Mutual Coupling Between Antennas

A guide to mutual coupling between various types of antennas in arrays such as wires, apertures and microstrip patches or antennas co-sited on platforms

Mutual Coupling Between Antennas explores the theoretical underpinnings of mutual coupling, offers an up-to-date description of the physical effects of mutual coupling for a variety of antennas, and contains techniques for analysing and assessing its effects. The book puts the topic in historical context, presents an integral equation approach, includes the current techniques, measurement methods, and discusses the most recent advances in the field.

With contributions from noted experts on the topic, the book reviews practical aspects of mutual coupling and examines applications that clearly demonstrate where the performance is impacted both positively and negatively. Mutual Coupling Between Antennas contains information on how mutual coupling can be analysed with a wide range of methods from direct computer software using discrete methods, to integral equations and Greens function methods as well as approximate asymptotic methods. This important text:

• Provides a theoretical background for understanding mutual coupling between various types of antennas
• Describes the interaction that occurs between antennas, both planned and unplanned
• Explores a key aspect of arrays in any wireless, radar or sensing system operating at radio frequencies
• Offers a groundbreaking book on antenna mutual coupling

Written for antenna engineers, technical specialists, researchers and students, Mutual Coupling Between Antennas is the first book to examine mutual coupling between various types of antennas including wires, horns, microstrip patches, MIMO antennas, co-sited antennas and arrays in planar or conformal configurations.

