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**Updated with color and gray scale illustrations, a companion website housing supplementary material, and new sections covering recent developments in antenna analysis and design**This book introduces the fundamental principles of antenna theory and explains how to apply them to the analysis, design, and measurements of antennas. Due to the variety of methods of analysis and design, and the different antenna structures available, the applications covered in this book are made to some of the most basic and practical antenna configurations. Among these antenna configurations are linear dipoles; loops; arrays; broadband antennas; aperture antennas; horns; microstrip antennas; and reflector antennas. The text contains sufficient mathematical detail to enable undergraduate and beginning graduate students in electrical engineering and physics to follow the flow of analysis and design. Readers should have a basic knowledge of undergraduate electromagnetic theory, including Maxwell’s equations and the wave equation, introductory physics, and differential and integral calculus. * Presents new sections on flexible and conformal bowtie, Vivaldi antenna, antenna miniaturization, antennas for mobile communications, dielectric resonator antennas, and scale modeling * Provides color and gray scale figures and illustrations to better depict antenna radiation characteristics * Includes access to a companion website housing MATLAB programs, Java-based applets and animations, Power Point notes, Java-based interactive questionnaires and a solutions manual for instructors * Introduces over 100 additional end-of-chapter problems __Antenna Theory: Analysis and Design, Fourth Edition__ is designed to meet the needs of senior undergraduate and beginning graduate level students in electrical engineering and physics, as well as practicing engineers and antenna designers.**Constantine A. Balanis** received his BSEE degree from the Virginia Tech in 1964, his MEE degree from the University of Virginia in 1966, his PhD in Electrical Engineering from The Ohio State University in 1969, and an Honorary Doctorate from the Aristotle University of Thessaloniki in 2004. From 1964 to 1970, he was with the NASA Langley Research Center in Hampton, VA, and from 1970 to 1983, he was with the Department of Electrical Engineering of West Virginia University. In 1983 he joined Arizona State University and is now Regents' Professor of Electrical Engineering. Dr. Balanis is also a life fellow of the IEEE. ANTENNA THEORY 3 Contents 9 Preface 15 About the Companion Website 21 1 Antennas 23 1.1 Introduction 23 1.2 Types of Antennas 25 1.2.1 Wire Antennas 25 1.2.2 Aperture Antennas 25 1.2.3 Microstrip Antennas 27 1.2.4 Array Antennas 27 1.2.5 Reflector Antennas 28 1.2.6 Lens Antennas 28 1.3 Radiation Mechanism 29 1.3.1 Single Wire 29 1.3.2 Two-Wires 32 1.3.3 Dipole 35 1.3.4 Computer Animation-Visualization of Radiation Problems 35 1.4 Current Distribution on a Thin Wire Antenna 37 1.5 Historical Advancement 40 1.5.1 Antenna Elements 41 1.5.2 Methods of Analysis 42 1.5.3 Some Future Challenges 43 1.6 Multimedia 43 References 44 2 Fundamental Parameters and Figures-of-Merit of Antennas 47 2.1 Introduction 47 2.2 Radiation Pattern 47 2.2.1 Radiation Pattern Lobes 48 