PHASE-LOCK BASICS

Second Edition

WILLIAM F. EGAN, Ph.D. Lecturer in Electrical Engineering Santa Clara University

<<>>IEEE The Institute of Electrical and Electronics Engineers, Inc., New York

B I C I N T I N N I AU J I in

; 1 8 0 7 jj

\ ®WILEY \ I 2 0 0 7 \

WILEY-INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

CONTENTS

PREFACE TO THE SECOND EDITION xix

PREFACE TO THE FIRST EDITION xxi

SYMBOLS LIST AND GLOSSARY xxv

PART 1 PHASE LOCK WITHOUT NOISE

1 INTRODUCTION 3

1.1 What is a Phase-Locked Loop (PLL)? / 3 1.2 Why Use a Phase-Locked Loop? / 3

1.3 ScopeofthisBook / 4 1.4 Basic Loop / 5 1.5 Phase Definitions / 6 1.6 Phase Detector / 8

1.7 CombinedGain / 10 1.8 Operating Range / 11 1.9 Units and the Laplace Variable s I 13

VÜi CONTENTS

2 THE BASIC LOOP 15

2.1 Steady-State Conditions / 15 2.2 Classical Analysis / 16

2.2.1 Transient Response / 17 2.2.2 Modulation Response / 19

2.3 Mathematical Block Diagram / 21 2.4 Bode Plot / 24 2.5 Note on Phase Reversais / 26

2.6 Summary of Transient Responses of the First-Order Loop / 26

3 LOOP COMPONENTS 29

3.1 Phase Detector / 29 3.1.1 Flip-Flop Phase Detector / 29 3.1.2 Exclusive-OR Gate Phase Detector / 30 3.1.3 Charge-Pump Phase Detector / 31 3.1.4 Sinusoidal Phase Detector / 33

3.1.4.1 Phase Detection in a Simple Mixer / 34

3.1.4.2 Balanced Mixers / 35

3.1.4.3 Analog Multipliers / 39

3.1.4.4 Integrated Circuit Doubly Balanced Mixer / 40

3.2 Voltage-Controlled Oscillator (VCO) / 41 3.3 Loop Filter / 44

3.3.1 Passive Loop Filter / 45

3.3.2 Filters Driven by Current Sources / 45 3.3.2.1 Integrator / 47

3.3.2.2 Integrator-and-Lead Filter / 48

3.3.2.3 Lag (Low-Pass) Filter / 48

3.3.2.4 Lag-Lead Filter / 49

3.3.2.5 DC Transimpedance / 49

3.3.3 Active Filters / 51 3.3.3.1 Op-Amp Considerations / 52

3.3.3.2 Unintended Poles / 53

3.3.3.3 Reducing the Size ofC/54

3.3.3.4 Impractical Loop Filters / 55

3.4 Filter Reference Voltage / 55 3.5 Note on the Form of the Filter Equation / 56 3.6 Capacitors in Loop Filters / 56 3.7 Higher-Order Filters / 57

CONTENTS IX

i3.A Appendix: Integrated Circuit Doubly Balanced Mixer—Details / 57 i3.B Appendix: Op-Amps in Loop Filters / 57

4 LOOP RESPONSE 59

4.1 Loop Order and Type / 59

4.2 Closed-Loop Equations / 60 4.3 Open-Loop Equations—Lag-Lead Filter / 62 4.4 Loop with a Lag Filter / 65 4.5 Loop with an Integrator-and-Lead Filter / 66 4.6 Summary of Equations / 67 4.7 Unity-Gain Bandwidth in Highly Damped Loops / 67

5 LOOP STABILITY 71

5.1 Observing the Open-Loop Response / 71 5.2 Methods of Stability Analysis and Measures of Stability / 72

5.2.1 BodePlo t /73 5.2.2 NyquistPlot / 74 5.2.3 Evans Plot (Root Locus) / 76

5.3 Stability of Various PLL Configurations / 76

5.3.1 First-Order Loop / 77 5.3.2 Second-Order Loop / 77 5.3.3 Third-Order Loop / 77 5.3.4 Controlling Stability over Wide Gain Ranges / 79

