tim Hi I mm ■ ran Mi ■ mkM ^H Uh.Wy': Hi i ■ HHHfii L&ttj T I B HAHY OF THE UNIVERSITY Of ILLINOIS 621.365 I!G55te no. 50-57 cop. 2 fciTriii'iLummfi Digitized by the Internet Archive in 2013 http://archive.org/details/analysisdesignof52carr Westinghouse Electric Corporation Air Arms Division Attn: Librarian (Antenna Lab) P. ; Box 746 Wheeler Laboratories Attn; Librarian (Antenna Lab) Box 561 Smithtown, New York Electrical Engineering Research Laboratory University of Texas Box 8026, Univ„ Station Austin, Texas University of Michigan Research Institute Electronic Defense Group Attn: Dr. J „ A„ M. Lyons Ann Arbor. Michigan Dr. Harry Letaw, Jr. Raytheon Company Surface Radar and Navigation Operations State Road West Wayland, Massachusetts Dr„ Frank Fu Fang IBM Research Laboratory Poughkeepsie, New York Mr. Dwight Isbell 1422 11th West Seattle 99. Washington Dr. A, K. Chatterjee Vice Principal Birla Engineering College Pilani, Rajasthan India New Mexico State University Head Antenna Department Physical Science Laboratory University Park, New Mexico Bell Telephone Laboratories, Inc. Whippany Laboratory Whippany. New Jersey Attn: Technical Reports Librarian Room 2A-165 Robert C. Hansen Aerospace Corporation Box 95085 Los Angeles 45, California Dr. D E„ Royal Ramo-Wooldridge, a division of Thompson Ramo Wooldridge Inc. 8433 Fallbrook Avenue Canoga Park ? California Dr. S„ Dasgupta Government Engineering College Jabalpur, M„P„ I ndi a Dr. Richard C Becker 10829 Berkshire Westchester, Illinois AN I ENNA LABORATORY Technical Report No. 52 ANALYSIS AND DESIGN OF THE LOG-PERIODIC DIPOLE ANTENNA by ~;v- Robert L. Carrel Contract AF33 (616) -6079 Project No. 9-(13-6278) Task 40572 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ABSTRACT A mathematical analysis of the logarithmically periodic dipole class of frequency independent antennas, which takes into account, the mutual coupling between dipole elements, is described. The input impedance, directivity, and bandwidth, as well as the input current and voltage of the several elements, are calculated. A new concept, the bandwidth of the active region, is formulated and is used to relate the size and operating bandwidth of the antenna. The limiting values of the various parameters that describe the antenna are explored. The results from the mathematical model are shown to be in good agreement with measurements. A step by step procedure is presented which enables one to design a log-periodic dipole antenna over a wide range of input impedance, bandwidth, directivity, and antenna size. 5RARY i ACKNOWLEDGEMENT The author wishes to thank all the members of the Antenna Laboratory Staff for their help and encouragement. The guidance of his advisor, Professor G. A : Deschamps, and the continued interest of Professor P. E. Mayes are particularly appreciated. This work would not have been possible without the timely invention of the log-periodic dipole antenna by D. E. Isbell, whose counsel during the Initial phase of this research was most helpful. The author is also fortunate to have been associated with V, H. Rumsey and R. H. DuHamel during the time of their original contributions to the field of frequency independent antennas. Thanks are also due to Ronald Grant and David Levinson, student technicians who built the models and performed many of the measurements. This work was sponsored by the United States Air Force, Wright Air Development Division under contract number AF33 (616) -6079, for which the author is grateful. CONTENTS iv Introduction Page 1 Formulation of the Problem 15 2.1 2.2 2.3 Description of the Log-Periodic Dipole Antenna 15 Separation of the Problem into Two Parts 20 2.2.1 The Interior Problem 21 2.2.1,1 The Feeder Admittance Matrix 25 2 ,21 .2 The Element Impedance Matrix 26 2.2.2 The Exterior Problem 33 Use of the Digital Computer in Solving the Mathematical Model 37 Results and Analysis 39 3.1 The Transmission Region 3.1.1 Computed and Measured Results 3.1.2 An Approximate Formula for the Constants of an Equivalent Line in the Transmission Region 3.2 The Active Region 3„2 1 Element Base Current in the Active Region 3.2.2 Width and Location of the Active Region 3.3 The Unexcited Region 3.4 The Input Impedance 3.4,1 General Characteristics of LPD Input Impedance 3 4 2 Input Impedance as a Function of t and <7 3.4.3 Input Impedance as a Function of Z and h/a 3.5 The Far Field Radiation 3.5.1 Radiation Patterns 3.5.1.1 The Characteristic Pattern as a Function of t and t a 0.95, CT = 0.0564, N = 13, Z Q = 100, Z = short at h-,/2, h/a = 177 44 17, Computed and measured amplitude and phase of the transmission wave vs. relative distance from the apex at frequency f„ , ,_ ; t = 0.95, CT = 0.0564, N = 13, Z q = 100, 1^ = short at h /2, h/a = 177 45 vii ILLUSTRATIONS (Continued) Figure Page 18. Computed and measured amplitude and phase of the transmission wave vs, relative distance from the apex at frequency fg 3/4; T = 0,95, G = 0,0564, N = 13, Z Q = 100, Z^ = short at h-,/2, h/a =177 46 19. Computed and measured amplitude and phase of the transmission a1 't h/a = 177 47 20. Relative velocity of transmission wave vs. T and O computed from the approximate formula 49 21. Computed and measured amplitude and phase of the transmission wave vs, relative distance from the apex at frequency f„; T = 0.888, a = 0,089, N s 8, Z q = 100, Z^ a short at h-j/2, h/a = 125 50 22. Relative velocity of transmission wave as a function of the relative phase velocity along the feeder with the elements removed, 51 23. Computed base impedance Z b vs, element number for an eight element LPD at frequency f. 56 24. Computed and measured amplitude and phase of the element base current vs„ relative distance from the apex, at fre- quency f 3 ; t = 0.95, CT = 0.0564, Z = 100, h/a = 177. Z^ = 58 circuit at h-, '2 59 25, Computed and measured amplitude and phase of the element base current vs, relative distance from the apex, at frequency f . ; T = 0.95, 0" = 0.0564, Z Q = 100, h/a = 177, Z T = short 26, Computed and measured amplitude and phase of the element base current vs. relative distance from the apex at frequency f 3 1/2' T ~ °' 95 .- a = °.° 564 - z = 10 °; h/a = 177 , Z t = short circuit at h /2 60 27, Computed and measured amplitude and phase of the element base current vs. relative distance from the apex, at frequency f 3 3/4' T = °' 95 ' ° = 0,0564, Z Q = 100, h/a = 177, Z^ = short circuit at h-,/2 61 28, Computed and measured amplitude and phase of the element base current vs, relative distance from the apex^ at frequency f . ; T = 0,95, Q = 0.0564, Z Q = 100, h/a = 177, Z^ = short circuit at h r /2 62 viii ILLUSTRATIONS (Continued) Figure Page 29. Relative amplitude of base current in the active region vs. element number, frequencies fj thru f 6 . T = 0.888, O = 0.089, N = 8, Z Q a 100, h/a a 125, Z T = short at hj/2 63 30. Computed relative phase velocity of the first backward space harmonic in the active region vs. 0" for several values of T . 65 31. A typical curve of base current vs, distance from the apex, showing the quantities used in the definition of the bandwidth and location of the active region 68 32. Bandwidth of the active region, B , vs., 0" and T 70 ar' 33.. Shortening factor S, vs. Z and h/a 72 34. Radiating efficiency of the active region vs, relative length of the longest element 75 35 „ Radiating efficiency of the active region vs. feeder impedance and t 76 36 . Input impedance vs, frequency of an eight element LPD 79 37. Input impedance showing periodic variation with frequency 82 38, Input impedance R vs. a and t for Z = 100 and h/a = 177 83 o o 39. Difference between the approximate discrete formula and approximate distributed formula for R , vs the distance between elements as a percent of the latter 85 40, Computed SWR vs, O and T, for Z = 100 and h/a = 177 86 o 41, Input impedance R Q vs, feeder impedance Z T = 0,888, 0" = 0,089, N = 8 h/a a 125 88 42. Input impedance R^ vs, Z^ and 0"' with h/a = 177, from the approximate foi >rmul* 43, Input impedance R vs, h/a and 0", Z = 100, from the approxi- mate formula 90 44a, Input impedance T = 0,888, 0" = 0,089, N = 8, Z = 50, Z T = 50 at frequencies f 3 , f^, f^ j, f and f 92 44b. Input impedance t = 0.888, CT = 0,089, N = 8, Z^ = 50, Z = short at h,/2, at frequencies f_, f„, f,_ and f ■L ' o 4 5 ' 93 45„ Average characteristic impedance of a dipole Z vs, height to radius ratio h/a 94 ILLUSTRATIONS (Continued) Figure Page 46. Relative feeder impedance Z /R vs, relative dipole impedance Z /R , from the approximate formula 95 a o 47. A frequency independent 4:1 balun transformer for use with LPD antennas 97 48 „ An example of radiation patterns computed by ILLIAC, t = 0.888, = 0089, N = 8, Z - 100, Z^ = short at h ,/2 99 o T 1 49. Computed patterns^ t = 0„888, 0" = 0„089, Z Q = 100, Z = short at h^/2, showing no difference between patterns for N = 5 and N = 8 100 50. Computed half power beamwidth vs„ frequency; T = 0.888, 0" = 0,089, N = 8, Z s 100!^ Z = short at h /2 101 51, Computed and measured patterns; t = 0.888, (7 = 0,08 9, Z = 100, Z = short at h /2 103 52 „ Computed and measured patterns t = 888, O = 0.089, Z = 100, Z a short at h /2 104 53. Computed and measured patterns; T = 0.888, O = 0.089, Z = 100, Z T = short at h /2 105 54., Computed and measured patterns; T = 0.888, C = 0,08 9, Z = 100, Z T = short at h /2 106 55, Computed and measured patterns, T = 0,888, 0" = 0.08 9, Z = 100, Z = short at h /2 ° 107 56, Computed and measured patterns; T = 0.888, 0" = 089, Z = 100, Z T = short at h 2 . 108 57, Computed and measured patterns; t - 98, O = 0,057, N = 12, Z = 100, Z^ = short at h 2 109 o 7 T 1 58 , Computed and measured patterns; t = 0.98, G = 0.057, N = 12 Z = 100, Z = short at h /2 110 59, Computed and measured patterns, T = . 98 , (J = 0.057, N = 12, Z q = 100. Z T = short at h /2 111 60, Computed and measured patterns; t = 8 , Q - 0.137 N = 8 Z q = 100, Z T = short at h /2 112 61, Computed and measured patterns; t = 0.8, Q - 0,137, N = 8, Z = 100, Z T = short at h/2 113 o ' i 1 62 , Computed and measured patterns; t = 0,8, O = 0.137, N = 8 Z q = 100, Z T = short at h/2 114 ILLUSTRATIONS i Continued) Figure Page 63. Computed E-plane half-power beamwidth vs. T and 0"; Z = 100, Z, ■ short at h.,/2. h/a = 177 116 r i 64.. Computed H-plane half -power beamwidth vs. T and 0"; Z = 100, Z = short at h /2, h/a = 177 117 65. Computed contours of constant directivity vs. T } 0", and a. ; Z = 100, Z. n a short at h.,/2, h/a = 177 118 o r i 7 66. Measured patterns; T = 0.7, (J = 0.206, Z = 100, Z = short at h . 2, v = 6 h/a = 177 120 67. Computed and measured H-plane patterns; t = 0.7, 0" = 0,206, Z = 100, Z = 100, N = 6, h/a = 177 121 68 Computed pattern front to back ratio vs, (J and T; Z = 100, Z = short at h /2 122 69. Computed and measured patterns; t = 0,888, ff = 0,089, Z = 150, Z = short at h /2 123 70. Computed and measured patterns; T - 0,888, 0" = 0.089, Z = 150, Z s short at h /2 124 71. An example of computed and measured directivity vs. h/a 126 72. Computed far field phase as a function of frequency, illu- strating the phase rotation phenomenon 130 73. Coordinate system for phase center computations 132 74. A typical evolute of an equiphase contour, plotted on a wavelength scale 135 75. Typical frequency variation of the relative distance from the apex to the phase center 136 76. Measured and computed location of the phase center with reference to the active region 138 77. Location of the phase center in wavelengths from the apex 139 78. Coordinate system for the computation of the phase tolerance 140 79. Nomograph, CT = 1/4(1 - T)cot a 148 80. Nomograph^ B = 1.1 + 7.7 (1 - t) 2 cot a 149 81. Nomograph, L/X = 1/4(1 - — -) cot a 150 & ' max B ILLUSTRATIONS (Continued) Figure Page 82„ Nomograph, N = 1 + (log B /log — ) 151 5 T 83, The LPD realized by the design procedure of Section 4.2.3 157 84, Measured standing wave ratio vs. frequency of the design model 158 85.. Measured E- and H-plane half-power beamwidth and directivity of the design model 159 86., Computed and measured patterns of the design model 160 87. Two LPD antennas in cascade 162 88 An LPD antenna etched from double copper-clad Rexolite 164 89, Measured patterns of an LPD antenna which was etched from double copper-clad Rexolite 165 90, Measured patterns of an LPD antenna which was etched from double copper-clad Rexolite 166 91, Computation time vs. argument x for the series and continued fraction expansion of Kixi = Ci(x) f j Si (x) 175 92, A picture of one of the antennas used for near field measurements 183 93 v Details of the probe used for measuring the voltage between the feeder conductors 184 94 Details of the probe used for measuring the dipole element current 186 95 c A block diagram of the amplitude measuring circuit 188 96. A block diagram of the phase measuring circuit 189 97. Phasor relations and the nulls obtained for values of | E T | / j Er> ! for two methods of measuring relative phase, E is the test signal, E is the reference signal 190 R 98, A picture of the equipment arrangement used in the near field measurements 192 99, Details of the symmetrical feed point, showing the reference plane for impedance measurements 194 100. Antenna positioner and tower at the University of Illinois Antenna Laboratory 196 1. INTRODUCTION The object oi this work is to provide a mathematical model of the log-periodic dipole antenna which contains the essential features of the practical antenna and which is amenable to solution. The need for such a model occurs for two reasons. First,, the principles of log-periodic antenna design have evolved from the interpretation of laboratory measure- ments, without the benefit of mathematical analysis. A rigorous formulation of these principles is clearly called for. Second, the task of extending the state of the design art of log-periodic antennas is formidable if carried out on a wholly experimental basis „ Even an approximate analysis is quite useful if it lends direction to an experimental program. The conclusions resulting from the solution of the mathematical model proposed herein are sufficiently general to lend insight into the operation of log-periodic dipole antennas, and the results are applicable to the design of LP dipole antennas which must meet given electrical specifications, Throughout this work it becomes necessary to define as precisely as possible certain concepts relating to wideband antennas, such as "broadband", "frequency independent", ''active region", and "end effect". Some of the terms have been objects of disagreement over the past years, and while the definitions herein may not settle the issue, they can provide a I | common ground of understanding for this work. The terms "wideband" and "broadband" have become so much a part of the engineering vernacular that they express only a notion and must be qualified each time they are used. In the following paragraphs the term "frequency independent" is used. Strictly speaking, there are no frequency independent antennas. However, it 2 is proposed that a more liberal definition be adopted — one which applies to a special class of antennas. By frequency independence as applied to an