LL - L &<2~> Qc cop7"[ <^^^X^^U^y ANTENNA LABORATORY Technical Report No. 54 THE COUPLING AND MUTUAL IMPEDANCE BETWEEN BALANCED WIRE-ARM CONICAL LOG-SPIRAL ANTENNAS by John D. Dyson Contract AF33(657)-8460 Project No. 6278, Task No. 40572 June 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION WRIGHT-PATTERSON AIR FORCE BASE, OHIO ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA, ILLINOIS Antenna Laboratory Technical Report No. 54 THE COUPLING AND MUTUAL IMPEDANCE BETWEEN BALANCED WIRE-ARM CONICAL LOG-SPIRAL ANTENNAS by John D . Dyson Contract AF33 (657)8460 Project N6278,, Task No. 40572 June 1962 Sponsored by: Aeronautical Systems Division ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA, ILLINOIS I ! C ! r r i i r r i i i i PREFACE This work was presented in a somewhat abbreviated form as a paper at the IRE International Convention, March 1962. "The Coupling and Mutual Impedance Between Conical Log-Spiral Antennas in Simple Arrays ', Proc. of the 1962 IRE International Convention, Vol. I. ABSTRACT rhe radiation characteristics of the conical logarithmic spiral antenna make it attractive as an element for circularly polarized arrays. The de- sign of such arrays requires a knowledge of the radiation coupling and mu- tual impedance between these log-spiral elements as a function of rotation *? well as element spacing, and for some array configurations a knowledge of the location of the phase center of the elements, A general consideration of the conical log-spiral antenna as a locally periodic, slow wave structure is outlined,, and in so doing a quasi-empirical formula for the location of the antenna phase center is developed. This investigation of the mutual impedance between the log-spiral an- tennas has shown it to be very low. Information on the mutual impedance and on the coefficient of coupling between antennas is supplied for two dif- ferent geometrical array configurations and for various antenna parameters . Pattern modification due to the presence of parasitic elements has been in- vestigated ,, For element to element spacing of at least one half wavelength, this consists primarily of an increase in beamwidth . Radiation patterns for several arrays are shown. ACKNOWLEDGEMENT The author would like to acknowledge, many helpful discussions with members of the University of Illinois Antenna Laboratory^ in particular Professors G. A, Deschamps and P. E. Mayes, and Mr. 0, L. McClelland. In addition,, he is pleased to acknowledge the assistance of the mem- bers of the laboratory in the measurements. Mr, R„ E. Griswold made the extensive coupling and mutual impedance measurements „ CONTENTS Page 1 ., Introduction 1 2, Theoretical Considerations 4 2.1 Coupled Antennas 4 2.2 The Phase Center 8 3, Experimental Considerations 19 4, Measurements 25 4.1 Coefficient of Coupling 25 4,1 ol Parallel Arrays 25 4do2 Conical Arrays 29 4.2 Mutual Impedance 34 4,2ol Parallel Arrays 34 4,2»2 Conical Arrays 45 4.3 Radiation Patterns 45 4„3ol Parasitic Arrays 45 4,3,2 Driven Arrays 62 5, Conclusions 75 References 76 Appendix A 78 I I I I I I I I I I I r r r ILLUSTRATIONS Figure Number Page 1 Possible Arrays of the Conical Log-Spiral Antenna 2 2 Equivalent Network for Coupled Antennas 5 3 Truncated Portion of Conical Log-Spiral Antenna 9 4 Brillouin Diagram for Bifilar Helix 11 5 Measured Phase Centers for Several Wire Arm Conical Log-Spiral Antennas 16 6 Approximate Circumference in Wavelengths at the Phase Center of Balanced Wire Arm Conical Log- Spiral Antennas (N - 2 } m = 1) 17 A Conical Antenna with Associated Coordinate System 20 8 Photograph of Conical Log-Spiral Antennas 21 9 Phase Center Location in Wavelengths from the Vertex of 15 Conical Wire-arm Antenna Versus Spiral Angle 23 10 Phase Center Location in Wavelengths from the Vertex of a 15 Conical Wire-Arm Antenna as a Function of Frequency, a = 73 23 11 Experimental Set Up for Measuring the Complex Coef- ficient of Coupling and Mutual Impedance 24 12 Magnitude of Coupling Coefficient Measured Between Parallel Antennas as a Function of Spacing Between Antennas in a Parallel Array 26 13 Coupling as a Function of Rotation in a Parallel Array 27 14 Coupling between Contrawound Antennas Directed at Each Other to Indicate Dipole Coupling Effect 28 15 Coupling between the n and n-5-2 Element in a Parallel Array 30 16 Coupling between Contrawound Antennas as a Function of Spacing in a Parallel Array 31 17 Coupling between Contrawound Antennas in a Parallel Array as a Function of Relative Rotation 32 Figure Number Page 18 Magnitude of Coupling Coefficient Measured between Antennas in a Conical Array as a Function of the Array Angle 33 19 Coupling as a Function of Rotation in a Conical Array 35 20 Coupling Between the n and n+2 Element in a Conical Array 36 21 Coupling Between Contrawound Antennas