CONTENTS Note Concerning the Numbering of Equations, Figures, and References; Notation, xxi Introduction: Theory, 1 A Bridge from Mathematics to Engineering in Antenna Isolated Antennas 1. Free Oscillations, 3 2. Forced Oscillations and the Poynting-Vector Theorem, 3 3. Cylindrical Antennas and Equivalent Transmission Lines, 4 4. The Antenna as a Boundary-Value Problem, 5 5. Retarded Potentials and Integral Equations, 7 Coupled Antennas 6. The emf Method, 8 7. The Retarded-Potential Method, 8 8. Antenna Over Conducting Earth, 9 Elements of a Consistent Theory 9. Theory and Practice, 10 I. Essentials of Electromagnetic Theory, 12 1. Density Functions, 12 2. Essential Density Functions, 12 3. The Equation of Continuity, 12 4. The Field Equations and the Boundary Conditions, 13 5. Auxiliary Field Vectors and Constants, 13 6. The Force and Torque Equations, 14 7. Potential Functions; Potential of Axial Current in Cylinder, 14 8. Polarization and Magnetization Potentials, 20 9. Integrals of the Field Equations, 21 10. Energy Functions, 21 11. Simple Media, 21 12. The Magnetic Dipole, 22 II. Linear Radiators as Circuit Elements, 24 Antenna as Transmitting System 1. Theory and Experiment: Discussion of the Problem, 24 2. Definition of an Antenna, 25 3. Antenna as a Circuit Element, 26 4. Transmitting System with Simple External Generator, 26 xi
xii CONTENTS Antenna and Transmission Line 5. Two-Wire Line with Symmetric Impedance as End-Load, 31 6. Two-Wire Transmission-Line Theory; Generalized Equations, 33 7. Approximate Solution of the Generalized Transmission-Line Equations, 44 8. Antenna Terminating Two-Wire Line, 50 9. Antenna as Mid-Point Load for Symmetrically Driven Line; Antenna as End Load with Stub Support, 55 10. Antenna Center-Driven from a Two-Wire Line with Minimum Coupling; Image Methods of Driving an Antenna Using Open-Wire and Coaxial Lines, 59 Hallen's Theory of Cylindrical Antennas 11. Cylindrical Antenna, 69 12. Formal Solution of Differential Equation for Vector Potential of Antenna. General Symmetry Conditions, 72 13. Vector and Scalar Potentials for the Symmetric Antenna, 74 14. Hallen's Integral Equation, 76 15. Series Solution of the Integral Equation, 81 Distributions of Current and Charge 16. The Zeroth-Order Solutions for Current and Charge, 86 17. Simplification for Small Base Separation, 87 18. Hallen's Expansion, 89 19. Hallen's Solution: Evaluation of First-Order Integrals, 94 20. Expansion of King and Middleton; Parameters, 101 21. King-Middleton Solution: General Formula for Current, 107 22. Axial Distribution of Current, 110 23. Axial Distribution of Charge, 119 24. The Instantaneous Current and Charge, 123 25. An Alternative Method of Solving Hallen's Integral Equation, 123 26. Experimental Determination of the Distributions of Current and Charge, 127 Theoretical Impedance and Admittance of Cylindrical Antenna 27. General Formula for Impedance and Admittance of Center- Driven Antenna, 141 28. Impedances of Zeroth and Modified Zeroth Order, 144 29. Admittance and Impedance Factors for Small Base Separation, 149 30. Impedance and Admittance for Zero Base Separation; First- and Second-order Solutions, 151 31. Impedance and Current for the Electrically Short Antenna, 184 32. Impedance of Antenna with Small Base Separation, 193
CONTENTS xiii Apparent Impedance of Cylindrical Antenna as Load on a Transmission Line 33. Antenna with Positive or Negative Capacitance in Shunt, 193 34. Apparent Load Impedance of Antenna; Summaries of Theoretical and Experimental Results, 198 35. Impedance Measurements with Open-Wire Lines; the Image- Plane Line, 215 36. Impedance Measurements with a Long, Two-Wire Line, 219 37. Impedance Measurements with a Vertical Coaxial Line; Effect of Dielectric Bead, 221 38. Effect of Transmission-Line Dimensions on Impedance of Antennas Measured with Coaxial Line, 227 Other Formulations of the Cylindrical Antenna Problem 39. Storer's Variational Modification of Hallen's Integral Equation, 237 40. The emf Method and the Poynting-Vector Method for Determining the Impedance of Cylindrical Antennas, 258 41. The Integral-Equation Formulation