antenna,, it is meant that the observable characteristics of the antenna such as the field pattern and input impedance vary negligibly over a band of frequencies within the design limits of the antenna, and that this band may be made arbitrarily wide merely by properly extending the geometry of the antenna structure. The ultimate band limits of a given design are determined by non-electrical restrictions; size governs the low frequency limit, and precision of construction governs the high frequency limit. The idea of frequency independent antennas is Oased upon the familiar operation of scaling and the principle of similitude „ It is well known that the performance of a lossless antenna remains unchanged if its dimensions in terms of wavelength are held constant Thus s if all dimensions of a lossless antenna are decreased by a factor T < l ( and the frequency is increased by 1/ T , the fields about the two antennas are similar^ that is, they differ at most by a constant fac r or, Consider a class of structures which are made up of an infinite number of interconnected 'cells 7 ' such as shown in Figure 1. Each cell is similar to its neighbor by a constant scale factor T., Structures of this class are called self-similar because they possess the unique property of transforming into themselves under a uniform expansion by T or an integral power of T, if each cell represents an electromagnetic apparatus, the performance of the structure remains the same for all frequencies related by f = f T p . p = 0, + 1, + 2, • • (1) If the structure is a source or sink of electromagnetic waves, then each -c Figure 1. An interconnection of scaled cells resulting in a self-similar structure 4 cell may contain lumped or distributed generators or loads. To preserve similitude, a generator at frequency f in cell n must "scale" into a generator at frequency Tf in cell n + 1. If it is undesirable to move the generators about, an excitation independent of scaling can be obtained by placing a generator at the small end of the structure. Only the latter method of excitation will be considered. Such self-similar structures exhibit what is called log-periodic performance. Although the patterns and input impedance may vary with frequency^ the variation must be periodic with the logarithm of frequency. In order for the pattern and input impedance to be independent of frequency, the variation in performance over the period, log T , must be negligible. To be a practical antenna, the infinite structure must be truncated. That is, the scaling must start with a given small cell and must stop at a given large cell. The requirement that the truncated structure must duplicate the performance of the infinite structure places certain restric- tions on the nature of the cells in the aggregate. At a given frequency the electrically small cells must behave as a transmission line. Truncation at the small end is equivalent to the elimination of a section of line; the net effect is a shift in the location of the generator. Tne electrically large cells must be unexcited, so that their presence or absence makes no difference in' the electromagnetic performance at the given frequency. When this is true, the fact that the structure has an end will not be observable, and the "end effect" is said to be eliminated. The foregoing restrictions on the small and large cells require that most of the energy be radiated from a limited number of adjacent, "medium-sized" cells. These 5 cells constitute the "active region". Thus three regions may be associated with the infinite, self-similar structure — the "transmission region", the active region", and the "unexcited region",. The key problem, therefore, is to determine which finite structures, if any, exhibit performance which approaches that of the infinite structure over a design band. Let the steps be traced which led to the discovery of several types of frequency independent antennas, leading up to the log-periodic dipole antenna. Prior to 1954 much effort had been expended in attempts at antenna broadbanding. These efforts, for the most part, applied the comparatively advanced knowledge of broadband circuit theory to basically 2 3 narrow band antennas. Some notable examples resulted ' . However, conventional antennas resisted efforts to extend the usable bandwidth ratio beyond two or three to one. In the fall of 1954 Rumsey broke the bandwidth barrier in antenna theory and practice with his "angle method" . He stated that if the shape of an antenna were such that it could be specified entirely in terms of angles it would exhibit constant input impedance and patterns independent of frequency because no fixed length is involved in its description. The infinite bi-conical and bi-fin structures shown in Figure 2 are frequency 5,6 independent, ' but infinite structures are not practical antennas. If one truncates these structures, the frequency independent behavior is lost; the patterns vary with frequency. The variation with frequency is a manifestation of the "end effect", that is, the effect of radiated and/or reflected current at the discontinuity introduced by the truncation. Rumsey suggested that the log-spiral curve defined by r = exp(k<£) could LjJ o o I m UJ i- Figure 2. Infinite bi-cone and bi-fin structures 7 be used to define another infinite structure in terms of angles only. He also showed that the log-spiral iamily of surfaces are the only surfaces for which an expansion is equivalent to a rotation,. For a rotation about the 6=0 axis this can be expressed symbolically as r = f(9,

=, e- ac f( e,

=90° Figure 3. A balanced planar log-spiral antenna The shaded portion represents one cell 9 current on the structure. He first considered planar structures which, if extended to infinity, were self -complementary „ This was in order to assure a frequency independent input impedance. Figure 4 shows one such structure which consists of a plurality of teeth and slots cut in a bi-fin in such a manner that the widths of successive teeth and slots form a geometric progression. This self-similar structure was called a log- periodic antenna because its geometry repeats periodically with the logarithm of the distance from the apex. The truncated structure exhibited patterns and impedance which varied periodically with the logarithm of frequency; and for a wide range of parameters the variation over a period was negligible, yielding frequency independent operation. The teeth and slots accomplished the necessary reduction of end effect. The radiation pattern of the planar LP is characterized by a bi-directional beam centered on the 6=0 axis. The antenna is horizontally polarized when oriented as shown in Figure 4 t Thus the polarization of the planar LP is orthogonal to the polarization of the smooth bi-fin,, An attempt at providing a uni-directional pattern led to the non- 10 planar LP structures which Isbell investigated. The antenna shown in Figure 5 exhibits a horizontally polarized uni-directional beam off the tip end. Again, a lack of end effect is observed and the patterns are frequency independent over a range of the design parameters. In the LPD antenna of Isbell , shown in Figure 6, dipole elements replace the teeth of the non-planar LP and a constant impedance two-wire feeder replaces the central bi-fin section. One observes the frequency independent behavior of the LPD antenna over large ranges of the design parameters. 4> = c Figure 4. A planar log-periodic antei 1 1 4> = o° tf>=90' = O C Figure 5. A non-planar log-periodic antenna L3 Table 1 summarizes the preceding discussion by a classification of self- similar structures. There are some finite structures that exhibit only log-periodic electrical characteristics and others that have log-periodic geometry but neither frequency independent nor log-periodic performance. Experience has shown that the latter category contains many members. The log-periodic dipole antenna was selected for analysis because it is made of conventional linear dipole elements, a fact which allows one to replace the tubular conductors with filamentary currents. In this work electromagnetic field theory is used to calculate the self and mutual impedances of the several dipole elements from an assumption of a sinusoidal form of current on each element. Circuit techniques are used to find the voltage and current at the terminals of each dipole element and the antenna input impedance. Once the element current is known, field techniques are again used to calculate the radiation pattern. The organization of this work is as follows; The preceding section was an introduction to the idea of frequency independent antennas. In Section 2 the mathematical model of the LPD antenna is formulated using the self and mutual impedances of the several dipole elements. The expressions for input impedance, and voltage and current at the base of each element are determined. The equations for the radiated field and the phase center are also set up. In Section 3 the computed and measured results are displayed and analyzed. Criteria for "optimum" LPD antennas are established. Section 4 presents design information, combining the computed and measured results in simple formulas and nomographs. Section 5 summarizes the work. Appended is a section which considers the computation of the equations of Part 2, and a section devoted to the measurement techniques used in this research. 15 2 FORMULATION OF THE PROBLEM 2.1 Description of the Log-Periodic Dipole Antenna The log-periodic dipole antenna, snown pictorially in Figure 7 and described by Figure 8, consists of a plurality of parallel, linear dipoles arranged side by side in a plane. The lengths of successive dipole elements form a geometric progression with the common ratio T < l . T is called the scale factor,, A line through the ends of the dipole elements on one side of the antenna subtends an angle a with the center line of the antenna at the virtual apex 0, The spacing factor CJ is defined as the ratio of the distance between two adjacent elements to twice the length of the larger element, and is a constant for a given antenna. The geometry of the antenna relates O to T and a. er = A(i . T) co t a (3) The largest element is called element number 1 The half length of element n is denoted by h Therefore, n ' h = h r n ~ X o (4) n 1 The distance d from element n to element n + 1 is given by d = d t n 1 IE a is the radius of element number n, the a s s are given by n ; n & 3 (5) n 1 ;The ratio of element height to radius is the same for all elements in a given antenna and will be denoted by h/a The elements are energized from a balanced, constant impedance feeder, (6) 17 DIRECTION OF BEAM 1 i 1 i . ' L i 1 I • ' I < • • 1 Xn . K Xn-i " JL n-i " X 2^ = Cr h n s V2 co) METHOD OF FEEDING Figure 8. A schematic of the log-periodic dipole antenna, symbols used in its description including 18 adjacent elements being connected to the feeder in an alternating fashion. Due to the alternating manner in which the elements are connected to the feeder, one cell of tne LPD antenna consists of two adjacent dipoles and two sections of feeder, Thus T as defined above is the square root of the cell scaling factor,, Ideally, ^be feeder should be conical or stepped, to preserve the exact scaling from one cell to the next However, it has been found in practice that two parallel cylinders can sat 1 sf actorily replace the cones as long as the cylinder radius remains small compared to the shortest wavelength of operation Tne element feeder configuration is shown in Figure 9 It is seen that the elements do not lie precisely in a plane; the departure therefrom is equal to tne feeder spacing,, which is always small The antenna may oe energized from a balanced twin line connected at the junction of tne feeder and tne smallest element Alternatively a coaxial line may be inserted through tne back of one of the hollow feeder conductors. The shield of the coax is connected to Its naif of the feeder at the front of tne antenna, the central conductor of tne coax is connected to the otber side of the feeder as shown in Figure 9 In r he latter method the antenna becomes its own balun because tne currents on the feeder at the large end of the antenna are negligible, as will be demonstrated later. Due to the diminution of current at the large eno the impedance Z wnich terminates the feeder at that point is immaterial For def mi teness , in most models Z will be taken equal to the characteristic impedance Z T o of the feeder The propagation constant of the feeder alone is 6 and may be different from tne free space propagation constant 6 if dielectric is used, FEED POI NT- Figure 9. Connection of elements to the balanced feeder and feed point details 20 When the antenna is operated at a wavelength within the design limits, that is approximately 4n N < X- < 4h ± (7) where N is the total number of elements, a linearly polarized undirectional beam is observed in the direction of the smaller elements. It is found that for any frequency within the design band there are several elements of nearly half -wavelength dimensions. The current in these elements is large compared to the current on the remainder of the elements; these elements contribute most of the radiation, and form the so-called "active region". As the frequency is decreased from f to T f , the active region shifts from one group of elements to the next. In most cases the variation in performance over a log-period is negligible and frequency independent operation results. Since the LPD antenna is a truncated section of the infinite structure, the performance of the antenna approaches that of the infinite structure only to the extent that a properly constituted active region exists on the antenna. The active region becomes deformed as it begins to include the smallest or largest, element on the antenna. When this happens, the upper or lower frequency limit is reached, and it is this phenomenon which determines the useful bandwidth of the antenna. 2.2 Separation of the Problem into Two Parts The problem may be divided into two parts for the purpose of simplifying the analysis. Finding the voltages and currents along the feeder constitutes the interior part of the problem, and finding the field of the dipole elements constitutes the exterior part of the problem. 21 Since the feeder has transverse dimensions which are small compared to wavelength, its principal function is to guide and distribute the energy to the radiating elements. There is negligible inductive and capacitive coupling from the feeder to the shunting elements because the fields due to the currents and charges on the feeder are very small at the location of each dipole element. In the exterior problem, the magnitude and phase of the far field radiation produced by the currents on the elements are of primary interest. I The E- and H-plane beamwidths, directivity, front to back ratio, and side lobe level can be determined from the radiation pattern. The phase center can be determined from the phase of the far field. 