in a Conical Array as a Function of the Array Angle 37 22 Coupling Between Contrawound Antennas in a Conical Array as a Function of Relative Rotation 38 23 Mutual Impedance as a Function of Spacing in a Parallel Array , a - 60° 39 24 Mutual Impedance as a Function of Spacing in a Parallel Array, a - 73 40 25 Mutual Impedance as a Function of Spacing in a Parallel Array, a = 83° 41 26 Mutual Impedance as a Function of Rotation in a Parallel Array, a = 60° 42 27 Mutual Impedance as a Function of Rotation in a Parallel Array, a = 73° 43 28 Mutual Impedance as a Function of Rotation in a Parallel Array, a - 83° 44 29 Mutual Impedance Vs Array Angle in a Conical Array, a = 60° 47 30 Mutual Impedance Vs Array Angle in a Conical Array, a - 73° 48 31 Mutual Impedance Vs Array Angle in a Conical Array, a 83° 49 32 Mutual Impedance Vs Relative Rotation in a Conical Arrxv, a - 60° 50 33 Mutual Impedance Vs Relative Rotation in a Conical Array , a = 73° 51 34 Mutual Impedance Vs Relative Rotation in a Conical Arr^y, a 83° 52 Figure Number Page 35 Electric Field Patterns of Conical Log-Spiral Antenna With and Without a Parasitic Antenna Present in a Parallel Array, a = 60° 53 36 Electric Field Patterns of Conical Log-Spiral Antenna With and Without a Parasitic Antenna Present in a Parallel Array,, a - 73 54 37 Electric Field Patterns of Conical Antenna With and Without a Parasitic Antenna Present in a Parallel Array, a = 83 55 38 Electric Field Patterns of Conical Antenna With and Without a Parasitic Antenna Present in a Conical Array, a «s 60° 56 39 Electric Field Patterns of Conical Antenna With and Without a Parasitic Antenna Present in a Conical Array, a = 73° 57 40 Electric Field Patterns of Conical Antenna With and Without a Parasitic Antenna Present in a Conical Array, a = 83° - 58 41 Three Element Parallel Parasitic Array, Variable Spacing, § = cj> =s fy , a = 73 (linear array in cj) = plane! 59 42 Three Element Conical Parasitic Array, Variable Array Antle. 2 = $ , a = 73° 60 43 Radiation Patterns of Two Element Conical Parasitic Array as a Function of Rotation of Parasitic Element, a = 73°, <\> = 18° 61 44 Half Power Beamwidth of the Element Function cos 8 63 45 Calculated Array Pattern for Two Element Parallel In-phase Array, S = A/2, a - 73 64 46 Radiation Patterns of Two Element Parallel Array, Variable Spacing „ (j> = § } a = 73 (linear array in = (j> . a = 73 (linear array in <|> = plane) 67 49 Geometry of Two Element Conical Array 68 50 Radiation Pattern for Two Element Conical In-phase Array, S - \/2 (^ =- 34°), a = 73° 69 51 Radiation Patterns for Two Element Conical Array, Variable Array Angle, A = A, a = 73 (linear array in cb = plane) 71 52 Radiation Patterns for Two Element Conical Array, ' o Variable Orientation of One Element, S = \/2, a = 73 (linear array in cb s plane) 72 53 Radiation Patterns for Three Element Conical Array, Variable Array Angle, cb - cb - cb , a = 73 (linear array in (j> = plane) 73 54 Half -power Beamwidth of Two Element Parallel and Conical Arrays 74 1. INTRODUCTION There are many applications which require a circularly polarized field with narrower beamwidths and higher gains than can be secured with the indi- vidual planar and conical logarithmic spiral antennas which have been developed to date . Many of these applications have been filled by constructing arrays of helical antennas. For wide-band applications, it would appear that a con- siderable advantage could be obtained from the extremely wide bandwidth of the conical log-spiral antenna as an array element . Thus, although the array factor of conventional arrays would be a limiting factor, the element factor could be made essentially independent of frequency over a consider- able band of frequencies. In addition to bandwidth, the conical log-spiral antenna is unique in that the radiated fields are essentially circularly polarized over considerable angles off the axis of the antenna. There are various geometrical configurations that can be used when 2 arraying the conical antennas. Four of these are shown in Figure 1. The parallel array and the collinear array would have a frequency dependent array pattern; however, the conical array and coaxial arrays could theo- retically be designed to be independent of frequency. The geometry of the conical array is that used by DuHamel and Berry when arraying log- 3 periodic elements. This geometry limits the array to a few log-spiral elements; therefore, to obtain very narrow beamwidths from an array the con- ventional parallel configuration may offer the only solution. In addition to the use of arrays as primary radiators, consideration has recently been given to using an array of two contrawould conical log-spiral antennas as 4-6 the feed for large parabolic tracking antennas. The design of arrays of conical