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Principal Antenna Theory and Design, 2nd Edition
Antenna Theory and Design, 2nd EditionWarren L. Stutzman, Gary A. Thiele
Highly respected authors have reunited to update the well known 1981 edition which is still hailed as one of the best in its field. This edition includes recent antenna innovations and applications. It features a succinct treatment of the finite difference, time domain (FDTD) computational technique. It is also the first text to treat physical theory of diffraction (PTD).
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Antenna Theory and Design SECOND EDmON Warren L. Stutzman Gary A. Thiele JOHN WILEY & SONS, INC. New York. Chichester. Weinheim • Brisbane. Toronto. Singapore ACQUISmONS EDITOR Charity Robey MARKETING MANAGER . Katherine Hepburr PRODUCTION EDITOR Ken Santor I ILLUSTRATION COORDINATOR Jaime Perea COVER DESIGN Carol Grobe The cover art is the radiation pattem'of a six-wavelength travelling wave line source and was created by George Ruhlmann. This book was set in 10/12 Times Ten by University Graphics, Inc. and printed and bound by Hamilton Printing Company. The cover was printed by Phoenix Color Corp. Recognizing the importance of preserving what has been written, it is a policy of John Wiley & Sons, Inc. to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end. I The paper in this book was manufactured by a mill whose forest management programs include sustained yield harvesting of its timberlands. Sustained yield harvesting principles ensure that the number of trees cut each year does not exceed the amount of new growth. Copyright © 1998 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508) 750-8400, fax (508) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail; email@example.com. \ C\ ~ \ Library of Congress Cataloging in Publication Data: Stutzman, Warren L. Antenna theory and design / Warren L. Stutzman, Gary A. Thiele.2nd ed. . em. p. Includes bibliographical references and index. ISBN 0-471-02590-9 (cloth: alk. paper) 1. Antennas (Electronics) I. Thiele, Gary A. II. Title. 97-35498 TK787~.§.,S19 1998 621:381'4-dc21 CIP / ":\ -::t2__ ~ Printed in the United States of America 109876 543 2 1 This book is dedicated to our wives, Claudia and Jo Ann and to our children Darren and Dana Eric, Scott, and Brad Preface Since the first edition of Antenna Theory and Design was published in 1981, there have been major expansions of application areas for antennas, mainly in wireless communications. In addition, in recent years new areas important to antennas have emerged into prominence. This second edition has been expanded to include new areas in antennas. Coverage of microstrip antennas (Sec. 5.8) as well as the use of antennas in systems and measurements (Chapter 9) have been added. In addition, the treatments of array antennas (Chapter 3), broadband antennas (Chapter 6), and aperture antennas (Chapter 7) have been expanded. Also, since the first edition was written there have been major advances in Computational Electromagnetics (CEM), due in part to the use of more sophisticated antennas and antenna systems. The second edition expands on the Method of Moments in Chapter 10, introduces a succinct treatment of the Finite Difference-Time Domain (FD-TD) technique for antennas in Chapter 11, and adds the topic of the Physical Theory of Diffraction (PTD) to high frequency methods in Chapter 12. The objective in the second edition has been to preserve the simplicity of the first edition, while adding modem topics. This book is a textbook and finds its widest use in the college classroom. Thus, the primary purpose is to emphasize the understanding of principles and the development of techniques for examining and designing antenna systems. Handbooks are available to supplement the fundamentals and antennas discussed here. We have found that the first edition is in wide use by practicing engineers as well as students. This is because of the applied nature of the material and the treatment of basic topics that are directly useable for analyzing practical antennas. This is illustrated by the material in Chapters 1 to 6 and 9, which do not rely heavily on mathematics and use calculus sparingly. Antenna Theory and Design covers antennas from three perspectives: antenna fundamentals, antenna techniques, and the design of popular antennas. The first four chapters stress antenna fundamentals. Since the student has probably had little exposure to antennas, many fundamentals are presented in Chapter 1. The emergence of antenna theory from Maxwell's equation!) is developed, along with a physical explanation of how antennas radiate. The four types of antenna elements (electrically small, resonant, broadband, and aperture) are introduced. The discipline of antennas has its own terminology that is quite different from other areas of engineering, so Chapter 1 includes definitions of many antenna terms. Chapter 2 examines simple radiating systems, such as dipoles, in order to solidify the principles of Chapter 1 and to equip the reader to move forward with analysis of antenna systems, such as arrays, that are treated in Chapter 3. Arrays are covered early in the book to introduce the relationship between the current distribution on an antenna and its spatial radiation characteristics using elementary mathematics. In addition, arrays are widely used in practice today. The discrete approach to antennas (arrays) is followed in Chapter 4 with line source antennas, which introduce the continuous form of antennas. Chapters 5 to 7 give details on commonly used antenna elements. Chapter 5 vii viii Preface surveys the resonant antenna elements encountered in practice, including dipoles, yagis, and microstrip patches. Chapter 6 covers broadband antennas such as helix, spiral, and log-periodic antennas. Chapter 7 treats aperture antennas. Emphasis in these chapters is on the operating principles using the fundamentals introduced in the Chapters 1 to 4 and on design guidelines. As appropriate, data are presented using numerical or experimental models, or computations based on theoretical formulations. In addition, empirical formulas are often presented for easy evaluation of performance parameters. The synthesis of arrays and continuous antennas is presented in Chapter 8 for shaped main-beam or low side-lobe applications. The use of antennas as devices in systems is covered in Chapter 9, along with antenna measurements. Chapters 10 to 12, as noted above, introduce CEM techniques for evaluating simple antenna elements as well as large complex antenna systems. Here, as ih all the book, actual code statements are not listed. The wide variety of computing environments and the availability of high-level mathematics applications packages makes this inappropriate and unnecessary. Instead, some key computational and visualization antenna software packages are made available on the World Wide Web (see Appendix G). It is important to be aware of the background that is assumed for this book. It is not necessary that the reader have complete mastery of the following subjects, but exposure to these topies is very helpful. A basic course in electromagnetics, such as is commonly required in engineering and physics, is assumed. Mathematics used often includes complex numbers, trigonometry, vector algebra, and the major coordinate systems (rectangular, cylindrical, and spherical). Vector calculus is used at various points and scalar integration is frequently used. This book can be readily adapted to various academic programs at both introductory and advanced levels. For a first course, the text is usually used in a senior elective or entry level graduate course. A one-semester introductory course usually covers Chapter 1 to 6. For a master's degree-level course, parts of Chapters 7, 8, and 9 can be added. In a one-quarter senior course, material in the latter parts of Chapters 3, 4, or 5 can be eliminated. A second course can focus on advanced design, synthesis, and systems using Chapters 7, 8, and 9. Alternatively, a second course can specialize on computational methods using Chapters 10 to 12. Several features have been included to aid in learning and in preparation for further self study. Defined terms follow the IEEE standard definitions. Literature references found at the end of each chapter provide sources for further reading. In addition, the bibliography in Appendix H lists literature sources by technical topic. The appendices also include information on the radio spectrum, data on materials, and important mathematical relations. The authors are indebted to the many individuals who provided invaluable technical assistance to this second edition. The reviewers of the entire manuscript (two of whom also reviewed the first edition) gave essential input on the organization of the book and on several technical issues. Many students offered critical remarks during classroom testing of the manuscript. In addition, special thanks are owed to those who gave detailed evaluations, including Keith Carver (Secs. 5.8 and 6.2), David Jackson (Sec. 5.8), Ahmad Safaai-Jazi (Sees. 6.2 and 8.4), Dave Olver (Chapter 11), Buck Walter (Secs. 4.4 and 10.12), Gerald Ricciardi (Sec. 5.8), Marco Terada (Sec. 7.6) and Krish Pasala (Sec. 12.15). One author (Gary Thiele) extends special thanks to his son, Eric T. Thiele, for many long, valuable discussions on FD~TD, Preface ix for reviewing Chapter 11, and for generating the numerical data used in several illustrations in Chapter 11. Finally, we recognize our wives, Claudia and Jo Ann, for enduring countless hours of neglect during the preparation for both editions. The same recognition goes to our children, Darren and Dana, and Eric, Scott, and Brad. Warren L. Stutzman Gary A. Thiele j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j Contents Chapter 1 • Antenna Fundamentals and Definitions 1 1.1 '" 1.2 X. 1.3 1.4 1.5 1.6 1.7 Introduction 1 How Antennas Radiate 4 Overview of Antennas 8 Electromagnetic Fundamentals 12 Solution of Maxwell's Equations for Radiation Problems 16 The Ideal Dipole 20 Radiation Patterns 24 1.7.1 Radiation Pattern Basics 24 1.7.2 Radiation from Line Currents 25 1.7.3 Far-Field Conditions and Field Regions 28 1.7.4 Steps in the Evaluation of Radiation Fields 31 1.7.5 Radiation Pattern Definitions 33 1.7.6 Radiation Pattern Parameters 35 1.8 Directivity and Gain 37 1.9 Antenna Impedance, Radiation Efficiency, and the Short Dipole 43 1.10 Antenna Polarization 48 References 52 Problems 52 Chapter 2 • Some Simple Radiating Systems and Antenna Practice 56 2.1 2.2 2.3 Electrically Small Dipoles 56 Dipoles 59 Antennas Above a Perfect Ground Plane 63 2.3.1 Image Theory 63 2.3.2 Monopoles 66 2.4 Small Loop Antennas 68 2.4.1 Duality 68 2.4.2 The Small Loop Antenna 71 2.5 Antennas in Communication Systems 76 2.6 Practical Considerations for Electrically Small Antennas 82 References 83 Problems 84 Chapter 3 • Arrays 3.1 3.2 3.3 3.4 3.5 The Array Factor for Linear Arrays 88 Uniformly Excited, Equally Spaced Linear Arrays 99 3.2.1 The Array Factor Expression 99 3.2.2 Main Beam Scanning and Beamwidth 102 3.2.3 The Ordinary Endfire Array 103 3.2.4 The Hansen-Woodyard Endfire Array 105 Pattern Multiplication 107 Directivity of Uniformly Excited, Equally Spaced Linear Arrays Nonuniformly Excited, Equally Spaced Linear Arrays 116 87 112 xii Contents 3.6 Mutual Coupling 121 3.6.1 Impedance Effects of Mutual Coupling 122 3.6.2 Array Pattern Evaluation Including Mutual Coupling 125 3.7 Multidimensional Arrays 128 3.8 Phased Arrays and Array Feeding Techniques 130 3.8.1 Scan Principles 130 3.8.2 Feed Networks for Beam Scanning 133 3.8.3 Scan Blindness 135 3.9 Perspective on Arrays 136 References 136 Problems 137 Chapter 4 • Line Sources 143 4.1 4.2 4.3 The Uniform Line Source 143 Tapered Line Sources 152 0{ Fourier Transform Relations Between the Far-Field Pattern and the Source Distribution 157 ,,4.4 Superdirective Line Sources 159 References 163 Problems 163 Chapter 5 • Resonant Antennas: Wires and Patches 164 5.1 ~ Dipole Antennas 165 5.1.1 Straight Wire Dipoles 165 5.1.2 The Vee Dipole 173 5.2 Folded Dipole Antennas 175 5.3 Feeding Wire Antennas 180 5.4 Yagi-Uda Antennas 187 5.5 Comer Reflector Antennas 196 5.6 Wire Antennas Above an Imperfect Ground Plane 198 5.6.1 Pattern Effects of a Real Earth Ground Plane 198 5.6.2 Ground Plane Construction 203 5.7 Large Loop Antennas 205 5.8 Microstrip Antennas 210 5.8.1 Microstrip Patch Antennas 210 5.8.2 Microstrip Arrays 216 References 218 Problems 219 Chapter 6 • Broadband Antennas 6.1 