2.2.1 The Interior Problem Insofar as the interior problem is concerned, the connection of the dipole elements to the feeder is equivalent to the parallel connection of two N terminal-pair circuits. One circuit consists of the feeder with alternating^ properly spaced taps which represent the terminals to which the elements are eventually attached. The feeder circuit is shown schematically in Figure 10b, and includes the arbitrary terminating impedance Z . The other circuit, shown schematically in Figure 10a, represents the behavior of the dipole elements as viewed from their input terminals. Let Y^ be the admittance matrix of the' feeder circuit. Then F It, = Y t, V (8) F F F where I and V are column matrices which represent the N driving currents and F F ANTENNA ■Al ff'«t 11 ELEMENTS r Al & A3 '& & %2 ^\3 'an ' r AN 0. ELEMENT CIRCUIT LI JM 'L2|f| j L3 |< |« — d, — 14$— d a H<3 — *l b. FEEDER CIRCUIT ^AIX OTA?, i /// w/7 ] v/y S/yy }{ r Ay VI vi r r ^AN\ 6$ < c. COMPLETE CIRCUIT Figure 10. Schematic circuits for the LPD interior problem 23 response voltages of the feeder circuit. Let Y be the admittance matrix of the element circuit. Then T A " Va (9 » where I and V are column matrices which represent the driving currents and response voltages of the element circuit. If the corresponding terminals of the feeder and element circuits are connected in parallel, a new circuit is obtained as shown in Figure 10c. The new response voltage matrix is equal to either V or V since they are equivalent. The new driving current matrix is now the sum of I. and I due to conservation of current A F t a node. If Equations (8) and (9) are added, 1 " J A + h " VA + Vf • (10) V is set equal to V. and factored, FA ; 1 ■ (Y A + V V A • (11) I } the base current at the dipole element terminals, is of primary interest Therefore 1 ■ (Y A + VVA (12) where Z = Y . Multiplying Z inside the parenthesis results in I = . (22 > Figure 11. Geometry and notation used in the calculation of mutual impedances 29 The expression for the parallel component of electric field due to sinusoidal current distribution in antenna 1 is given by E = 30 I zl 1 max e" j3r i e " jSr 2 2j C ° S 3h i e J3r ° r i J r 2 (23) Inserting (22) and (23) into (21) gives the mutual impedance referred to the base of the antenna, h 1 max 2 max I i^oryo)- J sin S(h 2 -|z|) f -iBr -iBr . e J 1 . e Jl : -J -J 2j cos Bh e~ J r o From Figure 11, r = /d 2 + o •i- F 7 ^ i zV 2 2 r 2 = V d + (h l + Z) 2 (24) (25) Under the assumption of sinusoidal currents the maximum currents are related to the base currents by 1(0) = I, sin Bh, 1 1 max 1 1(0) = I sin Bh 2 2 max p 2 (26) Therefore (24) can be rewritten as 30 A r -je- jBr i . -jBr Z n = -30 esc Bh n esc Bh sin B(h n - I z j) • - -^ - 12 1 2 J 2 L r r " h 2 (27) 2j cos Bh ie J on r o J Integration of (27) yields an expression for the mutual impedance in terms of cosine integral and sine integral functions. 60 12 cos w - cos — < e [K(u o )-K(u i )-K(u 2 ) j 4- e [K(v q )-K(v i )-K(v 2 ) / ~ jW 2 [K(u )-K(u )-K(v Q )] + e L K(v / )-K(v')-K(u o )] (28) o x ^ o x ^ ) 4 COS W g ] |> + 2K(w ) I cos w o 1 The * denotes the complex conjugate of the expression in the braces. Here K(x) - Ci (x) + j Si (x), (29) where Ci (x) and Si (x) are the cosine integral and sine integral functions of the real argument x! for definitions see Appendix A. Also u o = B ; /? I (n i I n 2 ) 2 - (h 2 4 h 2 ) )/7~~^~ v Q = B . \/6~ + (h x + h 2 ) 2 * (h x % h 2 ; u^ = B Vd 2 4- (h i - hg) 2 - (h ± - h 2 ) 31 B [ /d 2 4- (h x - h 2 ) 2 + (h x - h 2 ) ] u x = B [ /d 2 + h x 2 - h x ] v x = 3 [ /d 2 4- h L 2 + h^ (30) u 2 = B [ /d 2 4- h 2 2 - h 2 ] v 2 = 6 [ V^"^ 2 2 + V w x = B(h x 4- h 2 ) w 2 = S (h 1 - h 2 ) w = 3d o where B is the free space propagation constant, d is the separation of the two dipoles, and h f and h are the half-lengths of dipoles one and two, respectively. The self impedance of a dipole antenna can be calculated in a manner similar to that employed in tne calculation of mutual impedance. The self reactance of an antenna depends on the induction and electrostatic fields close to the antenna, which in turn depend on the details of the antenna geometry. Figure 12 is pertinent to this discussion. The current is assumed to be concentrated at tne center. Then the "average" distance S from a point on the cylinder P to a point on a typical ring Q is some- / 13 what greater than the distance S from P to the center of the ring 32 RADIUS a Figure 12. Geometry and notation used the calculation of self impedances .' = /777 /2 2 2a (1 - cos a and may be neglected if z » a. When z ^ a, the current and the field which it produces are in phase. Since the reactance depends on the out of phase components, the contribution to the self reactance is small. Hence the approximate expression for S is given by S Si V2a 2 + z 2 (33) for all z. If r , r } and r of Equation 25 are replaced by 2a 2 + z 2 o (34) and S x = pa 2 + (h i - z) 2 , S = V2a 2 + (h + z) 2 i respectively, it will be found that the final expression for the self impedance of a thin dipole antenna is then given by Equation (28)with h and h set equal to the half length of the dipole and d replaced by /~2a. 2.2.2 The Exterio r Problem Once the element base currents I are known from the solution of the interior problem, the far field components can be calculated. The coordinate 34 system of Figure 13 is used in this section. The antenna is oriented with the dipole elements parallel to the z direction, and the tip of the antenna points in the positive x direction. The z component of vector potential due only to currents on the dipole elements is first determined. The far-field spherical components are then found by a simple transformation. The vector potential A at a distant point P(r. Q, (41) £ ri O Z ^ and E a (6. CP = 0, 7T) = P„(9) = jB" G sin 9 f (9, C0= 0, ir) (42) y e o z Here P and P denote tne principal H~plane and E-plane patterns respectively, H E In the E-plane pattern takes on the values or it depending on whether the x coordinate of the point of observation is positive or negative. The magnitude of the far field components are then given by h |P„HH \ ^ JB X n C ° S ^ f ' i Cz W , (43) J H ' I | n= 1 n and h I lr> /-fl tiht R r J'3& x sin cos W I f i jBz'cos 9 , / P„ o.Vf ~ sin t) £ e J n i (z je dz E | n=l n J-h (44) n The distribution of current i (z) is assumed to be sinusoidal and is n related to the element base current i A oy An J i . sin Bin - I z i (z> = An . » (45) n sin Bh Performing the integration and simplifying yields 37 p H m N l (l - cos Bh ) tjB v- An n L, = e n=l sin Bh n x cos

l 1 I - 10 — -15 -20 ? — -25 ■30 0.70 0.80 0.90 1.00 1 .10 1.20 1.30 1.35 DISTANCE FROM APEX Figure 15. Computed and measured amplitude and phase of the transmission line voltage vs. Relative Distance from the Apex at Frequency f ; T = 0.95. h?a = 177 0.0564, N = 13, Z 100 short at h 1/2' 43 T = 0.95, (7 ^ 0.0564, a = 12.5°, h/a = 177, and Z = 100 ohms. The voltage is essentially constant from the feed point at x/X. = 0.675 to the beginning of the active region at x/\ = 1.00. This indicates that the transmission wave propagates with little reflection or attenuation. Since the small elements are closely spaced and fed out of phase, their contribution to the radiated field is negligible, and they act as small shunt capacitors. For x/\ > 1.00, the feeder voltage decreases rapidly, due to the coupling of energy into the elements of nearly half wavelength dimensions in the active region. Figures 16, 17, 18 and 19 are for frequencies f . , f . f . and f respectively, on the same model. The shape of these curves is the same, since the distance from the apex is normalized with respect to wavelength. The feeder on this model was terminated in a short circuit at a constant distance h /2 from the largest element. That the shape of the curves did not change with frequency, even though a frequency sensitive termination was used, shows the lack of end effect on this antenna. End effects will be discussed in Section 3.3. The phase of the feeder voltage is plotted in Figures 15, 16, 17, 18 and 19. The phase is essentially linear up to x/\ = 1.00. This also suggests that the transmission wave propagates away from the feed point with negligible reflection. \The computed input standing wave ratio for this antenna was 1.15:1 with respect to 65 ohms and the measured value was 1.17. The low VSWR is also indicative of a small reflected wave. For the case of low VSWR, the slope of the curve of phase versus distance is inversely proportional to the v t relative velocity of propagation of the transmission wave, — . v is the 7 c t phase velocity of the transmission wave and c is the velocity of light in v free space. For this antenna — = 0.63. 44 + 200 tr + 100 Ul o Id in < I a. ■100 ■200 ~_^ O 0^°^^ _ - ^ ^^Q^^o AMPLITUDE—^ A 'PHASE ^X — OA COMPUTED a\ o ) - MEASURED ^ _ A *\ POSITION OF ELEMENTS n i r i, i i , i ^ 1 1 1 1 1 i i - -5 -10 ■20 -25 30 0.70 0.80 0,90 1.00 1.10 1.20 1.30 1.35 DISTANCE FROM APEX Figure 16, Computed and measured amplitude and phase of the transmission line voltage vs. relative distance from the apex at frequency f_ , .; T = 0,95 C v - ■ 177 0,0564, N = 13, Z = 100, Z = short at 45 "D^vQ Q^-^Q Q - >.o AMPI ITIinF > _ \° + 200 A + 100 ^^^ /PHASE - - -100 O A COMPUTED ^^ MEASURED \ - -200 13 1. POSITION OF ELEMENTS-^ i r .1 1 1. 1 1. ^ 1 A^ ■ 1 1 1 i — -5 --10 --25 30 Q. 20 J 0.70 0.80 0.90 I .00 I .10 1.20 1.30 1.35 DISTANCE FROM APEX -^ Figure 17. Computed and measured amplitude and phase of the transmission line voltage vs. relative distance from the apex at frequency f : T = 0.95, 0" = 0.0564, N = 13, Z = 100, Z^ = short at h ._, h/a = 177 ° X 1/2 46 ■200 100 -100 -200 O A COMPUTED MEASURED .POSITION OF ELEMENTS i i i ) i i i i i i y^ ° o AMPLITUDE 2 \n - 200 A - -100 _ ^*"Aw /PHASE - ^v A -100 _ O A COMPUTED \ MEASURED — 200 .POSITION OF ELEMENTS) I 3 1 , l> 1 , 1 1,1 (\ A 1 A h a\ 1 A 1 > . 1 - r- — -10 CD O 15 uj Q Z> a. ■20 < -30 0.70 0.80 0.90 1.00 1. 10 1.20 1.30 1.40 DISTANCE FROM APEX - X - Figure 19. Computed and measured amplitude and phase of the transmission line voltage vs. relative distance from the apex at frequency f T = 0.95, 0" = 0.0564, N = 13, Z = 100, Z = short at h h/a =177 o T 1/2- 48 In Section 3.1.2 an approximate formula for the constants of an equivalent line in the transmission region is derived. The graph of Figure 20, based on the approximate formula, shows that the transmission wave phase velocity depends primarily on the relative spacing 0*. For small spacing the loading effect of the elements is appreciable; relative phase velocities less than 0.6c Have been observed. Since v is less than c, the wavelength of the transmission wave ^ is less than the free space value. ^ rather than ^ must be used if one is to compute the electrical length of any part of the transmission region. tfear field measurements made on a second model are shown in Figure 21. For this 8 element antenna T - 0.888, 0" = 0.089, a = 17.5°, h/a = 125, and Z .= l00 ohms. The graphs of the magnitude and phase are generally the o same as for the previous model. However, the linear portion of the phase curve is smaller, because less elements were used. A phase velocity of 0.75c is given by the slope of the left-most linear portion of the curve. The. measured values of phase velocity are plotted on the graph of Figure 20. The slow wave in the transmission field was observed in every computed model. The range of parameters of the computed models was 0.7 < T < . 98 and 0.03^ CT^o.23. Measurements on a different type of LP antenna have been made by Bell, 14 Elfving, and Franks at Sylvania Electronic Defense Laboratories. Their * results also demonstrate the slow wave nature of the transmission field. Several computations were made to determine the effect of changing the phase velocity of the unloaded feeder, to simulate the use of a dielectric material in the feeder configuration. Figure 22 shows that the relative velocity of the transmission wave decreases as the relative ■1!) 0.9 - 0.8 0.7 0.6 0.05 0.10 0.15 0.20 0.25 Figure 20 Relative velocity of transmission wave vs. T and 0" with Z Q / Z £ computed from the approximate formula (60) page 54. 0.33. 50 200 + 100 - en I 100 - ■200 O A COMPUTED MEASURED POSITION OF ELEMENTS L r I I iL I 30 0.4 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE FROM APEX -y Figure 21. Computed and measured amplitude and phase of the transmission line voltage vs. relative distance from the apex at frequency f ; T = 0.888, 0- = 0.089, N = 8, Z = 100, Z m = short at h. h/a = 125 1/2' 0.7 - 0.6 - 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 RELATIVE PHASE VELOCITY ALONG ISOLATED FEEDER Figure 22. Relative velocity of transmission wave as a function of the relative phase velocity along the feeder with the elements removed. UNIVERSITY OF ILLINOIS LIBRARY 52 velocity along the unloaded feeder decreases. The range of feeder velocity shown corresponds to a relative dielectric constant 1 < € < 2.78. Tne r LPD performs satisfactorily under these conditions; the only noticable change in the computed models was an increase in the input impedance and a shift of the active region towards the shorter elements. These effects are the same as those resulting from an increase in the characteristic impedance of the feeder, (See Sections 3.4.3 and 3.5.1.2). The feasibility of using a dielectric in the antenna structure was shown in a model that was constructed from double copper-clad Rexoiite using printed circuit techniques. It exhibited uniform directivity over the design band, This antenna is discussed in Section 5.2, A discussion of the effects of changing the cnaracteristic impedance of the feeder and the h/a ratio will be taken up in Section 3,4, because these factors can be used to control the input impedance 3.1.2 An Approximate Formula for tne Constants of an Equivalent Line in the Transmission Region. The transmission region displays some of the characteristics of a uniform transmission line. The small,, non-radiating elements load the feeder to produce the slow wave. Furthermore, tnis loading appears to be uniform because the magnitude of the voltage is constant and the phase is linear throughout the transmission region. This uniform loading is due to the shunting capacitance of each element. To the first approximation, the capacitance of a small dipole is proportional to its length, and on an LPD^ the spacing d at element j is also proportional to the length of element j, thus, the added capacitance per unit length is constant. Consider the approximate formula for the input impedance of a small dipole 53 antenna of half-length h, Z = - jZ cot Bh, (48) where B is the free space propagation constant. Z is called the average characteristic impedance of a dipole antenna, Z = 120 (In h/a - 2.25) (49) a 15 This is a modification of a formula in Jordan which was derived by Siegel and Labus . The original formula for Z contains a term which depends on the height of the dipole relative to wavelength; the factor 2.25 in Equation (49) represents an average height. Replacing the cotangent function in Equation(48) by its small argument approximation, the capacitance of the nth dipole is given by h C = --S-, (50) n cZ ' a where c is the velocity of light in vacuum. Using the mean spacing at dipole n d n d = i/d d n = — , (51) mean y n n-1 ^~ ' the capacitance per unit length is given by C h {T A C = — ±- = -5— (52) length cd Z n a But h /d is related to the spacing factor 0" by 4 h d (53) 54 hence Using J L C = 1/c and substituting, one finds R = Z . V^tT (57) where o V T m = 1 + - -V (58) a Similarly, the propagation constant for the equivalent line is given by B = 8 fm. (59) The relative phase velocity of the transmission wave is given by (60) c B t /m The graph of Figure 20 was plotted according to Equation (60). In Section 3,4, Equation (57) is used to approximate the input impedance of tne LPD. In summary, all tne available data suggests that the region which contains electrically small elements acts as a uniform transmission line which is 55 matched to the active region. Tnis is why the front end of an LPD can be truncated without adverse effects on the frequency independent character- istics. 