Coupled-Oscillator Based Active-Array Antennas

Foreword xi Preface xiii Acknowledgments xvii Authors xix PART I: THEORY AND ANALYSIS 1 Chapter 1 Introduction Oscillators and Synchronization 3 1.1 Early Work in Mathematical Biology and Electronic Circuits 3 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 Analytical Description and Operating Principles 27 2.1 Fundamental Equations 28 2.2 Discrete Model Solution (Linearization and Laplace Transformation) 31 2.3 Steady-State Solution 37 2.4 Stability of the Phase Solution in the Full Nonlinear Formulation 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 External Injection 103 4.2 Cartesian Coupling in the Continuum Model with External Injection 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 Delay 159 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, and Modulation 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 Model 235 7.6 An Alternative Perturbation Model for Coupled-Oscillator Systems 238 7.7 Matrix Equations for the Steady State and Stability Analysis 242 7.8 A Comparison between the Two Perturbation Models for Coupled Oscillator 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-Oscillator Arrays 263 8.1 Introduction to Numerical Methods 264 8.2 Obtaining Periodic Steady-State Solutions of Autonomous Circuits in Harmonic-Balance Simulators 270 8.3 Numerical Analysis of a Voltage-Controlled Oscillator 272 8.4 Numerical Analysis of a Five-Element Linear Coupled-Oscillator Array 278 8.5 Numerical Analysis of an Externally Injection-Locked Five-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 Arrays 297 9.1 Preliminary Concepts of Convex Optimization 297 9.2 Beamforming in COAs 301 9.3 Stability Optimization of the Coupled-Oscillator Steady-State Solution 308 9.4 Multi-Beam Pattern Generation Using Coupled-Oscillator Arrays 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 Directions 321 REFERENCES 325 ACRONYMS AND ABBREVIATIONS 341 INDEX 345

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