2.2.2 Isotropic, Directional, and Omnidirectional Patterns 52 2.2.3 Principal Patterns 52 2.2.4 Field Regions 53 2.2.5 Radian and Steradian 55 2.3 Radiation Power Density 57 2.4 Radiation Intensity 59 2.5 Beamwidth 62 2.6 Directivity 63 2.6.1 Directional Patterns 69 2.6.2 Omnidirectional Patterns 73 2.7 Numerical Techniques 77 2.8 Antenna Efficiency 82 2.9 Gain, Realized Gain 83 2.10 Beam Efficiency 87 2.11 Bandwidth 87 2.12 Polarization 88 2.12.1 Linear, Circular, and Elliptical Polarizations 90 2.12.2 Polarization Loss Factor and Efficiency 93 2.13 Input Impedance 97 2.14 Antenna Radiation Efficiency 101 2.15 Antenna Vector Effective Length and Equivalent Areas 103 2.15.1 Vector Effective Length 103 2.15.2 Antenna Equivalent Areas 105 2.16 Maximum Directivity and Maximum Effective Area 108 2.17 Friis Transmission Equation and Radar Range Equation 110 2.17.1 Friis Transmission Equation 110 2.17.2 Radar Range Equation 112 2.17.3 Antenna Radar Cross Section 114 2.18 Antenna Temperature 118 2.19 Multimedia 122 References 125 Problems 127 3 Radiation Integrals and Auxiliary Potential Functions 149 3.1 Introduction 149 3.2 The Vector Potential A for an Electric Current Source J 150 3.3 The Vector Potential F for A Magnetic Current Source M 152 3.4 Electric and Magnetic Fields for Electric (J) and Magnetic (M) Current Sources 153 3.5 Solution of the Inhomogeneous Vector Potential Wave Equation 154 3.6 Far-Field Radiation 158 3.7 Duality Theorem 159 3.8 Reciprocity and Reaction Theorems 160 3.8.1 Reciprocity for Two Antennas 162 3.8.2 Reciprocity for Antenna Radiation Patterns 163 References 165 Problems 165 4 Linear Wire Antennas 167 4.1 Introduction 167 4.2 Infinitesimal Dipole 167 4.2.1 Radiated Fields 167 4.2.2 Power Density and Radiation Resistance 170 4.2.3 Radian Distance and Radian Sphere 172 4.2.4 Near-Field (kr ≪ 1) Region 173 4.2.5 Intermediate-Field (kr > 1) Region 174 4.2.6 Far-Field (kr ≫ 1) Region 175 4.2.7 Directivity 176 4.3 Small Dipole 177 4.4 Region Separation 180 4.4.1 Far-Field (Fraunhofer) Region 182 4.4.2 Radiating Near-Field (Fresnel) Region 184 4.4.3 Reactive Near-Field Region 185 4.5 Finite Length Dipole 186 4.5.1 Current Distribution 186 4.5.2 Radiated Fields: Element Factor, Space Factor, and Pattern Multiplication 186 4.5.3 Power Density, Radiation Intensity, and Radiation Resistance 188 4.5.4 Directivity 194 4.5.5 Input Resistance 195 4.5.6 Finite Feed Gap 197 4.6 Half-Wavelength Dipole 198 4.7 Linear Elements Near or On Infinite Perfect Electric Conductors (PEC), Perfect Magnetic Conductors (PMC) and Electromagnetic Band-Gap (EBG) Surfaces 201 4.7.1 Ground Planes: Electric and Magnetic 202 4.7.2 Image Theory 204 4.7.3 Vertical Electric Dipole 205 4.7.4 Approximate Formulas for Rapid Calculations and Design 213 4.7.5 Mobile Communication Devices and Antennas for Mobile Communication Systems 214 4.7.6 Horizontal Electric Dipole 217 4.8 Ground Effects 225 4.8.1 Vertical Electric Dipole 226 4.8.2 Horizontal Electric Dipole 227 4.8.3 PEC, PMC and EBG Surfaces 229 4.8.4 Earth Curvature 233 4.9 Computer Codes 238 4.10 Multimedia 238 References 240 Problems 242 5 Loop Antennas 257 5.1 Introduction 257 5.2 Small Circular Loop 258 5.2.1 Radiated Fields 258 5.2.2 Small Loop and Infinitesimal Magnetic Dipole 263 5.2.3 Power Density and Radiation