mesa crystals are not as good as for “flat

Figure 2.17  Illustration of inverted mesa HFF crystal unit.

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blank” AT-cut crystals. The typical aging rate is ±1 ppm for the flat blank versus ±2 to ±3 ppm per year for the inverted mesa crystal. To specify an inverted mesa crystal, state the following: “Mode: Inverted Mesa Fundamental” or “Mode: Inverted Mesa Third Overtone.” Inverted mesa crystals can be specified as series-resonant or parallel resonant. Inverted mesa crystals are only available as AT cuts and not BT cuts or SC cuts. The inverted mesa crystal is many times more expensive than its flat blank counterpart. The heavy price increase is due to the complex nature in edging the mesa region correctly. Swept quartz is also usually used in manufacturing the inverted mesa crystal unit, which contributes to the higher price. 2.16.24  Specifying a Tuning Fork Crystal

A tuning fork crystal gets its name because the crystal blank is designed like a tuning fork as illustrated in Figure 2.18. This is a very clever structure that allows the making of miniature crystals at 32.768 kHz. It was an important evolution in crystal design, which allowed the boom in electronic watches. An AT-cut crystal if made at 32.768 kHz would not fit in a modern wristwatch. The frequency of 32.768 kHz for a tuning fork crystal was chosen because dividing it by 2, 15 times, results in 1 Hz (215 = 32,768). This design type makes it possible to fabricate crystals in the kilohertz range in tiny sizes [3].

Figure 2.18  Tuning fork (U-shaped) crystal design and its “flexural” mode of vibration.



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There is big advantage in current consumption in designing an oscillator with tuning fork-type crystals in kilohertz versus an AT-cut crystal in the megahertz region. The difference is like tenths of micro-amps versus milli-amps. For this reason, battery power application would use a tuning fork crystal oscillator at 32.768 KHz and then internally multiply it to a higher frequency. The tuning fork crystal frequency-temperature-characteristics are that of a parabola with a temperature coefficient of −0.035ppm/°C2 and a turnover temperature about 25°C as shown in Figure 2.19, the only temperature curve available for tuning fork type crystals. The designer cannot specify a frequency-temperature characteristic that is different from the parabola. There is no other choice. It is like the color choices in which the Ford model T was first offered; it comes in every color as long as it is black. Besides the 32.768-kHz nominal frequency, tuning fork type crystals can be made in the range of 30 to 500 kHz. To specify the frequency-temperature characteristic, one can only state the following: “Frequency-Temperature Stability: −0.035ppm/°C2 typical” and “Turnover Temperature: +25°C ± 5°C” or “Temperature Coefficient: −0.035ppm/°C2 typical” and “Turnover Temperature: +25°C ± 5°C.” A tolerance can also be added to the temperature coefficient instead of stating “typical.” If a tolerance needs to be added, then state the following as an example: “Temperature Coefficient: −0.035 ± 0.006 ppm/°C2.”

Figure 2.19  Tuning fork frequency-temperature characteristics.

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Understanding Quartz Crystals and Oscillators

To specify the load capacitance of a tuning fork crystal, tuning fork crystals have many holder options but only a few capacitive load choices such as 6 pF, 8 pF, and 12.5 pF. The reason for this is that tuning fork crystals are made in tremendously high volumes in the 32.768-kHz frequency and offering more choices is not cost effective. However, this limited choice should not be problem for the oscillator designer. Simply design the oscillator to present one of these load capacitance options. To specify the load capacitance value of a tuning fork crystal at 32.768 kHz, choose one of the three de facto options of load capacitance: 6 pF, 8 pF, or 12.5 pF. To specify the tuning fork ESR, unlike the resistance of an AT-cut fundamental-mode crystal, the ESR of a tuning fork is much higher. A typical maximum ESR specification value for a tuning fork crystal unit can be 40 kiloohms. These very high ESR values for the tuning fork mean that the oscillator circuitry will, in effect, be a high impedance circuit. This results in this oscillators being very susceptible to moisture. In fact, for reliable operation, I suggest conformal coating of the tuning fork oscillator section. To specify the resistance of a tuning fork crystal state the following: “ESR: 40 kilo-ohms maximum.” To specify the drive level of a tuning fork crystal, recall that the drive level in power for a crystal is P = I 2R, where the value of R is the ESR of the tuning fork, which can be 40 kilo-ohms. Consequently, it takes a very small current through the crystal to exceed the mechanical elastic limits and fracture the quartz. The large ESR values forces a maximum drive level of 1 µW for commercially available tuning fork crystal units at 32.768 kHz. To specify the drive level of a tuning fork crystal unit at 32.768 kHz, one can only state the following value: “Drive Level: 1 µW maximum.” 2.16.25  Specifying Strip Crystals

A strip crystal is a crystal unit that uses a rectangular or square blank inside the holder. Almost all the high-volume, low-end crystals in the world are strip crystals. Quartz bars are grown in specific shapes to take advantage of the strip blank geometry to minimize material loss. A strip crystal, if not properly designed as to its length to width ratio, can create many problems such as activity dips/perturbations and spurious. These problems exuberate for very small crystal unit designs, especially at high fundamental mode frequencies (30 MHz and higher). In a specification, the term “strip crystal” is not used; instead, it is a consequence of the holder needed.



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71

2.16.26  Specifying Mechanical Shock Resistance

Crystal units are fragile and care must be taken as not to drop them by themselves or when part of any equipment. They can break easily, especially highfrequency fundamentals. A typical shock specification for commodity crystals is 1,500g. If in its environment, the crystal will be experiencing shock forces, and then add your shock requirement as for example: “Mechanical Shock Resistance: 1,500-g minimum” and “Condition: 0.5 mS, half sine wave, 3 axes.”