3.2 The Active Region The active region whose lengths border on a half ^wavelength at a given frequency, and the portion of feeder to which these elements are attached. It is this part of , the antenna which determines the characteristics of the radiated field. This section presents calculated and measured results which show how the power in the feeder wave is divided among the radiating elements. A useful concept, the bandwidth of the active region, is formulated and its functional dependence on the several LPD parameters is given. 3.2.1 Element Base Current in the Active Region The dipole elements in the active region transfer the power from the transmission wave to the radiated field. Figure 23 shows the base impedances of the dipole elements in an 8-element LPD operating at f . Base impedance here means the ratio of voltage to current at the base of each element, when the antenna is fed in the usual manner. The base impedances of elements 4 and 5 are predominately real, so conditions are favorable for the coupling of energy from the feeder onto the radiating elements in the active region. The small elements 6, 7, and 8 are capacitive land therefore loosely coupled to the feeder. The large elements 1, 2, and 3 are inductive and also loosely coupled. The base impedance of all computed models in the range 0.8 <*t < o . 98 and 0.03 < 0" < 0.23 behaved in a similar manner. For t < 0.8, the base impedance of only one antenna element was predominately real at any given frequency; for these models the performance was not frequency independent. Im (Z b > O i 20 - 300 - 200 - 100 03 1 D 1 ' , **W -200 -100 < 07 100 05 - -100 - -200 SO 200 Figure 23. Computed base impedance Z vs. element number for an eight element LPD at frequency f . 57 Figure 24 is a graph of tne relative amplitude of element base current as a function of x/\, which is the normalized distance from the apex of the antenna. The location of each element is indicated in the figure. The lines which connect the values of current at the discrete location of each element are for clarity of presentation only. A loop-probe technique was used to measure the base current, as explained in Appendix B. The element base currents in the active region rise to a peak in the element which is somewhat shorter than a half -wavelength . As frequency is changed, the shape of this curve remains unchanged as shown in Figures 25, 26, 27 and 28. That is, the active region moves along the antenna as frequency is changed, but its distance in wavelengths from the apex remains constant. Figure 29 shows the computed magnitude of the base currents in the active region of an eight element antenna at frequencies related by T . These curves are identical except for f and f At these frequencies the active region becomes deformed as it begins to include the largest or smallest element on the antenna. When this happens, the lower or upper frequency limit is reached. The phase of the current from element to element in the active region is also plotted in Figures 24 through 28. This is the phase which has to be considered when computing the radiation pattern. Since the phase can be determined only within a multiple of 2^", many phase velocities are compatible with a given phase progression. In Figures 24 through 28, the slope of the phase curve in the active region was chosen to yield the largest phase velocity compatible with the given phase progression, This phase velocity is approximately equivalent to that of the first backward space harmonic of a periodic structure 'made up of cells identical to the central cell of the active region. Mayes, Deschamps and Patton have explained the operation of unidirectional -10 -15 -20 •180 90 -90 -180 A MEASURED COMPUTED O -POSITION OF ELEMENTS^, I I 1 I . 1 \LlA L_U L \ Jj I L 0.70 0.80 0.90 1.00 1. 10 1.20 1.30 DISTANCE FROM APEX Figure 24. Computed and measured amplitude and phase of the element base current vs. relative distance from the apex, at frequency f. 0.95, 0" = 0.0564, Z = 100, h/a = 177, Z = short circuit at h /2 . - 5 10 -15 -20 180 +90 ■90 180 J L A MEASURED COMPUTED O (-POSITION OF ELEMENTS") 0.70 0.80 0.90 1.00 1. 10 1.20 1.30 DISTANCE FROM APEX Figure 25. Computed and measured amplitude and phase of the element base current vs. relative distance from the apex, at frequency f 0.0564. 100, h/a 177. 3 V 0.95, short circuit at h /2 ■10 -15 20 + 180 + 90 -90 -180 l I .1 I IL_1 A MEASURED COMPUTED O POSITION OF ELEMENTS ll L ±1 Ll 0.70 0.80 0.90 1.00 1. 10 1.20 1.30 DISTANCE FROM APEX-— Figure 26. Computed and measured amplitude and phase of the element base relative distance from the apex, at frequency f. O = 0.0564, Z 100. h/a 177, short circuit at h /2. current 0.95. 61 — -5 ■10 -15 -20 1-180 +90 ■90 -180 A MEASURED COMPUTED O POSITION OF ELEMENTS _L_L ll I Li I .M 0.70 0.80 0.90 1.00 1.10 1.20 1.30 DISTANCE FROM APEX Figure 27. Computed and measured amplitude and phase of the element base current vs. relative distance from the apex, at frequency f. 0" = 0.056' 100, h/a = 177, Z short circuit A%/2. 0.95, 62 +90 -180 DISTANCE FROM APEX -r- Figure 28. Computed and measured amplitude and phase of the element base current vs. relative distance from the apex, at frequency f 0" = 0.0564, Z = 100, h/a = 177, \ 4' 0.95, 63 1.0 f, h h h h U 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 i I 1 1 1 1 ELEMENT NUMBER Figure 29. Relative amplitude of base current in the active region vs. element number, frequencies f thru f . t _ 0.888, 0" = 0.089, N = 8, Z = 100, h/a = 125, Z m = short at h ,/2. 64 frequency independent antennas in terms of backward wave radiation. The computed phase velocity of the first backward space harmonic as a function of 0" is shown in Figure 30 for several values of T. These curves represent average values taken over several frequencies. As the spacing increases, the phase velocity increases. For low values of T the relative phase velocity increases rapidly with increasing 0", indicating the possibility of radiation broadside to the antenna. Several models have exhibited broadside radiation patterns, these are discussed in Section 3.5, The mutual impedance of the elements in the active region plays a fundamental role in determining the amplitude and phase of the element currents. To find if any of the mutual terms in the element impedance matrix could be neglected, several tests were made in which the range of the mutual coupling was changed. If z .is the mutual impedance between element i and element j, the range is given by the number I i-j j . Thus range means all mutual terms are zero, range 1 means all mutual terms excepting those for adjacent elements are zero, etc Limiting the mutual effect to range 2 causes distortion of the computed patterns. The average input impedance level remains about the same, but the input standing wave ratio increases from its actual value for full range coupling. This means that one must take into account interaction at range 3 or greater to determine the relative excitation of each element. 3.2.2 Width and Location of the Active Region For a given antenna, the usable bandwidth for frequency independent operation depends on the relative distance the active region can move before it becomes distorted by the smallest or largest element. Thus the width of 65 UJ (T UJ X > UJ o < < ui x 1 ^ o q Q 0.9 > o 0.7 uj values which are unquestionably outside tne range of operation. Actually, the characteristics change slowly, and only by applying a tolerance to the pattern, or input impedance, can a figure be chosen wnich represents the operating bandwidth of an LPD. 67 It was impractical to hunt for the limits of operation of all the computed models; this would have involved testing each model at many closely spaced frequencies. Instead, empirical values of the bandwidth and location of the active region were determined in the following manner. Figure 31 is a sketch of a typical curve of base current versus distance from the apex. As the high frequency limit is approached, the active region moves toward the apex and the amplitude of the current in the shortest element increases. When this current increases to within 10 db of the maximum, it is observed that the input VSWR begins to depart from its mid-band value. This occurs somewhat before a significant change in the pattern is observed. Therefore x , the location of the -10 db point nearest the apex, can be taken as a definition of the high frequency edge of the active region. At the low frequency limit, distortion of the active region is accompanied by an increase in the H-plane beamwidth and a change of impedance. The beamwidth was chosen as a criterion because the low frequency patterns are always smooth and single-lobed, facilitating unambiguous measurement. When the current in the longest element increases to I , an amount sufficient 1 to increase the H-plane beamwidth by 10 percent, the low frequency limit is said to be reached. The distance from the apex to the I point, x , depends i on Z and h/a but is substantially independent of "*" and 0". For the cases with Z = 100 ohms and h/a = 177, x„ is equal to x\ , , the distance to the half- o 'Jo V2 1 wave element. As the feeder impedance Z is increased or the ratio h/a is o | decreased on a given model, the active region is found to move toward the apex in a manner where x is the distance to the -10 db point farthest from the apex, such that x /x and x /x , remain constant. As a consequence, Jo ' ' the location of the half-wavelength element with respect to the active region 68 CD Si CD -m si C «H •H O O C .G O to -h ^ cti X O CD O a ih cfl T5 CD C 4-> e s o -o «H £ ■o CD C o rt C £2 • O CO > C O 3 m U CD •a CD CO CD aa 3anindwv iN3uano 3sva changes 69 Knowing the high and low frequency edges of the active region, the band- dth of the active region is given by x B = -H , (63) ar x In many practical applications the requirements will be less stringent and the bandwidth of the active region can be decreased accordingly. In a previous paper x was taken as the low frequency edge. The old definition was subsequently found to result in B ? s greater than necessary for values of t less than 0.875. A graph of B versus 0" f or several values of T is ar shown in Figure 32. The circles are computed values and the straight lines are based on an empirical formula fitted to the computed results, B s 1.1+ 30.7 C(i-T) (64) ar The empirical formula agrees with the computed and measured results for all but the lowest values of T , so its use should be restricted to T > 0.875. For a fixed T , the bandwidth of the active region increases as 0" increases. This is an important design consideration, because the size of an antenna to cover a given band increases as B increases. ar The relationship between x. and x\ ,„ takes the form lo V2 x io = S(Z o , h/a) x X/2 (65) Due to the geometry of the antenna, an identical relation exists between the half-wavelength transverse dimension and the length of the largest required element in the active region. i io = S(Z o , V a)| (66) 7 2.6 2.5 2.4 2.3 2.2 2.1 g o LU 2.0 a: 1.9 1.8 1.7 1.5 1.4 1.3 1.2 I.I O COMPUTED EMPIRICAL FORMULA X =0.8 X = 0.875 X = 0.92 X = 0.95 X = 0.97 RELATIVE SPACING 0" Figure 32. Bandwidth of the active region, B . vs . 0" and T, 71 Thus S represents a shortening factor, and it serves to locate the active region with respect to the naif -wavelengtn transverse dimension. Figure 33 is a graph of the computed shortening factor as a function of Z for several o values of h/a. S was found to be essentially independent of T and 0" in the range 0.8 < T < 0.95, and 075 < °" < 0.21. In the computed models, low values of h/a which correspond to thick elements were not extensively investigated, in these cases the approximations of the theory are not well satisfied. The curves have been extrapolated to low values of h/a by the inclusion of measured results. The S factor is significant; for high values of feeder impedance the shortening can be equivalent to scaling the antenna by a factor T , resulting in the saving of one element at the large end of the antenna. Work done by Isbell on models in which the radii of the elements were held constant, so that h/a varied from 20 to 100, showed that the largest element was roughly 0.47^- at the low frequency cut-off, and this agrees with the trend observed in Figure 33 . Another L9 researcher in the field agrees with the location of the active region, but finds that the stated values of B are somewhat high if the front- ar to-back ratio of the pattern is used as a criterion. At any rate, the above results can be used as a guide in the design of LPD antennas for specific applications. 3.3 The Unexcited R egion The unexcited region consists of all dipole elements larger than a half- wavelength at a given frequency, and the portion of feeder to which these elements are attached, This section shows that an unexcited region exists on an LPD because of the efficient manner in which the power in the feeder wave is radiated by the active region. In this case the operation of the LPD antenna is unaffected by the truncation at the large end. The results of 72 1. 10 1.08 1.06 1.04 1.02 1.00 go or o 0.98 h- CJ \K. ■xr- o o o o o o CD O O o o o o O O OJ ° 58 OJ OJ Z 3~IOdl(] J0 3DNVC]3dlAII DllSld31DVdVHD 39VU3AV / 1 /] / o b o b / b CM / "b/ / / 95 N. K O tO Q < tf N. OeS N H 3 Cl) S h c rt «H X3 0) U) a •P £ cd b h X V TJ u a; p, 0) a cti 0) <1> > .c •H +j +J m H rH 0cor^>^ yT / \ ^s. X ( / \ ( / \ v i / \ / \ / \n / ^^ X \ // \ N = 5 N = 5 E-PLANE H- PLANE COMPUTED Figure 49. Computed patterns, t = 0.888, 0" = 0.089, Z = 100, Z = short at h /2, showing no difference between patterns for N = 5 and N = 8. 180 170 160 150 140 130 120 h 110 100 - 90 80 70 60 50 40 30 20 10 H- PLANE E- PLANE Figure 50. h U h k LOGARITHM OF FREQUENCY Computed half power beamwidth vs. frequency; T = 0. N = 8, Z = 100S2, Z = short at h /2. h 0" = 0.089. 