Description
Latest research and applications of analytical models and designs of phased array antennas and systems
Phased Array Antennas presents precise analytical models and designs of phased array antennas and systems in a logical and comprehensive manner starting from fundamental principles of phased array radiation. Numerous relevant and practical design examples and detailed derivations of theorems and concepts are included to make the book as self-contained as possible.
This new edition includes information on recent developments in the phased array arena published in journals and conference proceedings, a chapter devoted to metamaterial analysis from fundamental principles of electromagnetism, and the most updated phased array structures and beam formers.
The book is divided into three sections. The first section is mostly devoted to the development of the Floquet model-based approach for infinite and finite phased arrays. This section begins with an introduction of the traditional approach and its drawbacks. The second section presents applications of the Floquet modal analysis to important phased array structures. The third section covers several important aspects of a phased array design including array tolerance analysis.
Phased Array Antennas includes information on sample topics such as:
- Scan characteristics of a pencil beam array in light of gain, grating lobe, beam-width beam squint, and mitigation of beam-squint issue
- Scan characteristics of array-fed confocal reflectors
- The relationship between a Floquet mode and observable array parameters such as active element pattern, mutual coupling, scan blindness
- Generalized Scattering Matrix (GSM) approach to analyze multilayer array structures, including stacked patches, ring-slot radiators, FSS and screen polarizers
- Finite array modeling including the edge effects on input match and radiation pattern
- Characteristics of step and multi-flared horns for enhanced radiation efficiency
- Frequency selective surfaces, meander line polarizer screens, printed reflect-array antennas, metamaterial and their fundamental properties
- Beam formers, beam shaping algorithms, array system analysis and array tolerance analysis
Phased Array Antennas is an excellent reference for advanced graduate students who are seeking professional careers in antenna and microwave engineering as well as antenna design engineers and technical consultants seeking to understand and apply the latest Floquet model for analyzing practical phased array structures.
Table of Contents
About the Author xix
Foreword xxi
Preface xxiii
Acknowledgments xxix
About the Companion Websitexxxi
1 Phased Array Fundamentals: Pattern Analysis and Synthesis 1
1.1 Introduction 1
1.2 Array Fundamentals 1
1.2.1 Element Pattern, Directivity, and Gain 2
1.2.2 Co-polarization and Cross-polarization 4
1.2.3 Array Pattern 6
1.2.4 Array Gain 9
1.2.5 Maximum Array Gain Theorem 10
1.2.6 Array Taper Efficiency 12
1.3 Pencil Beam Array 13
1.3.1 Scan Loss and Beam Broadening 13
1.3.2 Scan Array Design Consideration 16
1.3.3 Grating Lobes 17
1.3.3.1 Planar Array 19
1.3.4 Fixed-Value Phase Shifter Versus True Time Delay Phase Shifter 23
1.3.5 Phase Quantization Effects 25
1.3.6 Two-Level Hybrid Beam Forming Network (BFN) 28
1.4 Linear Array Synthesis 33