6.2 6.3 6.4 6.5 Traveling-Wave Wire Antennas 225 Helical Antennas 231 6.2.1 Normal Mode Helix Antenna 232 6.2.2 Axial Mode Helix Antenna 235 Biconical Antennas 240 6.3.1 Infinite Biconical Antenna 240 6.3.2 Finite Biconical Antenna 242 6.3.3 Discone Antenna 243 Sleeve Antennas 246 6.4.1 Sleeve Monopoles 246 6.4.2 Sleeve Dipoles 248 Principles of Frequency-Independent Antennas 250 225 Contents xiii 6.6 Spiral Antennas 252 6.6.1 Equiangular Spiral Antenna 252 6.6.2 Archimedean Spiral Antenna 254 6.6.3 Conical Equiangular Spiral Antenna 257 6.6.4 Related Configurations 258 6.7 Log-Periodic Antennas 259 References 270 Problems 272 Chapter 7 • Aperture Antennas 275 7.1 7.2 Radiation from Apertures and Huygens' Principle 275 Rectangular Apertures 284 7.2.1 The Uniform Rectangular Aperture 285 7.2.2 Tapered Rectangular Apertures 289 7.3 Techniques for Evaluating Gain 291 7.3.1 Directivity 292 7.3.2 Gain and Efficiencies 294 7.3.3 Simple Directivity Formulas 296 7.4 Rectangular Hom Antennas 299 7.4.1 H-Plane Sectoral Hom Antenna 300 7.4.2 E-Plane Sectoral Hom Antenna 306 7.4.3 Pyramidal Hom Antenna 310 7.5 Circular Apertures 316 7.5.1 The Uniform Circular Aperture 316 7.5.2 Tapered Circular Apertures 319 7.6 Reflector Antennas 322 7.6.1 Parabolic Reflector Antenna Principles 322 7.6.2 Axisymmetric Parabolic Reflector Antenna 329 7.6.3 Offset Parabolic Reflectors 334 7.6.4 Dual Reflector Antennas 335 7.6.5 Cross-Polarization and Scanning Properties of Reflector Antennas 338 7.6.6 Gain Calculations for Reflector Antennas 342 7.6.7 Other Reflector Antennas 347 7.7 Feed Antennas for Reflectors 349 7.7.1 Field Representations 349 7.7.2 Matching the Feed to the Reflector 350 7.7.3 A General Feed Model 352 7.7.4 Feed Antennas Used in Practice 354 References 356 Problems 358 Chapter 8.. Antenna Synthesis 8.1 8.2 8.3 The Synthesis Problem 365 8.1.1 Formulation of the Synthesis Problem 365 8.1.2 Synthesis Principles 367 Line Source Shaped Beam Synthesis Methods 368 8.2.1 Fourier Transform Method 368 8.2.2 Woodward-Lawson Sampling Method 370 Linear Array Shaped Beam Synthesis Methods 373 8.3.1 Fourier Series Method 373 8.3.2 Woodward-Lawson Sampling Method 376 8.3.3 Comparison of Shaped Beam Synthesis Methods 377 365 xiv Contents 8.4 Low Side-Lobe, Narrow Main Beam Synthesis Methods 378 8.4.1 Dolph-Chebyshev Linear Array Method 378 8.4.2 Taylor Line Source Method 384 8.5 Perspective 390 References 390 Problems 391 Chapter 9 • Antennas in Systems and Antenna Measurements 395 9.1 \ 9.2 9.3 9.4 9.5 9.6 Receiving Properties of Antennas 395 Antenna Noise Temperature and Radiometry 400 Radar 403 Reciprocity and Antenna Measurements 404 Pattern Measurement and Antenna Ranges 409 Gain Measurement 415 9.6.1 Gain Measurement of CP Antennas 416 9.6.2 Gain Estimation 418 9.7 Polarization Measurement 418 9.7.1 Polarization Pattern Method 419 9.7.2 Spinning Linear Method 420 9.7.3 Dual-Linear Method 421 9.8 Field Intensity Measurements 422 References 423 Problems 424 Chapter 10 • CEM for Antennas: The Method of Moments 10:1 10.2 10.3 10.4 10.5 10.6 10.7 Introduction to Computational Electromagnetics 427 Introduction to the Method of Moments 429 Pocklington's Integral Equation 430 Integral Equations and Kirchhoff's Network Equations 432 Source Modeling 435 Weighted Residuals and the Method of Moments 440 Two Alternative Approaches to the Method of Moments 445 10.7.1 Reaction 445 10.7.2 Linear Algebra Formulation of MoM 447 10.8 Formulation and Computational Considerations 449 10.8.1 Other Expansion and Weighting Functions 450 10.8.2 Other Electric Field Integral Equations for Wires 451 10.8.3 Computer Time Considerations 454 10.8.4 ToepIitz Matrices 455 10.8.5 Block Toeplitz Matrices 455 10.8.6 Compressed Matrices 456 10.8.7 Validation 457 10.9 Calculation of Antenna and Scatterer Characteristics 457 10.10 The Wire Antenna or Scatterer as an N-Port Network 460 10.10.1 Series Connections 460 10.10.2 Parallel Connections 462 10.11 Antenna Arrays 465 10.11.1 The Linear Array 466 10.11.2 The Circular Array 467 10.11.3 Two-Dimensional Planar Array of Dipoles 470 10.11.4 Summary 471 10.12 Radar Cross Section of Antennas 472 427 Contents XV 10.13 Modeling of Solid Surfaces 477 10.13.1 Wire-Grid Model 477 10.13.2 Continuous Surface Model 482 10.14 Summary 487 References 487 Problems 488 Chapter 11 • CEM for Antennas: Finite Difference Time Domain Method 493 11.1 Maxwell's Equations for the FD-TD Method 495 11.1.1 Three-Dimensional Problem Formulation 496 11.1.2 Two-Dimensional Problem Formulation 496 11.1.3 One-Dimensional Problem Formulation 497 11.2 Finite Differences and the Yee Algorithm 498 11.3 Cell Size, Numerical Stability, and Dispersion 505 11.4 Computer Algorithms and FD-TD Implementation 508 11.5 Absorbing Boundary Conditions 511 11.6 Source Conditions 515 11.6.1 Source Functionality 515 11.6.2 The Hard Source 517 11.6.3 The Soft Source 517 11.6.4 Total-FieldlScattered-Field Formulation 519 11.6.5 Pure Scattered-Field Formulation 522 11.7 Near Fields and Far Fields 522 11.8 A Two-Dimensional Example: An E-Plane Sectoral Horn Antenna 524 11.9 Antenna Analysis and Applications 531 11.9.1 Impedance, Efficiency, and Gain 532 11.9.2 The Monopole over a PEC Ground Plane 533 11.9.3 The Vivaldi Slotline Array 538 11.10 Summary 542 References 542 Problems 543 Chapter 12 • CEM for Antennas: High-Frequency Methods 545 12.1 GeometricalOptics 546 12.2 Wedge Diffraction Theory 552 12.3 The Ray-Fixed Coordinate System 561 12.4 A Uniform Theory of Wedge Diffraction 564 12.5 E-Plane Analysis of Horn Antennas 568 12.6 Cylindrical Parabolic Antenna 571 12.7 Radiation by a Slot on a Finite Ground Plane 574 12.8 Radiation by a Monopole on a Finite Ground Plane 577 12.9 Equivalent Current Concepts 578 12.10 A Multiple Diffraction Formulation 581 12.11 Diffraction by Curved Surfaces 584 12.12 Extension of Moment Methods Using the Geometrical Theory of Diffraction 589 12.13 Physical Optics 597 12.14 Method of Stationary Phase 601 12.15 Physical Theory of Diffraction 604 12.16 Cylindrical Parabolic Reflector Antenna-PTD 610 12.17 Summary 612 References 613 Problems 614 xvi Contents Appendix A • Tables of Commonly Used Frequencies A.1 A.2 A.3 A.4 Appendix B Radio Frequency Bands 621 Television Channel Frequencies 621 Mobile Telephone Bands 622 Radar Bands 622 • Data Material and Other Constants B.1 B.2 B.3 623 Conductivities of Good Conductors 623 Wire Data 623 Dielectric Constant: Permittivity 624 Permeability 624 Velocity of Light 624 Intrinsic Impedance of Free Space 624 BA B.5 B.6 Appendix 621 C • Vectors C.1 C.2 C.3 625 Unit Vector Representations 625 Vector Identities 625 Vector Differential Operators 626 Appendix D • Trigonometric Relations 628 Appendix E • 630 Appendix F • Useful Mathematical Relations F.1 F.2 F.3 G• G.1 G.2 G.3 GA G.5 Appendix H Index 631 Dirac Delta Function 631 Binomial Theorem 631 Bessel Functions 631 Some Useful Integrals 632 FA Appendix Hyperbolic Relations Computing Packages 633 General Antenna Pagkage: APV 633 Array Plotting Package: PCARRPAT 633 Wire Antenna Code: WIRE 634 Parabolic Reflector Antenna Code: PRAC 634 Diffraction Codes 634 • Bibliography 636 643 Chapter 1 Antenna Fundamentals and Definitions 1.1 INTRODUCTION Communication between humans was first by sound through voice. With the desire for long distance communication came devices such as drums. Then, visual methods such as signal flags and smoke signals were used. These optical communication devices, of course, utilized the light portion of the electromagnetic spectrum. It has been only very recent in human history that the electromagnetic spectrum, outside the visible region, has been employed for communication, through the use of radio. The radio antenna is an essential component in any radio system. An antenna is a device that provides a means for radiating or receiving radio waves. In other words, it provides a transition from a guided wave on a transmission line to a "freespace" wave (and vice versa in the receiving case). Thus, information can be transferred between different locations without any intervening structures. The possible frequencies of the electromagnetic waves carrying this information form the electromagnetic spectrum (the radio frequency bands are given in Appendix A). One of humankind's greatest natural resources is the electromagnetic spectrum and the antenna has been instrumental in harnessing this resource. A brief history of antenna technology [1-4] and a discussion of the uses of antennas follow. Perhaps the first radiation experiment was performed in 1842 by Joseph Henry of Princeton University, the inventor of wire telegraphy. He "threw a spark" in a circuit in an upper room and observed that needles were magnetized by the current in a receiving circuit located in the cellar. This experiment was extended to a distance of over a kilometer. Henry also detected lightning flashes with a vertical wire on the roof of his house. These experiments marked the beginning of wire antennas. Based on his observations in 1875 that telegraph key closures radiate, Thomas Edison patented a communication system in 1885 that employed top-loaded, vertical antennas. The theoretical foundations for antennas rest on Maxwell's equations, which James Clerk Maxwell (1831-1879) presented before the Royal Society in 1864, that unify electric and magnetic forces into a single theory of electromagnetism. Maxwell also predicted that light is explained by electromagnetics and that light and electromagnetic disturbances both travel at the same speed. In 1887 the German physicist Heinrich Hertz (1857-1894) was able to verify 1 2 Chapter 1 Antenna Fundamentals and Definitions experimentally the claim of Maxwell that electromagnetic actions propagate through air. Hertz discovered that electrical disturbances could be detected with a single loop of the proper dimensions for resonance that contains an air gap for sparks to occur. The primary source of electrical disturbances studied by Hertz consisted of two metal plates in the same plane, each with a wire connected to an induction coil; this early antenna is similar to the capacitor-plate dipole antenna described in Section 2.1 and was called a "Hertzian dipole." Hertz also constructed loop antennas. Motivated by the need for more directive radiation; he also invented reflector antennas. In 1888 he constructed a parabolic cylinder reflector antenna from a sheet of zinc; see Fig. 1-la. It was fed with a dipole along the focal line and operated at 455 MHz. Guglielmo Marconi (1874-1937), an Italian inventor, also built a microwave parabolic cylinder reflector in 1895 for his original code transmission at 1.2 GHz. But his subsequent work was at lower frequencies for improved communication range .. The transmitting antenna for the first transatlantic radio communication in 1901 consisted of a 70-kHz spark transmitter connected between the ground and a system of 50 wires, forming a 48-m tall fan monopole; see Fig. 1-lb. The antenna resembles a variation of the discone antenna described in Sec. 6.3. The receiving antenna was supported by kites. Although Marconi is credited as the pioneer of radio, Mahlon Loomis (18261886), a dentist and inventor in Washington, DC, received a U.S. patent in 1872 for an "Improvement in Telegraphying" in which he described the use of an "aerial" to radiate and recieve "pulsations." In October 1866, Loomis demonstrated his wireless signaling system to U.S.