2.17  Crystal Unit Handling Precautions The major precaution that author offers is that quartz crystal resonators are mechanically very fragile. A crystal has a good chance of breaking by just dropping it from your hand without throwing. Crystals should not be placed onto boards with “Chip-Shooter” type pick-and-place machines. I have first-hand knowledge on how these types of machines break quartz-based components during picking and placing. The crystal should be placed slowly and should not be slammed down. For through-hole type crystals, the bending/forming of the leads to make the unit surface mountable should be left to the manufacturer. Bending these leads can compromise the hermetic package, causing an accelerated frequency aging. 2.17.1  High-Temperature Storage Precautions

If you need to bake the crystal unit or any equipment containing a crystal (with the power on or off ), the bake temperature should not be near the original crystal blank attachment curing temperature. For example, if the crystal unit manufacturer cured the internal epoxy/cement holding the crystal blank to the package at 150°C, then this temperature should be avoided for any extended amount of time. How much time? Hours are too much time. A solder reflow at 260°C for 10 seconds maximum is no problem. The problem is that reaching the epoxy cure temperature can induce outgassing, which will result in accelerated aging. A safe temperature that I recommend is +105°C or lower to bake any crystal unit for an extended period of time. 2.17.2  Electrostatic Discharge (ESD) Precautions

Crystals are not ESD-sensitive and hence no special precautions need to be taken regarding ESD. Now, its construction is like a physical capacitor, which will break down at some extreme voltage. However, it is not ESD-sensitive like a semiconductor device.

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2.18  Crystal Specification Template Table 2.4 is a crystal specification template as a starting point. Add or delete to the template as needed for specific situations. Some MIL standards have been added to complement the requirements listed earlier.

Table 2.4 Crystal Specification Template Nominal frequency: Mode (fundamental, third, fifth): Type of cut (AT, BT, or SC): Load capacitance (“Series” or x pF): Frequency stability: ±x ppm maximum Temperature coefficient (if tuning fork): Operating temperature range: Frequency/calibration tolerance: ±x ppm maximum Motional resistance R1 (if “series” crystal): ESR (if “parallel” crystal): Shunt capacitance C0: x pF maximum Frequency aging: ±x ppm maximum first year, ±x ppm maximum per year thereafter Holder type: Correlation drive level: x µW Maximum drive level: x µW maximum Motional capacitance C1: x fF minimum C0/C1 ratio: x maximum Pullability: Trim sensitivity: Insulation resistance: 500 mega-ohms minimum at 100 volts Mechanical: Shock: MIL-STD-883, Method 2002, Condition B Solderability: MIL-STD-883, Method 2003 Vibration: MIL-STD-883 Method 2007, Condition A Solvent resistance: MIL-STD-202, Method 215 Resistance to soldering heat: MIL-STD-202, Method 210, Condition B Environmental: Gross leak test: MIL-STD-883, Method 1014, Condition C Fine leak test: MIL-STD-883, Method 1014, Condition A Thermal shock: MIL-STD-883, Method 1011, Condition A Moisture resistance: MIL-STD-883, Method 1004



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73

References [1] Parzen, B., with A. Ballato, Design of Crystals and Other Harmonic Oscillators, New York: John Wiley & Sons, 1983. [2] Bottom, V. E., Introduction to Quartz Crystal Unit Design, New York: Van Nostrand Reinhold Company, 1982. [3] Vig, J. R., “Quartz Crystal Resonators and Oscillators, Frequency Control and Timing Applications—A Tutorial,” www.ieee-uffc.org, January 2007. [4] IEC 444-6, “Part 6: Measurement of Drive Level Dependence (DLD),” International Electrotechnical Commission, www.iec.ch, 1995. [5] Rose, D., “Load Resonant Measurements of Quartz Crystals,” Application note, Saunders and Associates, 2004. [6] Ballato, A., “Effect of Second Rotation on Frequency-Temperature Characteristics of ATCut Crystals,” U.S. Army Electronics Command, 1976. [7] Toyocom Corp., “Quartz Crystal Units,” Application note, 2005. [8] Shockley, W., D. R. Curran, and D. J. Koneval, “Trapped Energy Modes in Quartz Crystal Filters,” J. Acoustical Soc. Amer., Vol. 41, 1967, pp. 981–993. [9] Guttwein, G. K., T. J. Lukaszek, and A. D. Ballato, “Practical Consequences of Modal Parameter Control in Crystal Resonators,” Proc. 21st Ann. Symp. on Frequency Control, 1967, pp. 115–137. [10] Meeker, T. R., “Theory and Properties of Piezoelectric Resonators and Waves,” in E. A. Gerber and A. Ballato, (eds.), Precision Frequency Control, Vol. 1, New York: Academic Press, 1985, pp. 48–118. [11] Brendel, R., et al., “Investigations in Low Drive Level Sensitivity of Quartz Resonators Affecting Their Motional Parameters,” 20th European Frequency and Time Forum, March 2006. [12] MIL-STD-202, “Test Methods for Electronic and Electrical Component Parts,” Defense Logistics Agency, www.dscc.dla.mil. [13] MIL-STD-883, “Test Method Standard Microcircuits,” Defense Logistics Agency, www. dscc.dla.mil. [14] EIA-512, “Standard Methods for Measurement of the Equivalent Electrical Parameters of Quartz Crystal Units, 1 kHz to 1GHz,” Electronic Industries Alliance, www.eia.org. [15] MIL-PRF-3098J, “Performance Specification Crystal Units, Quartz General Specification For,” Defense Logistics Agency, www.dscc.dla.mil. [16] IEC-444-1, “Measurement of Quartz Crystal Unit by Parameters by Zero Phase Technique in π-Network. Part 1: Basic Method for the Measurement of Resonance Frequency and Resonance Resistance of Quartz Units by Zero Phase Techniques in π-Networks,” International Electrotechnical Commission, www.iec.ch.

3 Advanced Quartz Crystal Resonator Topics 3.1  Introduction Quartz crystal resonators can exhibit many irregularities that can be quite complex to quantify. For example, quartz resonators motional parameters can behave nonlinearly with drive level, environmental conditions, and aging. Some of the key parameters that this chapter will cover are resonator noise and Q as a function of drive level. The quartz resonator noise and Q will directly impact the short-term stability of the crystal oscillator. It is not always possible to distinguish and separate the oscillator noise from the resonator noise. Even though the quartz resonator is a passive component, it does create flicker (1/f  ) noise and random walk (1/f  2) that are functions of the drive level [1].