1.4.1 Array Factor: Schelkunoff’s Polynomial Representation 33
1.4.2 Binomial Array 35
1.4.3 Dolph–Chebyshev Array 38
1.4.3.1 Beam Width and Array Taper Efficiency 43
1.4.4 Taylor’s Line Source Synthesis 44
1.4.4.1 Dolph–Chebyshev Pattern for Continuous Source 45
1.4.4.2 Source Distribution 46
1.4.4.3 Taylor’s Line Source 47
1.4.4.4 Beam Width and Aperture Efficiency 50
1.4.5 Bayliss Difference Pattern Synthesis 50
1.5 Planar Aperture Synthesis 54
1.5.1 Taylor’s Circular Aperture Synthesis 55
1.5.2 Bayliss Difference Pattern Synthesis 59
1.6 Discretization of Continuous Source 61
1.7 Array-Fed Reflector 64
1.7.1 Scan Characteristics 65
1.7.2 Array Size and Array Orientation 67
1.7.3 Optimum Array Shape for Shaped Scan Coverage 69
1.8 Confocal Array-Fed Reflector 70
1.8.1 Radiation Characteristics 71
1.8.1.1 Feed- and Aperture-Field Distributions of Parabolic Reflector 71
1.8.1.2 Aperture Distribution of CAFR 73
1.8.2 Scan Gain With and Without Iris 73
1.9 Summary 76
References 77
Bibliography 78
Problems 78
2 Introduction to Floquet Modes in Infinite Arrays 83
2.1 Introduction 83
2.2 Fourier Spectrum and Floquet Series 84
2.2.1 Fourier Transform 84
2.2.2 Periodic Function: Fourier Series 85
2.2.3 Floquet Series 87
2.2.4 Two-Dimensional Floquet Series 91
2.2.4.1 Rectangular Grid 93
2.2.4.2 Equilateral Triangular Grids 93
2.3 Floquet Excitation and Floquet Modes 94
2.3.1 Main Beam and Grating Beams 96
2.4 Two-Dimensional Floquet Excitation 97
2.4.1 Circle Diagram: Rectangular Grid 99
2.4.2 Circle Diagram: Isosceles Triangular Grid 101
2.5 Grating Beams from Geometrical Optics 103
2.6 Floquet Mode and Guided Mode 104
2.7 Summary 108
References 108
Problems 109
3 Floquet Modal Functions 115
3.1 Introduction 115
3.2 TEz and TMz Floquet Vector Modal Functions 115
3.2.1 TEz Floquet Modal Fields 116
3.2.2 TMz Floquet Modal Fields 119
3.3 Infinite Array of Electric Surface Current on Dielectric Coated Ground Plane 121
3.3.1 TEzmn and TMzmn Modal Source Decomposition 122
3.3.2 TEzmn Fields 124
3.3.3 TMzmn Fields 126
3.3.4 Floquet Impedance 127
3.4 Determination of Blind Angles 129
3.5 Active Element Pattern 133
3.5.1 Array Pattern Using Superposition 135
3.5.2 Array Pattern Using Floquet Modal Expansion 136
3.5.3 Active Element Gain Pattern 138
3.6 Array of Rectangular Horn Apertures 141
3.6.1 Waveguide Modes and Modal Fields 142
3.6.2 Waveguide Modes to Floquet Modes 143
3.6.3 Reflection and Transmission Matrices 144
3.6.4 TE 10 Incident Mode 150
References 153
Bibliography 153
On Printed Dipole Elements and Arrays 153
On Waveguide Arrays 154
Problems 154
4 Finite Array Analysis 161
4.1 Introduction 161
4.2 Symmetry Property of Floquet Admittance 162
4.2.1 Floquet Admittance of Primary Source 164
4.3 Mutual Coupling 169
4.3.1 Mutual Impedance 169
4.3.2 Mutual Admittance 171
4.3.3 Scattering Matrix Elements 171
4.4 Array of Multimodal Sources 172
4.5 Mutual Coupling in Two-Dimensional Arrays 173
4.5.1 Rectangular Grid 173
4.5.2 General Grid 174
4.6 Active Input Impedance of Finite Array 176
4.6.1 Non-excited Elements Open-Circuited 176
4.6.2 Non-excited Elements Short-Circuited 177
4.6.3 Non-excited Elements Match-Terminated 177
4.7 Active Reflection Coefficient of Open-Ended Waveguide Array 178
4.7.1 Real Finite Array – Floquet, PWS, and MoM 180