· senators in the Blue Ridge Mountains of Virginia using wire supported by kites at both the transmitting and receiving antennas about twenty miles apart. The Russian physicist Alexander Popov (1859-1905) also recognized the importance of Hertz's discovery of radio waves and began working on ways of receiving (a) The 455-MHz cylinder reflector antenna invented by Hertz in 1888 Figure 1-1 Examples of early antennas. (b) The monopole transmitting antenna used by Marconi at 70 kHz for the first transatlantic radio communication 1.1 Introduction 3 them a year before Marconi. He is sometimes credited with using the first antenna in the first radio system by sending a signal over a 3-mile ship-to-shore path in 1897. However, it was Marconi who developed radio commercially and also pioneered transoceanic radio communication. Marconi may be considered to be the father of what was then called wireless. Since then the term "radio" has been used, but "wireless" has also returned to popular use. In 1912 the Institute of Radio Engineers was formed by the merger of the Wireless Institute and the Society of Radio Engineers. The importance of antennas is punctuated by the fact that the first article of the first issue of Proceedings of the l.R.E. was on antennas [5]. Antenna developments in the early years were limited by the availability of signal generators. Resonant length antennas (e.g., a half-wavelength dipole) of manageable physical size were possible about 1920 after the De Forest triode tube was used to produce continuous wave signals up to 1 MHz. Just before World War II, microwave (about 1 GHz) klystron and magnetron signal generators were developed along with hollow pipe waveguides. These led to the development of hom antennas, although Chunder Bose (1858-1937) in India produced the first electromagnetic hom antenna many years earlier. The first commercial microwave radiotelephone system in 1934 was operated between England and France at 1.8 GHz. The need for radar during the war spawned many "modem" antennas, such as large reflectors, lenses, and waveguide slot arrays [6]. Let us now direct our attention to the uses of antennas. Electromagnetic energy can be transported using a transmission line. Alternatively, no guiding structure is needed if antennas are used. For a transmitter-receiver spacing of R, the power loss of a transmission line is proportional to (e- aR?, where a is the attenuation constant of the transmission line. If the antennas are used in a line of sight configuration, the power loss is proportional to lIR2. Many factors enter into the decision of whether to use transmission lines or antennas. Generally speaking, at low frequencies and short distances transmission lines are practical. But high frequencies are attractive because of the available bandwidth. As distances become large and frequency increases, the signal losses and the costs of using transmission lines become large, and thus the use of antennas is favored. A notable exception to this is the fiber optic transmission line, which has very low loss. Transmission lines offer the advantages of not being subject to interference that is often encountered in radio systems and added bandwidth is achieved by laying new cable. However, there are significant costs and construction delays associated with cable. In several applications, antennas must be used. For example, mobile communications involving aircraft, spacecraft, ships, or land vehicles require antennas. Antennas are also popular in broadcast situations where one transmit terminal can serve an unlimited number of receivers, which can be mobile (e.g., car radio). Nonbroadcast radio applications such as municipal radio (police, fire, rescue) and amateur radio also require antennas. Also, personal communication devices such as pagers and cellular telephones are commonplace. There are also many noncommunication applications for antennas. These include remote sensing and industrial applications. Remote sensing systems are either active (e.g., radar) or passive (e.g., radiometry) and receive scattered energy or inherent emissions from objects, respectively. The received signals are processed to infer information about the objects or scenes. Industrial applications include cooking and drying with microwaves. Other factors that influence the choice of the type of transmission system include 4 Chapter 1 Antenna Fundamentals and Definitions historical reasons, security, and reliability. Telephone companies began interconnecting multiple transmit-receive terminals with transmission lines before radio technology was available. Currently, domestic telephone companies employ microwave radio and fiber optic transmission lines for long distance telephone calls. Satellite radio links are used heavily for international telephone calls. In addition, satellite-based communication systems are the primary means of distributing television program material to affiliate stations. Television programs distributed directly to consumers by satellite is increasing worldwide. Also, very small aperture terminals (VSAT) for satellite radio systems are widely used in private data networks to interconnect, for example, chain retail stores. Transmission lines inherently offer more security than radio. However, radio links using digital communications can be secured with coding techniques. Also, additional security is unnecessary for most communication systems. Another factor to be considered is reliability. For example, radio signals are affected by environmental conditions such as structures along the signal path, the ionosphere, and weather. Furthermore, interference is always a threat to radio systems. On the other hand, cables are vulnerable to being damaged by earthquakes or being accidentally dug up. All these factors must be examined together with the costs associated with using transmission lines and antennas. Cable systems often require expensive land purchase or lease. Every year radio equipment decreases in cost and increases in reliability. This tends to tip the scale in favor of radio systems. Cable and radio communication systems will continue to be used in the future with the choice depending on the specific application. For high reliability, both cable and radio are employed to provide diversity. As we shall see, antennas cannot be miniaturized and replaced by a chip as often happens in electronics. Although long-used types of antennas will remain in use far into the future, new applications will require innovative antenna systems. For example, the demand for more communications is leading to personal communications systems (PCS) where each person will be freed of wire connections by carrying a small radio that can be used anywhere on the globe. The first of the seven IEEE New Technology Directions Committee grand challenges for electrotechnology is, "To make any person anywhere in the world reachable at his or her discretion, at any time by communication methods independent of connecting wires and cables." A "wireless" society is possible only through the use of antennas. Indeed, the future of antennas is very bright. The next two sections of this first chapter provide a basic understanding of how antennas operate and an overview of the types of antennas encountered in practice. The remainder of Chap. 1 is devoted to developing the fundamental principles and terminology used throughout the book. References to specific literature works are found at the end of the chapter where the citation occurs. In addition, a complete bibliography is found in Appendix H. About 150 books are listed by their topical coverage to aid the student in locating further details. The IEEE definitions of antenna terms [Ref. 1 in App. H] are followed very closely in this book. L2 HOW ANTENNAS RADIATE Before we proceed with a mathematical development of antennas that is necessary for engineering design, it is instructive to explain the basic principles of radiation. Radiation is a disturbance in the electromagnetic fields that propagates away from the source of the disturbance so that the total power associated with the wave in a 1.2 How Antennas Radiate 5 lossless medium is constant with radial distance. This disturbance is created by a time-varying current source that has an accelerated charge distribution associated with it. We, therefore, begin our discussion of radiation with a single accelerated charge. Consider a single electric charge moving with constant velocity in the z-direction of Fig. 1-2. Prior to arrival at point A, the static electric field lines extend radially away from the charge to infinity and move with the charge. At point A, the charge begins to be accelerated (i.e., velocity is increased) until reaching point B, where it continues on at the acquired velocity. The static electric field (often called the Coulomb field) originates at the charge and is directed radially away from the charge. The radial field lines outside the circle of radius rAin Fig. 1-2 originated when the charge was at point A. The circle of radius rB is centered on point B, which is the charge position at the end of the acceleration period M. Interior to rB, the electric field lines extend radially away from point B. The distance between the circles is that distance light would travel in time at, or ar = rB - rA = at/c. Since the charge moves slowly compared to the speed of light, az « ar and the circles are nearly concentric; the distance az in Fig. 1-2 is shown large relative to ar for clarity. The electric field lines in the ar region are joined together because of the required continuity of electric field lines in the absence of charges. This region is obviously one of disturbed field structure. This disturbance was caused by acceleration of the charge, which ended a time rBlc earlier than the instant represented in Fig. 1-2. This z Figure 1·2 Illustration of how an accelerated charged particle radiates. Charge q moves with constant velocity in the + z-direction until it reaches point A (time t = 0), after which it accelerates to point B (time t = At) and then maintains its velocity. The electric field lines shown here are for a time rB/c after the charge passed pointE. 6 Chapter 1 Antenna Fundamentals and Definitions disturbance expands outward and has a transverse component E" which is the radiated field that persists as the disturbance propagates to infinity. This example illustrates that radiation is a disturbance. It is directly analogous to a transient wave created by a stone dropped into a calm lake, where the disturbance of the lake surface continues to propagate radially away from the impact point long after the stone has disappeared. If charges are accelerated back and forth (i.e., oscillate), a regular disturbance is created and radiation is continuous. Antennas are designed to support charge oscillations. The directional properties of radiation are evident in the accelerated charge example. The disturbance in Fig. 1-2 is maximum in a direction perpendicular to the charge acceleration direction, and we shall see in this chapter that maximum radiation occurs perpendicular to a straight wire antenna. We can now explain how an actual antenna operates. To do this, we begin with the open-circuited transmission line of Fig. 1-3, which has a standing wave pattern with a zero current magnitude at the wire end and nulls every half wavelength from the end. The currents are in opposite directions on the wires, as indicated by arrows in Fig. 1-3a. In transmission lines, the conductors guide the waves and the power resides in the region surrounding the conductors as manifested by the electric and magnetic fields. The fields for the open-ended transmission line are shown in Fig. 1-3a. The electric fields originate from or terminate on charges on the wires and are 1+ 1 1 Y ~-- J. 1 1 1 1 1 J. 1 1 1 1 1 o (a) Open-circuited transmission line showing currents, charges, and fields. The electric fields are indicated with lines and the magnetic fields with arrow heads and tails, solid (dashed) for those arising from the top (bottom) wire. (b) Peak currents on a half-wavelength dipole created by bending out the ends of the transmission line. Figure 1-3 Evolution of a dipole antenna from an open-circuited transmission line. 1.2 How Antennas Radiate 7 perpendicular to the wires. The magnetic fields encircle the wires. Note that all fields reinforce between the wires and cancel elsewhere. This is true for a wire spacing that is much smaller than a wavelength, as is usually the case. If the ends of the wires are bent outward as shown in Fig. 1-3b, the reinforced fields between the wires are exposed to space. Note that the currents on the vertical wire halves, which are each a quarter wavelength in this case, are no longer opposed as with the transmission line, but are both upwardly directed. In reality, the currents on the dipole are approximately sinusoidal as shown, but the transmission line currents are not pure standing waves due to the improved impedance match presented by the antenna compared to the open circuit. The situation of Fig. 1-3b is for a peak current condition. As time proceeds and current oscillations occur, disturbances are created that propagate away from the wire, much as the accelerated single charge. The time dynamics of the fields associated with an oscillating dipole charge distribution are shown in Fig. 1-4 [7]. This is similar to the electrostatic dipole with equal, but opposite signed, separated charges. In this case, the charge distributions oscillate at frequency f. As the charge distributions at the ends oscillate, a current flows between them that is uniform with distance. This is the ideal dipole of Sec. '- 1.6. In Fig. 1-4, an oscillating current of frequency f (and period