Preface xv

Acknowledgments xvii

List of Contributors xix

Notation xxi

1 Introduction 1
Trevor S. Bird

1.1 Aims and Scope 1

1.2 Historical Perspective 3

1.3 Overview of Text 4

References 7

2 Basics of Antenna Mutual Coupling 9
Trevor S. Bird

2.1 Introduction 9

2.2 Electromagnetic Field Quantities 9

2.2.1 Definitions 9

2.2.2 Field Representations in Source-Free Regions 11

2.3 Mutual Coupling Between Elementary Sources 12

2.3.1 Radiation 12

2.3.2 Generalized Infinitesimal Current Elements 14

2.3.3 Mutual Coupling Between Infinitesimal Current Elements 15

2.4 Network Representation of Mutual Coupling 18

2.4.1 Extension to Combination of Elements 18

2.4.2 Mutual Impedance and Admittance Matrix Formulation 19

2.4.3 Scattering Matrix Representation 20

2.5 Radiation from Antennas in the Presence of Mutual Coupling 23

2.5.1 Far-Field Radiation 23

2.5.2 Magnetic Current Only 25

2.5.3 Electric Current Only 25

2.6 Conclusion 26

References 26

3 Methods in the Analysis of Mutual Coupling in Antennas 27
Trevor S. Bird

3.1 Introduction 27

3.2 Mutual Coupling in Antennas with Continuous Sources 30

3.2.1 Impedance and Admittance with Continuous Sources 30

3.2.2 Reaction 31

3.2.3 Definition of Circuit Quantities 32

3.3 On Finite and Infinite Arrays 34

3.3.1 Finite Array Analysis by Element-by-Element Method 35

3.3.2 Infinite Periodic Array Analysis 36

3.4 Integral Equation Methods Used in Coupling Analysis 36

3.4.1 Introduction 36

3.4.2 Green’s Function Methods 37

3.4.2.1 Free-Space Green’s Function for Harmonic Sources 38

3.4.2.2 Free-Space Green’s Function for Transient Sources 40

3.4.2.3 Fields with Sources 40

3.4.3 Solution by Weighted Residuals 43

3.5 Some Other Methods Used in Coupling Analysis 46

3.5.1 Unit Cell Analysis in Periodic Structure Method 46

3.5.2 Mode Matching Methods 51

3.5.3 Moment Methods 52

3.5.4 Method of Characteristic Modes 52

3.5.5 Minimum Scattering Element Method 53

3.6 Practical Aspects of Numerical Methods in Mutual Coupling Analysis 54

3.6.1 Introduction 54

3.6.2 Numerical Quadrature 55

3.6.3 Matrix Inversion 56

3.7 Conclusion 58

References 59

4 Mutual Coupling in Arrays of Wire Antennas 63
Trevor S. Bird

4.1 Introduction 63

4.2 Formulation of the Problem 63

4.2.1 Moment Method 66

4.2.2 Moment Method Solution for the Dipole 67

4.3 Mutual Impedance 68

4.3.1 Closed Form Expressions for Mutual Impedance 70

4.3.2 Asymptotic Approximations to Mutual Impedance 73

4.4 Arrays of Wire Antennas 76

4.4.1 Full-Wave Dipole Above a Perfect Ground 77

4.4.2 The Yagi–Uda Array 80

4.4.3 7 X 7 Array of Closely Packed Elements 83

4.5 Concluding Remarks 84

References 84

5 Arrays of Planar Aperture Antennas 87
Trevor S. Bird

5.1 Introduction 87

5.2 Mutual Coupling in Waveguide and Horn Arrays 88

5.2.1 Integral Equation Formulation 88

5.2.2 Modal Representation 91

5.2.3 Modeling of Profiled Horns and Mode Matching 94

5.2.4 Asymptotic Approximation of Mutual Admittance 97

5.3 Coupling in Rectangular Waveguides and Horns 99

5.3.1 Self-Admittance of TE 10 Mode 102

5.3.2 Example of Mutual Coupling Between Different-Sized Waveguides 104

5.3.3 Application to Horns 106

5.3.4 Waveguide-Fed Slot Arrays 111

5.3.5 Asymptotic Approximation of Coupling in Rectangular Apertures 112

5.3.6 Coupling in Horns Approximated with Quadratic Phase 114

5.4 Coupling in Arrays of Coaxial Waveguides and Horns 114

5.4.1 Self-Admittance of TE 11 Mode in Coaxial Waveguide 118

5.4.2 TEM Mode Coupling in Coaxial Waveguide 120

5.4.3 Asymptotic Approximation of Coupling in Coaxial Waveguide Apertures 123

5.4.4 Coaxial and Circular Aperture Array Examples 127

5.5 Mutual Coupling Between Apertures of General Cross-Section 129

5.5.1 Elliptical Apertures 129

5.5.2 General Apertures 134

5.6 Coupling in Apertures Loaded with Dielectrics and Metamaterials 135

5.6.1 Dielectric-Loaded Apertures 136

5.6.2 Metamaterial-Loaded Apertures 142

5.7 Concluding Remarks 148

References 148

6 Arrays of Microstrip Patch Antennas 153
Trevor S. Bird

6.1 Introduction 153

6.2 Representation of Mutual Coupling Between Patch Antennas 155

6.2.1 E-Current Model of Coupling 159

6.2.2 Cavity Model (H-Model) of Coupling 162

6.2.3 Full-Wave Solution 165

6.3 Applications of Microstrip Arrays 167

6.3.1 Mutual Coupling Between Microstrip Patches 167

6.3.2 Steering by Switching Parasitic Elements 167

6.3.3 A Metasurface from Microstrip Patches 170

6.4 Concluding Remarks 174

References 174

7 Mutual Coupling Between Antennas on Conformal Surfaces 177
Trevor S. Bird

7.1 Introduction 177

7.2 Mutual Admittance of Apertures on Slowly Curving Surfaces 178