4.8 Radiation Patterns of Finite Array 182
4.8.1 Non-excited Elements Open-Circuited 182
4.8.2 Non-excited Elements Short-Circuited 185
4.8.3 Non-excited Elements Match-Terminated 185
4.9 Radiation Patterns of Open-Ended Waveguide Array 185
4.10 Active Element Patterns of Real Finite Array 187
4.10.1 Waveguide Array on Infinite Ground Plane 187
4.10.2 Waveguide Array on Finite Ground Plane 189
4.10.2.1 Array-Aperture Field Excited by a Single Element 189
4.10.2.2 The Active Element Pattern 191
4.10.2.3 Array Pattern of a Real Finite Array 196
4.11 Array with Nonuniform Spacing 197
4.12 Finite Array Analysis Using Convolution 198
4.12.1 Convolution Relation for Aperture Field 198
4.12.2 Mutual Impedance 199
References 200
Bibliography 201
Problems 202
5 Array of Subarrays 207
5.1 Introduction 207
5.2 Subarray Analysis 208
5.2.1 Subarray Impedance Matrix: Eigenvector Approach 209
5.3 Subarray with Rectangular Grid 213
5.4 Subarrays with Arbitrary Grid 214
5.5 Subarray and Grating Lobes 216
5.6 Active Subarray Pattern 218
5.7 Four-element Subarray Fed by a Power Divider 223
5.7.1 E-Plane Subarray 224
5.7.2 H-Plane Subarray 224
5.8 Subarray Blindness 227
5.9 Subarray with Sequentially Rotating Elements 228
5.9.1 Directivity Analysis 229
5.9.2 Mutual Coupling Effects on Directivity 230
5.10 Concluding Remarks 234
References 234
Bibliography 235
Problems 237
6 The GSM Approach for Multilayer Array Structures 243
6.1 Introduction 243
6.2 Generalized Scattering Matrix Approach 244
6.3 GSM Cascading Rule 245
6.4 Transmission Matrix Representation 249
6.5 Building Blocks for GSM Analysis 250
6.5.1 Dielectric Layer 250
6.5.2 Dielectric Interface 252
6.5.3 Array of Patches 253
6.5.3.1 Simplification of Integrals 258
6.5.3.2 Basis Functions 260
6.6 Equivalent Impedance Matrix of Patch Layer 261
6.7 Stationary Character of MoM Solutions 265
6.7.1 Stationary Expression 265
6.7.2 GSM from Stationary Expression 269
6.8 Convergence of MoM Solutions 272
6.8.1 Number of Coupling Modes (M) 273
6.8.2 Failure of MoM Analysis 274
6.8.3 Lower Limit for Number of Expansion Modes (Nff) 276
6.8.4 Number of Basis Functions 280
6.9 Advantages of GSM Approach 280
6.10 Other Numerical Methods 281
References 281
Bibliography 282
Problems 283
7 Analysis of Microstrip Patch Arrays 287
7.1 Introduction 287
7.2 Probe-Fed Patch Array 288
7.2.1 Generalized Impedance Matrix of Probe Layer 288
7.2.2 Input Impedance 292
7.2.3 Impedance Characteristics 293
7.2.4 Active Element Patterns 297
7.2.5 Multilayer Array and Bandwidth Limits 299
7.3 Electromagnetically Coupled Patch Array 302
7.4 Slot-Fed Patch Array 303
7.4.1 Microstripline to Slot Transition 304
7.4.2 Input Impedance 309
7.4.3 Active Element Patterns 310
7.5 Stripline-Fed Slot-Coupled Array 314
7.6 Finite Patch Array 315
7.7 Ring-slot Element 317
7.7.1 Input Match and Scan Performance 319
7.7.2 Bandwidth Improvement 321
References 321
Bibliography 322
Problems 323
8 Array of Waveguide Horns 327
8.1 Introduction 327
8.2 Linearly Flared Horn Array 328
8.2.1 Reflection Coefficient 329
8.2.2 Active Element Pattern 332
8.3 Grazing Lobes and Pattern Nulls 334
8.3.1 Aperture Admittance 334
8.3.2 Equivalent Circuit 336
8.3.3 Reflection Loss at Grazing Lobe Condition 337
8.4 Surface and Leaky Waves in Array 340
8.4.1 Surface Wave 342
8.4.2 Leaky Wave 347