3.2  Flicker Noise There are two primary sources of noise in crystal oscillators. There is white noise that is uniform and broadband and a low-frequency 1/f  noise called flicker noise whose amplitude varies inversely with frequency. Flicker noise is not well understood, but it exists in all active components and some passive devices such as crystal resonators. Flicker like fluctuations are also widely found in nature. It has been observed in biology, physics, and even music [2].

75

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Understanding Quartz Crystals and Oscillators

Flicker noise appears on the output signal as modulation sidebands, both in phase and amplitude. Flicker noise is dominant at frequencies primarily below the half-bandwidth of the resonator. The resonator Q determines this halfbandwidth and thus the rate of change of phase, meaning that the higher the resonator Q is, the higher the noise rate decay is. This emphasizes maintaining a high loaded Q oscillator design to reduce noise. The fractional frequency 1/f noise in crystal oscillators is generated by energy dissipated in the resonator. Current flow losses through the resonator generate frequency fluctuations that are changed into 1/f phase fluctuations due to the transportation delay [3]. Flicker noise can be successfully reduced by using negative feedback and/ or unbypassed emitter resistors in the sustaining and buffer stages. A selection of low flicker noise components (even within the same component value) is another option. For example, changing out the transistor or crystal until the desired phase noise performance is achieved, if at all possible. This is because 1/f flicker noise can vary orders of magnitudes between identical components. Flicker noise perturbs the phase of the oscillator feedback path, which forces the resonator to compensate with a frequency shift. Thus, phase fluctuations are converted to frequency fluctuations. This action converts the flicker noise into an 1/f  3 effect [3]. In general, any noise that perturbs the phase within the loop will be converted to frequency fluctuations.

3.3  Introduction to Fluctuation Equations In a RLC circuit, the phase shift is given by  imZ  ϕ = arctan   reZ 



(3.1)

where imZ and reZ are the imaginary and real parts of the impedance Z. It can be shown that [4]



∆ϕ =

1  imZ  1+   reZ 

2

⋅

∆imZ 2 ( f reZ

)

(3.2)

where ∆imZ 2(f  ) is the mean square fluctuation of the reactance Z at frequency f. The power spectral density (PSD) of the phase fluctuations can now be defined as [4]



Advanced Quartz Crystal Resonator Topics



Sϕ

77

2 ∆ϕ ( f )) ( (f ) =

(3.3)

BW

where BW is a normalizing bandwidth. Inserting (3.1) and (3.2) into (3.3) results in

(

sin 2 ( 2 ϕ) ∆imZ ( f Sϕ ( f ) = ⋅ 4 imZ 2

))

2

⋅

1 BW

(3.4)

If the phase fluctuations ϕ are small, then (3.4) can be approximated by

Sϕ ( f ) ≅ ϕ

2

(⋅ ∆imZ ( f ))2 ⋅ imZ

2

1 BW

(3.5)

When the RLC circuit is at resonance, (3.2) also reduces to ∆ϕ =



∆imZ ( f reZ

)

(3.6)

At resonance, the imaginary part of the impedance equation cancel, that is, imZ = 0, and using the fact that Q = 1/wCR = wL/R near resonance, the phase fluctuations (PM noise) can be approximated by [4]



(

 ∆C ( f Sϕ ( f ) ≈ Q ⋅   C2  2

))

2

2 ∆L ( f ))  ( ⋅ + 2

L

1  BW 

(3.7)

where ∆C 2(f ) and ∆L2(f ) are the mean square fluctuations in the capacitor C and inductor L, respectively. It was assumed that the capacitor and inductor fluctuations are independent of each other.

3.4  Quartz Resonator Flicker Noise Model A noise model of the quartz resonator has been developed [5] by assuming that all of the motional parameters are flicker noisy. Let the double-sided power spectral density flicker noise of each motional component equal to:

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Understanding Quartz Crystals and Oscillators



S R ( f ) = βR f

−1



S L ( f ) = βL f

−1



SC ( f ) = βC f

−1



SC 0 ( f ) = βC 0 f



(3.8)



(3.9)



(3.10)

−1



(3.11)

where SR(f ), SL(f ), SC(f ), and SC 0 (f )are the two-sided PSD of the motional resistance, motional inductance, motional capacitance, and shunt capacitance, respectively, with β being the proper flicker coefficient and f being the Fourier frequency. The motional resistance has an additional thermal additive noise, which has the one-sided PSD of STR =



2kT Pr

(3.12)

where k is the Boltzmann constant, T is the temperature in Kelvin, and Pr is the power dissipated in the motional resistance. Next we define the resonator x-factor as 2pfSR1C0 and the reduced fractional frequency n(f ) = Qu(fos – fS – f )/fS where Qu is the unloaded Q of the resonator and fs is the series resonance frequency. To convert the fluctuations of the individual motional components to the phase fluctuations of the entire resonator, the following is used [6] K RB =



1 − 4 υx (1 − 2 υx )2 + x 2  1 + 2 υ (1 − 2 υx ) − x  2  

2x (1 − 2 υx ) + 2 υ (1 − 2 υx ) − x  (1 − 2 υx ) − x 2    = Qu (3.14) 2 2 (1 − 2 υx )2 − x 2  1 + 2 υ (1 − 2 υx ) − x    2



K XB

(3.13)



Advanced Quartz Crystal Resonator Topics









K 0B = x 1 + 2 υ (1 − 2 υx ) − x 

K R ϕ = −K XB

Qu

K X ϕ = Qu

2

79

2 υ (1 − 2 υx ) − x 2

(1 − 2 υx )2 + x 2   

(1 − 2 υx )2 + x 2 2 1 + 2 υ (1 − 2 υx ) − x  1 − 4 υx

1 + 2 υ (1 − 2 υx ) − x 

K 0ϕ = −

2

(3.15)



(3.16)



(3.17)

x

(1 − 2 υx )2 + x 2

(3.18)

where KRB, KXB = KCB = KLB , and K0B are the conversion coefficients of the amplitude fluctuations. Similarly, KRϕ, KXϕ = KCϕ = KLϕ, and K0ϕ are the conversion coefficients of the phase fluctuations. Finally, the double-sided phase PSD of the resonator is given by [6],