Resistance 263 5.2.4 Near-Field (kr ≪ 1) Region 267 5.2.5 Far-Field (kr ≫ 1) Region 267 5.2.6 Radiation Intensity and Directivity 268 5.2.7 Equivalent Circuit 269 5.3 Circular Loop of Constant Current 272 5.3.1 Radiated Fields 272 5.3.2 Power Density, Radiation Intensity, Radiation Resistance, and Directivity 274 5.4 Circular Loop with Nonuniform Current 281 5.4.1 Arrays 288 5.4.2 Design Procedure 289 5.5 Ground and Earth Curvature Effects for Circular Loops 290 5.6 Polygonal Loop Antennas 291 5.7 Ferrite Loop 292 5.7.1 Radiation Resistance 292 5.7.2 Ferrite-Loaded Receiving Loop 293 5.8 Mobile Communication Systems Applications 294 5.9 Multimedia 294 References 297 Problems 299 6 Arrays: Linear, Planar, and Circular 307 6.1 Introduction 307 6.2 Two-Element Array 308 6.3 N-Element Linear Array: Uniform Amplitude and Spacing 315 6.3.1 Broadside Array 319 6.3.2 Ordinary End-Fire Array 321 6.3.3 Phased (Scanning) Array 324 6.3.4 Hansen-Woodyard End-Fire Array 326 6.4 N-Element Linear Array: Directivity 334 6.4.1 Broadside Array 335 6.4.2 Ordinary End-Fire Array 337 6.4.3 Hansen-Woodyard End-Fire Array 339 6.5 Design Procedure 340 6.6 N-Element Linear Array: Three-Dimensional Characteristics 341 6.6.1 N-Elements Along Z-Axis 341 6.6.2 N-Elements Along X- or Y-Axis 342 6.7 Rectangular-to-Polar Graphical Solution 344 6.8 N-Element Linear Array: Uniform Spacing, Nonuniform Amplitude 345 6.8.1 Array Factor 347 6.8.2 Binomial Array 348 6.8.3 Dolph-Tschebyscheff Array: Broadside 352 6.8.4 Tschebysheff Design: Scanning 366 6.9 Superdirectivity 367 6.9.1 Efficiency and Directivity 368 6.9.2 Designs with Constraints 368 6.10 Planar Array 370 6.10.1 Array Factor 370 6.10.2 Beamwidth 376 6.10.3 Directivity 381 6.11 Design Considerations 382 6.12 Circular Array 385 6.12.1 Array Factor 385 6.13 Multimedia 389 References 389 Problems 390 7 Antenna Synthesis and Continuous Sources 407 7.1 Introduction 407 7.2 Continuous Sources 408 7.2.1 Line-Source 408 7.2.2 Discretization of Continuous Sources 409 7.3 Schelkunoff Polynomial Method 409 7.4 Fourier Transform Method 414 7.4.1 Line-Source 414 7.4.2 Linear Array 417 7.5 Woodward-Lawson Method 420 7.5.1 Line-Source 421 7.5.2 Linear Array 425 7.6 Taylor Line-Source (Tschebyscheff-Error) 426 7.6.1 Design Procedure 428 7.7 Taylor Line-Source (One-Parameter) 430 7.8 Triangular, Cosine, and Cosine-Squared Amplitude Distributions 437 7.9 Line-Source Phase Distributions 438 7.10 Continuous Aperture Sources 439 7.10.1 Rectangular Aperture 440 7.10.2 Circular Aperture 440 7.11 Multimedia 442 References 442 Problems 443 8 Integral Equations, Moment Method, and Self and Mutual Impedances 453 8.1 Introduction 453 8.2 Integral Equation Method 454 8.2.1 Electrostatic Charge Distribution 454 8.2.2 Integral Equation 461 8.3 Finite Diameter Wires 461 8.3.1 Pocklington’s Integral Equation 462 8.3.2 Hallén’s Integral Equation 466 8.3.3 Source Modeling 467 8.4 Moment Method Solution 470 8.4.1 Basis (Expansion) Functions 471 8.4.2 Weighting (Testing) Functions 475 8.5 Self-Impedance 477 8.5.1 Integral Equation-Moment Method 477 8.5.2 Induced EMF Method 480 8.6 Mutual Impedance Between Linear Elements 485 8.6.1 Integral Equation-Moment Method 487 8.6.2 Induced EMF Method 489 8.7 Mutual Coupling in Arrays 496 8.7.1 Coupling in the Transmitting Mode 496 8.7.2 