7.2.1 Green’s Function Formulation for Curved Surfaces 178

7.2.2 The Cylinder 179

7.2.3 The Sphere 182

7.3 Asymptotic Solution for Fields Near Convex Surfaces 184

7.3.1 Review of Literature for Convex Surfaces 184

7.3.2 Asymptotic Solution for the Surface Fields 186

7.4 Mutual Coupling of Apertures in Quadric Surfaces 187

7.4.1 Closed-Form Expressions for Mutual Coupling Between Rectangular Waveguides in a Cylinder 188

7.4.2 Expressions for Mutual Coupling Between Circular Waveguides in a Sphere 194

7.4.3 Mutual Coupling Between Microstrip Patches on a Cylinder 197

7.5 Extension of Canonical Solution to Large Convex Surfaces with Slowly Varying Curvature 201

7.6 Applications of Coupling on Curved Surfaces 210

7.6.1 Mutual Coupling in a Waveguide Array on a Cylinder 210

7.6.2 Mutual Coupling Between Monopoles on a Cylinder 211

7.6.3 Mutual Coupling Between Waveguides on an Ellipsoid 215

7.7 Conclusion 216

References 217

8 Mutual Coupling Between Co-Sited Antennas and Antennas on Large Structures 221
Derek McNamara and Eqab Almajali

8.1 Preliminaries and Assumptions 221

8.1.1 The Problem at Hand 221

8.1.2 Course Adopted 223

8.2 Full-Wave CEM Modeling View of a Single Antenna 223

8.3 Full-Wave CEM Modeling View of Coupled Antennas in the Presence of a Host Platform 225

8.3.1 Field Point of View 225

8.3.2 Two-Port Network Parameter Point of View 227

8.4 Useful Expressions for Coupling in the Presence of a Host Platform 230

8.4.1 Motivation 230

8.4.2 Reciprocity and Reaction Theorems Revisited 230

8.4.3 Generalized Reaction Theorem 233

8.4.4 Expressions for Mutual Impedance and Open Circuit Voltage 234

8.4.5 Power Coupling 235

8.5 Supplementary Comments on CEM Modeling Methods 236

8.6 Full-Wave CEM Modeling of Coupled Antennas on a Platform – The Ideal 243

8.7 Reduced Complexity Antenna Electromagnetic Models 244

8.7.1 Necessity for Simplified Antenna Models 244

8.7.2 Huygens’ Box Model 244

8.7.3 Spherical Wave Expansion Models 246

8.7.4 Infinitesimal Dipole Models 246

8.7.5 Planar Aperture Models 247

8.7.6 Point Source Models 247

8.8 CEM Modeling of Coupled Antennas on a Platform – Pragmatic Approaches 247

8.9 Co-Sited Antenna Coupling Computation Examples 249

8.10 Concluding Remarks 251

References 251

9 Mutual Coupling and Multiple-Input Multiple-Output (MIMO) Communications 257
Karl F. Warnick

9.1 Introduction 257

9.2 Previous Work on Mutual Coupling and MIMO 258

9.3 Basics of MIMO Communications 260

9.3.1 MIMO Channel Capacity 261

9.3.2 Eigenchannels and the Water-Filling Solution 261

9.3.3 Eigenchannels in MIMO Systems and Beamforming Arrays 262

9.3.4 Reference Planes and the Intrinsic Channel Matrix 263

9.4 Mutual Coupling and MIMO Transmitting Arrays 264

9.4.1 Radiated Electric Field and Embedded Element Patterns 265

9.4.2 Pattern Overlap Matrix, Conservation of Energy, and Mutual Coupling 266

9.4.3 Gain and Directivity in the Overlap Matrix Formulation 267

9.4.4 Overlap Matrix for Isotropic Radiators 268

9.4.5 Mutual Coupling for Closely Spaced Elements, Superdirectivity, and Q-Factor Bounds 268

9.4.6 EEPs, Mutual Coupling, and Minimum Scattering Antennas 269

9.4.7 Mutual Coupling and Interactions Between Elements 269

9.4.8 Transmitter Power Constraint 271

9.4.9 Impedance Matching at the Transmitter 271

9.5 Mutual Coupling and MIMO Receiving Arrays 273

9.5.1 Receive Array Signal and Noise Model 273

9.5.2 Receive Array Thévenin Equivalent Network 274

9.5.3 Loaded Receive Array Output Voltages 275

9.5.4 External Noise and Loss Noise 276

9.5.5 Signal Correlation Matrix 277

9.5.6 Signal Correlation in a Rich Multipath Environment 277

9.5.7 Mutual Coupling, Noise Matching, and Equivalent Receiver Noise 278

9.5.7.1 Active Impedances for Receiving Arrays 279

9.5.7.2 Equivalent Receiver Noise Temperature and Active Impedance Matching 280

9.5.7.3 Noise Matching Efficiency 281

9.6 Conclusion 282

References 283

10 Mutual Coupling in Beamforming and Interferometric Antennas 287
Hoi Shun Antony Lui and Trevor S. Bird

10.1 Introduction 287

10.2 The Array Manifold 288

10.3 Direction-of-Arrival Algorithms 288

10.3.1 Matrix Pencil Method for Direction of Arrival Estimation 290

10.4 Maximum Gain Design for Single and Multiple Beams 292

10.4.1 Penalty Function Optimization of Array Parameters 296

10.4.2 Method of Successive Projections 298

10.4.3 Comparison of Penalty Functions and Successive Projections 299

10.5 Direction-of-Arrival Estimation 302

10.5.1 No Coupling Situation 303

10.5.1.1 Cramer-Rao Lower Bound 303

10.5.1.2 Four-Element Linear Arrays with Different Apertures (Two Incoming Signals) 304