of L. V. King, 259 42. Storm's Method of Undetermined Coefficients, 261 III. Circuit Properties of Arrays of Linear Radiators, 263 Theory of Two Identical, Parallel, Nonstaggered, Antennas Center-Driven 1. General Discussion of the Problem, 264 2. Coupled Antennas with Small Base-Separations; Arbitrary Driving Voltages, 264 3. Symmetrically Driven Antennas, 267 4. Antisymmetrically and Arbitrarily Driven Antennas, 269 5. Closely Spaced Antennas: Cage Antenna; Two-Wire Line, 273 6. First-Order Distributions of Current in Symmetrically and Antisymmetrically Driven Antenna, 275 7. Impedances of Symmetrically and Antisymmetrically Driven Antennas; Antenna Parallel to Conducting Plane, 277 8. Impedance of Antennas with Arbitrary Voltages: Self- and Mutual Impedances, 286 9. Coupling and End Effects for Antennas with Individual Feed Lines, 312 10. H-Arrays, 318 11. Driven Antenna with Parasite, 322 12. Transmission-Line Radiators: Antenna with Parasite; Folded Dipole, 334 13. Experimental Determination of Self- and Mutual Impedance of Parallel Antennas, 346
xiv CONTENTS Theory of Identical, Parallel, Nonstaggered Antennas 14. General Theory, 351 15. Three Antennas at the Vertices of an Equilateral Triangle, 361 16. Four Antennas at the Corners of a Square, 366 17. Cage Antennas, 369 18. Current and Impedance of Antenna with Corner Reflector, 370 19. Parallel Arrays with All Elements Driven; Broadside and End- Fire Arrays; Circuit Properties, 371 20. Parallel Arrays with Parasitic Elements, 377 Arrays with All Units in Neutral Planes 21. Two Mutually Perpendicular Antennas, 378 22. Turnstile Antennas and Arrays, 381 V-Antennas 23. Integral Equation for Symmetric, Center-Driven V-Antenna, 381 24. Solution of the Integral Equation for the V-Antenna, 384 25. Impedance and Current for V-Antenna, 386 26. Crossed Antennas, 396 Asymmetrically Driven Linear Antennas 27. Current and Impedance for Asymmetrically Driven Cylindrical Antennas, 397 28. Functions and Parameters for Asymmetrically Driven Antennas, 402 29. Approximate Impedance of Asymmetrically Driven Antennas, 403 30. Theory of the Sleeve Dipole, 407 31. Half-Dipole with Multielement Counterpoise; Ground-Plane Antenna, 418 Collinear Arrays 32. Three-Element Collinear Array; General Analysis of Central Antenna, 422 33. Three-Element Collinear Array; General Analysis of Outer Antennas, 430 34. Collinear Array of Three Half-Wave Dipoles, 436 35. The Center-Driven Collinear Array with Phase-Reversing Stubs, 442 36. Experimental Study of the Collinear Array; Lumped Capacitive Coupling, 447 IV. The Receiving Antenna as a Circuit Element, 456 General Theory of Two Different, Arbitrarily Oriented Antennas 1. Integral Equations with Arbitrary Spacing, 456 2. Specialization of the General Case to Great Separation Equations for Vector Potential, 459
CONTENTS xv The Center-Loaded Receiving Antenna 3. General Formula for Current in the Center-Loaded Antenna, 461 4. Equivalent Circuit of a Receiving Antenna; Effective Length, 466 5. Zeroth-Order Solution for Center-Loaded Receiving Antenna, 468 6. The Expansion Parameter for the Receiving Antenna, 470 7. First-Order Current in Center-Loaded Receiving Antenna, 473 8. Experimental Determination of Distribution of Current and Impedance for a Base-Loaded Receiving Antenna on a Conducting Plane, 479 9. Effective Length, 486 10. Power in the Load; Effective Cross Section, 496 Parasitic Antennas as Scatterers and Reflectors in the Radiation Zone 11. Reradiation from Parasitic Antennas General Theory, 501 12. Current Distribution in Reradiating Antenna, 503 13. Reflecting or Back-Scattering Cross Section of Reradiating Antennas, 506 14. Experimental Determination of Reflecting Cross Section, 512 15. Two Parallel Nonstaggered Receiving and Scattering Antennas, 517 V. The Electromagnetic Field of Center-Driven and Multiple Half-Wave Antennas, 522 Infinitely Thin Center-Driven Antennas Cylindrical 1. The Vector Potential, 523 2. The Magnetic Field, 524 3. The Electric Field, 526 4. The Radiation Field, 528 5. Field Patterns, 530 Theory of Thin Center-Driven and Multiple Half-Wave Confocal Coordinates Coordinates Antennas 6. The Complete Field of Multiple Half-Wave Antennas in Confocal Coordinates, 532 7. The Radiation Field of a Multiple Half-Wave Antenna, 535 8. Phase Relations in the Field of Multiple Half-Wave Antennas, 538 9. Field of Antiresonant Antenna and Antennas of Arbitrary Length, 546 10. Graphs of Instantaneous Electric and Magnetic Fields for Multiple Half-Wave Antennas, 549 11. Correlation of Theoretical Near-Zone Field of Multiple Half- Wave Antenna with Experiment, 555