Coupling in the Receiving Mode 498 8.7.3 Mutual Coupling on Array Performance 498 8.7.4 Coupling in an Infinite Regular Array 498 8.7.5 Active Element Pattern in an Array 500 8.8 Multimedia 502 References 502 Problems 504 9 Broadband Dipoles and Matching Techniques 507 9.1 Introduction 507 9.2 Biconical Antenna 509 9.2.1 Radiated Fields 509 9.2.2 Input Impedance 512 9.3 Triangular Sheet, Flexible and Conformal Bow-Tie, and Wire Simulation 514 9.4 Vivaldi Antenna 518 9.5 Cylindrical Dipole 522 9.5.1 Bandwidth 523 9.5.2 Input Impedance 523 9.5.3 Resonance and Ground Plane Simulation 525 9.5.4 Radiation Patterns 525 9.5.5 Equivalent Radii 526 9.6 Folded Dipole 527 9.7 Discone and Conical Skirt Monopole 534 9.8 Matching Techniques 535 9.8.1 Stub-Matching 535 9.8.2 Quarter-Wavelength Transformer 536 9.8.3 Baluns and Transformers 543 9.9 Multimedia 545 References 546 Problems 547 10 Traveling Wave and Broadband Antennas 555 10.1 Introduction 555 10.2 Traveling Wave Antennas 555 10.2.1 Long Wire 557 10.2.2 V Antenna 565 10.2.3 Rhombic Antenna 570 10.3 Broadband Antennas 571 10.3.1 Helical Antenna 571 10.3.2 Electric-Magnetic Dipole 581 10.3.3 Yagi-Uda Array of Linear Elements 581 10.3.4 Yagi-Uda Array of Loops 601 10.4 Multimedia 602 References 602 Problems 604 11 Frequency Independent Antennas, Antenna Miniaturization, and Fractal Antennas 613 11.1 Introduction 613 11.2 Theory 614 11.3 Equiangular Spiral Antennas 615 11.3.1 Planar Spiral 616 11.3.2 Conical Spiral 620 11.4 Log-Periodic Antennas 620 11.4.1 Planar and Wire Surfaces 621 11.4.2 Dipole Array 624 11.4.3 Design of Dipole Array 630 11.5 Fundamental Limits of Electrically Small Antennas 636 11.6 Antenna Miniaturization 641 11.6.1 Monopole Antenna 642 11.6.2 Patch Antennas 648 11.6.3 Antenna Miniaturization Using Metamaterials 648 11.7 Fractal Antennas 649 11.8 Multimedia 655 References 655 Problems 657 12 Aperture Antennas 661 12.1 Introduction 661 12.2 Field Equivalence Principle: Huygens’ Principle 661 12.3 Radiation Equations 667 Summary 669 12.4 Directivity 670 12.5 Rectangular Apertures 670 12.5.1 Uniform Distribution on an Infinite Ground Plane 672 12.5.2 Uniform Distribution in Space 683 12.5.3 TE10-Mode Distribution on an Infinite Ground Plane 685 12.5.4 Beam Efficiency 688 12.6 Circular Apertures 689 12.6.1 Uniform Distribution on an Infinite Ground Plane 691 12.6.2 TE11-Mode Distribution on an Infinite Ground Plane 693 12.6.3 Beam Efficiency 697 12.7 Design Considerations 697 12.7.1 Rectangular Aperture 699 12.7.2 Circular Aperture 700 12.8 Babinet’s Principle 702 12.9 Fourier Transforms in Aperture Antenna Theory 706 12.9.1 Fourier Transforms-Spectral Domain 706 12.9.2 Radiated Fields 707 12.9.3 Asymptotic Evaluation of Radiated Field 711 12.9.4 Dielectric-Covered Apertures 716 12.9.5 Aperture Admittance 717 12.10 Ground Plane Edge Effects: The Geometrical Theory of Diffraction 724 12.11 Multimedia 729 References 729 Problems 731 13 Horn Antennas 741 13.1 Introduction 741 13.2 E-Plane Sectoral Horn 741 13.2.1 Aperture Fields 741 13.2.2 Radiated Fields 744 13.2.3 Directivity 750 13.3 H-Plane Sectoral Horn 755 13.3.1 Aperture Fields 755 13.3.2 Radiated Fields 756 13.3.3 Directivity 760 13.4 Pyramidal Horn 765 13.4.1 Aperture Fields, Equivalent, and Radiated Fields 766 13.4.2 Directivity 770 13.4.3 Design