S ϕ ( f ) ≅  βR K R2ϕ ( f ) − 2 β X K X2 ϕ ( f ) f  

−1



(3.19)

The noise contribution of C0 has been ignored as it is practically negligible [6]. Note that the thermal noise of the motional resistance has also been ignored. The double-sided amplitude PSD is calculated with

2 2 SB ( f ) ≅  βR K RZ ( f ) − 2 β X K XZ ( f ) − βC 0K 02Z ( f ) f

−1

(3.20)

We can convert (3.19) and (3.20) to be single-sided with

S ( f ) + S ( − f ) S (f ) =  2

(3.21)

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Understanding Quartz Crystals and Oscillators

In conclusion, we can then plot the logarithmic measure of S(f ) with

L [ dBc Hz ] = 10 log S ( f )

(3.22)

3.5  Quartz Resonator Drive Level Sensitivity In Chapter 2 we defined drive level dependency (DLD) or drive level sensitivity (DLS). In short, the crystal resonator motional parameters and/or noise changes with drive level. An important paper [7] has given great insight into the cause mechanism responsible for DLD/DLS. It is well known in the crystal industry that particles on the surface of the crystal are one of the root causes of DLD/DLS. How even a single miniscule particle can do this is elegantly explained in [7]. Following that line of thought, let us envision a particle on the surface of quartz crystal unit as depicted in Figure 3.1. A tightly bound particle would act as mass loading and force a negative frequency shift which can be confirmed by measurements. However, the particle can be bounded to the surface by an elastic force. The force can be from a thin sticky coating of oil or resin. Other attractive forces are possible such as electrostatic and capillary as examples. With any of the above attractive forces, the surface moves back and forth under shear motion. The effect is that the particle behaves as a very small oscillating system (Figure 3.2) as the surface beneath it moves back and forth. A mechanical model can now be arranged to analyze the effect of the bounded particle as illustrated in Figure 3.3. The small spring-mass on the right represents the particle while the larger mass is the resonator under shear motion. The shear motion of the resonator acts as the external driving force F on the system. Applying Newton’s equation of motion for the two masses takes the form of a differential system. That is [7],

Mu + Ku + k (u − x ) = F0e j wt

Figure 3.1  Bounded particle on crystal surface held by an elastic force.

(3.23)



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81

Figure 3.2  Bounded particle behaves like a very small oscillating system.

Figure 3.3  Mechanical model of bounded particle on crystal surface as a coupled system.



mx + k (x − u ) = 0

(3.24)

where k and K are spring constants and M and m are the mass of the quartz and particle respectively. Also, F0ejwt signifies sinusoidal motion. The resonant frequency of the coupled system can be found by searching for a harmonic solution. However, this is beyond the scope of this book. The solution can be expressed as a function of the resonant frequencies of the isolated spring-mass system and of the stiffness ratio (k/K ) of the springs. The conclusion found in [7] is that weak particle binding raises the resonant frequency, and for strong coupling the frequency is decreased. The above is only one scenario for particle contamination/attachment. Particles can also get trapped in surface pits which need to be analyzed with a different mechanical model. Such model and others are covered in [7]. A different approach to resonator surface contamination was taken by Young and Vig [8]. In this paper, they derived an equation relating the spectral density of frequency fluctuations relating the rates of adsorption and desorption of contaminating particles. In this study, one conclusion was that the phase noise is not a simple function of the percent area contaminated. They also concluded that spectral density of frequency fluctuations is inversely proportional to the fourth power of the resonator thickness.

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3.6  Resonator Q and 1/f Noise Versus Drive Level Before discussing Q and 1/f versus drive level, there does exist a relationship between resonator Q and its 1/f noise. In other words, flicker noise fluctuations are dependent on resonator Q. Most available data points to a 1/Q 4 dependence, but some of the best resonators show a range between 1/Q 2 and 1/Q 5 [9]. 3.6.1  1/f Flicker Noise Versus Drive Level

A crystal resonator is a mechanical vibrating system in which the vibration amplitude is directly proportional to motional arm current. It is no surprise therefore that the 1/f fluctuations would be a function of the resonator drive level. One can imagine driving the resonator so hard that it becomes chaotic and generate excess noise. The problem with studying the resonator for flicker noise in the oscillator is that one cannot easily separate the noise contribution from the circuitry. A technique that tests the quartz resonator independently of the oscillator was used by Gagnepain [10] to quantify the 1/f noise versus drive level. Briefly, the test setup uses two identical crystals driving a double balance mixer in quadrature as shown in Figure 3.4. The frequency of the high purity driving source is set to the series frequency of the crystals. This test setup measures the phase fluctuations induced by the two resonators’ resonance frequency fluctuations. This test setup requires the two resonators to be matched closely in frequency and Q. Gagnepain’s results are summarized here [10]: 1. At low power levels the data clearly showed a 1/f noise dependence on the resonator’s unloaded Q following a 1/Q 4 law. 2. At medium power levels nonlinear effects noticed were due mainly to the higher-order elastic constants. The induced phase noise for the

Figure 3.4  Passive quartz crystal noise measuring bridge setup.



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83

lower Fourier frequency components were increased by the crystal nonlinearities. 3. At high power levels, the quartz resonators exhibited large instabilities and chaotic behavior. In addition at high power level, thermal effects added to the instability of the resonator. These high power levels also generated very large white noise.