10.5.1.3 Fixed Aperture Uniform Linear Arrays with Different Numbers of Elements (Two Incoming Signals) 306

10.5.1.4 Fixed Aperture Uniform Linear Arrays with Different Number of Elements (Three Incoming Signals) 308

10.5.2 Perturbation Due to Mutual Coupling 308

10.5.2.1 Eight-Element Linear Arrays with Different Apertures (Three Incoming Signals) 310

10.5.2.2 Fixed Array Aperture with Different Numbers of Elements (Two Incoming Signals) 316

10.6 Conclusion 319

References 320

11 Techniques for Minimizing Mutual Coupling Effects in Arrays 325
Hoi Shun Antony Lui and Trevor S. Bird

11.1 Introduction 325

11.2 Mutual Coupling in Transmitting and Receiving Arrays 326

11.2.1 The Mutual Coupling Path 326

11.2.2 Moment Method Analysis 327

11.3 Typical Methods for Minimizing Mutual Coupling 330

11.3.1 Aperture Field Taper 331

11.3.2 Electromagnetic Fences 331

11.3.3 Other Approaches to Compensation 331

11.4 Techniques for Practical Mutual Coupling Compensation 332

11.4.1 Conventional Mutual Impedance Method 332

11.4.2 Full-Wave Method 335

11.4.3 Receiving-Mutual-Impedance Method 337

11.4.3.1 Determination of the Receiving Mutual Impedance 340

11.4.3.2 Comparison Between Different Mutual Impedances and Direction-Finding Applications 343

11.4.4 Calibration Method 347

11.4.5 Compensation Through Beamforming Network 348

11.4.6 Compensation in the Aperture 349

11.5 Concluding Remarks 354

References 355

12 Noise Performance in the Presence of Mutual Coupling 357
Christophe Craeye, Jean Cavillot and Eloy de Lera Acedo

12.1 Generalities About Noise in Receiving Arrays 357

12.2 Coupling of Noise Originating from LNAs 359

12.3 Coupling of Noise Originating from Lossy Antenna Arrays 362

12.4 Coupling of Noise Originating from the Far-Field Environment 363

12.5 Conclusion 366

References 367

13 Methods for Analyzing Mutual Coupling in Large Arrays 369
Christophe Craeye and Ha Bui Van

13.1 Goals of Numerical Mutual Coupling Analysis 369

13.2 Periodic Method of Moments 372

13.3 Iterative Solution Techniques 374

13.4 Macro Basis Functions 376

13.5 Pattern Transformations 380

13.6 Optimization 382

13.7 Conclusion 383

References 384

14 Measurement of Mutual Coupling Effects 389
Alpha O. Bah and Trevor S. Bird

14.1 Introduction 389

14.2 Instrumentation 389

14.3 Basic Measurement of Static Element Coupling and Radiation 391

14.3.1 Measurement of Coupling Coefficients 391

14.3.1.1 Input Reflection Coefficient and Insertion Loss 392

14.3.1.2 Mutual Coupling Coefficients 393

14.3.2 Measurement of Element Radiation 393

14.4 Measurement of Active Element Coupling and Array Radiation 398

14.4.1 Measurement of Active Element Patterns 398

14.4.2 Measurement of Array Radiation Patterns 399

14.4.2.1 Pattern Multiplication Method 400

14.4.2.2 The Unit Excitation Active Element Pattern Method 402

14.4.2.3 The Average Active Element Pattern Method 402

14.4.2.4 The Hybrid Active Element Pattern Method 403

14.4.3 Measurement of Input Mismatch and Coupling 403

14.4.3.1 Mutual Coupling Coefficient Method 404

14.4.3.2 Directional Coupler Method 405

14.4.3.3 Power Divider Method 406

14.4.4 Measurement of Gain 407

14.5 Conclusion 409

References 410

Appendix A Useful Identities 413
Trevor S. Bird

A. 1 Vector Identities 413

A. 2 Geometric Identities 414

A. 3 Transverse Representation of the Electromagnetic Field 415

A. 4 Useful Functions 415

A. 5 Complex Fresnel Integrals 416

A. 6 Hypergeometric Function 417

References 417

Appendix B Bessel and Hankel Functions 419
Trevor S. Bird

B.1 Properties 419

B.2 Series Involving Bessel Functions 422

B.3 Integrals of Bessel Functions 422

B.4 Lommel-Type Integrals 424

References 424

Appendix C Properties of Hankel Transform Functions 425
Trevor S. Bird

References 426

Appendix D Properties of Surface Fock Functions 429
Trevor S. Bird

D. 1 Definitions 429

D. 2 Soft Surface Functions (m > 0) 429

D. 3 Hard Surface Fock Functions (m < 0) 431

D. 4 Hard Surface Fock Function of the First Kind 432

References 432

Appendix E Four Parameter Noise Representation of an Amplifier 433
Christophe Craeye, Jean Cavillot and Eloy de Lera Acedo

Reference 434

Appendix F Equivalent Noise Currents 435
Christophe Craeye, Jean Cavillot and Eloy de Lera Acedo

Reference 436

Appendix G Basic Reciprocity Result 437
Christophe Craeye, Jean Cavillot and Eloy de Lera Acedo

Appendix H On the Extended Admittance Matrix 439
Christophe Craeye and Ha Bui Van

Index 441

Trevor S. Bird, PhD, is Principal Antengenuity, Distinguished Visiting Professor University of Technology, Sydney, and Adjunct Professor Macquarie University, Australia.