T = 11f) was turned on a quarter period before t = O. The upward-flowing current creates an excess of charges on the upper half of the dipole and a deficit of charges on the lower half. Peak charge buildup occurs at t = 0 as shown in Fig. 1-4a and produces a voltage between the dipole halves. The positive charges on the top are attracted to the negative charges on the bottom half of the dipole, creating a current. The current _/ ..,~ (a) t = 0 (b) t= TI4 (c) t= TI2 Figure 1-4 Electric fields of an oscillating dipole for various instants of time. The oscillations are of frequency f with a period of T = lIf. 8 Chapter 1 Antenna Fundamentals and Definitions is maximum at t = T/4 as shown in Fig. 1-4b, at which time the charges have been neutralized and there are no longer charges for the termination of electric field lines, which form closed loops near the dipole. During the next quarter cycle, negative charges accumulate at the top end of the dipole as shown in Fig. 1-4c. Near the dipole, the fields are most intense normal to the oscillating charges on the dipole, just as we found with the single accelerated charge. As time progresses, the electric field lines detach from the dipole, forming closed loops in space. Viewed in terms of current, the conduction current on the antenna converts to a displacement current in space, consisting of the longitudinal fields near the antenna and solenoidal (loops) fields away from the antenna. Thus, current continuity is satisfied. This process continues,producing radiation via electric field components that are transverse to the radial direction and propagate to large distances from the antenna. We shall see in Sec. 1.6 that the mathematical solution of the (oscillating) dipole produces the property required for successful radiation: fields decrease with distance as 1fr. In contrast, the electric field of an electrostatic dipole decreases as 1fr 3 • The time-space behavior of fields from an antenna is revisited in Sec. 11.8 and 11.9. L3 OVERVIEW OF ANTENNAS An antenna acts to convert guided waves on a transmission structure into free space waves. Figure 1-3 illustrates a parallel wire transmission line feeding a half-wave dipole antenna. The official IEEE definition of an antenna follows this concept: "That part of a transmitting or receiving system that is designed to radiate or receive electromagnetic waves." Most antennas are reciprocal devices and behave the same on transmit as on receive. Antennas are treated as transmitting or receiving as appropriate for the particular situation. In the receiving mode, antennas act to collect incoming waves and direct them to a common feed point where a transmission line is attached. In some cases, antennas focus radio waves just as lenses focus optical waves. In all cases, antennas have directional characteristics; that is, electromagnetic power density is radiated from a transmitting antenna with intensity that varies with angle around the antenna. In this section, we introduce the parameters used to evaluate antennas and then discuss the four types of antennas. The parameters are defined and developed in more detail in the remainder of this chapter after the brief overview given here. Antenna performance parameters are listed in Table 1-1. The radiation pattern (or simply, pattern) gives the angular variation of radiation at a fixed distance from an antenna when the antenna is transmitting. Radiation is quantified by noting the value of power density S at a fixed distance r from the antenna. When receiving, the antenna responds to an incoming wave from a given direction according to the pattern value in that direction. The typical pattern in Fig. 1-5 shows the pattern main beam and side lobes. This directive antenna, with a single narrow main beam, is used in point-to-point communications. In some applications, the shape of the main beam is important. On the other hand, an omnidirectional antenna with constant radiation in one plane is used in broadcast situations. An antenna is essentially a spatial amplifier and directivity expresses how much greater the peak radiated power density is for an antenna than it would be if all the radiated power were distributed uniformly around the antenna. Fig. 1-5 shows the radiation pattern of a real antenna compared to an isotropic spatial distribution; also see Fig. 1-20. The spatial enhancement that can be achieved by an antenna is evident. Gain G is directivity reduced by the losses on the antenna. 1.~ Overview of Antennas 9 Table 1·1 Antenna Performance Parameters • Radiation Pattern F(O, tfJ): Angular variation of radiation around the antenna, including: Directive, single or multiple narrow beams Omnidirectional (uniform radiation in one plane) Shaped main beam • Directivity D: Ratio of power density in the direction of the pattern maximum to the average power density at the same distance from the antenna. • Gain G: Directivity reduced by the losses on the antenna. • Polarization: The figure traced out with time by the instantaneous electric field vector associated with the radiation from an antenna when transmitting. Antenna polarizations: Linear, Circular, Elliptical • Impedance ZA: Input impedance at the antenna terminals. • Bandwidth: Range of frequencies over which important performance parameters are acceptable. • Scanning: Movement of the radiation pattern in space. Scanning is accomplished by mechanical movement or by electronic means such as adjustment of antenna current phase. • System Considerations: Size, weight, power handling, radar cross section, environmental operating conditions, etc. The third parameter, polarization, describes the vector nature of electric fields radiated by an antenna and is discussed in detail in Sec. 1.10. The figure traced out with time by the tip of the instantaneous electric field vector determines the polarization of the wave. A straight wire antenna radiates a wave with linear polarization parallel to the wire. Another popular polarization is circular. In general, polarization is elliptical. Dual polarized antennas enable the doubling of communication capacity by carrying separate information on orthogonal polarizations over the same physical link at the same frequency. The input impedance of an antenna is the ratio of the voltage to current at the antenna terminals. The usual goal is to match antenna input impedance to the characteristic impedance of the connecting transmission line. Bandwidth is the range of frequencies with acceptable antenna performance as measured by one or more of the performance parameters; see Chap. 6 for commonly used definitions of bandwidth. Finally, it is often desired to scan the main beam of an antenna over a region of space. This can be accomplished by moving an antenna, by electronic scanning or by a combination of mechanical and electronic scanning. Pattern F(8, q,) of a real antenna I I I I I ," ,.' ~~ , \ , ----~ \ Isotropic ~" pattern ... Side lobe .... _ T Sj ... __ fIII" / ' D=-S S j Figure 1·5 Illustration of radiation pattern F(8, ¢) and directivity D. The power densities at the same distance are Sand Sj for the real and isotropic antennas, respectively. 10 Chapter 1 Antenna Fundamentals and Definitions • Electrically small antennas: The extent of the antenna structure is much less than a wavelength A. Properties: Very low directivity Low input resistance High input reactance Low radiation efficiency Examples: o Short dipole Small loop • Resonant antennas: The antenna operates well at a single or selected narrow frequency bands. Properties: Low to moderate gain " Real input impedance Narrow bandwidth Examples: i _b i I 2 ~-~Half-wave dipole --~~ Microstrip patch ~ J 1I I Yagi • Broadband antennas: The pattern, gain, and impedance remain acceptable and are nearly constant over a wide frequency range, and are characterized by an active region with a circumference of one wavelength or an extent of a half-wavelength, which relocates on the antenna as frequency changes. Properties: Low to moderate gain Constant gain Real input impedance Wide bandwidth Examples: Spiral Figure 1·6 Types of antennas. Log periodic dipole array 1.3 Overview of Antennas 11 • Aperture antennas: Has a physical aperture (opening) through which waves flow. Properties: High gain Gain increases with frequency Moderate bandwidth Examples: Hom Reflector Figure 1-6 (continued). There are trade-offs between parameter values. Usually, performance cannot be improved significantly for one parameter without sacrificing one or more of the other parameter levels. This is the antenna design challenge. Antennas can be divided into four basic types by their performance as a function of frequency. These are introduced so that the common features can be grasped early in the study of antennas. When you encounter a new antenna, try to determine which type it is. The antenna types in Fig. 1-6 are listed in the order that they are commonly used across the radio spectrum; see Appendix A for lists of frequency bands. This discussion serves as an overview and should be referred to from time to time as your knowledge of antennas builds. Electrically small antennas are used at VHF frequencies and below. Resonant antennas are mainly used from HF to low GHz frequencies. Broadband antennas are mainly used from VHF to middle GHz frequencies. Aperture antennas are mainly used at UHF and above. Electrically small (or simply, small) antennas are much less than a wavelength in extent. Recall from electromagnetics that wavelength ..\. is related to frequency f through the speed of light c as A = clf. Electrically small antennas are simple in structure and their properties are not sensitive to construction details. The vertical monopole used for AM reception on cars is a good example. It is about 0.003..\. long and has a pattern that is nearly omnidirectional in the horizontal plane. This is often a desirable property, but its low input resistance and high input reactance are serious disadvantages. Also, small antennas are inefficient because of ohmic losses on the structure. . Resonant antennas are popular when a simple structure with good input impedance over a narrow band of frequencies is needed. It has a broad main beam and low or moderate (a few dB) gain. The half-wavelength long dipole is a prominent example. Many applications require an antenna that operates over a wide frequency range. A broadband antenna has acceptable performance as measured with one or more parameters (pattern, gain, and/or impedance) over a 2: 1 bandwidth ratio of upper to lower operating frequency. A broadband antenna is characterized by an active region. Propagating (or, traveling) waves originate at the feed point and travel without radiation to the active region where most of the power is radiated. A broadband antenna with circular geometry has an active region where the circumference is one wavelength and produces circular polarization. An example is the spiral antenna I 12 Chapter 1 Antenna Fundamentals and Definitions illustrated in Fig. 1-6 that can have a 20: 1 bandwidth. A broadband antenna made up of linear elements or straight edges has an active region where these elements are about a half-wavelength in extent and produces linearly polarized radiation parallel to the linear elements. Since only a portion of a broadband antenna is responsible for radiation at a given frequency, the gain is low. But it may be an advantage to have gain that is nearly constant with frequency, although low. Also, the traveling wave nature of a broadband antenna means that it has a real-valued input impedance. Aperture antennas have an opening through which propagating electromagnetic waves flow. A horn antenna is a good example; it acts as a "funnel," directing the waves into the connecting waveguide. The aperture is usually several wavelengths long in one or more dimensions. The pattern usually has a narrow main beam, leading to high gain. The pattern main beam narrows with increasing frequency for a fixed physical aperture size. Bandwidth is moderate (as much as 2:1). L4 ELECTROMAGNETIC FUNDAMENTALS This and the next section present a brief review of electromagnetic field principles and the solution of Maxwell's equations for radiation problems. Any basic electromagnetic fields textbook can be consulted for the details on these topics. The fundamental electromagnetic equations are! a~ Vx~=-- (1-1) a v x ~ = -~ at + 1T (1-2) at v· ~ = PT(t) v·~ (1-3) (1-4) =0 a v ·1T = -at PT(t) (1-5) The first four of these differential equations are frequently referred to as Maxwell's equations and the last as the continuity equation. The curl equations together with the continuity equation are equivalent to the curl and divergence equations. In timevarying field problems, the curl equations with the continuity equation is the most convenient formulation. Each of these differential equations has an integral counterpart. If the sources PT(t) and 'T(t) vary sinusoidally with time at radian frequency w, the fields will also vary sinusoidally and are frequently called time-harmonic fields. The fundamental electromagnetic equations and their solutions are considerably simplified if phasor fields are introduced as follows: 2 ~ = Re(Ee jw'), ~ = Re(He jwt), etc. (1-6) where phasor quantities E, H, D, B, Pro and JT are complex-valued functions of spatial coordinates only (Le., time dependence is not shown). Using the phasor ITime-varying quantities will be denoted with script quantities, for example, ~ = ~(x, y, z, t). z.rhe student is cautioned that some authors use e- jw" which leads to sign differences in subsequent developments. 