5.3.5 Representing Delay / 80 5.4 Computing Open-Loop Gain and Phase / 81

5.5 Phase Margin Versus Damping Factor / 86

i5.M Appendix: Stability Plots Using MATLAB® / 86

6 TRANSIENT RESPONSE 87

6.1 Step Response / 87 6.1.1 Form of the Equations / 87 6.1.2 Step Response Equations / 89

6.2 Envelope of the Long-Term Step Response / 92

6.3 Response to Ramp Input / 96 6.4 Response to Parabolic Input / 98 6.5 Other Responses / 100 6.6 Note on Units for Graphs / 101 6.7 Equivalent Circuit / 102

X CONTENTS

6.8 General Long-Term (Steady-State) Response Characteristics / 102

6.9 Open-Loop Equations in Terms of Closed-Loop Parameters / 103

6.10 More Complex Loops and State-Space Analysis / 103 6.10.1 Basic Equations / 104 6.10.2 Initial Conditions / 107

6.11 An Approximate Solution Using State-Space Variables / 108

6.12 Effect of an Added Pole / 108 i6.M Appendix: Transient Responses Using MATLAB / 109

7 MODULATION RESPONSE 111

7.1 Phase and Frequency Modulation / 111 7.2 Modulation Responses / 113

7.3 Responses in a First-Order Loop / 113 7.4 Transfer Functions in a Second-Order Loop / 116

7.4.1 Output Response / 117 7.4.2 Error Response / 118 7.4.3 Responses Near u>L / 118

7.4.4 Phase or Frequency at Inputs and Outputs / 119 7.5 Transient Responses Between Various Points / 120 7.6 Magnitude and Phase of the Transfer Functions / 120

7.6.1 Output Responses / 121 7.6.2 Error Responses / 121

7.6.3 Effect ofor / 122 7.6.4 Responses for f < 1 / 123 7.6.5 Responses for £ > 1 / 125

7.7 Related Responses / 127 7.8 Modulation and Demodulation in the Second-Order Loop / 128

7.8.1 Frequency Demodulation / 128 7.8.2 Phase Demodulation / 130

7.8.3 Frequency Modulation / 130 7.8.4 Phase Modulation / 131 7.8.5 Extending the Modulation Frequency Range / 131

7.8.5.1 Frequency Modulation / 131

7.8.5.2 Phase Modulation / 133 7.9 Measurement of Loop Parameters for a — 0 or 1 from

Modulation Responses / 134

CONTENTS Xi

7.10 Effect of an Added Pole / 134 7.M Appendix: Modulation Response Using MATLAB® / 135

8 ACQUISITION 137

8.1 Overview / 137

8.2 Acquisition and Lock in a First-Order Loop / 141 8.2.1 Transient Time / 143 8.2.2 Acquisition / 144

8.3 Acquisition Formulas for Second-Order Loops with Sine Phase Detectors / 146

8.4 Approximate Pull-In Analysis / 149

8.4.1 Basic Equations / 150 8.4.2 General Analysis / 153 8.4.3 Pull-In Range / 157 8.4.4 Approximate Pull-In Time / 158

8.5 Phase-Plane Analysis / 161

8.6 Pull-Out / 164 8.7 Effect of Offsets / 164 8.8 Effect ofComponent Saturation / 165 8.9 Hangup / 166 8.10 Simulation of the Nonlinear Loop / 166 8.A Appendix: Summary of Acquisition Formulas for

Second-Order Loops / 167 i8.M Appendix: Nonlinear Simulation / 167 8.S Appendix: Acquisition Spreadsheet / 167 8.V Appendix: Some Values in Terms of a, {,,

and a>„ I 167

9 ACQUISITION AIDS 171

9.1 Coherent Detection—Lock Indicator / 171 9.1.1 During Acquisition / 172

9.1.2 During Sweep, Locked Loop / 173 9.2 Changing Loop Parameters Temporarily / 174

9.2.1 Coherent Automatic Gain Control / 174 9.2.2 Filter Modifikation / 175 9.2.3 Comparison of Two Types of Parameter

Modifications / 176 9.3 Automatic Tuning of«c , Frequency Discriminator / 177 9.4 Acquisition Aiding Logic / 179

