71-22,477 FREELAND, John, 1941- YAGI-UDA ARRAYS OF V DIPOLES. The Ohio State University, Ph.D., 1971 Engineering, electrical University Microfilms, A XEROX Company, Ann Arbor, Michigan © Copyright by John Freeland 1971 YAGI-Um ARRAYS OF V DIPOLES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By John Freeland, B.E.E., M.Sc. * * * * * The Ohio State University 1971 Approved by Department of Electrical Engineering PLEASE NOTE: Some pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS. ACKNOWLEDGMENTS I wish to express my gratitude to Professor Richmond for his patience during the ten or eleven drafts through which this dissertation went, and to Professors Roger C. Rudduck and Daniel B. Hodge of the reading committee for their objective appraisel of my efforts. I am also grateful to my parents, Mr. and Mrs. Albert Freeland, for their steadfast confidence in my ability to finish the academic program which I began eleven years ago. 11 VITA December 29, 19^1 • • Born - Linton, Indiana 196k • ••••••• B.E.E., The Ohio State University, Columbus, Ohio 1968 •••••••• M.Sc., The Ohio State University, Columbus, Ohio PUBLICATIONS "An Evaluation of Several Approximate Methods for Analyzing the Scattering from Strips (Plates)," URSI-FGAP Digest, 196 6 Fall Meeting, p. 90, December, 1 9 6 6. "Strip Current Approximation by Fresnel Integrals," PGAP Digest and Technical Program, 1 9 6 7 International Antenna and Propagation Symposium, pp.23^-237, October 1967. "Ifedified Single Diffraction," URSI-PGAP Digest, 1 9 6 8 Fall Meeting, p. 79, September 1968. FIELDS OF STUD! Major Field: Electrical Engineering Studies in Electromagnetics. Professor Jack H. Richmond Studies in Antenna Theory. Professor Carlton H. Walter Studies in Communication Theory, Professor Claude E Warren iii t a b l e o f c o n t e n t s Page ACKNOWLEDGMENTS............................... ii VITA ........................................... ill LIST OF F I G U R E S .......................... v Chapter I INTRODUCTION................... 1 II SURVEY OF EARLIER INVESTIGATIONS ON THIN WI R E S ........................ h III THE ZERO REACTION INTEGRAL EQUATION . lU IV COMPARISON OF THREE INTEGRAL EQUATIONS FOR THIN LINEAR DIPOLES ........ 21 V PIECEWISE SINUSOIDAL CURRENT EXPANSION . 27 VI NUMERICAL ANALYSIS FOR THIN V DIPOLES . 37 VII CURRENTS, FIELDS, AND INPUT IMPEDANCE OF 0.5A, l.oA, ana 1.5A THIN V D I P O L E S .................... 50 VIII DESCRIPTION AND APPLICATION OF A REACTION ANTENNA THEORY TO YAGI-UDA ARRAYS OF 0.5A LINEAR DIPOLES ..... 57 IX APPLICATION OF THE REACTION ANTENNA THEORY TO YAGI UDA ARRAYS OF 1.5 A V D I P O L E S ............... 65 X S U M M A R Y ...................... 71 GRAPHS....................... 73 APPENDIX...................... 113 REFERENCES.................... 123 lv LIST OF FIGURES Figure Page 1 A current J flovs on the surface S of a vire T ......... .......... 1^ 2 Six segment solution for a linear dipole# 29 3 Six segment solution for a V dipole . 38 ^ Magnitude of current on 0.50A linear dipole 7^ 5 Phase of current. (See Fig. U) . 75 6 Admittance versus V for 0.50A V dipole . 76 7 Impedance versus Y . (See Fig. 6) . 77 8 Electric field magnitude in the E-plane for a 0 .5 0A linear d i p o l e ............ 78 9 Magnitude of current on 1.00A linear dipole ...................... 79 10 Phase of current. (See Fig. 9) • • • • . 80 11 Admittance versus 7 for 1.00 A V dipole . 81 12 Impedance versus *P. (See Fig. 11) . 8 2 13 Electric field magnitude in the E-plane for a 1.00A linear dipole ....... 83 lh Magnitude of current on 1.50A linear dipole .............. 84 15 Phase of current. (See Fig. lk) .... 85 ▼ Figure Page 16 Magnitude of current on 1.50A V dipole for S'= 180° and 11 5 ° ................ 66 17 Phase of current. (See Fig. 16) . 87 18 Admittance versus S'for 1.50A V dipole • 8 8 19 Impedance versus S'. (See Fig. 18) . 89 20 Electric field magnitude in the E-plane for a 1.50 A linear di p o l e ............. 90 21 Directivity versus S' for V dipoles . 91 22 Comparison of 1.50 A V dipole radiation patterns in the E-plane for S' 3 115°, NSEGs 2 and 1 2 ...................... 92 23 H-plane patterns. (See Fig. 22) • • • • 93 2^ Yagi-Uda array of half wave elements . • 9^ 25 Comparison of two segment reaction antenna theory solution with Green’s measured pattern for his four element Yagi-Uda array of 0.50A linear dipoles . 95 26 Comparison of two and six segment reaction antenna theory solutions for the array of Fig. 25. E-plane patterns . 96 27 E-plane patterns. (See Fig. 26 ) . • • • 97 28 Comparison of two segment reaction antenna theory solution with Green’s measured pattern for his ten element Yagi-Uda array of 0.50A linear dipoles . 