8.4.3 Super Gain Phenomenon 349
8.5 Wide-Angle Impedance Matching (WAIM) 351
8.5.1 WAIM: Admittance Perspective 353
8.6 Multimodal Rectangular/Square Horn Elements 357
8.6.1 Potter Horn 357
8.6.2 High-efficiency Horn 358
8.7 Multimodal Step-Circular Horn 360
8.8 Muti-flared High-efficiency Horns 361
8.9 Excitation of Waveguide Horn 362
References 365
Bibliography 366
Problems 367
9 FSS, Polarizer, and Reflect-array Analysis 371
9.1 Introduction 371
9.2 Frequency-Selective Surface 371
9.2.1 Reflection and Transmission Characteristics 372
9.2.2 Cross-polarization Performance 376
9.2.3 FSS-Loaded Antenna 378
9.3 Screen Polarizer 382
9.3.1 Analysis 382
9.3.2 Meander Susceptance 384
9.3.3 Reflection Coefficient and Axial Ratio 384
9.3.4 Scan Characteristics 387
9.4 Printed Reflect Array 389
9.4.1 Phase Characteristics 390
9.4.2 Design and Performance 393
9.4.3 Circular Polarization 396
9.4.4 Bandwidth Enhancement 399
9.4.5 Contour-Beam Reflect Array 402
References 402
Bibliography 402
Problems 403
10 Metamaterial Analysis and Fundamental Properties 407
10.1 Introduction 407
10.2 Origin of Constitutive Parameters: Maxwellian Point of View 409
10.3 Solutions of Maxwell’s Equation in a Periodic Array Medium 411
10.3.1 Transmission Matrix and Eigenmode 412
10.3.2 Properties of the Eigenvectors 413
10.3.2.1 Symmetrical Obstacles 414
10.3.2.2 Orthogonality of Eigenvectors 414
10.3.2.3 Eigenvector Component Ratio for 2D PEC Obstacles 417
10.3.2.4 More on Orthogonality for 2D obstacles 417
10.4 Constitutive Parameters for 2D Obstacles 418
10.4.1 Numerical Results 423
10.5 Constitutive Parameters for 3D Obstacles 427
10.5.1 Numerical Results 429
10.6 On the Uniqueness and Structure of [ε] Tensor 430
10.7 The Lorentz Model Versus Floquet Model 434
10.8 Floquet Versus Lorentz: Test on Metamaterial Slab 436
10.8.1 Characteristic Modes 437
10.8.2 Mode Matching 438
10.9 Floquet Versus Lorentz: Nonresonant Obstacles 441
10.10 Mode Degeneracy and Non-Maxwellian Medium 443
10.10.1 Finite Slab with SRR Obstacles 449
10.11 Concluding Remarks 451
References 452
Bibliography 454
Problems 456
11 Multilayer Array Analysis with Different Periodicities and Cell Orientations 459
11.1 Introduction 459
11.2 Layers with Different Periodicities: Rectangular Lattice 460
11.2.1 Patch-Fed Patch Subarray 463
11.3 Nonparallel Cell Orientations: Rectangular Lattice 465
11.3.1 Patch Array Loaded with Screen Polarizer 468
11.4 Layers with Arbitrary Lattice Structures 473
11.5 Summary 475
References 475
Bibliography 475
Problems 476
12 Shaped Beam Array Design: Optimization Algorithms 479
12.1 Introduction 479
12.2 Array Size: Linear Array 480
12.3 Element Size 483
12.4 Pattern Synthesis (Woodward–Lawson’s Method) 486
12.5 Gradient Search Algorithm 489
12.5.1 Mathematical Foundation 490
12.5.2 Array Synthesis Using GSA 491
12.5.3 Phase-only Optimization 494
12.5.4 Contour Beams Using Phase-only Optimization 496
12.6 Conjugate Field Matching Algorithm 500
12.7 Successive Projection Algorithm 505
12.7.1 Successive Projection and Conjugate Field Matching 508
12.8 Projection Matrix Algorithm 509
12.8.1 Implementation of the Algorithm 510
12.8.2 Convergence of Solution 511
12.8.3 Consideration for Numerical Computations 513
12.9 Other Optimization Algorithms 514
12.10 Design Guidelines of Shaped Beam 515