XÜ CONTENTS

9.5 Sweeping coc, Type 2 Loop / 180 9.5.1 Maximum Sweep Speed, Closed-Loop Sweeping / 180 9.5.2 Open-Loop Sweeping / 181 9.5.3 Combined Techniques / 185

9.6 Sweep Circuits / 185 9.6.1 Switched Current Source / 185 9.6.2 Automatic Sweep Circuit—Sinusoidal / 186 9.6.3 Automatic Sweep Circuit—Nonsinusoidal / 186

9.A Appendix: Maximum Sweep Rate, Open-Loop vs. Closed-Loop / 187

10 APPLICATIONS AND EXTENSIONS 189

10.1 Higher-Order Loops / 189 10.2 Generalized Voltage-Controlled Oscillator / 189

10.2.1 Frequency Synthesis, Frequency Division / 190

10.2.1.1 Stability / 191

10.2.1.2 Transient Response / 191

10.2.1.3 Response to Noise / 192

10.2.2 Heterodyning (Frequency Mixing) / 192 10.3 Long Loop / 193 10.4 Carrier Recovery / 195

10.4.1 Biphase Costas Loop / 195 10.4.2 JV-Phase Costas Loop / 196 10.4.3 Multiply and Divide / 196

10.5 Data Synchronization / 197 10.5.1 Early-Late Gate Bit Synchronizer / 197 10.5.2 Synchronizing to a Pseudorandom Bit Sequence / 198 10.5.3 Delay-and-Multiply Synchronizer / 200

10.6 Clock and Data Timing Control / 200 10.6.1 Phase-Locked Loops / 200 10.6.2 Delay-Locked Loops / 202

10.6.2.1 For Clock Synchronization / 202

10.6.2.2 For Data Synchronization / 203

10.6.3 Combined Loops / 204 10.7 All-Digital Phase-Locked Loop (ADPLL) / 204

10.7.1 Basic Digital Implementation / 205

10.7.1.1 The Loop/ 205

10.7.1.2 Sampling and Stability / 206

CONTENTS XÜi

10.7.1.3 Choice ofValues / 209

10.7.1.4 Higher-OrderLoops / 210

10.7.2 OA, NCO,DDS / 211 10.7.3 Implementing an ADPLL by Pulse Addition and

Removal / 212

10.7.3.1 Transfer Function / 213

10.7.3.2 Tuning Range / 214

10.8 Summary / 214 10.A Appendix: Exact Analysis of a Special-Case Third-Order Loop / 215

10.A.1 Loop Response / 217 10.A.2 Final Values / 219 10.A.3 TripleRoots / 220 10.A.4 Step Response / 220

10.A.5 Modulation Response / 224 ilO.B Appendix: Costas Loop forNPhases / 227 ilO.C Appendix: Symbol Clock Recovery / 227 ilO.D Appendix: ADPLL by Pulse Addition and Removal,

Additional Material / 228

10.F Appendix: Fractional-/V and Sigma-Delta / 228

ilO.M Appendix: MATLAB® Simulations / 228

ilO.T Appendix: Combined PLL and DLL / 229

PART 2 PHASE LOCK IN NOISE

11 PHASE MODULATION BY NOISE 233

11.1 Representation of Noise Modulation / 233 11.2 Processing of Noise Modulation by the

Phase-Locked Loop / 236 11.3 Phase and Frequency Variance / 237 11.4 Typical Oscillator Spectrums / 238 11.5 Limits on the Noise Spectrum—Infinite Variances / 240 11.6 Power Spectrum / 242

11.6.1 Spectrum for Small m I 242

11.6.2 Single-Sideband Density / 243 11.6.3 When is the Modulation Small? / 245 11.6.4 Modification of Spectral Shape at Higher Modulation

Index / 246 11.6.5 Scr ip ta / 247

XJV CONTENTS

11.7 Frequency Multiplication and Division / 247 11.8 Other Representations / 248 ill.H Shape of Output Spectrum / 249 1 l.S Appendix: Spreadsheets for Integrating Densities / 249