3.6.2  Resonator Q Versus Drive Level

The quality factor Q provides a measure of the energy losses including mounting losses and imperfections in the quartz, surface stresses, and any additional things that serves to dissipate energy in the crystal [11]. Because the series resistance can change with drive level, it follows directly that the Q will also change with drive level. A crystal with DLD/DLS issues can show a series resistance starting a low value at low drive levels increasing to a maximum and then back down as the drive level is further increased. Therefore, any conditions including drive level that affects the losses in the resonator will directly impact the resonator Q. 3.6.3  Resonator Q Versus Manufacturing Defects

Empirical data in my workplace has shown that surface imperfections and defects in the quartz itself result in a lowering of the resonator Q. What follows is a short story to emphasize this. We have been manufacturing a certain crystal oscillator for many years. One day, most of the oscillators were no longer meeting the phase noise requirement. Right away, we suspected the Q of the crystal but the Q data was the same as before. The next step was to look into the oscillator circuitry to figure out any changes with the components. This was a quick and simple determination by getting a previous batch of oscillators and simply switching the crystal with the new batch. To our surprise, the problem persisted with the new batch of crystals. Therefore, it was not the circuitry. Now we were confused. The Q on the new and old batch of crystals measured the same on the crystal tester: the same crystal design and the same electrode size, yet something about the new batch of crystals the oscillator did not like. After further investigation, it turns out that the new batch of crystal was not polished to the same extent as the old batch of crystals. Manufacturing had used a coarser grid to polish the blanks simply because they had run out of the finer abrasive. The big question that one might ask is: Why did the measured Q value remain essentially the same? The crystal engineer stated that he measured the Q of the crystal with the coarser abrasive and the Q remain the same and therefore

84

Understanding Quartz Crystals and Oscillators

continued building the entire lot. The answer is twofold. First, the test equipment that measures the crystal Q does so from a calculation of the motional parameters at a certain drive level (typically 100 µW). The two problems with this are that it is simply a mathematical calculation from the motional parameters derived from a network analyzer test system and not from a working oscillator. Second, in the actual oscillator crystal, the drive level was different (about 300 µW). In conclusion, the coarser surface of the crystal unit resulted in worse output phase noise of the production oscillator. The more disordered surface contributed to surface stresses and energy dissipation resulting in additional noise. It has been determined that the following issues can affect the resistance and Q of quartz crystal units [11]: • Quartz material defects and/or impurities; • Crystalline orientation of surface; • Quartz plate diameter; • Final plate thickness and frequency; • Initial surface roughness; • Surface cleanliness; • Etchant composition; • Etch batch temperature; • Etch bath agitation; • Etching time; • Electrode design; • Mounting method including type of epoxy; • Ambient temperature.

3.7  The Effect of Acceleration on Quartz Resonators A crystal resonator subjected to a steady acceleration will have a different frequency than when at rest (zero acceleration). The frequency shift will be proportional to the magnitude of the acceleration. This acceleration sensitivity varies in value depending upon the direction acting on the physical resonator. Because of this, the frequency shift during acceleration is written as a function of the scalar product of two vectors [12]. Let f (a ) be the resonant frequency of the resonator experiencing an acceleration of a and Γ the acceleration-sensitivity vector (also known as g-sensitivity vector). The frequency of the resonator under acceleration can then be written as



Advanced Quartz Crystal Resonator Topics



(

)

f (a ) = f 0 1 + Γ · a

85

(3.25)

where f0 is the frequency of the resonator at rest. In other words, Γ is the fractional frequency sensitivity of the resonator, a is the applied acceleration vector magnitude, and Γa is the vector product of the two. Equation (3.25) tells us that the frequency of a resonator under acceleration is maximum when the acceleration is parallel to the acceleration-sensitivity vector and minimum when the acceleration is antiparallel to acceleration-sensitivity vector. Another important result of (3.25) is that the frequency shift, f (a ) – f0, is zero for any acceleration in the plane normal to the accelerationsensitivity vector [12]. Measuring the individual mutually orthogonal g-sensitivity in the x, y, and z axes, the maximum g-sensitivity vector, Γmax can be determined without any prior knowledge of the quartz resonator as:

Γ max = g x2 + g y2 + g z2

(3.26)

Knowing the magnitude and angular orientation of Γmax, the expected effect of externally applied acceleration in any direction can be calculated. 3.7.1  Gravitational Acceleration

A resonator even if at rest is being accelerated by gravity, which is exerts a magnitude of acceleration of 1g. This gravitational acceleration of 1g has a magnitude of 9.80 m/sec2 at sea level. The acceleration direction due to gravity is towards the center of the earth. If an oscillator at rest is turn over by 180° it will experience 2g of acceleration. This is because the 1g at rest and the 1g at turnover are additive. This can be conducted all three axis while measuring the frequency of the oscillator to obtain Γmax. This is called the 2-g tip-over test. 3.7.2  Sinusoidal Acceleration/Vibration

A crystal oscillator experiencing a sinusoidal acceleration will produce discrete sidebands at ±fv from the carrier (fc) where fv is the vibration frequency. The phase noise performance under sinusoidal vibration is given by [13]



 Γ ⋅ a ⋅ fc  L (fv ) = 20 log  in dBc  2fv 

(3.27)

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Understanding Quartz Crystals and Oscillators

In (3.27), “a” is now the peak of the acceleration magnitude and Γ is the total g-sensitivity. Example 3.1

A 25-MHz noise-free crystal oscillator with an output level of +5 dBm and a g-sensitivity of |Γ| = 1.2 ppb/g is subjected to a 1-kHz sinusoidal acceleration of 2.5g in magnitude. Plot the frequency spectrum of the oscillator under these conditions. Solution:

Using (3.27), we have:



(

)

(

 1.2 × 10 −9 (2.5) 25 × 106 L (1000) = 20 log  2 × 1000  

)  = −88.5 dBc  

The spectrum plot of the oscillator is shown in Figure 3.5. Note that sidebands are generated even though the oscillator is noise free. 3.7.3  Random Acceleration/Vibration

Acceleration spectral density (ASD) or power spectral density (PSD) is typically how random vibration is specified. The root mean square acceleration (Grms) is the square root of the area under the ASD/PSD curve in the frequency domain. Grms is a single number that defines the overall energy or acceleration level of

Figure 3.5  Solution to Problem 3.1. Sidebands generated by 1-kHz sinusoidal acceleration.



