Coupled–Oscillator Based Active–Array Antennas

Describing an innovative approach to phased–array control in antenna design

This book explores in detail phased–array antennas that use coupled–oscillator arrays, an arrangement featuring a remarkably simple beam steering control system and a major reduction in complexity compared with traditional methods of phased–array control. It brings together in one convenient, self–contained volume the many salient research results obtained over the past ten to fifteen years in laboratories around the world, including the California Institute of Technology′s Jet Propulsion Laboratory.

The authors examine the underlying theoretical framework of coupled–oscillator systems, clearly explaining the linear and nonlinear formalisms used in the development of coupled–oscillator arrays, while introducing a variety of state–of–the–art methodologies, design solutions, and tools for applying this control scheme. Readers will find:

Numerous implementation examples of coupled–oscillator array prototypes A continuum model that permits application of diffusion theory to the analysis of phase dynamics A demonstration of the array behavior through experimental results that validate the linearized theory Examples of how incorporating coupling delay restores causality, including the latest published results Guidance on how to accurately analyze and optimize coupled–oscillator arrays using modern simulation tools A review of current developments, including the design of compact couple–oscillator array antennas

Complete with 150 diagrams and photographs, Coupled–Oscillator Based Active–Array Antennas is a highly useful tutorial for antenna designers and a valuable reference for researchers and engineers wishing to learn about this cutting–edge technology.

Foreword xi

Preface xiii

Acknowledgments xvii

Authors xix

PART I: THEORY AND ANALYSIS 1

Chapter 1 Introduction Oscillators and Synchronization3

1.1 Early Work in Mathematical Biology and Electronic Circuits3

1.2 van der Pol′s Model 5

1.3 Injection Locking (Adier′s Formalism) and Its Spectra(Locked and Unlocked) 7

1.4 Mutual Injection Locking of Two Oscillators 21

1.5 Conclusion 26

Chapter 2 Coupled–Oscillator Arrays–Basic AnalyticalDescription and Operating Principles 27

2.1 Fundamental Equations 28

2.2 Discrete Model Solution (Linearization and LaplaceTransformation) 31

2.3 Steady–State Solution 37

2.4 Stability of the Phase Solution in the Full NonlinearFormulation 41

2.5 External Injection Locking 46

2.6 Generalization to Planar Arrays 50

2.7 Coupling Networks 54

2.8 Conclusion 66

Chapter 3 The Continuum Model for Linear Arrays 67

3.1 The Linear Array without External Injection 68

3.2 The Linear Array with External Injection 81

3.3 Beam–Steering via End Detuning 93

3.4 Beam–Steering via End Injection 95

3.5 Conclusion 102

Chapter 4 The Continuum Model for Planar Arrays 103

4.1 Cartesian Coupling in the Continuum Model without ExternalInjection 103

4.2 Cartesian Coupling in the Continuum Model with ExternalInjection 109

4.3 Non–Cartesian Coupling Topologies 118

4.4 Conclusion 137

Chapter 5 Causality and Coupling Delay 139

5.1 Coupling Delay 139

5.2 The Discrete Model with Coupling Delay 141

5.3 The Continuum Model with Coupling Delay 146

5.4 Beam Steering in the Continuum Model with Coupling Delay159

5.5 Conclusion 173

PART II: EXPERIMENTAL WORK AND APPLICATIONS 175

Chapter 6 Experimental Validation of the Theory 177

6.1 Linear–Array Experiments 177

6.2 Planar–Array Experiments 188

6.3 Receive–Array Experiments 201

6.4 Phase Noise 210

6.5 The Unlocked State 213

6.6 Conclusion 215

PART III: NONLINEAR BEHAVIOR 217

Chapter 7 Perturbation Models for Stability, Phase Noise, andModulation 219

7.1 Preliminaries of Dynamical Systems 220

7.2 Bifurcations of Nonlinear Dynamical Systems 226

7.3 The Averaging Method and Multiple Time Scales 230

7.4 Averaging Theory in Coupled Oscillator Systems 231

7.5 Obtaining the Parameters of the van der Pol Oscillator Model235

7.6 An Alternative Perturbation Model for Coupled–OscillatorSystems 238

7.7 Matrix Equations for the Steady State and Stability Analysis242

7.8 A Comparison between the Two Perturbation Models for CoupledOscillator Systems 246

7.9 Externally Injection–Locked COAs 247

7.10 Phase Noise 250

7.11 Modulation 256

7.12 Coupled Phase–Locked Loops 258

7.13 Conclusion 261

Chapter 8 Numerical Methods for Simulating Coupled–OscillatorArrays 263

8.1 Introduction to Numerical Methods 264

8.2 Obtaining Periodic Steady–State Solutions of AutonomousCircuits in Harmonic–Balance Simulators 270

8.3 Numerical Analysis of a Voltage–Controlled Oscillator272

8.4 Numerical Analysis of a Five–Element LinearCoupled–Oscillator Array 278

8.5 Numerical Analysis of an Externally Injection–LockedFive–Element Linear Coupled–Oscillator Array 286

8.6 Harmonic Radiation for Extended Scanning Range 288

8.7 Numerical Analysis of a Self–Oscillating Mixer 291

8.8 Conclusion 296

Chapter 9 Beamforming in Coupled–Oscillator Arrays297

9.1 Preliminary Concepts of Convex Optimization 297

9.2 Beamforming in COAs 301

9.3 Stability Optimization of the Coupled–OscillatorSteady–State Solution 308

9.4 Multi–Beam Pattern Generation Using Coupled–OscillatorArrays 311

9.5 Control of the Amplitude Dynamics 315

9.6 Adaptive Coupled–Oscillator Array Beamformer 317

9.7 Conclusion 320

Chapter 10 Overall Conclusions and Possible Future Directions321

REFERENCES 325

ACRONYMS AND ABBREVIATIONS 341

INDEX 345

RONALD J. POGORZELSKI, PhD, is Senior Research ScientistEmeritus and the former supervisor of the Spacecraft AntennaResearch Group at the Jet Propulsion Laboratory. He has authored orcoauthored over 100 technical publications and presentations andwas editor of the IEEE Transactions on Antennas andPropagation.

APOSTOLOS GEORGIADIS, PhD, is a Senior Research Associateat Centre Tecnològic de Telecomunicacions de Catalunya (CTTC)in Barcelona, Spain, where he is involved in active antennas andantenna arrays as well as RFID technology and energyharvesting.