antennas requires a knowledge of the radiation coupling between antennas to determine the effect on the element pattern, and the mutual impedance between antennas to compute the element terminal impedance. Since a simple rotation of these antennas about their axes allows phasing of the array, a knowledge of the radiation coupling and mutual impedance as a function of rotation, as well as a function of element spacing, is desirable. The determination of the element to element separation requires a knowledge of the location of the element phase center. The purpose PARALLEL ARRAY CONICAL ARRAY COAXIAL ARRAY COLLINEAR ARRAY Figure L Possible Arrays of the Conical Log-Spiral Antenna of this paper is to supply this information for parallel and conical linear arrays over a restricted, although representative, range of antenna para- meters . Experimental information is also supplied on the radiation patterns, half-power beamwidth, and axial ratio of typical two and three element parallel and conical arrays . THEORETICAL CONSIDERATIONS 2ol Coupled Antennas Consider a two element array in which currents I and I flow as a result of voltages V and V applied across the feed terminals of the two 1 2 elements. As far as these terminal voltages and currents are concerned, the two antennas may be represented by the general four-terminal network in Figure 2a . In this network, Z is the mesh impedance of mesh 1 (less the genera- tor impedance) and is the impedance of antenna 1 with antenna 2 present and open circuited (not equal to the self -impedance of antenna 1 when isolated). Thus, we have V l = Z ll h + Z 12 h V 2 - Z 21 h + Z 22 *2 (1) where Z is the mutual impedance between the antennas. In the usual case where the medium is reciprocal, Z = Z . If V = 0, the input presented to the generator at reference plane A by antenna 1 with antenna 2 present S"--^ Rearranging; we can express this ratio as V l 2 I --Z u x l v;,i 2 ' x 2 (O Figure 2. Equivalent Network for Coupled Antennas The complex number k is defined to be the coefficient of coupling between 7 8 antennas one and two ' .. For identical antennas Z = Z and Z 12 - k Z u (5) If antenna 1 is driven and antenna 2 open circuited, I = 0, and it can be shown that k = / (6) I 1 Thus the measurement procedure is reduced to the determination of one im- 9 pedance and the complex ratio of two voltages . This method has advantages of accuracy and ease of measurement over the short circuit-open circuit method or the symmetric-antisymmetric method when measuring low values of mutual impedance o These advantages have been pointed out by Stratoti and Wilkinson who measured the coupling between short helices. The impedance of tfte antenna and its mutual impedance with the other antenna, referred to the feed terminals, are the basic quantities of interest However for the conical log-spiral, which radiates a broad lobe off the apex of the cone, it is not practical to attempt to measure the magnitude and phase of the voltage across the feed terminals which are situated at this apex. Thus it is necessary to measure the coupling and mutual impedance at some length of a transmission line and, taking into account line loss, trans- form these quantities down this line to the antenna terminals-. Consider the equivalent network in Figure 2(b) „ Although it is not convenient to establish and measure the open circuit voltage at reference plane A (apex terminals of antenna 2), it is still desirable to use the impedance-voltage-ratio method, hence we force I ', the current at the desired reference plane B to be zero. For I * = 0, and Z = Z , the terminal impedance Z and &> X X A A XX the mutual impedance Z can be expressed in terms of the measurable quan- tities Z' , Z ! and Z' at reference planes B and B . and the charac- teristic impedance and length of the transmission lines involved, by the following equations: JZ o tan pi a z Ir ig- -J z o tan p»i z o- j ( z ;i- 12 ? 22/ an pi jZ o Cot P i 2 L Z« -JZ q tan P i 1 o V jZ ll tan (^ 11 jZ Q (tan (3i 2 +Cot pi g ) ~ z ir Jz o tan ^1 V Jz ii tan f"l z ;r zf- -J z o tan p'i z o- j(z n- 5f- )ta " f"i (7) and 2 2 Z 'ifJ Z o tan P'l Z = Z - iZ Z Cot Si - z z — 12 11 J 11 o P 2 11 o Z -jZ' tan pi o 11 ^ 1 Z' 1 -jZ o tan P i i o Z -jZ tan pi o 11 1 jZ Q Cot pi 2 (8) If i = i_ and Z ' = Z ' only one impedance measurement is required. These equations are developed in the appendix- Since in practice the antennas will be fed by a transmission line of arbitrary length, ralations transforming the terminal impedances Z and Z away from the antenna terminals are also given in the appendix = The coupled energy measured at the antenna terminals may be only a partial indication of the interaction of antennas operated in an array. This is true in particular for traveling wave antennas. Recently Rupp has proposed