1.4 Electromagnetic Fundamentals 13 definitions of the electromagnetic quantities from (1-6) in (1-1) to (1-5) and eliminating the e jwt factors that appear on both sides of the equations yields vxE V = -jwB x H= jwD + J T (1-7) (1-8) V.D=PT (1-9) V· B = 0 (1-10) V· J T = -jWPT (1-11) The time derivatives in (1-1) to (1-5) have been replaced by a jw factor in (1-7) to (1-11) and time-varying electromagnetic quantities have been replaced by their phasor counterpart. This process is similar to the solution of network equations where the time-dependent differential equations are Laplace-transformed and the time derivatives are thus replaced by jw (or s). Equations (1-7) to (1-10) are often referred to 'as the time-harmonic form of Maxwell's equations because they apply to sinusoidally varying (i.e., time-harmonic) fields. H more than one frequency is present, the time-varying forms of the electromagnetic quantities can be found by inverse transforms after (1-7) to (1-11) have been solved for the phasor quantities as a function of radian frequency w. This is again analogous to the procedure used to solve network problems. Fortunately, this is not usually necessary in antenna problems since the bandwidth of the signals is usually very small. In the typical case, a carrier frequency is accompanied by some form of modulation giving a spread of frequencies around the carrier. For analysis purposes, we use a single frequency equal to the carrier frequency. Thus, all subsequent material in this book (except in Chap. 11) will assume time-harmonic fields. The total current density JT is composed of an impressed, or source, current J and a conduction current density term uE, which occurs in response to the impressed current: JT = uE + J (1-12) The role played by the impressed current density is that of a known quantity. It is quite frequently an assumed current density on an antenna, but as far as the field equations are concerned, it is a known function. The current density uE is a current densityilowing on a nearby conductor due to the fields created by source J and can be ~mputed after the field equations are solved for E. In addition to conductivity u, a material is further characterized by permittivity e and permeability JL, where3 D = eE (1-13) and B = JLH We now rewrite the field equations in preparation for their solution. Substituting (1-12) and (1-13) into (1-8) gives " , Vx H= jW( e + j:)E + J = jwe'E + J (1-15) 3In general, Band IL can be complex, but in most antenna problems they can be approximated as real . constants. 14 Chapter 1 Antenna Fundamentals and Definitions where we have defined 8' = 8 - j(ulw). For antenna problems, we are usually solving for the fields in air surrounding an antenna where u = 0 and 8' = 8. We therefore use 8 instead of 8' in subsequent developments. However, if the conductivity is nonzero, 8 can be replaced by 8' = 8 - j( ulw). Note also that E and Hare the fields of primary interest in antennas. They are properly referred to as electric and magnetic field intensities and have units of Vim and Nm, respectively. For conciseness, it is common to refer to them simply as electric and magnetic fields. Let p be the source charge corresponding to the source current density J. Then using (1-12) to (1-14) in (1-7) and (1-9) to (1-11), and repeating (1-15), we have [see Prob. 1.4-2 (for 1-18)] vx E = -jwJLH (1-16) V x H = jW8E + J P V· E =- (1-17) (1-18) 8 V·H = 0 V· J (1-19) = -jwp (1-20) These are the time-harmonic electromagnetic field equations with source current density J and source charge density p shown explicitly. Sometimes, it is convenient to introduce a fictitious magnetic current density M. Then (1-16) becomes V x E = -jwJLH - M (1-21) Magnetic currents are useful as equivalent sources that replace complicated electric fields. The solution of the fundamental electromagnetic equations is not complete until the boundary conditions are satisfied. A sufficient set of boundary conditions in the time-harmonic form is fi X (H2 - HI) = J s (1-22) (E2 -E1 )xfi=Ms (1-23) where the electric and magnetic surface currents Js and Ms flow on the boundary between two homogeneous media with constitutive parameters 810 JL10 U10 and 82, JL2, U2· Ms is zero unless an equivalent magnetic current sheet is used. The unit normal to the boundary surface fi is directed from medium 1 into medium 2. The cross products with the unit normal form the tangential components to the boundary, and these equations can be written as H tan2 E tan2 = Htan 1 + Is = Etan1 + Ms (1-24) (1-25) These boundary conditions are derived from the integral form of (1-17) and (1-21). If one side is a perfect electrical conductor, the boundary conditions become (1-26) (1-27) The tangential boundary conditions on the magnetic field intensity are illustrated in Fig. 1-7 for the general case and for the case where medium 1 is a perfect con- 1.4 Electromagnetic Fundamentals 15 Medium 2 (a) General case. (b) One medium a perfect conductor. Figure 1·7 Magnetic field intensity boundary condition. ductor. It is important to note that all field quantities in the boundary condition equations are evaluated at the boundary and the equations apply to each point along the boundary. Also derivable from Maxwell's curl equations is a conservation of power equation, or Poynting's theorem. Consider a volume v bounded by a closed surface s. The complex power Ps delivered by the sources in v equals the sum of the power Pf flowing out of s, the time-average power Pdav dissipated in v, plus the time-average stored power in v: P s = Pf + Pd + J'2w(Wm - We ) BV BV BV (1-28) The complex power flowing out through closed surface s is found from Pf = ! Jr 2:tr. E x H* • ds (1-29) where ds = dso and 0 is the unit normal to the surface directed out from the surface. Note that E and H are peak phasors, not rms, leading to 112 in power expressions. The integrand inside this integral is defined as the Poynting vector: s =!E x H* (1-30) 2 which is a power density with units of W/m • The time-average dissipated power in volume v bounded by closed surface s is (1-31) The time-average stored magnetic energy is Wmav =~ fff ~ 1L1~2 dv (1-32) IJ The time-average stored electric energy is (1-33) 16 Chapter 1 Antenna Fundamentals and Definitions If the source power is not known explicitly, it can be calculated from the volume current density as follows: (1-34) If magnetic current density is used, the term H* • M is added to the integrand in the preceding equation. From (1-29), we see that the integral of the complex Poynting vector ~E X H* over a closed surface s gives the total complex power flowing out through the surface s. It is assumed that the complex Poynting vector represents the complex power density in watts per square meter at a point. Then the complex power through any surface s (not necessarily closed) can be found by integrating the complex Poynting vector over that surface. We are particularly interested in real power (the real component of the complex power that represents the electric and magnetic field intensities being in-phase). The real power flowing through surface s is p = Re(~f S· dS) = ~ Re(~f ExH* • dS) (1-35) The reference direction for this average power flow is that of the specified unit normal II contained in ds = dSll. 1.5 SOLUTION OF MAXWELL'S EQUATIONS FOR RADIATION PROBLEMS This section develops procedures for finding fields radiated by an antenna based on Maxwell's equations. Subsequent antenna analysis in this book begins with these basic relations and it is not necessary to return to Maxwell's equations. The antenna problem consists of solving for the fields that are created by an impressed current distribution J. In the simplest approach, this current distribution is obtained during the solution process. How to obtain the current distribution will be discussed at various points in the book, but for the moment suppose we have the current distribution and wish to determine the fields E and H. As mentioned in the previous section, we need only work with the two curl equations of Maxwell's equations as given by (1-16) and (1-17). These are two coupled, linear, first-order differential equations. They are coupled because the unknown functions E and H appear in both equations. Thus, these equations must be solved simultaneously. In order to simplify the solution for E and H with a given J, we introduce the scalar and vector potential functions <I> and A. The vector potential is introduced by noting from (1-19) that the divergence of H is zero: V·H =0 (1-36) Therefore, the vector field H has only circulation; for this reason, it is often called a solenoidal field. Because it possesses only a circulation, it can be represented by the curl of some other vector function as follows: 1 H=-V·A JL (1-37) 1.5 Solution of Maxwell's Equations for Radiation Problems 17 where A is the (magnetic) vector potential. To be more precise, (1-37) is possible because it satisfies (1-36) identically; that is, from (C-9) V • V x A == 0 for any A. The curl of A is defined by (1-37), but its divergence is yet to be specified for a complete definition of A. The scalar potential is introduced by substituting (1-37) into (1-16), which gives V x (E + jwA) = 0 (1-38) The expression in parentheses is an electric field, and since its curl is zero, it is a conservative field and behaves as a static electric field. The (electric) scalar potential eI> is defined from E + jwA = -Vel> (1-39) because this definition satisfies (1-38) identically, that is, from (C-I0) V x Vel> == 0 for any eI>. Solving (1-39) for the toW electric field gives E = -jwA - Vel> (1-40) which may be a familiar result. The fields E and H are now expressed in terms of potential functions by (1-37) and (1-40). If we knew the potential functions, then the fields could be obtained. We now discuss the solution for the potential functions. Substituting (1-37) into (1-17) gives = .!. V x V x A = jwsE + J JL Using the following vector identity, from (C-17), Vx H V x V x A == V(V. A) - VZA (1-41) (1-42) and (1-40) in (1-41) yields V(V. A) - VZA = jWJLs(-jwA - Vel» + JLJ (1-43) or (1-44) As we mentioned previously, the div~rge~ce of A is yet to be specified. A convenient choice would be one that eliminates: the third term of (1-44). It is the Lorentz condition (perhaps more properly attributed to L.! Lorenz rather than H. Lorentz [8]): V· A ,=. -jwJ.tsel>. (1-45) Then (1-44) reduces to (1-46) The choice of (1-45) leads to a de~oupling of variables: that is, (1-46) involves A and not eI>.' This is the vector wave equation. It IS a differential equation that can be solved for A after the impressed current J is ~pecified. The fields are easily found then from (1-37) and I .,. I . A - ]. V(V • A) E = -JW WJLS I (1-47) 18 Chapter 1 Antenna Fundamentals and Definitions where this equation was obtained from (1-40) and (1-45). Notice that only a knowledge of A is required. A more cumbersome approach would be to solve the scalar wave equation (1-48) in addition to the vector wave equation. It is left as a problem to derive (1-48). If this approach is used, E is found from (1-40). Note that pin (1-48) is related to J in (1-46) by the continuity equation of (1-20). The vector wave equation (1-46) is solved by forming three scalar equations. This begins by decomposing A into rectangular components using (C-18): (1-49) Rectangular components are used·because the unit vectors in rectangular components can be factored out of the Laplacian since they are not themselves functions of coordinates. This feature is unique to the rectangular coordinate system. Although A is always decomposed into rectangular components, the Laplacian of each component of A is expressed in a coordinate system appropriate to the geometry of the problem. The solution proceeds by substituting (1-49) into (1-46) and equating rectangular components: + {3 2A x = -pJx + {32A Y = - tM'y .. 