12 RESPONSE TO PHASE NOISE 251

12.1 Processing of Reference Phase Noise / 251 12.2 Processing ofVCO Phase Noise / 254 12.3 Harmful Effect of Phase Noise in Radio Receivers / 255 12.4 Superposition / 256

12.5 Optimum Loop With Both Input and VCO Noise / 257 12.6 Multiple Loops / 260 12.7 Effects of Noise Injected Elsewhere / 260 12.8 Measuring Phase Noise / 261

12.8.1 Using a Phase Detector / 263 12.8.1.1 Calibration / 263

12.8.1.2 Obtaining a Measurement Reference / 265

12.8.2 Using a Frequency Discriminator / 269 12.8.3 Using a Spectrum Analyzer or Receiver / 270

13 REPRESENTATION OF ADDITIVE NOISE 271

13.1 General / 271

13.2 Phase Modulation on the Signal / 273 13.3 Multiplicative Modulation on Quadrature Carriers / 275 13.4 Noise at the Phase Detector Output / 276 13.5 Restrictions on the Noise Models / 277 13.6 Does the Loop Lock to the Additive Noise? / 280

13.7 Other Types of Phase Detectors in the Presence of Noise / 281 13.7.1 Triangulär Characteristic / 282 13.7.2 Sawtooth Characteristic / 283

13.8 Modified Phase Detector Characteristic with Phase Noise / 283 il3.A Appendix: Decomposition of a Single Sideband / 286

14 LOOP RESPONSE TO ADDITIVE NOISE 287

14.1 Noise Bandwidth / 287 14.2 Signal-to-Noise Ratio in the Loop Bandwidth / 290 14.3 Loop Optimization in the Presence of Noise / 291

14.3.1 The Problem / 292

CONTENTS XV

14.3.2 Measures to be Used / 292 14.3.3 Optimum Loop for a Phase Step Input / 293 14.3.4 Optimum Loop for a Frequency Step Input / 294 14.3.5 Optimum Loop for a Frequency Ramp Input / 294 il4.A Appendix: Integration of Eq. (6.4a) / 296 il4.B Appendix: Loop Optimization in the

Presence of Noise / 296

15 PHASE-LOCKED LOOP AS A DEMODULATOR 297

15.1 Phase Demodulation / 297 15.1.1 Noise / 297

15.1.2 DistortionoftheDemodulated Signal / 299 15.1.3 Demodulation with a Linear Phase Detector

Characteristic / 300 15.2 Frequency Demodulation, Bandwidth Set by a Filter / 301 15.3 Frequency Discriminator, First-Order Loop / 305 15.4 Frequency Discriminator, Second-Order Loop / 306 15.5 Expected Phase Error / 307

15.6 Summary of Frequency Discriminator S/N I 308 15.7 Standard Discriminator and Click Noise / 310 15.8 Clicks with a PLL / 313 15.9 Noise in a Carrier Recovery Loop / 314 15.C Appendix: Spectrum of Clicks / 316 15.E Appendix: erfc / 317

16 PARAMETER VARIATION DUE TO NOISE 319

16.1 Preview / 319 16.1.1 Automatic Gain Control (AGC) / 319 16.1.2 Limiter / 320 16.1.3 Driving the Phase Detector Hard from

the Signal / 321 16.1.4 Effects of Variations / 322

16.2 AGC / 322

16.2.1 Types of Detectors / 322 16.2.2 Square-Law Detection / 323

16.3 Limiter / 325 16.3.1 Limiting in the Presence of Small Noise / 325 16.3.2 Limiting in the Presence of Large Noise / 326

16.4 Effects of Gain Variation on Loop Parameters / 329

XVI CONTENTS

16.5 Effect of AGC or Limiter on an Optimized Loop / 331 16. A Appendix: Modified Bessel Functions / 333

17 CYCLE SKIPPING DUE TO NOISE 335

17.1 Phase / 336 17.1.1 Fokker-Plank Method / 336

17.1.1.1 Assumption Regarding the Nature ofthe Noise / 337

17.1.1.2 First-Order Loop / 337

17.1.1.3 Second-Order Loop / 339

17.1.2 Experimental Results / 341 17.1.3 Simulation / 342

17.1.3.1 First-Order and a =0/342

17.1.3.2 Second-Order, High-Gain / 343

17.2 Cycle Skipping, Mean Time / 347 17.2.1 First-Order Loop / 347 17.2.2 Second-Order Loop, a = 0 / 348 17.2.3 Second-Order Loop, a = 1 / 3 4 8