Advanced Quartz Crystal Resonator Topics

87

random vibration. Random acceleration on a crystal oscillator will produce a spectral density increase in the phase noise instead of generating discrete sidebands as in the case of sinusoidal vibration. Hence, the acceleration magnitude “a” will have to be described differently. It is now defined as: a2 = the power spectral density of acceleration in Grms2/Hz versus frequency [Hz] or simply G 2/Hz versus frequency. Let us assume we have a band-limited white random excitation as shown in Figure 3.6. This is simplest case to consider. The overall Grms under the curve is simply the square root under it, that is

Grms = P ( f 2 − f 1 )

(3.28)

It is rare to encounter a simple white band-limited PSD specification. It is more common to encounter a certain slope from one frequency to the other. In Figure 3.7, we show the case negative slope of −3 dB/octave case. For this case the Grms is given by

Figure 3.6  Band-limited white PSD.

Figure 3.7  Negative slope PSD at −3 dB/octave.

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Understanding Quartz Crystals and Oscillators

Grms = P × f 1 × ln [ f 2 f 2 ]

(3.29)

In the case of positive slopes (Figure 3.8) from one frequency to the other, let us first define PS = (S/3) + 1 where S is the slope in dB/Octave. In this case Grms is equal to

Grms =

P × f 2   f1   × 1 −    PS   f2 

(3.30)

For the negative slope case (Figure 3.9) let NS = (S/3) − 1, then Grms is given by



NS P × f1   1    × 1 −  Grms = NS   ( f 2 f 1 )    

Figure 3.8  Positive slope PSD, general case.

Figure 3.9  Negative slope PSD, general case.

(3.31)



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3.8  Drive Level Dependency Testing To facilitate measuring, capturing, and/or documenting many of the crystal’s irregularities, Saunders and Associates [14] have included many test routines with their crystal test chambers. What follows is a short description of some of these test routines directly from their 350-A crystal test chamber manual. • DLD: Measures the ratio between the largest resistance, measured over a user-defined power range, and the resistance at nominal power. • DLD1: Measures the ratio between the largest resistance, measured over a user-defined power range, and the resistance at the lowest power level. • DLD2: Determines the difference between the maximum and minimum resistance measured over a specified power range. • RLD: Measures the crystal’s resistance over a specified power range. • FDLD: Determines the difference between the maximum and minimum frequency over a specified power range. • FDLD2: Determines the difference in parts per million between the maximum and minimum frequency at up to 20 user-specified power levels. Note that all the tests above can also be tested as temperature being a parameter. Many DLD/DLS issues only manifest themselves over a narrow temperature range.

References [1] Walls, F. L., et al, “Excess Noise in Quartz Crystal Resonators,” Proceedings of the 37th Annual Symposium on Frequency Control, 1983, pp. 218–225. [2] www.sholarpedia.org. [3] Kroupa, V. F., “Origin of 1/f Noise in Crystal Oscillators,” 2003 IEEE International Ultrasonic Symposium, 2003. [4] Ascarrunz, H. D., et al., “PM Noise Generated by Noisy Components,” Proceedings of the 1998 IEEE Frequency Control Symposium, June 1998, pp. 210–217. [5] Shmaliy, Y. S., “One-Port Noise Model of a Crystal Oscillator,” IEEE Translations on Ultrasonics, Ferroelectronics, and Frequency Control, Vol. 51, No. 1, January 2004. [6] Shmaliy, Y. S., et al., “Noise Conversion in Crystal Oscillators,” Proceedings of Frequency Control Symposium and PDA Exhibition, 2002, IEEE, December 6, 2012.

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[7] Brendel, R., et al., “Investigation in Low Drive Level Sensitivity of Quartz Resonator Affecting Their Motional Parameters,” 20th European Frequency and Time Forum, Braunschweig, Germany, March 2006. [8] Young, Y. K., and J. R. Vig, “Resonator Surface Contamination: A Cause of Frequency Fluctuations?” Proceedings of the 42nd Annual Symposium on Frequency Control, 1988, pp. 397–403. [9] Walls, F. L., et al., “A New Model of 1/f Noise in BAW Quartz Resonators,” IEEE Frequency Control Symposium, 1992. [10] Gagnepain, J. J., et al., “Excess Noise in Crystal Resonators,” IEEE paper, 1983. [11] Jones, K. H., “Effects of Initial Quartz Surface Finish and Etch Removal on Etch Figures and Quartz Crystal Q,” 41st Annual Frequency Control Symposium, 1987. [12] Filler, R. L., “The Acceleration Sensitivity of Quartz Crystal Oscillators,” 41st Annual Frequency Control Symposium, 1987. [13] MIL-PRF-55310, Defense Logistics Agency, www.dscc.dla.mil. [14] Saunders and Associates, www.saunders-assoc.com.

4 MEMS Resonators and Oscillators 4.1  Introduction It has long been a goal for IC manufactures to integrate the external quartz crystal oscillator, the dream of an all-silicon solution with the frequency accuracy and phase noise performance of the quartz-based oscillator. However, due to many differences in the technologies, this was not possible until microelectromechanical systems (MEMS) resonators were developed. MEMS resonators can be manufactured using the same IC processing technologies. In addition to resonators for oscillators, MEMS are being embedded in digital cameras, automotive air bags, gyroscopic sensors, and a host of other devices. MEMS promise a profound transformation of engineered mechanical systems by the size reduction and increased capabilities [1]. MEMS resonators have been in development since the 1960s [2], but viable commercial applications began to appear around 2000. The research conducted by the U.S. Defense Advanced Research Projects Agency (DARPA) provided the base technology from which start-up companies like Discera [3] and SiTime [4] began producing MEMS oscillators. SiTime and Discera introduced production MEMS oscillators in 2006 and 2007, respectively. This chapter is mainly a comparison between quartz crystal and MEMSbased oscillators at the time of this writing. Only MEMS resonators will be introduced here. For readers interested in learning more on MEMS, reference [1] is an excellent first textbook with which to start.