102 This trend is evidenced by all LPD antennas, and results from the strong I dependence of the H-plane pattern on the array factor of the antenna, whereas the more directive E-plane element pattern masks small changes in the array ■ factor. The graph of H-plane beamwidth shows that the active region spans several elements, because the pattern bandwidth is significantly less than the structure bandwidth. The graph also shows that the center of the active ! region is located somewhat ahead of the half-wavelength element, because the center of the pattern band is located at a lower frequency than the center of the structure band. Examples of measured and computed patterns are given in Figures 51 through 56 for t - 0.888 and 0" = 0.089; in Figures 57 through 59 for T = 0.98 and ;0" = 0.057, and in Figures 60 through 62 for T - 0.8 and 0" - 0.137. The measured cross polarization was found to be more than 20 db below the pattern maximum as long as the spacing between the feeder conductors is small compared to the shortest wavelength of operation. In every case the computed JE-plane beamwidth is narrower than the one measured, and the computed H-plane beamwidth is wider than the one measured. These errors tend to compensate each other in the computation of antenna directivity, so the comparison of computed and measured directivity is more accurate than the comparison of computed and measured beamwidth. The error is incurred because the actual current on the dipole elements apparently departs from the assumed sinusoidal form. * Actually, it can be seen from Equations (46) and (47) that the pattern cannot be expressed as the product of an element factor and an array factor, because of the differing element lengths. However, it is convenient to think in these terms here, owing to the dissmilarity between the E-plane pattern of a single element (a figure eight pattern) and the H-plane pattern (omnidirectional) . 103 X i /\ t / \ • / \ V / X \ x \ S\ • / \ 1/ \ // / 1 X / ' \ / I X / * \ 1 H X ^^X\ \ X Vx /\ \ / ' \ / • \ / 1 \ / 1 \ \ / [ \\ / ' x / / X / a \ ^ / \ / x ^ — ■ \ ' 1 x / / \ / / X • / ^'**\ 1 1 b /x / ' / XI' / XV * / X l / V\i il / \ ; 1/ \ i y \ 1/1 \ /Y/ \ Ajf / *K / ' 1 X / ' \ X / v V \ / N \ X / ^ X X \ x^^ X X>X V X \Vx s\\ \ /l \ / • \ / V \ /I' \ / /I 1 / /' \ • x / C f, N^ 1 1 d E- PLANE H- PLANE COMPUTED MEASURED Figure 51. Computed and measured patterns; T = 0.888, 0" - 0.089, Z = 100, Z = short at h /2 104 / \\l / V l\ / \ • 1 / \ / y \ /// /\fj / '{ \ / A \ / v \ \ / \\ \ 1 ^v \ \ N x\ >n \ /I ' \ / \ i \ / 1 1 \ / J i \ / /' \ / // h a f 2 ^^ ' b s // /\ I 1 / X 1 ' / X 1 1 / W l / M 1 * / Vi / i\\ / I \ X / v \ X 1 ^ J /I 1 \ / // \ / ji \ / /' \ / // \ h ^^~^ c f 3 ^^~ ^-"^^ d E-PLANE H-PL ANE ASURED 0.888, 1 /2 CO Figure 52. Computed and meas 0" = 0.089, Z = 1 jred patterns; t = 30, Z„, = short at h 105 < ft /X /' A A i\ x \ 1 1 X \ s )/ \ ^XX 1 / ' / / ' r\ / i I \ / i \ \ 1 V X^ \ Xv \\ \ \ x \ \X\ A ' \ X/ ' \ / 1 i \ X J t \ X / f \ X ^X ' \ u a 1 b X\ // / X / ' / X l / X i / \^ A X A x\ 1 /A ' X \ i\/ \ / Ok / • IX / 1 l\ / * \ X / v \ X / v \ X V X\ Jn \ Xl i \ x / ' \ / 1 > \ / / ' \ / y / \ f 4|^^ c 1 v *>« , is*y5 1 — d E-PLANE H- PL .ANE ASURED 0.888, i x /2 Figure 53. Computed and meas 0" = 0.089, Z = ] ME ured patterns; T = 00, Z = short at E-PLANE - COMPUTED H-PLANE — MEASURED Figure 54. Computed and measured patterns; T - 0.888, C = 0.089, Z = 100, Z = short at h /2 ' o ' T 1 107 E-PLANE - COMPUTED H-PLANE — MEASURED Figure 55. Computed and measured patterns; t - 0.888 0" = 0.089, Z E-PLANE - COMPUTED H-PLANE — MEASURED Figure 56. Computed and measured patterns; "■" = 0.888, 0" = 0.089, Z = 100, Z m = short at h/2 7 o T 1 109 A /' / \ V M / \ A / \ 1 / \ /// \ A // / \ // / \l[ / vv\ / v \ \ / A \ / \ \ f 3 ~~~ a f 3 ^^"^ b / r 1 ^\ 1 / \ ' 1 / \ ' // \ ' y \ / V\ / \ \ f 3^ C f 3-L ^—^ d E-PLANE H-PL ANE ASURED i, 0- - 0.057, Figure 57. Cc N mputed and measured = 12, Z = 100, Z patterns; T = . 91 = short at h /2 110 ^"/^ ^v^\ / ft s fi /\ h / \ r / \ V / h 1 l\ / \ J4 \ * . ^ v ^*— 'i 3 t E-PLANE - COMPUTED H-PLANE — MEASURED Figure 58 Computed and measured patterns N = 12, Z = 100, Z r 0.057, hort at h /2 Ill E-PLANE - COMPUTED H-PLANE — MEASURED Figure 59. Computed and measured patterns; t - 0.98, °" = 0.057, N = 12, Z = 100, Z = short at h /2 o T 1 112 E-PLANE - COMPUTED H-PLANE — MEASURED Figure 60. Computed and measured patterns; T = 0.8. 0" = 0.137, 113 ^^--^ »! \\ I / \ l l / \p 1 / V 'a \ \^\ /fir 1 f, I ~ v ^^^>> ^0\ s f / \\ \ /\ / / / ok A \ \ / \ V \ / ' \ / f \ / \\ \ / J/ \ / ^ \ \ / // \\ > / // v.>v A k ^fs / (1 A. i / \ ' / \ V 1 x\ \ / V ' \ / \ '1/ \ lY \ 1 ^^v ^ E-PLANE - COMPUTED H-PLANE — MEASURED Figure 61. Computed and measured patterns N = 8. Z = 100, Z_ 0" = 0.137, / (1 / \ f' / \ h / \\l / X* / Vv A / \ ' / \ ly \ /\ / / / ' P\ / 1 1 \ / M \ / M \ \ ^ X /\ ' \ / LI \ u ^^~ a u ' C y / j b >x E-PLANE H-PLANE CO Figure 62. Co N iviru i lu mputed and measurec = 8, Z = 100, Z m = o T patterns; t = 0.8, O = 0.137. short at h /2 115 It can be shewn that the E-plane pattern of a half-wave dipole antenna becomes narrower as the current distribution changes from sinusoidal to uniform. Thus the actual current distribution in the elements must be more nearly uniform than sinusoidal. 3.5.1.1 The Characteristic Pattern as a Function of T and 0" The scale factor T and the relative spacing 0" exercise primary control over the shape of the radiation patterns of LPD antennas. Figures 63 and 64 are plots of the computed E- and H-plane naif power beamwidth as a function of 0" for several values of T „ In these curves Z = 100 ohms and h/a = 177. o Each curve possesses a minimum in the range 12 < CT < 0.18. The H-plane beamwidth varies more than the E-plane beamwidth, it should be noted that the scale of the ordinate is different in the two figures. Using the graphs of Figures 63 and 64, the directivity in decibels can 21 be approximated using the formula from Kraus^ I ■ D - 10 1o «T5^5v ' <76) BW and BW are the half-power beamwidtbs in degrees. In Figure. 65 are E H > plotted contours of constant directivity in decibels, as a function of t and 0.875 the calculated F/'B is greater than 20 db. The F/B for T < 0.875 depends on the value of 0"; it attains a maximum for 0" near the optimum value shown in Figure 65. F/B decreases as the cut-off frequencies are approached. (Compare the patterns of Figures 51 through 56,) This decrease can be limited to some extent by adjusting the reactive determination Z . Isbell found that a short circuit at a distance h / 2 behind the largest element resulted in a minimum deviation of F/B from its mid-band value. 3.5.1.2 The Characteristic Pattern as a Function of Z and h/a o The computed and measured patterns of an LPD with T = 0.888, ® = 0.089, ;and Z = 150 ohms are shown in Figures 69 and 70. A comparison of the ^corresponding patterns of Figures 52, 53, and 55, for Z = 100 ohms shows 120 E-PLANE H-PLANE MEASURED Figure 66. Measured patterns; T = 0.7, 0" = 0.206, Z q = 100. Z = short at h /2, N = 6, h/a = 177 121 H-PLANE - COMPUTED H-PLANE — MEASURED Figure 67. Computed and measured H-plane patterns; t = 0.7. 0" = 0.206, Z = 100, Z = 100, N = 6, h/a = 177. 122 20 - ^^° N. T=0.8 18 - / Nd 16 — c 14 >v T = 0.7 \ y/ >v O 12 - X 10 8 6 1 1 1 1 1 1 1 1 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 RELATIVE SPACING (X Figure 68. Computed pattern front to back ratio vs. Z =100, o Z m = short T it h /2 123 ^r ft X / / /\ \ ' / X 1 ' / \i ' 1 / \ iff \ yli \ / / / \ • y / \ f 3 ° f 3 b /\ 1 ' / V l / \x /) / \ / l\ X / l \ X / \ \ / N X X \ \/ \ V> \ /I ' \ / I ' \ / 1 ' \ / J i \ / / i \ f4 C 1 ^^""—^ ^^^ d E-PLANE H-PL ANE 4SURED = 0.888, t h x /2 Figure 59. Computed and m 0" = 0.089, Z — ML sasured patterns; T = 150, Z = short a y — /' X (I s\ 1 ' 1 >\ ^ / V '1 X \ '// \ / vX \ 1 X J A \ \ / J ' \ / / i \ / J i \ / s / h ^^~ a f 5 ^^ b x // / \ 1 ' / n1 [ 1 Y^ 1 ^C*V v\ X I \ s\ 1 1 / \ / \/ \ '/J \ / / |k / * \ X / ' \ X 1 X X ] / // \ /J; f 6 ^^ C f 6 ^^ d E-PLANE H- PLANE Figure 7 0. Computed and me 0" = 0.089, Z = ' o asured patterns; T = 0.888, 150, Z = short at h /2 125 that a change in Z has little effect on trie cnaract< I i patterns In this model the average directivity decreased 'by less tnan one decibel as the feeder impedance increased froai 75 'o 300 or.ms On nine models with different t and C, similar results were oo«er\ed, antennas with 0" greater than optimum lost as much as one db over the range 100 * Z < 300. For small 0" the variation was less than 5 db, As an approximation suitable for design purposes, one may assume that the directivity contours of Figure 65 hold for all practical values of Z . The element thickness controls the directivity to some extent. Figure 71 shows that the average directivity decreases as the element height to radius ratio increases „ For this model, with T = 888 and 0" = 0.089, the decrease amounts to about 1.2 db over rne range 100 < h„ a < 100,000 Although the approximations of the theory are best satisfied with large h/a, in the frequency range for which the Antenna Laboratory is equipped, it was impossible to build models with h/a much greater than 800. Trie two laboratory models with t - 0.888 and 0" - 089 agree with the trend of the curve for h/a < 800, although the measured directivity wa => •a Z) 00 00 o < T3 < d£ 3 cc II X 1 u UJ f "90 A1IAI1D3UIQ 127 Table 2 presents a comparison ol the measured averagi directivity <>i ' several laboratorv models and Che corresponding directivity as read from the I graph of Figure 65, with a correction for the change in h.'a obtained from Figure 71. In general, the computed directivity is higher than the one measured, The mean error is 0.35 db and trie maximum error does not exceed 1 db. 3.5.2 The Far F ield Phase Ch aracterist ics The far field phase and the phase center are of special interest when one attempts to array several LPD anrennas to achieve higher directivity or specially shaped beams, or when an LPD is used as the primary feed in 22 ! lens or reflector systems . Tne relative phase In the principal planes of : the far field is given by the phase of the complex field quantities P u exp (-jBr) and P exp (-jBr) Written in polar form, H E p = j p j e J[F(+,f)-Br] (7?) Here P stands for eitner of the principal plane patterns and the phase F is a function of the relevant angular variable ^ and the frequency f. In the following sections F is used to demonstrate the phase rotation phenomenon. The phase center of an LPD antenna is defined and its dependence ; on the LPD parameters is determined. 3.5.2.1 The Phase Rotation Phenome no n 23 In 1958 DuHamel and Berry discovered the phase rotation phenomenon which is characteristic of LP antennas. They verified experimentally that if the phase of the electric field at a distant point is measured relative to the phase of the input current at the apex of an LP antenna, the phase of the received signal is delayed by 360 as the structure is expanded I through a "cell". The experiment was conducted by building several LP TABLE 2 Comparison of Measured and Computed Directivity 128 Measured Compute T a a Z o h/a BW E BW H D P 0.98 0.057 5.0 100 125 56.5 85.7 9.31 10.0 0.98 0.038 7.5 75 240/148* 54.1 85.7 9.50 9.5 975 0.0717 5.0 100 200/151* 52.8 76.6 10.08 10.0 0.95 0.0268 25.0 100 100/26* 68.7 114.2 7.20 — 0.93 0.125 8.0 65 66 60.0 85.0 9.08 9.8 0.92 0.200 5.7 100 142 52.7 69.1 10.53 9.9 0.92 0.160 7.1 100 177 57.0 80.2 9.56 10.0 0.92 0.150 7.6 300 60/20* 54.0 85.4 9.53 9.8 0.92 0.120 9.5 100 118 57.0 81.1 9.52 9.4 0.92 0.080 14.0 100 177 61.5 98.0 8.36 8.9 0.91 0.128 10.0 94 66 59.2 90.8 8.85 9.4 0.89 0.103 15.0 75 80/45* 67.0 106.3 7.63 8.6 0.888 0.089 17.5 100 80 61.8 101.9 8.18 8.7 0.888 0.089 17.5 100 177 63.4 106.6 7.86 8.5 0.888 0.089 17.5 150 177 65.2 112.0 7.52 8.3 0.86 0.080 23.6 200 50 64.3 112,1 7.57 7.9 0.85 0.216 10.0 75 80/49* 71.3 IS^G 7.14 7.3 J 0.84 0.068 30.5 215 50 67.8 126.6 6.83 7.5 0.81 0.364 7.5 75 80/43* 95.0 180.0 3.83 — 0.80 0.137 20.0 100 125 58.5 101.9 8.41 8.3 In these models, the element diameter was held constant; the h/a of the longest and shortest element is recorded. 129 antennas with indent ical T and a, the models differing from each other in size. If Kx is th< distance from the apex od the structur* to element n, n the expansion factoi tor each antenna is determined by assigning values 2 2 of K from 1 to 1. T , (The cell scaling factor for LPD antennas is T ). All other dimensions of the structure are also multiplied by K. Since an expansion of a log-periodic structure is equivalent to a change of frequency, the phase rotation phenomenon can also be observed by measuring the phase of a given antenna as frequency is increased from 2 f to f/ T } provided that the distance r from the apex to the far field 2 point is decreased from r to x r Tne phase of a far field component is given by the exponent of Equation (77), F(4',f)-Br. Under the stated conditions Br is held constant. The phase should be measured relative to the input current at the apex of the antenna, so the front truncation which occurs on most practical antennas must be taken into account for the reasons given in Section 3.1.1. The dotted curve of Figure 72 is for a computed •model with T - 0.875 and C = 0,067, from frequency f to f In this model the front truncation was 0.158^ at f and 0.206^ at f A , and the 2 4' computed values of F were corrected accordingly. The phase function F was found to be essentially proportional to the logarithm of frequency. The slight deviation from linearity was also observed by DuHamel and Barry. Since the phase is proporrional to the logarithm of frequency, or equivalently to the logarithm of the expansion factor K, it is possible to adjust the phase of an LP antenna in a manner which is independent of the pattern and input impedance. This property has been exploited in the design of phased arrays of LP antennas; for a detailed discussion and 130 180 -90 -180 4.75 >3 ^3.25 3.5 *3.75 M LOG OF FREQUENCY Figure 72. Computed far field phase as a function of frequency, illustrating the phase rotation phenomenon. L3] several examp 1 es, reft should be consulted.. 3.5. 2.2 titer It is well known thai the amplitude pattern oi th< far field is independent of the location of the origin of the coordinate system in which the antenna is situated. However, the phase function F is quite sensitive to the location of the reference point, and it follows that there may be a certain reference point which leads to a simplification of F. If there exists an origin which reduces F to a constant, then this origin is said to be the phase center of the antenna. Since this definition of phase center depends on the polarization of the field and the plane which contains the angular variable 4* , these two quantities must be specified whenever the concept of phase center is used For most antennas the phase is a function of 4 1 whatever the origin chosen, but over a limited range of 4 1 there may exist a point p such that F is practically constant. If p is chosen as the phase center for a given aspect angle 4* , then the range of 4 1 for which the fixed point p can be used as the phase center will depend on the allowable tolerance on F. To find the point p use is made of the evolute of a plane equiphase contour. The evolute is the locus of tne center of curvature of the contour, and the j center of curvature corresponds to the location of an origin which leads I ! to no change in the phase function over an increment a4^ It will be shown I : that the knowledge of F as a function of 4 1 for any origin near the antenna is sufficient to determine the evolute of a far field equi-phase contour. In the coordinate system of Figure 73 OP = r is the distance from j the origin to a point on an equi -phase contour S, The ray DP is normal to the tangent line of S at P, therefore DP must go through the center of curvature. 