17.2.4 Second-Order Loop, 0 < a < 1 / 350 17.2.5 Probability of Cycle Skipping in a Given Time / 350

17.3 Cycle Skipping, Mean Frequency / 350 17.4 Cycle Skipping with Mistuning / 353

17.4.1 Effect ofa I 353 17.4.2 Comparison to Other Results / 356

17.5 Summary / 356 17.5.1 Phase Variance / 356 17.5.2 Cycle Skipping / 357 17.5.3 Skip Frequency / 357 17.5.4 Mistuning in High-Gain Loops (f = 0.7) / 357

il7.D Appendix: Additional Data / 357

U7.M Appendix: MATLAB® Scripts / 357

18 NONLINEAR OPERATION IN A LOCKED LOOP 359

18.1 Notation / 359

18.2 Phase-Detector Output m / 360 18.3 Changes in the Output Spectrum / 361 18.4 Gain Suppression, Quasi-Linear Approximation / 361

18.4.1 Basics / 363

CONTENTS XVÜ

18.4.2 Phase Variance with Additive Noise / 365 18.4.3 Tracking the Carrier / 368

18.4.3.1 With Phase Modulation / 368

18.4.3.2 With Phase Modulation and VCO Noise / 369

18.4.4 Effect on Phase Offset / 372 18.4.4.1 With High-Frequency Phase Modulation / 372

18.4.4.2 With Additive Noise / 373

18.4.4.3 With Limiting / 375

18.4.5 Summary of Suppression / 375 il8.S Appendix: Additional Offset Simulation Data / 376

ACQUISITION AIDS IN THE PRESENCE OF NOISE 377

19.1 Sweeping with Piain Closed-Loop / 377 19.1.1 Maximum Sweep Rate in Noise / 377 19.1.2 Effect of Initial Mistuning / 380

19.1.3 Determining Successful Acquisition / 382 19.1.4 Optimum Sweep Parameters / 383

19.2 Reduction of Coherent Detector Output (Closed-Loop Sweeping) / 385 19.3 Closed-Loop Sweeping in Noise with Coherent Detector / 386

19.3.1 Successful Acquisition / 386 19.3.2 False Stops / 388

19.3.3 False Restart / 389

19.3.4 False Stop versus False Restart / 390

Ü9.M Appendix: MATLAB® Script, swpi / 393 Ü9.S Appendix: Optimum Sweep for a Fixed Noise Density / 393

BANDLIMITED NOISE 395

20.1 Signals Centered in a Noise Band / 395 20.1.1 Clicks with a First-Order PLL / 395 20.1.2 Simulation Results Compared to Measured Data / 396

20.1.3 SkipRate / 398 20.1.4 Output Phase Variance / 403 20.1.5 Effect of a Limiter / 404 20.1.6 Comparison to AGC / 406 20.1.7 Higher-Order Loops / 407 20.1.8 Summary, Symmetrical Narrowband Noise / 407

20.2 Eccentric Signals / 407 20.2.1 Expected Performance / 408

xviii CONTENTS

20.2.2 Simulating Eccentric Noise / 409 20.2.3 Some Simulation Results / 409

20.2.3.1 Cycle Skipping / 410

20.2.3.2 Phase Offset / 411

20.2.3.3 Output Variance / 411

20.3 Extension to Other Types of Interference / 411 i20.M Appendix: MATLAB® Scripts / 413

i20.O Appendix: Off set Interference Data Correlation / 413 i20.S Appendix: Band Limited Simulation Data / 413

21 FURTHER INFORMATION 415

21.1 Sources for Additional Studies in Phase Lock / 415 21.2 Sources Covering Phase-Locked Frequency Synthesis / 416 2 LA Appendix: Modulations and Spectrums / 416 21.B Appendix: Getting Files from the Wiley Internet Site / 421

REFERENCES INDEX

423

429