xvi CONTENTS Radiation Factors 12. Radiation Function, Radiation Resistance, Directivity of Center-Driven and Multiple Half-Wave Antennas, 560 13. The Approximate Representation of the Radiation Field of a Center-Driven Antenna; Effective Length of a Driven Antenna, 566 Center-Driven Antennas of Nonvanishing Cross Section 14. The Radiation Field of a Symmetric Center-Driven Antenna of Nonvanishing Radius Application of the Reciprocal Theorem, 568 15. The Complete Field of a Center-Driven Antenna of Small Radius Approximate Analysis, 571 16. The Radiation Field of a Center-Driven Antenna of Small Radius Approximate Analysis, 574 17. The Electric Field Near the End of an Antenna; Spark Discharges, 575 Center-Driven Antennas with Unequal Currents in the Halves 18. Radiation Field of an Unbalanced Center-Driven Antenna, 576 VI. Electromagnetic Fields of Antenna Arrays, 579 Uniform Parallel Arrays 1. The Diffraction Formula and the General Array Factor, 579 2. The Collinear Array; The Marconi-Franklin Antenna, 587 3. Parallel Arrays the Broadside and the End-Fire Array, 595 Parasitic Parallel Arrays 4. Field of Driven Antenna with Single Parasite Approximate Second-Order Theory, 622 5. Field of Driven Antenna with Several Parasites Zeroth-Order Solution; Arrays of Yagi-Uda Type, 635 6. Yagi-Uda Arrays Experimental Investigation, 645 7. Broadside Array with Parasitic Curtain, 647 Nonuniform Parallel Arrays 8. Complex Array Polynomials, 651 9. Binomial End-Fire Arrays, 652 10. Directive End-Fire Arrays with Assigned Nulls, 654 11. Nonuniform Broadside Arrays Derived from Uniform Arrays, 660 12. Real Polynomials for Symmetric Nonuniform Broadside Arrays, 661 13. Optimum Currents for Symmetric Broadside Arrays; Tchebyscheff Polynomials, 665 14. Closely Spaced Tchebyscheff Arrays, 676
CONTENTS xvii Arrays with Omnidirectional Properties 15. Circular Array, 679 16. Mutually Perpendicular Antennas and Turnstile Array, 684 17. Resonant V-Antenna, 687 The Reciprocal Theorem and the Properties of Arrays 18. Transmitting Arrays with Elements of Finite Cross Section, 689 19. Receiving Arrays, 690 VII. Antennas Over a Conducting Region, 695 1. Hertzian Potentials, 696 Vertical Dipoles Over Conducting Earth; General Formulation 2. The Polarization Potential of a Vertical Electric Dipole or of an Element of a Vertical Antenna, 698 3. The Magnetization Potential of a Vertical Magnetic Dipole or of an Element of a Horizontal Loop Antenna, 701 4. Representation of Spherical Waves as a Bundle of Plane Waves, 703 5. Boundary Conditions for Vertical Dipoles; Incident, Reflected, and Refracted Waves, 707 Far-Zone Fields of Vertical Dipoles Over a Conducting Earth 6. Asymptotic Integration of the Hertzian Field of a Dipole; Field Patterns, 715 7. Dipoles in Air Over Dielectrics and Conductors, 726 8. The Far-Zone Field of an Antenna with Sinusoidally Distributed Current Over a Conducting Earth, 743 Quasi-Near-Zone Fields of Vertical Dipoles Over a Conducting Earth 9. Van der Pol's Integrals for the Vertical Dipole Over a Plane Earth, 746 10. Approximate Integration of Van der Pol's Formulas, 754 11. Norton's Formulas for Practical Evaluation, 759 12. The Electromagnetic Field of a Vertical Dipole; Summary, 760 13. The Vertical Electromagnetic Field of a Vertical Dipole, 763 14. The Radial Electromagnetic Fields, 768 15. Complete Electromagnetic Fields of Vertical Dipoles, 770 16. Polarization and Tilt of the Surface Waves, 773 17. The Field of a Vertical Electric Dipole at Large Numerical Distances, 774 18. Quasi-Near-Zone Fields of Vertical Antennas with Sinusoidal Currents Over a Conducting Earth, 775