132 Figure 73. Coordinate system for phase center computations, L33 In the following developnn n i , point l) a i \ -d is found, bhen r La made very large so thai N Ls approxima ^ Knowing d and V for each point on the curve, a pencil oi Lines such as DP can be determined. The locus of the phase center or equivalently the evolute of S, is traced by the envelope curve of the rays L . Let an equi phase contour in the x, y plane be given by FOlO - Br = C, (78) where C is an arbitrary constant. The angle 6 is given by -rjj sin 4 1 + r cos 4 1 tan § = ~ (79) — tj cos 4" - r sin 4 1 From the geometry of Figure 73, d = -y tan ° -x (80) where x and y are the coordinates of point P. Substituting for x , y and o o o o tan Q, one finds d 1 liilzci , (81) (F-C) sin ^ - F cos 4 where the prime indicates differentiation with respect to 4 „ Changing to the variable u = 2lT cos ty, (82) and writing d in terms of wavelength, results in -dF/du (g3) udF/du 1 + sr- For the far field condition r is very large, hence d/^ is approximated by d _ _ dF ^ du (84) 134 Thus, djA is given by the slope of F(40 as a function of 2ir cos y\ d,A can be computed from the difference equation d. F(t\ + At 1 ) - FOlO K~ = 2ir[cos (t\ + At') - cos x 1 .] (85) where 4 1 . is a given value of H> and At is an increment. Once d is found as a function of r , rays such as DP can be drawn by setting Y = t" . The evolute of the equiphase contour is then the envelope curve of the rays. Figure 74 shows a pencil of rays and the corresponding evolute for an LPD with t = 0.92 and O = 0.12. Only half of the evolute is shown, o o because it is symmetric about the axis. For angles other than or 180 the phase center clearly lies off the axis of the antenna. However, for all antennas the departure from the axis is small in terms of wavelengths, i ° and it can be neglected for t 1 angles less than 70 . Each LPD antenna exhibits an evolute of the equiphase contour that is peculiar to itself, and no correlation was found between the shape of the evolute and the LPD parameters. i ° In what follows, the point p corresponding to t 1 = is chosen as the o phase center. Ideally, the distance in wavelengths from the apex to p should remain constant as frequency is changed by an integral power of T . Although a variation with frequency was observed, the computed results similar to those in Figure 75 show that the variation is small. The ordinate is x /^-, the distance from the apex to P in wavelengths, and the abscissa is the log of frequency. In all cases the variation was less than 5$>, so the average value can be used without significant loss 135 070 0.68 0.66 - ^ 0.64 _ Q. 0.62 H-PLANE , X o— — — Ld H O 0.60 E-PLANE LU UJ 0.58 < X Q. 0.56 0.54 0.52 0.50 1 1 1 1 1 3.25 3.50 3.75 LOGARITHM OF FREQUENCY Figure 75. Typical frequency variation of the relative distance from the apex to the phase center. 137 of accuracy. An example of the measured and computed location of the phase center is shown in Figure 7G . I >ive magnitude of the element base current is also plotted so that the location of the phase center with reference to the active region can be visualized. A graph of the computed location of the phase center for models with Z = 100 and h/a = 177 is shown in Figure 77. x A is the distance to the o p' H-plane phase center in wavelengths, for *\> = . The location of the phase center is independent of T and CT over the range of a shown. The E-plane phase center was found to lie ahead of the H-plane phase center by an amount which varied linearly from 0.95 x for a = 25 to x for o a = 2.5 . In all cases x < x\ / } which is the point where the transverse P .2 dimension is a half -wavelength. The shortening factor of Figure 33 can be used to find the change in x as Z or h/a is changed. Since no LP antenna exhibits a completely spherical equiphase surface, the question arises as to the utility of using point p as the phase center. This, of course, depends on the allowable tolerance between the actual phase at some angle 4*, and the phase computed on the basis of an hypothetical spherical wave which originates at the phase center. Figure 78 shows an equiphase contour S described by F(4 J )-Br = C, where 4 1 is the angle subtended at the origin 0. S is an arc of radius r -t- d with center at p , the o o chosen phase center. S represents an hypothetical circular equiphase contour of value F(o)-Br . The error A F is the distance between S and S o at the angle Y, and is given by AF F(o) - FQlO d sin 4* - sin 7 , *" = 2^ + X [1 sin <+ - Y) ] - (86) 138 0.7 0.8 0.9 1.0 I.I 1.2 1.3 1.4 1.5 1.6 1.7 1.8 I, X/ x FROM APEX Figure 76. Measured and computed location of the phase center with reference to the active region. COMPUTED POINTS r =0.8 • t= 0.875 A r =0.92 © t =0.95 X t = 0.97 LOCATION OF HALF- WAVELENGTH ELEMENT 8 139 20 22 24 26 10 12 14 16 oc DEGREES Figure 77. Location of the phase center in wavelengths from the ape) Figure 78. Coordinate system for the computation of the phase tolerance 141 For the far field condition r approaches infinity and "Y becomes equal to 4 1 , reducing (86) to If a tolerance q is given, a value ^ can be found such that q |^| < £- for |4«| < + q (88) This defines the range of 4 1 over which p is a useful phase center, for the tolerance q. Setting q = Vl6 and using Equation (8^, 4 1 was found by a trial and error process. Little correlation was found between y and the antenna q parameters, because the phase function F was different for each antenna. However, for all but two models in which cr is less than optimum, the value i o o of T was 90 for the E-plane pattern and greater than 90 for the H-plane pattern. The two exceptions were T = 0.95, 0" - 0.143, and T = 0.97, 0" = 0.175; both are high directivity models. For 0" greater than optimum, ^ decreased to 59 at T .= 0.875 and CT = 0.222, which was the lowest value q ' of 4* recorded. Thus, in all models in the range 0„8 < T < 0.97, and q ' ' 0.05 < CT < 0.222, the error in using p as the phase center is less than \/16 for all angles within the 3 db beamwidth. The values of x described herein can be used as a guide in the design of arrays of LPD antennas. However, due to the approximations of the theory and the sensitivity to error of phase, it is prudent to measure the location of the phase center in critical designs. An indication of the nature of the error involved in using an incorrect value of d can be determined from 142 Equation (87). If the error in d is Ad, then the phase error in wavelengths will be Ad(l - cos 40/V 143 4. THE DESIGN OF LOG-PERIODIC DIPOLE ANTENNAS This section first reviews the LPD parameters and their effect on the observed antenna performance. A procedure is outlined whereby the physical dimensions of an antenna which meets given electrical specifications can be determined by the use of graphs and nomograms. Finally, several novel designs are given which exploit certain properties of log-periodic dipole antennas. 4.1 Review of Parameters and Effects To varying degrees all the parameters which specify an LPD have an effect on the observed performance. Table 3 lists the parameters and qualitatively describes how each effects the performance. The principal LPD parameters and their range are listed in the first two columns. At the top of each succeeding column are the features of the antenna. The entries denote how the performance changes with an increase in the pertinent parameter, while all other factors are held constant. The directivity of an LPD depends primarily on the combination of t and 0" selected. Directivities from 7.5 to 12 db over isotropic have been measured. Since an increase in directivity implies an increased aperture, it is not surprising that high directivity models are characterized by small a and large L/^ . For a given t 0" and element thickness, max 7 ' ' the input impedance depends on the characteristic impedance of the feeder. Fortunately, the directivity is essentially independent of the feeder impedance. This makes it possible to design an antenna for a given directivity and then, in most cases, the input impedance can be adjusted to the required level. The exceptions occur on models with both low t and low Z . Under this condition the radiating efficiency of the active region Boom length lA max for a fixed Operating Bandwidth B 0) u a ti ft c 0) ■p ^ pa" CD en d as o CD U fan CJ d °i CD ft CD •a CD en ■p o CD U Q (D cn CD rH c CD in rH CD B o CD en CD CD a) U cn CD •a CD en H CD E o en a) ■a CD en rH CD a! U B O cn a> ■a Input Impedance (always less than Z ) o CD en i-i a ^H CD cfl u E O CO CD ■a CD en rH CD CB r. cn cd •a CD CD ID cn a! CD u d cn CD U cj d ID en CD r( Bandwidth of Active Region B CD en r. cj CD •a CD cn ri CD CD CD U CD m CD d ■p en X d -H CD ft ■D nj aj d a E CD TJ ft r* (SS h CD < CU ■a -p s d 3 ih ■H ,Q O -P ~ d ft -P O en ce! E ■O tl oo d CJ E r< CD O > ■a -p cu d 3 r 147 One of this set (^, f) leads to minimum boom length and another leads to a minimum number of elements, With these facts in mind a preliminary choice of t and 0" can be made from the graph of Figure 65. It is usually best to start with the optimum value of 0" and then proceed to lower values. Knowing t and 0", the value of the dependent variable a can be determined from (1 - T) tan « - 4 CT (89) or from the nomograph of Figure 79. The structure bandwidth B must be found to determine the boom length and the number of elements. B depends on the required operating bandwidth B and the bandwidth of the active region B , For the given values of T } \ 0" and a, B can be determined from Figure 32, or from the nomograph of Figure 80. B is then given by B = B B . (90) s ar Since the length of element number one is — — in the preliminary design, the geometry of the LPD antenna provides an expression for the boom length relative to the longest operating wavelength. ■y^- = i (1 - ~)cot a (91) max s where L is the boom length between the longest and shortest elements. A nomograph of 91 is given in Figure 81. The number of elements is found from the equation log B N = 1 + -7= (92) log 1/T a nomograph of which is given in Figure 82. -.70 \ »-l - 1.6- -.72 - 1.4- -.74 - 1.2- 3- -.76 - 1.0- -.78 - - .80- -.80 4- -.82 .60- - -.84 5- - .40- - -.86 6- - .30- - -.88 7- b .20- 8- *o -.90 e> " (T 2 .15- 9- -.91 P o ^ 10- LU l0 h- < O Q_ LU -.92 < CO LU .10- .09- or CD l2 _ LU l2 LU -.93 _I > .08- Q < h- .07- o CO < _l .06- 8 '*-- -.94 LU (T .05- 16- .04- 18- -.95 - 20- _ .03- - - 22- 24- -.96 .02- 26- - 28- 30- -.97 - .01- - 40- -.98 - 45- 148 Figure 79, Nomograph, 0" = —(1 - T) C ot a -2.5 16.0 - .70- 14.0- ,72 12.0 - .73- -3.0 74 10.0- 75 76 -3.5 8.0- 7.0- 77 78 6.0- 79 -4.0 .80 - 5.0- .81 - -4.5 4.0- .82- .83 - -5.0 .84- -5.5 3.0- 2.8- .85- -6.0 2.6- 2.4- .86- -6.5 2.2- .87- -7.0 00 2.0- .88- -7.5 1.9- ^— 18- -8.0 .89- -8.5 -9.0 **>:. -9.5 q: i.5- - -10 i 8-: L - ii Ll Q - ,2 Ll 1 ^ '■<- ] £,.30- -13 C < < 1.28 - J 1.26 - uj.93- ::a . Ll_ 1.24 - _i ) OI.22- 5 - - 16 -,- 1.20 - -«- : l 19- co.94- — 17 tr .18- Q||7- - - 18 - 19 -20 -21 -22 ^ 1.16- § U5 _ .95- < 1.140- - 00 -23 -24 1.130 - .96- -25 - -26 -27 1.120- -28 _ -29 -30 -32 1.1 10- .97- -34 -36 -38 - -40 1.105- -42 -44 ~ - .98- Figure 80. Nomograph, B ■ - 1.1 + 7.7 '1 - T) 2 C ot ff 149 -20 _ 2.5- : 5.0- -10 r 8.0 " -6.0 4.0- 3- r 4.0 3.0- - E-3.5 - - 4- -3.0 -2.8 - -2.6 2.0- - - 1.8- -2.4 . - 1.6- 5 - -2.2 - 1.4- - r 20 1.2- 6- r 1.9 - x i.o- - 1.8 ^.90- 7 - 5-1.7 m <^ .80- - -1 .70- 8- : X ^• 6 fe X .60- h- 9- -1.5 Q g .50- LU 1 .40- C0 10- LU LU < or DQ - 1.4 LU or h- O -1.3 3 O .30- o DQ LU > .20- 12- LU Q 14- q: b - 18 - h- < .16- 18- cn _J _ - LU .14- 20- - QT .12- 22- 2 .10- 24- - I.| .09- .08- 26- .07- 28- - .06- 30- i .05- .04- - i 40- .03- ' .02- 46- 150 'igure 81. Nomograph fr -)cot a -.98 160- 20 100- 10- 90- 9- - 80- 8- 70- 7- 60- 6- 50- 5- 45 -. - 40 -i 4 - 35 T 3.8- 3.6- -.96 30 4 3.4- 3.2- 25^ 3.0 - 2.8- 20- w 2£- z ^ CQ 2.4- 16 - Vo 0) 14- ^ 2.2- -.94 1- n " or 2 12- ^ 2.0 - o LU ^ h- s .'°- Q o LU 9- ^ 1.8- - i2 -.92 -J 8- < LU 7- DO LU _l < Ll o 6 - £,e- O 3 L_ an C/) LT 5- K - .90 v LU O CD ZD - ^ 4- Z> -z. CO " 3- -.85 - 2- 1.2- -.80 -.75 : , -.70 |- l.l- log B M 1 'i F igure 82. Nomograph, JN — X + - log - 151 152 It is likely that the first estimate of t and 0" will not minimize L or N Repeating the process for different values of t and 0" will establish the trend, and the minimum designs will become readily apparent. For some values of B , minimum L cannot be attained for values of T and 0" within s the Operating range of the graph of Figure 65. In these cases a compromise will have to be made. If a choice of (T 0") exists, and there is no apparent basis for a decision, it should be recalled that the SWR increases and the front to back ratio decreases as 0" departs from the optimum value, according to Figures 40 and 68. Some mention should be made of the secondary factors which affect the directivity Z and z Since the graph of Figure 65 is based on o a' Z = 100 ohms and Z = 350, an adjustment should be made if it is o a ' anticipated that Z and Z will depart from these values by more than a factor of 2. The exact value of Z is as yet undetermined. However, o ' it. is known that the feeder impedance is always greater than the resulting input impedance, so if R is greater than 100 ohms the directivity contours of Figure 65 will read a fraction of a decibel high. If the ratio h/a is much different than 177 a correction must be made. According to the curve of Figure 71, the directivity decreases by about 0.1 db for each doubling of h/a; for h/a > 177 the constant directivity contours of Figure 65 will read high. One may wish to include an additional safety factor, since a comparison of computed and measured directivity in Table 2 shows that the directivity computed from Figure 65 averages 0.35 db higher than the measured value. The corrected value of directivity, for use in Figure 65, is obtained by adding the above contributions to the required design value. 