that this measured characteristic be referred to as the "terminal coupling factor" to differentiate it from a "pattern modification factor". He aptly points out that since traveling wave antennas may couple to each other in a directional manner, a pair of such antennas may be considered to be a radiating directional coupler, Under certain conditions it would thus 8 be possible for more coupled energy to be radiated than transmitted to the terminals. For these reasons a study was made of the modification of the radiation pattern due to the presence of parasitic elements. This study will be the basis for a subsequent report, however representative radiation patterns indicating the general magnitude of pattern distortion are included in the present paper, 2,2 The Phase Center In the parallel array the effective spacing of the antennas is simply the separation between the axes of the cones . In the conical array the effective separation is the spacing in wavelengths between the phase cen- ters of the antennas. Thus a knowledge of the relative position of the phase center is required to calculate conical array patterns . In this section we shall outline a general application of the "backward-wave con- cept'" to the conical log-spiral antenna and in so doing develop a quasi- empirical formula for the location of the phase center of the radiated field, 12 Mayes, Beschamps, and Patton introduced the basic idea that the loga- rithmic periodic antenna could be considered to be a locally periodic struc- ture whose period varies slowly, increasing linearly with the distance to the point of excitation. It was pointed out that for normal operation there is a radiation region with a phase of excitation such that the structure will radiate toward the f eedpoint , Among such structures the backward wave zig- zag antenna is the counterpart of the log-periodic zig-zag, and the backward wave bifilar helix the periodic counterpart of the conical log-spiral antenna To apply this concept to the conical log-spiral antenna, it is instructive to consider some of the techniques which have been useful in the study of helices „ Consider the truncated portion^of 'the wire version of the conical log- spiral antenna shown in Figure 3, This structure is of the same basic geo- metry as the helix, and a local period of the structure can be defined in terms of a helix with parameters equivalent to the corresponding log-spiral parameters averaged over that period, The study of periodic structures such as the helix has been facilitated for a number of years by the use of Bril- louin or "k-p" diagrams to display the frequency variation of the propagation constant along the structure. Recently this approach has been applied to radiating structures } V, P c 27Ta cosa cosa X, ka Figure 3. Truncated Portion of Conical Log-Spiral Antenna 10 The solution of the determinantal equation for the real phase constants on the helix,, as a function of ka, requires that the fields go to zero ampli- tude as- the distance extends to infinity, As a result we require I (3 a I > ka (9) where (3 is the phase constant of the nth space harmonic on the structure, k is the phase constant of wave propagation in free space and "a" is the radius of the helix. Thus there are regions in the k-(3 diagram where the roots of the transcendental equation are real, corresponding to slow waves . Similarly this condition produces so-called "forbidden regions" in which solutions are complex, 15 The boundaries of these regions can be expressed by the inequalities *»» and ka . i^n I I I ka / . nn . cot s — cot 9 — cot 5 ka „ N cot % - 2 where for multi-wire helices in the higher modes m = -1 + nN (11) £ is the pitch angle, N is the number of wires or arms, and n is an integer. o For a two wire symmetrical helix with the arms excited with a 180 phase reversal, the k-(3 diagram takes the form of Figure 4. The parameters normally plotted for the helix are the pitch normalized to the wavelength in free space and to the wavelength of the surface wave, ©n the conical spiral the pitch distance on the surface is p", and ] ] ] ] "I J I ! o 12 p' M ka ka X "-" cotT ~ tan a (12) RL ~ (3a \ tan a s where a is the average radius of the period with pitch p", \ the wavelength in free space and \ the wavelength of the surface wave. These are the para- meters used in Figure 4 . 