1 Y V2AX V 2A where {32 (1-50) = alp-s. The real-valued constant (3 = wV/ii: (1-51) is recognized as the phase constant for a plane wave. The three equations in (1-50) are identical in form. After solving one of these equations, the other two are easily solved. We first find the solution for a point source. This unit impulse response solution can then be used to form a general solution by viewing an arbitrary source as a collection of point sources. The differential equation for a point source is (1-52) where '" is the response to a point source at the origin, and B( ) is the unit impulse function, or Dirac delta function (see Appendix F.l). In spite of the fact that the point source is of infinitesimal extent, its associated current has a direction. This is because in solving practical problems, the point source represents a small subdivision of current that does have a direction. If the point source current is taken as z-directed, then (1-53) Since the point source is zero everywhere except at the origin, (1-52) becomes (1-54) away from the origin. 1.5 Solution of Maxwell's Equations for Radiation Problems 19 This is the complex scalar wave equation or Helmholtz equation. Because of spherical symmetry, the Laplacian is written in spherical coordinates and I/J has only radial dependence. The two solutions to (1-54) are e-j {3r/r and e+j {3r/r. These correspond to waves propagating radially outward and inward, respectively. The physically meaningful solution is the one for waves traveling away from the point source. Evaluating the constant of proportionality (see Prob. 1.5-2), we have for the point source solution: (1-55) . This is the solution to (1-52) and is the magnitude and phase variation with distance r away from a point source located at the origin. If the source were positioned at an arbitrary location, we must compute the distance R between the source location and observation point P (see Fig. 1-8). Then (1-56) The point source solution is actually that of an ideal dipole and will be discussed more fully in the next section. For an arbitrary z-directed current density, the vector potential is also z-directed. If we consider the source to be a collection of point sources weighted by the distribution lz, the response A z is a sum of the point source responses of (1-56). This is expressed by the integral over the source volume v': Az = III -j{3R JLlz IJ' ~ R dv' (1-57) 7T. Similar equations hold for the x- and y-components. The total solution is then the sum of all components, which is A - i III = . -jfJR JLI ~7TR dv' (1-58) V This is the solution to the vector wave equation (1-46). The geometry is shown in Fig. 1-8. The coordinate system shown is used to describe both the source point and field point. r' is the vector from the coordinate origin to the source point, and rp is the vector from the coordinate origin to the field point P. The vector R is the vector Figure 1-8 Vectors used to solve radiation problems. 20 Chapter 1 Antenna Fundamentals and Definitions from the source point to the field point and is given by rp - r'. This geometry is standard and will be used here. We can summarize rather simply the procedure for finding the fields generated by a current distribution J. First, A is found from (1-58). The H field is found from (1-37). The E field can be found from (1-47), but frequently it is simpler to find E from (1-17) as 1 E=-. (VXH-J) ,we (1-59) in the source region, or from 1 E =- V jwe X H (1-60) if the field point is removed in distance from the source; that is, if J = 0 at point P. 1.6 THE IDEAL DIPOLE The principles presented in the previous section are used in this section to find the fields of an infinitesimal element of current. We shall use the term ideal dipole for a uniform amplitude current that is electrically small with az « A. It is ideal in the sense that the current is uniform in both magnitUde and phase over the radiating element extent. Such a discontinuous current is difficult to realize in practice; practical realizations that approximate the ideal dipole are presented in Sec. 2.1. The term current element is often used for the ideal dipole to describe its application as a section of a larger current associated with an actual antenna. Thus, any practical antenna can first be decomposed into filaments of continuous current that are then subdivided into ideal dipoles. The fields from the antenna are then found by summing contributions from the ideal dipoles. Other terms used for the ideal dipole are Hertzian electric dipole, electric dipole, infinitesimal dipole, and doublet. An electrically small, center-fed wire antenna has a current distribution that tapers to zero from the center to the ends of the wire. This short dipole antenna has the same pattern as an ideal dipole and is discussed further in Secs. 1.9 and 2.1. Consider an element of current of length az along the z-axis centered on the coordinate origin. It is of constant amplitude l. In this case, the volume integral of (1-58) for vector potential reduces to the one-dimensional integral4 A = ipJ 11Z'2 f -l1zl2 -jfJR _e_ dz' 47TR (1-61) The length az is very small compared to a wavelength and to the distance R; see Fig. 1-9. Since az is very small, the distance R from points on the current element +rile result in (1-61) could also be obtained by representing the current density on the dipole as J = I 8(x') 8(y')i for .:1z .:1z -"2 < z , <"2 Substituting this into (1-58) yields A = ipl . from which (1-61) follows. f ~ -~ f~ 8(x') dx' -~ 8(y') dy' f4.%/2 -j/3R - dz' -4.%12 _e 4'7TR 1.6 The Ideal Dipole z 21 p ~------------------y Figure 1-9 The ideal dipole. The current I is uniform, Az « A, andR = r. x to the field point approximately equals the distance r from the origin to the field point. Substituting r for R in (1-61) and integrating gives .. T tJlde -j{3r A A A=--~zz (1-62) 47Tr This is exactly true for a point current element and is approximately true for a small (tlz « A and tlz « R) but finite uniform current element. The vector potential A z for a point source was also derived in the previous section; see (1-55) in which I tlz = 1. For many current sources, we can readily make the substitution of r for R in the denominator of the integrand in (1-61), but usually cannot make the same substitution in the exponent. However, in the case of a very small source, we can use r for R in both the denominator and exponent. We are now ready to calculate the electromagnetic fields created by the ideal dipole. The magnetic field is found from (1-37) as H = 1:. V x A = J.L 1:. V x (Azz) (1-63) J.L If we apply the vector identity (C-16), the preceding equation becomes (1-64) The second term is zero because the curl of a constant vector is zero. Substituting (1-62) into (1-64), we have I tlze-jf3r) 4 Xz A H=V ( (1-65) 7Tr Applying the gradient in spherical coordinates from (C-33) gives if3r )A I tlz a (eH= ----- rXz 47T ar r I tl Z =-47T A [-]·{3e-if3r e -i{3r] ---rXz r r2 A A (1-66) From (C-3), we have i x z= i x (i cos (J - 6 sin (J) = -ci> sin (J (1-67) Substituting (1-67) into (1-66) gives H = I tlz 47T jf3r sin (J ci> [jf3r + l]er2 (1-68) 22 Chapter 1 Antenna Fundamentals and Definitions The electric field can be obtained from (1-60) as E = 1 az [jWJL + r 41T + 1-az 21T ~.l + _1_]e-j~r sin () 0 3 ~-;; r2 jWBr [j§- - + - - 1 1 ] e-]'~r cos jWB r3 1 B r2 (1-69) A ()r Here, (3 is given by (1-51) and is related to wavelength as ~ r-:- (3 = WV 21T A JLB = - (1-70) Note that if the medium surrounding the dipole is air or free space, (3 where JLo and Bo are the permeability and permittivity of free space. Eqs. (1-68) and (1-69) can be written as 1 az ( 1 ) e-j~r sin () cf» ](3r r + -. - H =- E = --]wJL 1 + - 41T j(3 1 1 A (1-71a) 1 1] [1 1] az. [ j(3r 41T = wY JLoBo> e-j~r. «(3r)2 r - - - - - sm 96 A (1-71b) 1 az e-j~r +--11 - - j - --cos9r 21T r (3r2 r If (3r is large (Le., (3r » 1, or r » A since (3 = 21T/A), then all terms having inverse powers ofj(3r are small compared to unity, and (1-71) reduces to 1 az e-j~ E = - 4 jWJL - - sin () 6 A 1T H = (1-72a) r A 1 '-lZ 'a e -j~r . 41T]I-' -r- sm () ;. (l-72b) 'I' These are the fields of an ideal dipole at large distances from the dipole. The ratio of these electric and magnetic field components is !: = w; = wJ;e = j§ = (1-73) 11 where 11 = ViJ8 is the intrinsic impedance of the medium (for free space 11 0 = 376.70. = 1201T 0.). This is a property of plane waves. Also, as we shall see, at large distances from any antenna the fields are related in this manner. Using the fields of (1-72) in (1-30) gives an expression for the complex power flowing density out of a sphere of radius r surrounding the ideal dipole: 1 S = -E x H* 2 az)2 e-j~r e+j~r = 2 41T jWJL -r- sin () 6 X (-jf3) -r- sin 9 cf» 1(1 =! 2 (1 A 2 az)2WJL(3 sin 41T r2 () r • A (1-74) 1.6 The Ideal Dipole 23 which is real-valued and radially directed, both characteristics of radiation. The total power flowing out through a sphere of radius r surrounding the ideal dipole using (1-29) is Pf = ff s· ds 1 (I !!Z)2 = 2 4'1T = r ~ e4~z WJLf3 f:~ dcp !o1T sin 3 () d() 4 WJLf32'1T 3" = WJLf3 (I !!zf (1-75) 12'1T This is a real quantity, and real power indica~es dissipated power. It is dissipated in the sense that it travels away from the source. In fact, the average power going out through a sphere of radius r can be found as indicated in (1-35) by taking the real part of (1-75), which leaves it unchanged. This power expression is independent of r, and thus if we integrate over a sphere of larger radius, we still have the same total power streaming through it. We refer to this type of power as radiated power. The fields in (1-72) are called radiation fields. The general field expressions of (1-71) are valid at any distance from an ideal dipole and are important in some applications and for understanding the input impedance of a dipole. For distances so close to the dipole that f3r « 1, or r « A, only the dominant terms with the largest inverse powers of r need be retained in each component of (1-71): I !!ze-i {3r. 4 2 sm () cl» '1Tr A H'" = E of • I!!z e-i {3r. = -rfl - - - sm 4'1Tf3 r3 (1-76a) I!!z e-i {3r 2'1Tf3 r3 () a - 1'1/ - - - - cos () r A • A (1-76b) These are referred to as the near fields of the antenna. Actually, the magnetic field of (1-76a) which varies as 1Ir2 is that of a short, steady or slowly oscillating current, that is, an induction field. The electric fields of (1-76b) vaiy as 1Ir3 and are those of an electrostatic or quasi-static dipole with charges of +q and -q spaced !!z apart. Note that the electric field components Et and E~f are in-phase, but are in phase-quadrature with the magnetic field H'J!, indicating reactive power. This can be shown directly using these near fields in the complex Poynting vector expression of (1-30): j'1/ (I-!!Z)2 . 2 ()rA + cos () sm . () aA) =--2 - 51 (sm f3 4'1T r (1-77) Note that this power density vector is imaginary and therefore has no time-average radial power flow. The radiation fields, in contrast, are in-phase giving a real-valued Poynting vector that is radially directed; see (1-72) and (1-74). The imaginary power density corresponds to standing waves, rather than traveling waves associated with radiation, and indicate stored energy as in any reactive device. The quadrature phase relationship between the electric and magnetic field components of (1-76) 24 Chapter 1 Antenna Fundamentals and Definitions indicates that energy is interchanged between these fields with time. That is, at one instant of time electric fields near the dipole are strong close to maximum charge regions and a quarter-period later energy is stored in the magnetic field, primarily close to the center of the dipole where the current is maximum. The imaginary power density in the near field is manifested by a reactive component of the antenna input impedance. The real part of the input impedance represents radiation if ohmic losses on the antenna structure can be neglected. Antenna impedance will be discussed further in Sec. 1.9. The power density associated with radiation exists everywhere and passes through the near field. The radiated power density from (1-74) and the near-field power density from (1-77) are both maXimum for () = 90°. The distance for which the maximum radiated and reactive powers are equal for the ideal dipole is r = Al21T. That is, interior to this radius reactive power dominates. This region is sometimes referred to as the radiansphere. At large distances from an antenna, called the far-field region, all power is radiated power. The far field is further characterized by the fact that the angular distribution around the antenna (i.e., the radiation pattern) is independent of distance from the antenna. Field regions and the distance away from an antenna where the far field begins are discussed further in Sec. 1.7.3. 