153 4.2.2 Designing for a Given Input Impedance Once the final values of T and 0" are found, the characteristic impedance '.. of the feeder Z must be determined so as to give the required input impedance R . The ratio h/a is determined from structural considerations, and ideally o should be the same for each element. Practically, the element diameters can be scaled in groups, and in the computation of input impedance the average h/a in a group should be used. The average characteristic impedance of a dipole element Z can be found from a Z = 120 (In h/a - 2,25), (93) a or from the graph of Figure 45. Inverting Equation(71), gives the characteristic impedance of the feeder relative to R Z o 1 R o V [ R oj + 1 , (94) where Z /R is the average characteristic impedance of a dipole element with a o / respect to the required input impedance R , and <-> is the mean relative spacing, a - Q/ /f". (95) A graph of (94) is given in Figure 46. The major parameters of the required design have now been determined. It remains to find the size of the first element relative to the maximum operating wavelength. As shown in Section 3.2.2, the longest element \ can be less than _E52£ for values of Z > 100 and h/a < 177. The half- 2 o length of the longest element is given by 154 \ max h 1 = S — — (96) where S is the shortening factor as read from the graph of Figure 33. Knowing T } 0" ; N ; Z and h , one can find the dimensions of all other parts of the antenna. The impedance Z which terminates the feeder has an effect only at the lowest operating frequencies. In practice, the feeder is terminated in a short circuit a distance of — — or less behind the largest element, o so that Z remains inductive at the lowest frequencies. 4.2.3 Application of the Design Procedure: An Example As an example of the design procedure, let the dimensions of an LPD antenna be determined such that the directivity is 9.5 db, the bandwidth is 1.75, and the input impedance is 80 ohms. Size limitations dictate an h/a of 118. This means that the corrected directivity for use in Figure 65 is between 9.4 and 9.5 db. Starting with a value of 0" greater than optimum (to clearly illustrate the trend), the set (t^0~) which yields a directivity of 9.4 db is recorded from Figure 65 in Table 5. The values of B , L/^ , and N were determined from the nomographs. ar ; max' The minimum-element design is (0.89, 0.165). The minimum boom length design cannot be attained for values of T and 0" within the operable range. A compromise between length and number of elements is given by T = 0.92 and 0" = 0.12. This design has L/\ = 0.89 and N=12. The average characteristic dipole impedance for h/a = 118 is about 300 ohms (from Figure 45) so Z /R = 3.75. a o 0" =0"/ rr is 0.125, and from the graph of Figure 46, Z /R is approximately o o 1.3, therefore the feeder impedance Z is 104 ohms. The half-length of element one is set equal to ^ /2 because the shortening factor is nearly unity max & J (see Figure 33) 1!,!, TABLE 5 Values of T a and a. which give 9.4 db directivity over a 1.75:1 band 0.92 5.. I 1.62 2.84 1.72 13+ 0.91 0.197 6.6 1.64 2.87 1.41 12 0.90 0.187 7.7 1.68 2.94 1.23 11+ 0.89 0.165 9.5 1.66 2.91 0.99 10 0.148 9.7 1.56 2.73 0.94 10+ 0.90 0.91 0.135 9.5 0.92 0.120 9.5 0.93 0.105 9.5 B a i B = BB S :i l' 1.62 2.84 1.64 2.87 1.68 2.94 1.66 2.91 1.56 2.73 1.47 2.57 1.40 2.45 1.33 2.33 1.27 2.22 1.21 2. 12 1.16 2.03 TABLE 6 max 1 72 1 41 1 23 99 94 92 89 86 .83 .74 .66 11- 12- 13- 0.94 0.090 9.5 1.27 2.22 0.83 14- 0.95 0.070 10.2 1.21 2.12 0.74 15+ 0.96 0.050 11.0 1.16 2.03 0.66 18+ Antenna Dimensions in Inches n T n-1 h n X n D n 2a n Diameter o wire used 1 1.0000 4.66 28.00 16.81 0.079 0.078 2 0.9200 4.29 25.76 14.57 0.073 0.072 3 0.8464 3.95 23.70 12.51 0.067 0.072 4 0.7787 3.63 21.80 10.61 0.062 0.065 5 0.7164 3.34 20.06 8.87 0.057 0.061 6 0.6591 3.07 18.45 7.26 0.052 0.050 7 0.6064 2.83 16.98 5.79 0.048 0.048 8 0.5578 2.60 15.62 4.43 0.044 0.040 9 0.5132 2.39 14.37 3.18 0.041 0.040 10 0.4722 2.20 13.22 2.03 0.037 0.038 11 0.4344 2.02 12.16 0.97 0.034 0.031 12 0.3996 1.87 11.19 0.032 0.031 156 In compiling a table of the dimensions of the antenna, it is best to start with a tabulation of powers of T which is accurate to at least four decimal places. The half lengths of the elements h^; the distance from the apex to each element, x ; the distance from the front of the antenna to each element, D ; and the element diameters are then computed, ; n as shown in Table 6. The half length of the largest element is 4.66 inches because the desired low frequency cut-off is 635 mcs. The feeder is spaced to give a characteristic impedance of 104 ohms according to the formula, Z = 120 cosh 1 £-, (97) o 2a' where b is the center to center spacing and 2a is the diameter of the feeder conductors. The above described antenna was constructed of coin silver tubing and copper wire, using silver-solder techniques. The resulting model is pictured in Figure 83. The measured input impedance clustered around a mean resistance level R of 73 ohms, a bit lower than the design value. This could be brought closer to 80 ohms by increasing the spacing between the feeders. The SWR with respect to 73 ohms is plotted in Figure 84. Taking an SWR of 1.3:1 to define the useful band, one finds that this antenna operates from £ Q5 to f g 2g , which gives a bandwidth of 1.82. The measured E-and H-plane half -power beamwidths and the directivity are shown in Figure 85 . The average directivity from f 1>05 to f is about 9,5 db, which just meets the requirement. A comparison 8 . 25 of measured and computed patterns for this antenna is shown in Figure 86. 157 158 T CM O 00 to 6 a continued fraction expansion was used. The crossover point x = 6 was determined by the computer time required to achieve 8 or 9 accurate significant digits, using either method. A plot of computer time vs. x is shown in Figure 91. The crossover point x = 6 was chosen from this plot. The series representations 12 1 II I 10 \ Ld 9 \ 2 O < CL 8 7 \ O u 6 - X. sef^U UJ > _i UJ 5 4 3 2 1 1 1 1 i i i i i i i 10 ARGUMENT x Figure 91. Computation time vs. argument x for the series and continued fraction expansion of K(x) = Ci(x) + j Si(x) 176 Ci(x) = C + *n(x) + E - * 1 * .* (100) oo n 2n (-1) x (2n) I 2n n=l where C = 0.57721 is Euler's constant, and oo (-i) n 2n + 1 SiM = nil (2„~+ 1) < (2n » 1) ' (101) Both the above series are uniformly convergent for all finite values of x. The convergence of these series is slow, and the number of terms required for a given accuracy increases with x. Fifteen terms were required for eight place accuracy with x equal to 6. However the terms of the series are monotonically decreasing and alternating in sign, so the error involved in using n terms is numerically less than the n +■ 1st term. This fact allows one to compute until a given accuracy is attained, and this advantage outweighs the fact that the convergence is slow. The two series of Equations 100 and 101 were computed together as oo . n K(x) = C + in(x) E, J , . (102) n=l n ' n To find the continued fraction expansion for K(x) observe that K(x) = Ci(x) f jSi(x) = Ei(jx) + j | (103) where -u -Ei(-z) = - du (104) J 5 As a function of the complex variable z, Ei(-z) has a logarithmic branch i . J point at z = 0, therefore the argument z =jzr e is restricted such that .26 -It < 6 < IT. A continued fraction expansion exists for (104), 177 oo f e V, 1112 2 3 J U Z+1 + Z+1 + Z+1 + e Z du = ; . y (105) 7 By means of an equivalence transf ormation (105) can be rewritten oo e Z I - du = - - (106) f e U ^ 1111111 I — du = — — - — — — — U Z+1 + Z + 1 + Z+1 + Z + J z 2 3 This is a continued fraction of Stieltjes. The general form of the Stieltjes continued fraction is 1 1_ _J_ 1_ _1_ 1_ kz + k+kz + k' +kz + k+... C } 1 2 3 4 5 6 The continued fraction (107) is uniformly convergent in the cut z-plane due to the fundamental convergence theorem of Stieltjes, which states that if the k are positive constants and the series £ k diverges the continued P P fraction is uniformly convergent over every finite closed domain of z whose distance from the negative half of the real axis is positive, and its value is an analytic function of z for all z not on the negative half of the real axis. Therefore K(x) may be computed from (106). By means of another equivalence transformation, 7T e JX 1 K(x) = j J + — • -4-v (108) 2 jx F(jx) where F(Jx) = x _ Vi£ _ Uf. _ 'Uf. _ 2^x _ 3^x ^ (10Q) The p approximate of F, A /B , is given by the fundamental recurrence P p formulas: 178 with A = b A + a A p + 1 p+1 p p+1 p - 1- B , = *> B + a B p+1 p+1 p p+1 p - 1' A = 1, B-1 = 0, A ( (110) (111) and = -(P + D/2 p jx a P = -P/2 Jx ; b P = 1. p = 1,3,5, 2,4,6, (112) The fundamental recurrence formulas allow one to compute the successive approximates of F rather than start with a given p and rationalize F . In the ILLIAC program for the continued .fraction expansion of K(x) ; it was found that nine place accuracy for Ci(x) or Si(x), whichever was larger, was achieved when B B < 10 (113) P - 1 for x > 1.5. The number of iterations required ranged from 73 for x = 1.5 to 15 for x = 9. The subroutine for K(x) was programmed to test x to determine the type of expansion to use and then to compute K(x) to nine sigificant digits. It is worthy of note that continued fraction expansions exist for many trans- cendental functions, some of which are given by Wall , and that in certain cases the continued fraction is more rapidly convergent than the corresponding series expansion. 179 A. 2 Matrix Operations The program which computes T = (U + Y Z ) was straightforward. < The prime difficulty in this case is the systematic addressing of the storage locations of the matrix elements. The solution of I = T I (114) employs the upper triangularization and back-substitution method. The steps in this method are as follows. First the augmented matrix T is set up, using T and I. T = t t t ■ t 11 12 13 IN t t t — - 21 22 23 *2N " l 2 t Nl t N2 t N3 'NN (115) Here the t .'s are the elements of T and the i.'s are the elements of I T is a matrix which represents the equation T T - I = A (116) By the elementary determinate operations, T can be upper triangularized, yielding 1 *12 t 13 Vn+1 1 t 1 23 2,N+1 1 t' N,N+1 (117) 180 i , i ; -a ' ' ' can ^ e successively determined by a series of back- A N Vl A N-2 substitutions, / 1a * t = ~ t N,N+l (118) / / 1 A at n " " t N-l, N+l N-l, N 1 A XT N-l ' ' N Several test matrices were employed to check the accuracy of this part of the program. It was found that nine significant digit accuracy was preserved in solving equations in four unknowns. Nine significant digits is the maximum accuracy that can be obtained using the ILLIAC in the floating-point mode. It was expected that the accuracy decreased as the number of equations increased. However, a sample calculation of an eight element antenna using the entire LPD analysis program, agreed in the first five significant digits with the computations performed on a desk calculator (which, incidently, took three weeks to perform) . 1K1 APPENDIX B MEASUREMENT CONSIDERATIONS This section discusses several types of measurements which were per- formed in the experimental phase of this research. The procedure used in the measurement of the near field amplitude and phase will be considered in detail, whereas the input impedance, radiation pattern, and phase center measurements will be given only summary remarks. Except for the near field phase measurements standard techniques were used, and a majority of problems that arose were concerned with the application of these techniques to the particular antenna structures under investigation. In any measurement involving log-periodic or log-spiral structures, one is faced with the problem of obtaining a great amount of data over a wide range of frequencies, if the frequency independence of the antenna is to be confirmed. The large bandwidth (4:1 or greater) over which the measurements must, be performed places certain requirements on the experimental set-up and on the construction of the model. The experimental set-up must be changed several times throughout the course of a measurement because of the relatively narrow bandwidth of some testing equipment. This means that certain pieces of equipment are operated under slightly differing conditions, hence the set-up should provide for easy calibration and tuning. Of equal importance is the actual fabrication of the model antenna. Construction tolerance should be figured on the basis of the shortest wave- length of operation. Thus, to some extent, the size of the antenna is determined by the accuracy which can be maintained in the model shop. The transverse dimensions of the feeder configuration of an LPD must remain small compared to the length of the shortest element. Nevertheless, the minimum 182 diameter of the feeder members is limited by the size of the internal feed coax. This again places a limit on the size of the antenna and the range of useful testing frequencies. The above requirements led to the choice of the 100 mcs to 400 mcs range for the models used for impedance and near field measurements. A picture of one of the impedance models is shown in Figure 92. The feeder diameter was 0.420 inches, just large enough to accommodate the teflon dielectr: RG-115 A/U coaxial cable which was chosen for its low loss and excellent uniformity. The pattern models were built for the 500 mcs to 2000 mcs range, the low frequency limit being dictated by the antenna pattern range facilities. B.l Near Field Measurements The near field measurements consisted of determining the amplitude and phase of the voltage between the feeder conductors and the current into each dipole element. The voltage between the feeder conductors was found by measuring the signal received by a small probe antenna as pictured in Figure 93. The coax which energizes the antenna model runs through one of the hollow feeders. The other feeder member is equipped with a milled slot which guides the probe assembly throughout the length of the antenna. The probe is connected to a length of RG 58/U coaxial cable which is contained in a thin walled tube. The end of the tube opposite the probe assembly is fitted with a pointer which moves along the length of a rule, permitting accurate and repeatable positioning of the probe. The input impedance of the antenna was found to change 1 . 