15 Sensiper has shown that a good approximation to the locus of phase constants on the helix for ka in the range of interest here, is a straight line drawn through the origin with slope 7^ = sin £ (13) and the boundaries of the m = 1 forbidden region. Such an approximation assumes propagation of energy with the speed of light along the arms . Al- though the determination of the propagation constant along the surface of the cylindrical helix applies to the infinite structure, it has been shown that the solution is useful for interpretation of the characteristics of the 16 17 finite monofilar endfire helix ' and the finite backward wave bifilar 18 helix with thin wire arms . On the conical spiral the ratio of the velocity of the slow wave to that in free space is given by ka Pi / 1 A >V - = j- = cos a (14) The radius "a" of the cylindrical helix is a constant, therefore as the frequency of operation is increased the parameter ka increases. Under the above approximation^ as ka is increased the propagation constant of the wave propagating along the surface of the structure away from the point of excitation increases and as it approaches that of the 1st backward space wave there will be strong coupling between these waves . When coupling L3 begins the propagation constant will depart from the straight line approxi mation. The slope of this curve at any point on the curve, is the ratio of the group velocity of the surface wave, v , to that in free space and hence when d(ka) ~£ = (15) d(|3a) v ' o propagation in the surface wave stops. This point corresponds to one end point of a stop band. For ka above this point the propagation constant is 19 complex . In the open structure this implies a transfer of energy to the backward space wave, and experimental evidence indicates that the effective phase center of the antenna is located in this region in normal backward wave radiation . As ka is increased further it is possible to force the beam to scan through the visible region of the spectrum. In the above discussion we have considered operating conditions with a change in ka . On a periodic structure such as the helix, with constant ra- dius and pitch angle, a change in ka implies a change in the frequency of operation. On the conical log-spiral antenna the spiral angle a is constant but the radius "a" is a linearly varying function of the distance from the apex along the conical surface. Thus a change of ka can result from either a change in position on the cone or from a change in frequency. Further, on the structure of infinite length any particular value of ka can be found for any frequency of operation. Thus, to consider the behavior of the struc- ture one of these two variables must be held fixed. Again, we will consider only those waves predicted by the straight line approximation above, since there is experimental evidence indicating that only the lowest order mode, consistent with the excitation and winding of the arms, contributes signifi- cantly to the radiation pattern of the two arm balanced conical antenna when ., .,20 operated such that there is negligible end effect If we assume operation at a fixed frequency, a wave propagating along the arms away from the apex implies an increasing ka . Hence at some point on the structure, the propagation constant of the surface wave associated with the flow of energy along the arms increases until there is coupling of energy to the backward space wave and rapid radiation. Experimental evidence 14 indicates that the position of the phase center, expressed in wavelengths to the apex, remains essentially fixed., Therefore since a/\ at the phase center remains constant, ka remains constant for all frequencies of opera- tion, excluding those frequencies where normal operation is distorted by the finite truncations of the antenna, i.e. end-effect. The operating point of the frequency independent antenna does not scan along the line ka = (3a cos a with a change in frequency; instead the parameters ka and (3a scale with frequency . In Figure 4 if we calculate an expression for the intersection of the asymptote ka = pa cos a (16) and the edge of the m = 1 forbidden region, we obtain the expression 1 + cos a (17) which is the circumference such that the antenna is phased to radiate back- ward* along the surface of the cone. However, a study of the symmetry of the conical antenna indicates that all radiation from that along the surface of the cone out to that at an angle of 9 from the surface will radiate in the o backward direction. Thus it can be shown that the average circumference, of the period such that the following relationship is satisfied is the approxi- mate region on the conical log-spiral structure that is phased for backward radiation . ! ln a ■ < ka < ^2-2 (18) 1 + cos a — — 1 + cos a cos o If the efficiency of excitation of the first backward space wave is high enough, it can be postulated in the light of the above discussion that the radiating region is' confined to approximately that circumference which satisfies Equation (18) . However, Sensiper has shown that the approximation due to the use of the asymptote and the region boundaries results in a ka For example see Figure 10, 15 that is slightly high since the coupling between these waves forces the curve