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4.2  Some MEMS Terminology The first word to introduce is transducer. A transducer converts power from one form to another. An electrical-mechanical transducer changes mechanical into electrical power or vice versa. The second important term is actuator. An actuator is a mechanical device that controls or creates mechanical motion. An electrical-mechanical actuator is a transducer that creates mechanical motion based on an electrical input. Electromechanical relays and motors are common examples. Our third term to introduce is a sensor. A sensor is any device that emits a signal in response to a physical stimulus. The old record player pick-up head is an example of an electromechanical sensor. In this case, the grooves in the record player (the mechanical stimulus) are converted to voltage or current. 4.2.1  Electromechanical Systems

The notation that follows is based on the notation used in [1]. Let us start with a simple electromechanical actuator with one input pair of electrical stimulus and a single degree of freedom mechanical motion. This is represented by the simple diagram of Figure 4.1. On the left side of the diagram, we have a timedependent current i(t) and a time-dependent voltage v(t). On the mechanical side, we introduce the notation f  e(t) for electrical force. The transducer exerts the electrical force on the external mechanical system along the mechanical variable x(t). The mechanical variable x(t) can be replaced by velocity, which is, x = dx / dt . Considering power flow will give us further insight into the physical interpretation of the force f e(t). The instantaneous mechanical power is [1]

pm (t ) = f

e

(t ) x

(4.1)

based on the instantaneous electrical input power to the device, which is

pe (t ) = v (t )i (t )

(4.2)

Figure 4.1  A lossless, basic electromechanical transducer with one electrical pair and one mechanical degree of freedom.



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4.3  MEMS Resonators MEMS resonators are miniaturized electromechanical structures that are made using the techniques of microfabrication. MEMS can vary in size from 1 micron or lower, all the way to several millimeters. Some MEMS have no moving elements, while others can have complex electromechanical systems with multiple moving elements. These moving elements can consist of beams, rings, and squares to generate megahertz frequencies. Figure 4.2 shows a MEMS resonator design from Discera that has been used in their MEMS oscillator offering. MEMS resonators are classified based on their mode of vibration [5]. Some of the modes are: 1. 2. 3. 4.

Flexural: 10 kHz to 10 MHz; Contour mode/lamb wave: 10 MHz to 10 GHz; Thickness extensional: 500 MHz to 20 GHz; Shear mode: 800 MHz to 2 GHz.

These can be further classified as capacitive-transduced and piezoelectrically-transduced MEMS. Capacitive transducers create an electrical force that is used to excite a particular resonance mode. This means that they are driven electrostatically by applying a dc bias voltage across a small gap between the resonator and its chamber [5]. However, some resonators that are electrostatically driven require large voltages (>10V) to be applied across the gap. Piezoelectric transducers take advantage of electromechanical coupling to excite the resonance mode. This can be accomplished by applying electricity to a coating of aluminum nitride that has been deposited on the resonator beam [6, 7]. The quartz crystal resonator is considered a piezoelectric transducer, also known as bulk acoustic wave (BAW) resonator.

Figure 4.2  A Discera MEMS resonator design. (Courtesy of Discera.)

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4.3.1  Quartz MEMS (QMEMS)

QMEMS are quartz-based resonators that use MEMS manufacturing techniques to reduce the form factor of conventional quartz crystal device. Seiko Epson Corporation [8] reported in 2008 that it perfected a semiconductorlike photolithography process that produces high-performance resonators in compact packages. They coined the fabrication technique QMEMS for quartz MEMS. The technique is solely based on quartz and not on silicon. Tuning-fork QMEMS crystals can be fabricated in such a small form factor that they can be integrated in very slim smart credit cards [8]. 4.3.2  MEMS Resonator Equivalent Circuit Model

A MEMS resonator can be modeled as a mass, spring, and damper system as illustrated in Figure 4.3. When this system is in motion, it can be described by Newton’s equation of motion in one dimension, which is given by

f (t ) = mx (t ) + bx (t ) + kx (t )

(4.3)

where f  (t) is the driving force acting on the mechanical system, k is the spring constant, m is the mass, and b is the damping coefficient. If f  (t) is assumed to be sinusoidal, then it can be written as

Figure 4.3  (a) MEMS resonator equivalent mechanical system; (b) Typical MEMS resonator equivalent one-port circuit model.



MEMS Resonators and Oscillators



f (t ) = Fe j wt

95

(4.4)

A sinusoidal force acting on the system will create a sinusoidal displacement and hence

x (t ) = Xe j wt + ϕ

(4.5)

This mechanical system should be familiar to us as the same mechanical equivalent system of a crystal resonator. Further derivation is beyond the scope of this chapter; however, [9, 10] have shown that in the electrical domain, it is identical to motional arm of a crystal resonator. That is, the series R-L-C circuit where the impedance is given by

Z ( j w) = Rm + j wLm +

1 j wC m

(4.6)

Note the slight subscript notation difference using m instead of 1 for the motional components. Table 4.1 summarizes the equivalents from the mechanical domain to the electrical domain. Given the reduced size of MEMS resonators, the external elements that provide physical support like the silicon and routing electrodes need to be accounted for. Therefore, the electrical equivalent model will include a parasitic resistance, Rs, and parallel resistance, R0. The MEMS resonator model including these parasitic components is shown in Figure 4.3. In some models additional parasitic capacitors are added depending on the specific design of the MEMS resonator. 4.3.3  Frequency-Temperature Performance of MEMS

A major drawback of some commercially available MEMS resonators is the large temperature coefficient of frequency (>25 ppm/°C). This large coefficient will necessitate some type of frequency compensation to compete with quartz Table 4.1 Mechanical to Electrical Variables Mechanical Variable Electrical Equivalent Force (F) Voltage (V) Velocity (v) Current (I) Damping Resistance (Rm) Spring compliance Capacitance (Cm) Mass Inductance (Lm)

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oscillators. A common method that MEMS oscillator vendors [4] have chosen for frequency compensation is to use a high-resolution fractional-N PLL in conjunction with a temperature sensor. This is illustrated in the block diagram of Figure 4.4. The digitally frequency compensation technique shown in Figure 4.4 can result in excellent medium-term frequency stability. Less than 1 ppm over −40°C to +85°C can easily be achieved. However, these techniques create poor short-term frequency stability. This is definitely the case in techniques with look-up tables to discipline the frequency on a periodic basis. Figure 4.5 illustrates this phenomenon. Frequency correction jumps like those in Figure 4.5 vary in step size and rate of occurrence. Many systems cannot tolerate such frequency jumps. For example, a frequency jump is a phase jump. Many PLLs will or can unlock every time their reference clock source undergoes frequency/phase jumps. 4.3.4  Phase Noise and Jitter Performance of MEMS Oscillators

Phase noise and jitter performance are where quartz-based oscillators outshine their MEMS counterparts. Due to their PLL-based designs, the phase noise of MEMS oscillators tend to be much higher than quartz-based designs. This difference can easily be seen in the phase noise plots as depicted in Figure 4.6. The MEMS phase noise plot shape is that of a characteristic PLL frequency synthesis. The quartz oscillator can have 20 dB or lower noise at some offsets, giving them superior noise performance. The integration of this noise will translate to higher jitter generation for the MEMS oscillators.