4$> as the position of the probe was changed from the front of the antenna to the back. This is due to a small change in the feeder impedance owing to the presence of the milled slot. The variation Figure 92. A picture of one of the antennas used for near field measurements GO < GO C cr Q_ >i Q c/) O Z> o o UJ L85 of the measured data was found to be independent o! the Length of the probe. which was varied from one < me quai u i inch. The measurement of the dipole element current was accomplished by the loop probe shown in Figure 94. The loop was designed to measure the magnetic field encircling the dipole, and was located as close to the base of the dipole as possible. Many difficulties were encountered in the selection of the proper loop size, spacing, and location. Shielding was necessary, as indicated by initial results with an unshielded loop. However, a completely shielded loop could not be constructed small and accurately, because there was no space to extend the pick-up coax through the guide assembly. The partially shielded loop in Figure 94 was selected as a compromise. The figure also shows the delectric disk which encloses the loop. This device was used to accurately position the loop at a constant distance from the dipole and in a plane containing the dipole element and the loop. The diameter of the loop and the dielectric was chosen such that a minimum amount of transverse motion was required in going from an element connected to one feeder to an element connected to the other. The configuration shown in Figure 94 was found to yield repeatable results although effects of undesired field components were never entirely eliminated. The largest extraneous component was thought to be due to the proximity of the current on the neighboring dipoles. It was estimated that this component was down at least 15 db from the desired component. The agreement between measured and computed results of the element current is therefore not as good as that of the feeder voltage. B.l.l Amplitude Measureme nts A block diagram of the circuit used for the amplitude measurements is 186 CD < Ld rr 00 en Q_ >- _i n go 1 _j ill ( ) z> u_ LJ i= x o o 1- ce LU CO 2 UJ o 3 o Ld o: CD LU O o % cr z Q_ =) LU Q O :> Lul CO cr Q < ~Z. 1- 3 -z. CO O o i CE cr i- O Ll 187 shown in Figure 95. The filtered output of the square wave modulated power oscillator is delivered to the antenna through a double stub tuner, which was adjusted for maximum signal to the probe. The square law detected out- put of the tuned probe is displayed on a SWR amplifier, peak tuned to the modulation frequency of 1000 cps. In performing the measurement, the probe was moved back and forth until the location of maximum signal was found. This point was taken as the db reference. A reading was taken every centimeter throughout the length of the antenna. A power output from the oscillator on the order of several watts was required for a dynamic range of 40 db above noise. B.1.2 Phase Measurements The phase measuring circuit is diagrammed in Figure 96. The principal features are the hybrid junction and the balanced input adapter. The measurement theory is as follows: A CW reference signal, whose phase can be adjusted by means of the slotted section and line stretcher, is injected into the series arm (4) of the hybrid junction. The modulated test signal feeds the shunt arm (3) . The RF phasor sum of these signals is impressed across the load connected to output (1) and the phasor difference appears at output (2) . The balanced input adapter takes the audio difference of these detected signals from the hybrid junction and also provides the bias voltage for the bolometers. As shown in Figure 97, a sharp null is observed o at the output of the balanced adapter when the reference signal is 90 out of phase with the test signal. Figure 97 compares the phasor relations and the nulls obtained of the balanced detection method and the single detection method. In the latter method only the difference channel would be used; the jnull is obtained when the reference and test signals are in phase, provided 188 A. X CD H if) -I §| O H Q "S £ s If* c/> q_ IO < ^ 189 |e r | = .5 i/0 1 E t"Er e t -Er Et+Er |e r | = .5 0=0° JErl |e r | = .5 <£=90° SINGLE DETECTION BALANCED DETECTION 2.0 1.5 a: UJ M.O UJ \ \ IetI |e r I =i .5 N ^ I i '/- ir > y V \ / -180 4> DEGREES 2.0 uf 1.5 "\ \ IetI |Er| y V \ \ 1 .5 ^v \ \ 1 / .25 *■*»■ "\ ^ P s + 180 90 180 j> DEGREES Figure 97. Phasor relations and the nulls obtained for values of | E T | /) E R | for two methods of measuring relative phase. E is the test signal, E is the reference signal. 191 both signals have equal amplitude. Since the amplitude of the test signal varies, this method would require a precision variable attenuator calibrated for amplitude and phase. The advantage of the balanced detection system is that a well defined null is obtained even when the test and reference signals o are of unequal amplitude. However a 180 ambiguity exists in the balanced detection method. This can be resolved by carefully following the incremental phase shift as the probe is moved away from the reference position. For phase measurements the probe signal, rather than the oscillator is modulated. This results in a sharper null than would otherwise be obtained. Even though the unmodulated reference signal, which may be larger than the test signal, is present at the detectors, the result is only a DC component of current to the transformer. The only error from the reference signal is due to leak-through in the hybrid junction. This is negligible; in commercially available instrument hybrids the decoupling is on the order of 50 db. The experimental set-up for the near field phase measurements is shown in Figure 98. All the equipment was mounted on a bench whidh could be rolled up to a second story window in the Antenna Laboratory, in order for the model under test to look into an uncluttered environment. The most predominant feature in the picture is the take-up reel for the cable which is connected to the probe. In the phase measurements it is important that the flexing of the cable be controlled and held to a minimum; the take-up reel serves this purpose. The following procedure was carried out to balance the hybrid circuit after each change of frequency. Refer to Figure 96. The connection at terminal (4) is broken and replaced with a matched load. The bias to 192 193 each bolometer was adjusted to 8.75 ma. One bolometer was then disconnected and the balancing potentiometer was adjusted for maximum output. The tuning stubs throughout the circuit were then tuned for maximum output. The other bolometer was reconnected and the first was disconnected. The balancing potentiometer was again adjusted for maximum output, and only the stubs associated with the operative bolometer were tuned, the others were left as previously adjusted. With both bolometers connected, the balancing potentiometer was adjusted to null the signal. This step had the effect of balancing out any leak- through component of the test signal; it also equalized any residual unbalance which might have existed between the two bolometers. The slotted line was reconnected, completing the tune-up procedure . The probe was positioned at a chosen point and the phase of the reference signal was changed to obtain a null. This was repeated every centimeter along the feeder of the antenna. By recording the phase change introduced by the slotted line, the relative phase of the test signal was determined. B.2 Impedance Measurements The input impedance of the LPD models was determined by the SWR and null shift method. The set-up incorporated a PRD standing wave indicator or alternatively a HP slotted line, and a tuned amplifier using a bolometer detector. The short circuit reference plane for the impedance measurements was taken at the front of the antenna, as shown in Figure 99. This choice of a reference plane lumps the gap or terminal impedance in with the antenna impedance. For this reason, extra care was taken in the construction of the feed region. After several trials it was found that the feed region REFERENCE PLANE METAL ELBOW FOR SYMMETRY FEED COAX CENTER CONDUCTOR Figure 99. Details of the symmetrical feed point, showing the reference plane for impedance measurements 195 configuration shown in Figure 99 was best from the standpoint of introducing a minimum unbalanced current component and a minimum gap reactance. Ideally, the reference measurement should be made at every frequency, just before or after the corresponding load measurement. This was not done because of the large number of test frequencies involved and because of the uncertainty in reconstructing an identical feed configuration each time. Instead, the electrical length of the coax from the reference plane to a reference on the measuring device was determined for all the frequencies within the band, prior to taking the load measurements. Several lengths of cable were tested; the one which yielded the most uniform results was used. B,3 Far Field Measurements The radiation patterns of the LPD models were recorded by the commercially equipped University of Illinois Antenna Laboratory pattern range facilities. The tower and an LPD model are pictured in Figure 100. Using a bolometer as the square law detector, the system was found to have a linear dynamic range of greater than 20 db. Since the LPD models tested had an operating bandwidth of 3:1 or greater, interference from other sources operating within this band sometimes presented a problem. Except for the signal from an S-band radar that was 250 yards away, the interference problem was eliminated by filtering and tuning techniques. The phase center measurements were accomplished using the balanced detector shown in Figure 96, except that the model under test assumed the role of the probe antenna. The test model was mounted on a tower in such a way that the axis of rotation could be moved toward or away from the transmitting antenna. The relative phase of the received signal was plotted as a function of the azimuth angle over the range + 40 degrees from the dead 196 Figure 100 . Antenna positioner and tower at the University of Illinois Antenna Laboratory 197 ahead position. The center of rotation of the model was varied from one plot to the next until a position was found such that the phase was most nearly constant over the range of the azimuth angle. The boresight and positioning accuracy was such that the location of the phase center could be determined within an error of + 0.125 inch, which was about 0.01 ^ at the testing frequencies. ANTENNA LABORATORY TECHNICAL REPORTS AND MEMORANDA ISSUED Contract AF33 (616; -310 "Synthesis of Aperture Antennas," Technical Report No. 1, C.T.A. Johnk, October, 1954.* "A Synthesis Method for Broad-band Antenna Impedance Matching Networks," Techni cal Report No. 2, Nicholas Yaru, 1 February 1955.* "The Asymmetrically Excited Spherical Antenna," Technical Report No. 3, Robert C. Hansen, 30 April 1955.* "Analysis of an Airborne Homing System," Tec hnical Report No. 4, Paul E. Mayes, 1 June 1955 (CONFIDENTIAL). "Coupling of Antenna Elements to a Circular Surface Waveguide," Technical Report No, 5, H. E. King and R, H. DuHamel, 30 June 1955.* "Axially Excited Surface Wave Antennas," Te chnical Report N o. 7, D. E. Royal, 10 October 1955. * "Homing Antennas for the F-86F Aircraft (450-2500mc), " Technical Report No. 8, P. E. Mayes, R. F. Hyneman, and R. C. Becker, 20 February 1957, (CONFIDENTIAL) "Ground Screen Pattern Range," Technical Memorandum No. 1^ Roger R. Trapp, 10 July 1955.* Contract A F 33 (616) -3220 "Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. Scott and R. H, DuHamel, 16 April 1956, "An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report ' No. 10, D. G. Berry and J. B. Kreer, 10 May 1956. i "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," Technical R eport No, 11, J. W. Duncan and R. H. DuHamel, 15 July 1956.* in Directional Characteristics of a U-Shaped Slot Antenna," Technical Report No. 12, Richard C Becker, 30 September 1956.** "impedance of Ferrite Loop Antennas," Technical Report No. 13, V. H. Rumsey and W. L„ Weeks, 15 October 1956. Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report ' No. 14, Richard F. Hyneman, 20 December 1956. '"Distributed Coupling to Surface Wave Antennas/" Technical R eport No. 15, Ralph Richard Hodges, Jr., 5 January 1957, "The Characteristic Impedance of the Fin Antenna of Infinite Length," Technica Report No, 16, Robert L. Carrel, 15 January 1957.* "On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17 Walter L c Weeks, 10 April 1957.* "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Sourc Antenna," Technical Report No. 18, Walter L. Weeks, 20 April 1957. "Broadband Logarithmically Periodic Antenna Structures " Technical Report No. R. H. DuHamel and D c E. Isbell, 1 May 1957. "Frequency Independent Antennas," Technical Report No. 20, V. H. Rumsey, 25 October 1957. "The Equiangular Spiral Antenna," Technical Report No 21, J. D. Dyson, 15 September 1957. "Experimental Investigation of the Conical Spiral Antenna," Tec hnical Report No. 22, R. L. Carrel, 25 May 1957."^ ** Coupling between a Parallel Plate Waveguide and a Surface Waveguide,' Techn ic Report No. 23, E. J c Scott, 10 August 1957 "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," Technical Report No. 24, J. W. Duncan and R, H, DuHamel, August 1957, "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary Cross Section," Technical Report No. 25 Robert 1. Carrel, August 1957. "Cavity -Backed Slot Antennas," Technical Report No . 26, R, J. Tector, 30 October 1957. Coupled Waveguide Excitation of Traveling Wave Slot Antennas, Technical 5£E££.Ui2j__2L W - L Week3 > 1 December 1957. "Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," Technical Report _No. 28 \_, W. L. Weeks, 20 December 1957. "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrar Cross Section," Technical Report No. 29, J. D. Dyson, 10 January 1958. "Non-Planar Logarithmically Periodic Antenna Structure,' 1 Technical Report No. . D. W. Isbell, 20 February 1958. "Electromagnetic Fields in Rectangular Slots, ' Technical Report No. 31, N. J. Kuhn and P. E. Mast, 10 March 1958. "The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder," Technical Report No. 32, J. W. Duncan, 25 May 1958. "A Unidirectional Equiangular Spiral Antenna," Technical Report No. 33, J. D. Dyson, 10 July 1958. "Dielectric Coated Spheroidal Radiators, Technical Report No, 34, W. L Weeks, 12 September 1958 "A Theoretical Study of the Equiangular Spiral Antenna,' 1 Technical R epor t No. 35, P. E. Mast, 12 September 1958 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Tech nica l Report^No^SG, R H. MacPhie, 30 April 1959. "On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and Infinite Ranges/"" ^Si£L££i._M p -°£lJl2.^_?L' Ra J Mittra ^ 15 Mav 1959. "Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical Report No. 38, W„ L Weeks, 5 June 1959. Log Periodic Dipole Arrays," Technical Report No. 39, D„ E„ Isbell, 1 June 1959. "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical 1 ?:2E2£iJl?__§£.' S - °asgupta and Y. T, Lo, 17 July 1959. 'The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical Re PPI?-^°.-.. iL K - G - Balmain, 1 August 1959. "The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem in a Waveguide," Technical Report No, 42, Raj Mittra, 1 October 1959, Impedance Properties of Complementary Multiterminal Planar Structures," Technical Report No. 43, G, A„ Deschamps, 11 November 1959. On the Synthesis of Strip Sources/" Technical ReportNo. 44, Raj Mittra, 4 December 1959, Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides, Technical Report No. 45, C a H Tang and Y. T„ Lo, 11 March I960. New Circularly Polarized Frequency Independent Antennas with Conical Beam or Omnidirectional Patterns," Technica l Report No. 46, J D. Dyson and P. E. Mayes, 20 June 1960. ~~ Logarithmically Periodic Resonant-V Arrays," Technical Report No. 47, P. E. Mayes and R. L Carrel, 15 July I960. A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Technical Report No. 48, Y. T. Lo. "Evaluation of Cross-Correlation Methods in the Utilization of Antenna System Technical Report No. 49, R H„ MacPhie, 25 January 1961. I ""Synthesis of Antenna Product Patterns Obtained from a Single Array, " Techn icl Report No. 50, R„ H. MacPhie. * Copies available for a three-week loan period ** Copies no longer available. AF 33<616)-6079 DISTRIBUTION LIST One copy each unless otherwise in dicat ed Armed Services Technical Information Agency Attn T1P-DR Arlington Hall Station Arlington 12 Virginia (10 copies) Aeronautical Systems Division Attn. 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