down. Thus, this intersection point can be considered as only an upper bound for the start of the backward wave radiating region. The measured .phase centers of several antennas with narrow constant width o o o arms and with cone and spiral angles varying from 15 < 2 < 30 and 60 < a o < 83 are plotted in Figure 5. These phase centers occur at a ka below the intersection point and their location on the k-(3 diagram may be approximated by a straight line. The orientation of this line is interesting. The distance between the intersections of the asymptote for that particular antenna and this line, and the edge of the m = 1 forbidden region is directly related to the beamwidth of the antenna. As this distance decreases more of the active region is moved into the visible region of the spectrum and more energy is radiated at an angle o from the surface. Wire antennas wound with an a of 60 have a wide beamwidth (on the order of 110 to 120 ) while those -with a = 83 have patterns with beamwidths on the order of 70 degrees . The line also implies that at a spiral o angle a of 45 the antenna at the phase center (and hence we could expect the major portion of the active region) is phased to radiate at an angle from the axis of the antenna. In agreement with this, it has been determined that thin o wire antennas wound with an a of 45 have multiple sidelobes and a major portion of the radiation directed in an endfire direction. The data plotted in Figure 5 is for wire antennas . From pattern informa- tion, and a previous study of the near fields on the planar antenna, we might expect the slope of the approximate locus of the phase centers for antennas with wider exponentially expanding arms, to become more negative and thus more nearly parallel to the region boundary, with a minimum change for large a. If we calculate an expression for the intersection of the phase center line and the asymptote, taking into account the cone angle we get the o approximate expression 1 .2 sin a /-, ^n ka ~ — — — (19) 1.4 + cos a cos Equations (17) and (19) have been plotted in Figure 6. Families of curves, 16 17 70° 75° 80° SPIRAL ANGLE a 85< Figure 6. Approximate Circumference in Wavelengths at the Phase Cente] of Balanced Wire Arm Conical Log-Spiral Antennas (N=2, m=l) 18 one for each value of could be plotted, but for antennas that are good o o unidirectional radiators the cone angle varies as 10 < 2 9 < 45 and — o — therefore ,996 < cos < ,924, Thus cos « l a nd it is felt that the — o — o ' single curves are sufficient. For normal frequency independent operation, the circumference at the phase center should lie within this bounded region. Present information indicates that it will lie on or near the lower boundary, A recent study of the near fields on the conical log-spiral antenna by McClelland indicates good agreement between the measured characteristics of the near fields and those predicted i in terms of the applications 21 of the above concepts , 19 3, EXPERIMENTAL CONSIDERATIONS The conical log-spiral antenna with controlling parameters and an asso- ciated coordinate system is shown in Figure 7, For the present investigation a series of antennas with parameters 20 = 15 , D . 17 .5 cm, d - 4.5 cm, h = 47.3 cm and a = 60, 73, and 83 were used. The antennas were constructed from 1/4 inch copper tubing supported by 1/2 x 1 inch wood struts as shown in Figure 8. The feed cable RG-141/U was carried through one of the arms to the apex of the cone . All measurements were made at a frequency of 610 mc which, for these antennas was near the low end of the range of frequencies where the antennas could be expected to operate in a frequency independent manner. Thus the active region was at the lower portion of the antenna and, in the case of parallel antennas, the coupling could be expected to be the tightest. The approximate half-power beamwidth of the radiation patterns of these antennas is shown in Table I . TABLE I HALF-POWER BEAMWIDTH OF RADIATION PATTERN OF CONICAL ANTENNA 28 =15, 1/4 copper arms, D = 17.5 cm, d = 4.5 cm, f = 610 mc a = plane. 20 Figure 7. A Conical Antenna with Associated Coordinate System 21 22 The apparent phase centers of the individual antennas referred to in Figure 5 were determined by mounting the antenna in question on a vertical rotating mast with the axis of the antenna perpendicular to the mast and measuring the phase of the received signal from a distant transmitting antenna, The center of rotation of the antenna was adjusted for a minimum phase varia- o tion over a 120 sector in the direction of the radiated beam. Over most of this sector an apparent phase center on the axis of the antenna was well defined. For the three antennas in Figure 8, the distance from the phase center to the true apex of the cone is shown in Figure 9 for the range of parameters