xviii CONTENTS Horizontal Dipoles Over a Conducting Earth 19. Hertzian Potentials of Horizontal Dipoles Over a Conducting Earth, 779 20. Far-Zone Hertzian Potentials of Horizontal Dipoles Over a Conducting Earth, 783 21. The Electromagnetic Field of a Horizontal Dipole Over a Conducting Earth, 785 22. Comparison of the Fields of Horizontal Electric Dipoles with Those of Vertical Dipoles, 791 23. Horizontal Antennas with Sinusoidal Currents Over a Conducting Earth, 794 24. Currents Excited on a Perfectly Conducting Plane by a Parallel Antenna, 794 Impedance and Radiation Resistance of Antennas Over Conducting Planes 25. Impedance of Antennas Over Infinite, Perfectly Conducting Planes, 799 26. Radiation Resistance of a Vertical Electric Dipole Over Plane Earth, 800 27. Radiation Resistance of a Horizontal Electric Dipole Over a Plane Earth, 804 28. Impedance of Base-Driven Antenna on a Ground Screen of Finite Size, 808 29. Impedance of Base-Driven Antenna on a Ground Screen of Finite Size on an Infinite Plane Imperfectly Conducting Earth, 815 VIII. The Antenna as a Boundary-Value Problem. 818 1. Hemispheroidal and Conical Antennas, 818 2. Equations for Spherical Waves with Rotational Symmetry, 820 Conical Antennas 3. Boundary Conditions and Equations for the Symmetric, Spherically Capped Biconical Antenna, 824 4. The Dominant Mode; Apparent Terminal Admittance, 826 5. Interior Complementary Modes; Input Impedance, 828 6. Exterior Complementary Modes; Matching of Fields, 830 7. Integrals of Products of Legendre Functions, 831 8. General Solution for Y la and the Infinite Set of Linear Equations, 832 9. Solution for the Apparent Terminal Admittance of a Biconical Antenna with Small Angles, 833 10. Impedance of a Biconical Antenna with Small Angle, 836
CONTENTS xix Cylindrical Antennas 11. Thin Antennas of Arbitrary Cross Section; Schelkunoff's Theory, 838 12. The Gap Problem: Cylindrical Antenna with Biconical Transmission Line, 844 13. Zuhrt's Analyiis of Cylindrical Antennas, 848 Appendix: Tables of Generalized Sine and Cosine Integrals, 857 Problems, 881 Bibliography, 901 List of Principal Symbols, 913 Index, 927
NOTE CONCERNING THE NUMBERING OF EQUATIONS, FIGURES, AND REFERENCES; NOTATION Chapters are numbered in roman; sections are numbered in arabic beginning with 1 in each chapter; equations are numbered consecutively (1), (2),... in each section, with no reference to the section number. When reference is made to an equation in the same section, only the equation number is given, e.g., (5). When reference is made to an equation in another section in the same chapter, the section and equation numbers are given in the form (7.14). When reference is made to an equation in another chapter, the chapter number, section number, and equation number are given, e.g., (II.7.14). Figures are given both section and figure numbers; thus, Fig. 6.2 is the second figure in Sec. 6. Reference to figures in another chapter includes the chapter number, e.g., Fig. II.6.6. Chapter and section numbers appear at the top of each page, and by reference to these numbers any equation or figure can be quickly found. Bibliographic references for each chapter are arranged alphabetically and numbered consecutively. General references and textbooks are included in the bibliography for Chapter I. Reference to the bibliography for the same chapter is by arabic number only, e.g., Schelkunoff 24 ; a reference to the bibliography for another chapter includes the chapter number, e.g., Hallen 1115. The notation distinguishes among vector, real scalar, and complex scalar quantities. In the text all vectors, whether real or complex, are in boldface roman type, except that a few unit vectors are represented by boldface Greek letters with circumflex; real scalars are in lightface italic or lightface Greek; complex scalars are in boldface italic or boldface Greek. A comparison of the several kinds of type used is available in the List of Symbols. In figures, vectors are indicated by arrowheads on the appropriate line segments; complex quantities are distinguished from real quantities by an underline. xxi
The Theory of LINEAR ANTENNAS