Figure 4.4  MEMS oscillator typical system architecture.



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Figure 4.5  Exaggerated depiction of medium- and short-term (zoom) frequency stability of a digital frequency compensating technique as used by some MEMS oscillators. The correction can be in the order of 1 ppm peak to peak.

Figure 4.6  Phase noise plot comparison of quartz versus MEMS oscillators.

4.4  MEMS Oscillators Versus Quartz Oscillators MEMS technology has promised reduction of size and weight, reduced power consumption, superior reliability, the ability to mass produce, lower cost, more robustness, lower lead times, and compatibility with IC technology. With such claims, companies like Discera, SiTime, and others are striving for a large piece of the lucrative crystal oscillator market and have succeeded in some markets like the disk drive industry and RF filters for cell phones. It is my opinion that the weakest performance of MEMS is their phase noise/jitter performance when going against quartz-based oscillators. Some MEMS oscillators do offer features and functionality not traditionally found in quartz oscillators. These include but are not limited to: 1. They have multiple independent output frequencies. These frequencies can be differential or single-ended output formats. 2. It is pin configurable. Output frequency and/or waveform can be changed with pin state.

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3. The I2C/SPI interface is configurable. Frequency and output formats (i.e., CMOS, LVPECL, LVDS) can be selected. 4. Some are field programmable while others can be configured on the fly by the users on their board. 5. Operating voltage can be selected. For example, +2.5V or +3.3V may be chosen and changed later if the need arises.

4.4.1  Performance Claims/Reports from MEMS Vendors

SiTime [4] has reported and/or has claimed that their MEMS oscillators are: 1. More robust than quartz oscillators able to withstand at least a 50,000g shock. Most crystal oscillators are rated to 1,500g. 2. More reliable than quartz oscillators. They report that their MEMS oscillators have a 500-million-hour mean time between failures (MTBF). That claim is 20 times better than the quartz oscillator that the industry typically reports. 3. Have the best frequency aging performance. They reported a figure of ±3.5 ppm for their MEMS as compared to 8 ppm for quartz oscillators for a 10-year period. 4. Best performance under vibration. They reported a figure of 30 times better than that of quartz-based oscillators. In fact, they claim 1 ppm/g for vibration frequencies between 15 Hz and 1 kHz. 5. Best power supply noise rejection. They attribute this to an internal differential architecture for common mode noise rejection. 6. Best in electromagnetic susceptibility (EMS). They reported that their product had up to 54 times more immunity to external electromagnetic fields than quartz oscillators. Claim 3 must compare a commodity crystal oscillator to its present MEMS oscillator offering. I have personally designed high-performance crystal oscillators with much lower aging rates than those claimed for the MEMS counterpart.



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4.4.2  Present Challenges Facing Commodity MEMS Oscillators

The MEMS oscillator industry has a very bright future ahead of it. However, they are facing some difficult challenges that will need to be overcome. It is my opinion that their biggest challenge is that they synthesize the output frequency from a few MEMS resonator designs. Having a PLL in between the MEMS resonator and the output does not generate a signal as good in quality as quartzbased designs that do not use PLLs. This translates to higher phase noise/jitter generation as compared to quartz-based oscillators. Phase noise/jitter is one of the key parameters in a system performance. At the time of this writing, the best phase jitter performance that I found in a MEMS oscillator data sheet is 300-fS RMS given an integration frequency from 12 kHz to 20 MHz. Quartzbased commodity oscillators can easily perform at 100 fS or lower at the same frequency. Second, the promise of quick turnaround is not an advantage in many circumstances. In fact, there are now configurable/programmable quartz-based oscillators (i.e., XpressO from Fox Electronics [11]) in the market that can meet or beat MEMS claims for fast turnaround. These configurable oscillators use a quartz crystal resonator instead of a MEMS resonator, so basically they are similar in that respect (resonator + PLL) to generate a range of output frequencies. Third, MEMS can withstand much higher shock forces than quartz oscillators, but a system is as strong as its weakest link. If I drop my laptop from a second-story building and it breaks into pieces, if the MEMS oscillator survived it, then what value did it add by being capable of handling 50,000g? My laptop is still broken and I have to buy a new one. In a toy that a kid throws around, it will be beneficial to use a MEMS oscillator versus a quartz oscillator. Fourth, price is another challenge for the present MEMS oscillator industry. Quartz-based oscillators have been around so long now that there is a plethora of low-cost manufacturers, especially in China. With regard to automation versus cheap labor, in today’s world, cheap labor is presently beating the MEMS automation. Table 4.2 summarizes some of the performance differences between MEMS and quartz-based oscillators.

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Table 4.2 Performance Comparison Between MEMS and Quartz Oscillators Parameter MEMS Oscillator Quartz Oscillator Comment Phase noise Fair Lowest MEMS keep improving and can meet many present standards Jitter Fair Lowest MEMS keep improving and can meet many present standards Defect rate Lowest Fair IC manufacturing technology is responsible for the MEMS very low defect rates Frequency-temperature Good Good/fair MEMS using stability digitally temperature compensation Short-term stability Poor Excellent MEMS can have frequency and phase jumps Long-term aging Good Good Shock resistance Excellent Poor Quartz crystals can be very fragile Lead time Best Can be very long Power consumption Lower Can be low Start-up time Longer