of interest, This distance in wavelengths is also plotted as a function of frequency, for one value of the spiral angle a, in Figure 10, The distance to the phase center in wavelengths remains essentially con- stant with a change in frequency of operation. The position of the phase center was initially measured for two ortho- gonally oriented incident waves (E and EJ , For these antennas, the apparent 8 y' phase center differed by considerably less than 1/10 wavelength for the two polarizations, Moreover, since the phase center for d. circularly polarized incident wave was measured to be the average of the two positions for ortho- gonal linearly polarized waves, measurements were confined to this condition, A schematic diagram of the experimental set up used to measure the coef- ficient of coupling and mutual impedance is shown in Figure 11, The voltage probes at plane B were loosely coupled to the transmission line and carefully balanced. The measurement of the input impedance, Z 1 } and the coefficient of coupling^ k', (equal to the complex ratio of V ' /V ' ) at reference plane B was by means of established methods » For a fixed arbitrary length of lossless transmission line Equation (8) will provide the terminal impedance Z , However for a systematic study of this impedance these equations involve a great deal of computation, If the transmission line can be any desired length, it is convenient to make it a multiple of \/2 , For i s I = n\/2 it can easily be shown that Z = Z' and Z 12 . Z ia . For the present study, the feed lines were constructed to be a multiple of a half-wavelength for all measurements except one, As a check on the transformation equations, Z ! was measured at the end of an arbitrary length of line and transformed to the apex terminals, These results were checked against measurements made with i = n\/2. 23 1.2 rpq "" " !l TTTT- -TTTTT-T I: 1 ! •::: :; j 1 "":|=:E| : ' ! 1.0 1 ~ B J , , ; , -i-i — ±::-±r::rfc — i »; ; ; ; i t~TT =illl OR ■J-H-L j i ;i" jpffi : . . . r T r " rra . . i // 1 1 i ; J . ' //< 0.6 0.4 xtth h H- . . . | : : 1 1 1 1 . . : 1 f-i-H r— H r _j _ i T —r— -J 1 , "Hi t : __ -q=j=F =: "-::: -i — \-\ — u: 77 M 1 ^A _ — UL 0.2 -ff} - X -H- H± .... J. ; . _. " W9m -ill:;::::: ItM- 1 60 70 80 = these "dipoles" were parallel and the total measured coupling is due to this dipole feed and the conical spiral arms. This is shown quite well in Figure 14, where contrawound antennas were pointed directly at each other and one rotated with respect to the other. Maximum coupling occurs when the dipole feeds are parallel and minimum coupling occurs when they are orthogonal, The truncated tip of these antennas was ,1\ at the frequency that all measurements were made,, and although the variation in coupling with rota- tion, and the general level of coupling as well, depends upon the size of this tip truncation, it was felt that these data are representative. The conical antenna will rarely be used in arrays at frequencies such that the tip truncation is greater than ,2\ and in most applications it will probably 26 -Q "D O a. O o -50 - s/x Figure 12. Magnitude of Coupling Coefficient Measured Between Parallel Antennas as a Function of Spacing Between Antennas in a Parallel Array 27 -20 - .a CD Q_ O O h4H a = 60 a = 83 a = 73 -70 I L 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 13. Coupling as a Function of Rotation in a Parallel Array 28 -10 a = 73° C0NTRAW0UND -20 — -30 -40 -50 — -60 -70 _J 1 . i 1 > 1 > 90 180 270 360 RELATIVE ORIENTATION OF ANTENNAS ■'igure 14. Coupling Between Contrawound Antennas Directed at Each Other to Indicate Dipole Coupling Effect 29 be less than .IX.. Thus the general level of coupling measured should be representative or even conservative . Although a rotation of these antennas to phase an array would cause as much as 10 db variation in the coupling;, the general level of coupling is so low that, as indicated later, the change in mutual impedance is small compared to the self impedance of the antennas . th Figure 15 indicates that the coupling from the n to the n + 1 and at least to the n ■;- 2 element must be considered, It is interesting to note o that there is almost a 90 shift between curves B and C, This shift varied o o o o from 90 for 83 antennas to only 20 for a = 60 antennas . The period of curve C shifts slightly along the horizontal axis with rotation of the center antenna^ indicating that there must be some coupling through or reradiation from the center antenna as well as direct coupling between the outside antennas Again the general level and variation with rotation is influenced by the "dipole feed" „ Terminal coupling between right and left hand wound antennas shown in Figure 16 is on the same order of magnitude as that between like antennas, with the coupling between the