12.11 Pattern Null Algorithms 516
12.11.1 Gram–Schmidt Algorithm 517
12.11.2 Projection Matrix Algorithm for Nulls 520
12.11.2.1 Sector Nulling 522
12.11.2.2 Wide-band Nulling 524
12.11.2.3 Nulling of Shaped and Low Side-lobe Beams 526
References 529
Bibliography 530
Problems 530
13 Beam Forming Networks in Multiple-Beam Arrays 533
13.1 Introduction 533
13.2 BFN Using Power Dividers/Combiners 534
13.3 Butler Matrix Beam Former 534
13.3.1 Orthogonal Beams 535
13.3.2 Fourier Transform and Excitation Coefficients 537
13.3.3 FFT Algorithm 538
13.3.4 FFT and Butler Matrix 542
13.3.5 Hybrid Matrix 544
13.3.6 Modified Butler BFN for Nonuniform Taper 545
13.3.7 Beam Port Isolation 546
13.3.8 Three-Dimensional BFN 548
13.4 Blass Matrix BFN 548
13.5 Rotman Lens 550
13.5.1 Rotman Surface Design 552
13.5.2 Numerical Results 555
13.5.3 Physical Implementation 557
13.5.4 Scattering Matrix 559
13.5.4.1 Spectral Domain Formulation 560
13.6 Digital Beam Former 566
13.6.1 Digital Phase Shifter 567
13.6.2 System Characteristics 569
13.7 Optical Beam Formers 569
References 570
Bibliography 571
Problems 571
14 Active Phased Array Antenna 575
14.1 Introduction 575
14.2 Active Array Block Diagrams 576
14.3 Aperture Design of Array Antenna 578
14.3.1 Number of Elements and Element Size 578
14.3.2 Radiating Element Design Consideration 580
14.4 Solid State Power Amplifier (SSPA) 582
14.4.1 System Characteristics 583
14.4.1.1 Input–Output Characteristics 583
14.4.1.2 The Power-Added Efficiency 584
14.4.1.3 Noise Figure 584
14.5 Phase Shifter 587
14.6 Time Delay Unit 588
14.7 Intermodulation Product 590
14.7.1 Estimation of SSPA Parameters from IM Data 592
14.7.2 IM Beam Locations 592
14.7.3 Multiple Channel Array: Noise Power Ratio (NPR) 594
14.7.4 AM–PM Conversion 596
14.8 Noise Temperature and Noise Figure of Antenna Subsystems 597
14.8.1 Antenna Noise Temperature 597
14.8.2 Noise Temperature and Noise Figure of Resistive Circuits 599
14.8.2.1 Thevenin’s Equivalent Circuit and Noise Temperature of a Resistor 599
14.8.2.2 Theorem of Equivalent Noise Temperature 600
14.8.2.3 Equivalent Noise Temperature of a Resistive Circuit at Different Temperatures 602
14.8.2.4 Noise Temperature of a Resistive Attenuator 603
14.8.2.5 Noise Temperature of a Power Divider/Combiner 606
14.9 Active Array System Analysis 607
14.10 Active Array Calibration 611
14.10.1 Two-Phase-State Method 611
14.10.2 Multiple-Phase-Toggle Method 612
14.10.3 Simultaneous Measurement: Hadamard Matrix Method 613
14.11 Concluding Remarks 615
References 616
Bibliography 617
Active Array Systems 617
SSPAs and Phase Shifters 617
Inter-modulation Product and Antenna Noise 619
Array Calibration 619
Problems 619
15 Statistical Analysis of Phased Array Antenna 621
15.1 Introduction 621
15.2 Array Pattern 622
15.3 Statistics of R and I 624
15.4 Probability Density Function of |F(u)| 629
15.4.1 Central Limit Theorem 630
15.4.2 PDF of R and I 632
15.4.3.1 PDF at Beam Peak 634
15.4.3.3 PDF at Null Locations and Raleigh PDF 637
15.5 Confidence Limits 638
15.5.1 Beam Peak 641
15.5.2 Side Lobe 641
15.5.3 Nulls 642
15.6 Element Failure Analysis 644
15.7 Concluding Remarks 648
References 650
Bibliography 650
Problems 650
Appendix A1 Shannon’s Sampling Theorem 653
Appendix A 2 A Proof of the Identity 655
Index 657