9 8 29 Comparison of two and six segment reaction antenna theory solutions for the array of Fig. 2 8, E-plane patterns . 99 30 H-plane patterns. (See Fig. 29) .... 100 vi Figure Page 31 Input Impedance versus frequency for a four element array of 0.5A linear dipoles ••••• ............ 101 32 Yagi-Uda array of 1.5A V dipoles • • • • 102 33 Two element Y-U-V array comparedwith Green*s four element Y-U array. E-plane patterns ....... ••••••• 103 3 I* H-plane patterns. (See Fig. 33) . • * . 10^ 35 Field magnitudes in the E-plane. (See Fig. 3 3 ) ......................... 105 36 Field magnitudes in the H-plane. (See Fig. 3 3 ) ......................... 106 37 Four element Y-U-V array compared vith Green's ten element Y-U array. E-plane patterns .......... 107 3 8 H-plane patterns. (See Fig. 37) • . • . 108 39 Field magnitudes in the E-plane. (See Fig. 3 7 ) ......................... 109 1*0 Field magnitudes in the H-plane, (See Fig, 37) 110 1*1 Input impedance versus frequency for a four element Y-U-V array. Directivity optimized ••••• 111 1*2 Input impedance versus frequency for a four element Y-U-V array. Input Impedance matched to 50 ohm line at the expense of directivity ••••••• 1 1 2 1*3 The electric field E = £ E + ^ E ... 115 - r |0 s 1*1* Two nohplanar skew segments ••••••• 11 6 1*5 The dependence of rR, ^o, s, and ©K on t . 117 vii CHAPTER I introduction In this dissertation Rumsey*s reaction concept (37) and a piecewise-sinusoidal current expansion (Schelkunoff and Friis (3 8 ) followed by Richmond (6 9)) are used to develop an efficient solution for perfectly conducting thin wire V dipoles and arrays of V dipoles. The theory was introduced by Richmond (35) in 1 9 6 9 for the study of closely coupled V dipoles; it is a generalization of Carter*s induced EMF method (lU) to the case in which several spatially-shifted overlapping dipole- mode currents are used in an expansion for the current on a thin wire. The unknown complex constants of the current expansion are determined by making the unknown field of the current expansion have the correct reaction with a set of test sources located inside the wire on its axis. The piecewise-sinusoidal current expansion consists of a linear combination of sine and cosine functions on each segment of wire. It retains the desirable properties of the two-segment sinusoidal approximation for the current on a dipole; for example, in the case of straight segments, the 1 2 electric and magnetic fields and the reaction integrals are available without numerical integration (6 9). In the language of numerical analysis, the unknown complex constants of the current expansion are determined by requiring the current to produce an electric field whose longitudinal (axial) component satisfies the condition inside the wire that a weighted average of it is zero on the axis of the wire. This follows from the zero reaction condition, which states that any test source inside a closed perfectly conducting surface S has zero reaction with the true sources on S. Solution of the integral equation is carried out by Galerkin*s method, which is described by Jones (63 )} in this method the current expansion functions and the test (weighting) functions are identical. In Chapter II we survey briefly the earlier investigations on thin wire antennas. A brief review of the literature on Yagi-Uda arrays is found in Section A of Chapter VIII. In Chapters III through VII the details of applying the reaction antenna theory to 0.5 a * 1.0X, and 1.5 X V dipoles are discussed, and groundwork is laid for appli­ cation of the theory to arrays. Emphasis is placed on the impedance and radiation properties of the V antenna for V angles greater than 30 °. In Chapter VTII the reaction antenna theory is tested on the conventional Yagi-Uda array of 0.5A linear dipoles. The rate of convergence is studied as the number of segments is increased. In Chapter IX a new array structure is designed on the basis of the results for the 1.5X V dipole discussed in Chapter VII. The Yagi-Uda array of 1.5X V dipoles is compared with the conventional array of half-wave linear elements, and the differences are discribed. When the spacings, lengths, and V angles are optimized, the array of 1.5 A v dipoles is found to have outstanding directivity characteristics. Chapter X summarizes the properties of the reaction antenna theory. CHAPTER II SURVEY OF EARLIER INVESTIGATIONS ON THIN WIRES Hallen*s integral equation, Pocklington’s integral equation, and the zero-reaction integral equation described in Chapter III are all Fredholm integral equations of the first kind; the unknown function is the current on the wire, and is the source of the radiated electromagnetic fields.