1.7 RADIATION PATIERNS We briefly introduce the radiation pattern in Sec. 1.3 as a description ofthe angular variation of radiation level around an antenna. This is perhaps the most important characteristic of an antenna. In this section, we present several definitions associated with patterns and develop the general procedures for calculating radiation patterns. 1.7.1 Radiation Pattern Basics A radiation pattern (antenna pattern) is a graphical representation of the radiation (far-field) properties of an antenna. We have seen that the radiation fields from a transmitting antenna vary inversely with distance, e.g., lIr. The variation with observation angles «(), cfJ), however, depends on the antenna. Radiation patterns can be understood by examining the ideal dipole. The fields radiated from an ideal dipole are shown in Fig. l-lOa over the surface of a sphere of radius r that is in the far field. The length and orientation of the field vectors follow from (1-72); they are shown for an instant of time for which the fields are peale The angular variation of E9 and Hcf> over the sphere is sin (). An electric field probe antenna moved over the sphere surface and oriented parallel to E9 will have an output proportional to sin (); see Fig. 9-7. Any plane containing the z-axis has the same radiation pattern since there is no cfJ variation in the fields. A pattern taken in one of these planes is called an E-plane pattern because it contains the electric vector. A pattern taken in a plane perpendicular to an E-plane and cutting through the test antenna (the xy-plane in this case) is called an H-plane pattern because it contains the magnetic field Hcf>. The E- and H-plane patterns, in general, are referred to as principal plane patterns. The E- and H-plane patterns for the ideal dipole are shown in Figs. l-lOb and l-lOc. These are polar plots in which the distance from the origin to the curve is proportional to the field intensity; they are often called polar patterns or polar diagrams. The complete pattern for the ideal dipole is shown in isometric view with a slice removed in Fig. l-lOd. This solid polar radiation pattern resembles a "doughnut" 1.7 Radiation Patterns 25 z z --------~~~--y sinO (a) (b) y z I -----+--~~~4-----x (c) (Ii) Figure 1·10 Radiation from an ideal dipole. (a) Field components. (b) E-plane radiation pattern polar plot of IE81 or IHc/>I. (c) H-plane radiation pattern polar plot of IE81 or IHc/>I. (d) Three-dimensional plot of radiation pattern. with no hole. It is referred to as an omnidirectional pauern since it is uniform in the xy-plane. Omnidirectional antennas are very popular in ground-based applications with the omnidirectional plane horizontal. When encountering new antennas, the reader should attempt to visualize the complete pattern in three dimensions. 1.7.2 Radiation from Line Currents Radiation patterns in general can be calculated in a manner similar to that used for the ideal dipole if the current distribution on the antenna is known. This is done by first finding the vector potential given in (1-58). As a simple example, consider a filament of current along the z-axis and located near the origin. Many antennas can be modeled by this line source; straight wire antennas are good examples. In this case, the vector potential has only a z-component and the vector potential integral is one-dimensionaI5 : A = z f -j(3R p. I(z') _e- snus result could also be obtained by usinglz(r'} 4'77R dz' (1-78) = I(z'} c5(x'} c5(y'} in (I-57), where dv' = dx' dy' dz'. 26 Chapter 1 Antenna Fundamentals and Definitions Due to the symmetry of the source, we expect that the radiation fields will not vary with cpo This is because as the observer moves around the source such that rand z are constant, the appearance of the source remains the same; thus, its radiation fields are also unchanged. Therefore, for simplicity we will confine the observation point to a fixed cp in the yz-plane (cp = 90°) as shown in Fig. 1-11. Then from Fig. 1-11, we see that r2=y2+z2 (1-79) z=rcosO (1-80) y = r sin 0 (1-81) H we apply the general geometry of Fig. 1-8 to this case, rp r' = z'i lead to R = rp - r' = yy + (z - z')i, and then R = r = yy + = Vy2 + (z - Z')2 = VyZ + Z2 - 2zz' + (z'i zi and (1-82) Substituting (1-79) and (1-80) into (1-81), to put all field point coordinates into the spherical coordinate system, gives R = {r2 + [-2r cos 0 z' + (Z')2]}1I2 (1-83) In order to develop approximate expressions for R, we expand (1-83) using the binomial theorem (F-4): R ~-!) 2 = (r 2)112 + -1 (r 2)-1I2 [-2r cos 0 z' + (Z')2] + L...L (r )-3/2 2 2 . [-2rcos Oz' + (Z')2]2 + ... (z ')2 sin2 0 (z ')3 sin2 0 cos 0 =r-z'cosO+ + + ... 2r 2r2 (1-84) The terms in this series decrease as the power of z' increases if z' is small compared to r. This expression for R is used in the radiation integral (1-78) to different degrees of approximation. In the denominator (which affects only the amplitude), we let R = r (1-85) We can do this because in the far field r is very large compared to the antenna size, so r » z' ~ z' cos 0. In the phase term - (3R, we must be more accurate when computing the distance from points along the line source to the observation point. z P(O,y, z) 1 1 1 1 1 1 _____________ 11 z-z' Y .~------------------------y x Figure 1-11 Geometry used for field calculations of a line source along the z-axis. 1.7 Radiation Patterns 27 The integral (1-78) sums the contributions from all the points along the line source. Although the amplitude of waves due to each source point is essentially the same, the phase can be different if the path length differences are a sizable fraction of a wavelength. We, therefore, include the first two terms of the series in (1-84) for the R in the numerator of (1-78), giving R "'" r - z' cos fJ (1-86) Using the far-field approximations (1-85) and (1-86) in (1-78) yields e-j/3(r-z' cos (J) e- j/3r A z = pJ(z') dz' = - - pl(z')e j/3Z' cos (J dz' 41Tr 41Tr f f (1-87) where the integral is over the extent of the line source. This may be recognized as a Fourier-transform-type integral; see Sec. 4.3. Next, the magnetic field is found using (1-37): 1 =- H 1· V x (Azz) =- V x A JL JL = 1:. V x JL (-A z sin fJO + A z cos (1-88) fJi) where (C-3) was used. Since A z is a function of rand fJ, the curl in spherical coordinates, as given by (C-35), leads to H All[a JL r ar . = «I» - _. - (-rA sm () - -a (A cos z afJ z fJ) ] (1-89) Substitution of (1-87) into the above gives H = .f, 1:. {-sin JL fJ 41Tr . [cos () f f j !.... JL I(z')ej/3z'cos(J dz'!... e- j/3r _ e- /3r ar 41Tr2 afJ j JL I(z ')e /3Z' cos (J dz' ] } j = .f,'! e- {3r {jf3 sin () JL 41Tr - ;: :fJ [cos fJ f f JL I(z')ej/3Z'cos(J dz' j JL I(z ')e /3z' cos (J dz' (1-90) J} The ratio of the first term to the second term above is of the order {3r. If {3r » 1, the second term is small compared to the first and can be neglected, as we did for the far-field approximation of the ideal dipole in Sec. 1.6. Thus, (1-90) becomes .{3 - j/3r .{3 H = .f, L sin fJ _eJL I(z ')e j /3Z' cos (J dz' = L sin fJ A .f, (1-91) f JL 41Tr JL z The electric field is found from (1-47), which is E = -jwA _ j V(V . A) WJLE Using (1-78) in (1-79) and retaining only the r- 1 term (and assuming f3r to the far-field approximation (1-92) » 1) lead (1-93) 28 Chapter 1 Antenna Fundamentals and Definitions Note that this result is the portion of the first term of (1-92) that is transverse to r because -jwA = -jw( -A z sin 00 + A z cos Or). This is an important general result for z-directed sources that is not restricted to line sources. The radiation fields from a z-directed line source (any z-directed current source in general) are H</> and E8> and are found from (1-91) and (1-93). The only remaining problem is to calculate A z , which is given by (1-57) in general and by (1-87) for z-dlrected line sources. The calculation of A z is the focal point of antenna analysis. We will return to this topic after pausing to further examine the characteristics of the far-field region. The ratio of the radiation field components as given by (1-91) and (1-93) yields wp, E8 = wp, /3 H</> = wv;i: H</> = "1H</> (1-94) where "1 = VT/iB is the intrinsic impedance of the medium. Thus, the radiation fields are perpendicular to each other and to the direction of propagation r and their magnitudes are related by (1-94). These are the familiar properties of a plane wave. They also hold for the general form of a "transverse electromagnetic (TEM) wave" that has both the electric and magnetic fields transverse to the direction of propagation. In general, radiation from a finite antenna is a special case of a TEM wave, called a "spherical wave," that propagates radially outward from the antenna and the radiation fields have no radial components. Spherical wave behavior is also characterized by the e-i /3rI4'TT'r factor in the field expressions; see (1-91). The e-i /3r phase factor indicates a traveling wave propagating radially outward from the origin and the 11r magnitude dependence leads to constant power flow just as with the infinitesimal dipole. In fact, the radiation fields of all antennas of finite extent display this dependence with distance from the antenna. Another way to view radiation field behavior is to note that spherical waves appear to an observer in the far field to be a plane wave. This "local plane wave behavior" occurs because the radius of curvature of the spherical wave is so large that the phase front is nearly planar over a local region. 1.7.3 Far-Field Conditions and Field Regions The results for the line current from the previous section are easily generalized to an arbitrary, but finite-sized, antenna. In the far field of an antenna, the fields exhibit· local plane wave behavior and have 11r magnitude dependence. In this section, we develop the conditions for determining the minimum distance from an antenna for far-field behavior. This begins with a geometric interpretation for far-field approximations. If parallel lines (or rays) are drawn from each point on a line current as shown in Fig. 1-12, the distance R to the far field is geometrically related to r by (1-86), which was derived by neglecting high-order terms in the expression for R in (1-84). The parallel ray assumption is exact only when the observation point is at infinity, but it is a good approximation in the far field. Radiation calculations often start by assuming parallel rays and then determining R for the phase by geometrical techniques. From the general source shown in Fig. 1-13, we see that R = r - r' cos a (1-95) 1.7 Radiation Patterns 29 \ \ z \ \ \ \ \ Tz· 1 \ \ \ Figure 1-12 Parallel ray approximation for far-field calculations of a line source. \ \ \ \ \~OS(J Using the definition of dot product, we have R r· r' rr' = r - r'-- or I R = r - r· r' (1-96) This is a general approximation to R for the phase factor in the radiation integral. Notice that if r' = z'i, as for line sources along the z-axis, (1-96) reduces to (1-86). The definition of the distance from the source where the far field begins is where errors due to the parallel ray approximation become insignificant. The distance \ \ \ \ \ Figure 1-13 Parallel ray approximation for far-field calculations of a general source. 30 Chapter 1 Antenna Fundamentals and Definitions where the far field begins, rtf, is taken to be that value of r for which the path length deviation due to neglecting the third term of (1-84) is a sixteenth of a wavelength. This corresponds to a phase error (by neglecting the third term) of 2'TT/A' Al16 = 'TT/8 rad = 22S. If D is the length of the line source, rtf is found by equating the maximum value of the third term of (1-84), which occurs for z' = D/2 and () = 90°, to a sixteenth of a wavelength: (D/2? 