Procedure 776 13.5 Conical Horn 778 13.6 Corrugated Horn 783 13.7 Aperture-Matched Horns 788 13.8 Multimode Horns 791 13.9 Dielectric-Loaded Horns 793 13.10 Phase Center 795 13.11 Multimedia 796 References 797 Problems 800 14 Microstrip and Mobile Communications Antennas 805 14.1 Introduction 805 14.1.1 Basic Characteristics 806 14.1.2 Feeding Methods 807 14.1.3 Methods of Analysis 809 14.2 Rectangular Patch 810 14.2.1 Transmission-Line Model 810 14.2.2 Cavity Model 820 14.2.3 Directivity 833 14.3 Circular Patch 837 14.3.1 Electric and Magnetic Fields—TMzmnp 838 14.3.2 Resonant Frequencies 839 14.3.3 Design 840 14.3.4 Equivalent Current Densities and Fields Radiated 841 14.3.5 Conductance and Directivity 843 14.3.6 Resonant Input Resistance 844 14.4 Quality Factor, Bandwidth, and Efficiency 845 14.5 Input Impedance 848 14.6 Coupling 849 14.7 Circular Polarization 852 14.8 Arrays and Feed Networks 854 14.9 Antennas for Mobile Communications 859 14.9.1 Planar Inverted-F Antenna (PIFA) 860 14.9.2 Slot Antenna 863 14.9.3 Inverted-F Antenna (IFA) 865 14.9.4 Multiband Antennas for Mobile Units 868 14.10 Dielectric Resonator Antennas 869 14.10.1 Basic DRA Geometries 870 14.10.2 Methods of Analysis and Design 871 14.10.3 Cavity Model Resonant Frequencies (TE and TM Modes) 872 14.10.4 Hybrid Modes: Resonant Frequencies and Quality Factors 874 14.10.5 Radiated Fields 877 14.11 Multimedia 880 References 884 Problems 889 15 Reflector Antennas 897 15.1 Introduction 897 15.2 Plane Reflector 897 15.3 Corner Reflector 898 15.3.1 90◦ Corner Reflector 900 15.3.2 Other Corner Reflectors 902 15.4 Parabolic Reflector 906 15.4.1 Front-Fed Parabolic Reflector 909 15.4.2 Cassegrain Reflectors 937 15.5 Spherical Reflector 942 15.6 Multimedia 945 References 945 Problems 947 16 Smart Antennas 953 16.1 Introduction 953 16.2 Smart-Antenna Analogy 953 16.3 Cellular Radio Systems Evolution 955 16.3.1 Omnidirectional Systems 955 16.3.2 Smart-Antenna Systems 958 16.4 Signal Propagation 961 16.5 Smart Antennas’ Benefits 964 16.6 Smart Antennas’ Drawbacks 965 16.7 Antenna 965 16.7.1 Array Design 965 16.7.2 Linear Array 966 16.7.3 Planar Array 967 16.8 Antenna Beamforming 968 16.8.1 Overview of Direction-Of-Arrival (DOA) Algorithms 969 16.8.2 Adaptive Beamforming 972 16.8.3 Mutual Coupling 975 16.8.4 Optimal Beamforming Techniques 977 16.9 Mobile Ad hoc Networks (MANETs) 982 16.9.1 Overview of Mobile Ad hoc NETworks (MANETs) 982 16.9.2 MANETs Employing Smart-Antenna Systems 983 16.10 Smart-Antenna System Design, Simulation, and Results 986 16.10.1 Design Process 986 16.10.2 Single Element—Microstrip Patch Design 987 16.10.3 Rectangular Patch 987 16.10.4 Array Design 989 16.10.5 4 × 4 Planar Array versus 8 × 8 Planar Array 991 16.10.6 Adaptive Beamforming 991 16.11 Beamforming, Diversity Combining, Rayleigh-Fading, and Trellis-Coded Modulation 994 16.12 Other Geometries 997 16.13 Multimedia 998 References 998 Problems 1002 17 Antenna Measurements 1003 17.1 Introduction 1003 17.2 Antenna Ranges 1004 17.2.1 Reflection Ranges 1005 17.2.2 Free-Space Ranges 1005 17.2.3 Compact Ranges 1008 17.2.4 Near-Field/Far-Field Methods 1014 17.3 Radiation Patterns 1022 17.3.1 Instrumentation 1023 17.3.2 Amplitude Pattern 1025 17.3.3 Phase Measurements 1025 17.4 Gain Measurements 1025 17.4.1 Realized-Gain Measurements 1028 