2rtf A --=- 16 (1-97) Solving for rtf gives (1-98) The far-field region is r ~ rtf and rtf is called the far-field distance, or Rayleigh distance. The far-field conditions are summarized as follows: 2D2 r>- (1-99a) A r»D r» A far-field conditions (1-99b) (1-99c) The condition r » D was mentioned in association with the approximation R = r of (1-85) for use in the magnitude dependence. The condition r » A follows from {3r = (2'TTrlA) » 1 that was used to reduce (1-90) to (1-91). Usually, the far field is taken to begin at a distance given by (1-98), where D is the maximum dimension of the antenna. This is usually a sufficient condition for antennas operating in the UHF region and above. At lower frequencies, where the antenna can be small compared to the wavelength, the far-field distance may have to be greater than 2D2/A in order that all conditions in (1-99) are satisfied. See Prob. 1.7-4. The concept of field regions was introduced in Sec. 1.6 and illustrated with the fields of an ideal dipole. We can now generalize that discussion to any finite antenna of maximum extent D. The distance to the far field is 2D 2/A. This zone was historically called the Fraunhofer region if the antenna is focused at infinity; that is, if the rays at large distances from the antenna when transmitting are parallel. In the farfield region, the radiation pattern is independent of distance. For example, the sin () pattern of an ideal dipole is valid anywhere in its far field. The zone interior to rtf, called the near field, is divided into two subregions. The reactive near-field region is closest to the antenna and is that region for which the reactive field dominates over the radiative fields. This region extends to a distance 0.62v'D 3/A from the antenna (see Prob. 1.7-7), as long as D » A. We mentioned in Sec. 1.6 that for an ideal dipole, for which D = Az « A, this distance is Al2'TT. Between the reactive near-field and far-field regions is the radiating near-field region in which the radiation fields dominate and where the angular field distribution depends on distance 1.7 Radiation Patterns 31 from the antenna. For an antenna focused at infinity, the region is sometimes referred to as the Fresnel region. We can summarize the field region distances for cases where D » A as follows: Distance from antenna (r) Region to O.62VD 3/>.. O.62VD 3/>.. to 2D2/>.. 2D2/>.. to 00 o Reactive near field Radiating near field Far field (1-IOOa) (1-IOOb) (1-IOOc) 1.7.4 Steps in the Evaluation of Radiation Fields The derivation for the fields radiated by a line source in Sec. 1.7.2 can be generalized for application to any antenna. Fortunately, the derivation itself need not be repeated each time an antenna is analyzed. That is, it is not necessary to return to Maxwell's equation with each new antenna system. Instead, we work from the results of the line source and its generalizations, which can be reduced to the threestep procedure detailed below: 1. Find A. Select a coordinate system most compatible with the geometry of the antenna, using the notation of Fig. 1-8. In general, use (1-58) with R = r in the magnitude factor and the parallel ray approximation of (1-96) for determining phase differences over the antenna. These yield A -j/3r = JL _e_ 41Tr III Je j /3i.r'dv' (1-101) Jz ej /3i.r' (1-102) v' For z-directed sources, j A = iJL e- /3r 41Tr III dv' v' For z-directed line sources on the z-axis, A -j(3r = iJL _e_ I 41Tr J(z')e j /3Z' cos 0 dz' (1-103) which is (1-87). 2. Find E. In general, use the component of E = -jwA (1-104) which is transverse to the direction of propagation r. This is expressed formally as E = -jwA - (-jwA. r)r = -jw(AoO + A4>~) (1-105) which.arises from the component of A tangent to the far-field sphere. For zdirected sources, (1-106) which is (1-93) 32 Chapter 1 Antenna Fundamentals and Definitions 3. Find H. In general, use the plane wave relation. H=.!iXE 71 (1-107) This equation expresses the fact that in the far field, the directions of E and H are perpendicular to each other and to the direction of propagation, and also that their magnitudes are related by 71. For z-directed sources, H - E9 "'-71 (1-108) which is (1-73). The most difficult step is the first, calculating the radiation integral. This topic will be discussed many times throughout the book, but to immediately develop an appreciation for the process, we will present an example. This uniform line source example also provides a specific setting for introducing general radiation pattern concepts and definitions. The Uniform Line Source The uniform line source is a line source for which the current is constant along its extent. If we use a z-directed uniform line source centered on the origin and along the z-axis, the current is I(z') = { x' = 0, y' = 0, IO 0 L Iz'I::s;2 (1-109) elsewhere where L is the length of the line source; see Fig. 1-14. We first findA z from (1-103) as follows: -jfJr A = f.L _e_ 41Tr z fLI2 -LIZ IoeifJz' cos 8 dz' 8- e-/{Jr [eifJ(LIZ) cos e-i{J(LlZ) cos =f.L-I 41Tr 0 jf3 cos 8 _ IoLe-i{Jr sin[(f3L12 cos 8] - f.L 41TT (f3L12) cos 8 . 8J (1-110) p Z / (J L / / / / / • z / 0/ T / / / L / T / f----t--I(z) L L -TL..-_...J -T (a) Antenna geometry (b) Current distribution Figure 1·14 The uniform line source (Example 1-1). 1.7 Radiation Patterns 33 The electric field from (1-106) is then _. . 8 A a E - JW sm zU - jwJL!oLe-ifjr 41Tr . sm 8 sin[(f3L12) cos 8] (L/2) f3 cos 8 a u (1-111) The magnetic field is simply found from this using Hcf> = E8/'T/. 1.7oS Radiation Pattern Definitions Since the radiation pattern is the variation over a sphere centered on the antenna, c/J variation of the field. It is convenient to normalize the field expression such that its maximum value is unity. This is accomplished as follows for a z-directed source that has only a O-component of E: r is constant and we have only 6 and . Eo F(6, c/J) = Eo(max) (1-112) where F(e, c/J) is the normalized field pattern and Eo(max) is the maximum value of the magnitude of Eo over a sphere of radius r. In general, E(J can be complex-valued and, therefore, so can F( 6, c/J). In this case, the phase is usually set to zero at the same point the magnitude is normalized to unity. This is appropriate since we are only interested in relative phase behavior. Pattern variation is, of course, independent of r. An element of current on the z-axis has a normalized field pattern from (1-72a) of j F(e) = (Il1z141T')jwJL(e- {3rlr) sin 6 = sin e (Jl1z141T')jwJL(e Jf3 r lr ) (1-113) and there is no c/J variation. The normalized field pattern for the uniform line source is from (1-111) in (1-112) F(6) = sin 6 sin[(,BLI2) cos 6] (f3L12) cos () (1-114) and again there is no c/J variation. The second factor of this expression is the function sin(u)lu and we will encounter it frequently. It has a maximum value of unity at u = 0; this corresponds to 6 = 90°, where u = (,BLl2) cos 6. Substituting 6 = 90° in (1-114) gives unity and we see that F(6) is properly normalized. In general, a normalized field pattern can be written as the product I F(6, c/J) = gee, c/J )f(6, c/J) I (1-115) where g(6, c/J) is the element factor and f(6, c/J) the pattern factor. The pattern factor comes from the integral over the current and is strictly due to the distribution of current in space. The element factor is the pattern of an infinitesimal current element in the current distribution. For example, we found for a z-directed current element that F(6) = sin 6. This is, obviously, also the element factor, so g(6) = sin 6 (1-116) for a z-directed current element. Actually, this factor originates from (1-93) and can be interpreted as the projection of the current element in the 6-direction. In other words, at 6 = 90° we see the maximum length of the current, whereas at 6 = 0° or 34 Chapter 1 Antenna Fundamentals and Definitions 1800 we see the endview of an infinitesimal current that yields no radiation. The sin (J factor expresses the fraction of the size of the current as seen from the observation angle (J. On the other hand, the pattern factor f( (J, cp) represents the integrated effect of radiation contributions from the current distribution, which can be treated as being made up of many current elements. The pattern value in a specific direction is then found by summing the parallel rays from each current element to the far field with the magnitude and phase of each included. The radiation integral of (1-101) sums the far-field contributions from the current elements and when normalized yields the pattern factor. Antenna analysis is usually easier to understand by considering the antenna to be transmitting as we have here. However, most antennas are reciprocal and thus their radiation properties are identical when used for reception; see Sec. 9.4. A typical power pattern is shown in Fig. 1-15 as a polar plot. The rays from various parts of an antenna arrive in the far field with different magnitude and phase due to variations in the current where the ray originated on the antenna and due to phase changes arising from path length differences to the far field. These rays interfere, as computed through the radiation integral, and produce a "lobing" effect. The radiation lobe containing the direction of maximum radiation is the major lobe, main lobe, or main beam. It is the most intense portion of the radiation pattern and is caused by the fact that the rays from various parts of the antenna arrive in the far field more nearly in-phase than they do for other directions. For a source with constant phase, all rays arrive in-phase in the direction normal to the antenna and the pattern is maximum there. For the ideal dipole, we have said that the source is so small that there are essentially no phase differences for rays along the source and thus the pattern factor is unity. For the z-directed uniform line source pattern of (1-114), we can identify the factors as g(O) = sin (J (1-117) and f(O) = sin[(f3L12) cos (f3L12) cos (J] (J Main lobe maximum direction 1.0 Half-power point (left) 0.5 Beamwidth between fIrst nulls (BWFN) Figure 1-15 A typical power pattern polar plot. (1-118) 1.7 Radiation Patterns 35 For long line sources (L » A), the pattern factor of (1-118) is much sharper than the element factor sin (), and the total pattern is approximately that of (1-118); that is, F( () = f( (). Hence, in many cases we need only work with f( (), which is obtained from (1~103). If we allow the beam to be scanned (this will be discussed in Sec. 1.7.6), the element factor becomes important as the pattern maximum approaches the z-axis. Frequently, the directional properties of the radiation from an antenna are described by another form of radiation pattern, the power pattern. The power pattern gives angular dependence of the power density and is found from the (), cJ> variation of the r-component of the Poynting vector. For z-directed sources, H</> = E(J/TJ so the r-component of the Poynting vector is (1I2)E(JJ4, = IE(J12/2TJ and the normalized power pattern is simply the square of its field pattern magnitude P(8) = IF(8)12. The general normalized power pattern is I P(8, cJ» = IF(8, cJ> WI (1-119) The normalized power pattern for a z-directed current element is P(8) = sin2 8 (1-120) and for a z-directed uniform line source is P(8) = {Sin 8 sin[(f3L12) cos 8]}2 (f3L12) cos 8 (1-121) Frequently, patterns are plotted in decibels. It is important to recognize that the field (magnitude) pattern and power pattern are the same in decibels. This follows directly from the definitions. For field intensity in decibels, IF(8, cJ> )ldB = 20 logIF(8, (1-122) cJ»1 and for power in decibels, P(9, c/»cm = 10 log P(9, c/» = 10 logIF(9, c/>)12 = 20 logIF(9, c/»I (1-123) and we see that I P«(), cJ»dB = IF«(), cJ> )ldB I (1-124) L 7.6 Radiation Pattern Parameters A typical antenna power pattern is shown in Fig. 1-15 as a polar plot in linear units (rather than decibels). It consists of several lobes. The main lobe (or main beam or major lobe) is the lobe containing the direction of maximum radiation. There is also usually a series of lobes smaller than the main lobe. Any lobe other than the main lobe is called a minor lobe. Minor lobes are composed of side lobes and back lobes. Back lobes are directly opposite the main lobe, or sometimes they are taken to be the lobes in the half-space opposite the main lobe. The term side lobe is sometimes reserved for those minor lobes near the main lobe, but is most often taken to be synonymous with minor lobe; we will use the latter convention. The radiation from an antenna is represented mathematically through the radiation pattern function, F(8, cJ» for field and P(8, cJ» for power. This angular distribution of radiation is visualized through various graphical representations of the 36 Chapter 1 Antenna Fundamentals and-Definitions pattern, which we discuss in this section. Graphical representations