17.4.2 Gain-Transfer (Gain-Comparison) Measurements 1031 17.5 Directivity Measurements 1032 17.6 Radiation Efficiency 1034 17.7 Impedance Measurements 1034 17.8 Current Measurements 1036 17.9 Polarization Measurements 1036 17.10 Scale Model Measurements 1041 17.10.1 Gain (Amplitude) Measurements, Simulations and Comparisons 1042 17.10.2 Echo Area (RCS) Measurements, Simulations and Comparisons 1043 References 1046 Appendix I: f(x) =sin(x)/x 1049 Appendix II: fN(x) =|sin(Nx)/N sin(x)| N = 1, 3, 5, 10, 20 1051 Appendix III: Cosine and Sine Integrals 1053 Appendix IV: Fresnel Integrals 1055 Appendix V: Bessel Functions 1057 Appendix VI: Identities 1063 Appendix VII: Vector Analysis 1067 Appendix VIII: Method of Stationary Phase 1077 Appendix IX: Television, Radio, Telephone, and Radar Frequency Spectrums 1083 Index 1087 EULA 1095 Updated with color and gray scale illustrations, a companion website housing supplementary material, and new sections covering recent developments in antenna analysis and design This book introduces the fundamental principles of antenna theory and explains how to apply them to the analysis, design, and measurements of antennas. Due to the variety of methods of analysis and design, and the different antenna structures available, the applications covered in this book are made to some of the most basic and practical antenna configurations. Among these antenna configurations are linear dipoles; loops; arrays; broadband antennas; aperture antennas; horns; microstrip antennas; and reflector antennas. The text contains sufficient mathematical detail to enable undergraduate and beginning graduate students in electrical engineering and physics to follow the flow of analysis and design. Readers should have a basic knowledge of undergraduate electromagnetic theory, including Maxwell’s equations and the wave equation, introductory physics, and differential and integral calculus. Presents new sections on flexible and conformal bowtie, Vivaldi antenna, antenna miniaturization, antennas for mobile communications, dielectric resonator antennas, and scale modeling Provides color and gray scale figures and illustrations to better depict antenna radiation characteristics Includes access to a companion website housing MATLAB programs, Java-based applets and animations, Power Point notes, Java-based interactive questionnaires and a solutions manual for instructors Introduces over 100 additional end-of-chapter problems Antenna Theory: Analysis and Design, Fourth Edition is designed to meet the needs of senior undergraduate and beginning graduate level students in electrical engineering and physics, as well as practicing engineers and antenna designers. Constantine A. Balanis received his BSEE degree from the Virginia Tech in 1964, his MEE degree from the University of Virginia in 1966, his PhD in Electrical Engineering from The Ohio State University in 1969, and an Honorary Doctorate from the Aristotle University of Thessaloniki in 2004. From 1964 to 1970, he was with the NASA Langley Research Center in Hampton, VA, and from 1970 to 1983, he was with the Department of Electrical Engineering of West Virginia University. In 1983 he joined Arizona State University and is now Regents' Professor of Electrical Engineering. Dr. Balanis is also a life fellow of the IEEE.