contrawound antennas actually a little higher for close spacing, The term contrawound is used in this paper to mean an antenna with a winding of the opposite sense to that of the antenna to which it is coupled o It is not used to mean one antenna with arms wound in both a right and left hand direction , The character of the curves is undoubtedly a function of the reflections at the site which return with the opposite sense of polarization, The dashed lines may be a more valid estimate of the true coupling. In Figure 17 the data measured as a function of rotation does not exhibit a well-defined character and again the variations are probably due in part to reflected energy, 4,1,2 Conical Arrays The coupling between conical log-spiral antennas in a conical array is considerably greater, as shown in Figure 18. Data was taken for a range of 4*, the array angle, For minimum 4 1 , approximately 17 , the struts of these antennas were just touching. In this position the separation angle between the surface of the cones was 2 degrees, Unlike the parallel case, the sur- faces of the cones are now closely coupled over the portion of the antenna that supports a strong surface wave. In addition the tip of the parasitic antenna is in strong fields radiated back toward the tip of the driven element 30 o - 10 -20 -30 -70 1 1 r a = 73 A I 2 3 B 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 15. Coupling Between the n* and n+2 Element in a Parallel Array 31 00 O Q_ o o 10 20 ■30 -40 50 -60 -70 i i i i I i i i r i — r C0NTRAW0UND A A -a = 60 I f f I l 1 I I I I I I I ! I ! I I_ ^5 1.0 1.5 2.0 S/X Figure 16. Coupling Between Contrawound Antennas as a Function of Spacing in a Parallel Array 32 u ■ , ■ | i C0NTRAW0UND I 1 i • - 10 — i. x , r 2 1 A ~ -20 — ( /x/ ) — .Q a= 60° _ "O 2 -30 — CL 3 -40 g^^ a = 83° S a = 73° O o - -50 — -60 — — -70 i l . 1 . i i 1 i 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 17. Coupling Between Contrawound Antennas in a Parallel Array as a Function of Relative Rotation 33 i — I — I — I — i — I — i — I — I — i — I — I — i — i — I — I — r -10 20 JQ e> 30 OL-40 => O o -50 ■60 70 I I I I i i I i i I i i L 30 60 90 120 150 180 i// IN DEGREES Figure 18. Magnitude of Coupling Coefficient Measured Between Antennas in a Conical Array as a Function of the Array Angle 34 Measured values of terminal coupling were approximately -18, -26, and -35 db for a of 60 , 73 , and 83 . As the radiation patterns indicate in a later section, in this case more of the coupled energy appears to be reradiated than reflected back to the feed. As the array angle was increased, coupling decreased to a minimum at o an angle of approximately 35 to 50 depending upon the spiral angle a. A further increase in 4* increased the coupling due to the fact that the antennas were tending to radiate more energy toward each other. In the i ° limit when ^ - 180 there is maximum coupling. This curve should be interpreted in conjunction with the radiation pat- terns which indicate that the array angle, 4^ will normally be limited to o around 60 or less to maintain reasonably low side lobes. Since S is a function of ty f the dual scales on several later figures (Figures 29, 30, 31) may be used as nomograms to convert to spacing in wavelengths . , Figure 19 indicates less variation with rotation than for the parallel configuration. The increase in variation with increasing a is due in part to the fact that these curves are taken with \/2 between phase centers, and hence the array angle (and of more significance, the separation angle between the surfaces of the cones) is smaller for the a = 83 antennas. Figure 20 indicates that in this array as well, the coupling to the second antenna in an array is on the same order, of magnitude as the coupling to the nearest neighbor. For small array angles the coupling between contrawound antennas in a conical array shown in Figures 21 and 22 is on the same general level as that for identical antennas. It however does not increase with an increase in the array angle since the two antennas are of opposite sense of polarization. 4.2 Mutual Impedance 4.2.1 .Parallel Arrays The real and reactive parts of Z and of the mutual impedance, Z , measured between parallel antennas are shown in Figure 23 through 28. As the coefficient, of coupling indicates, the mutual impedance is low, For an a equal to or greater than 73 the real and reactive parts of the mutual impedance between antennas with the same sense of winding are less than 1 ohm = For antennas with opposite sense of winding they are less than 2 ohms. 35 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 19. Coupling as a Function of Rotation in a Conical Array 36 O 20 -30 Qj -40 O O -50 -60 1 I | l l | I I | I i — | — I — i — [" i — r 70 I I i 1 i i 1 50ft J l I I I I l I I I I I I L 30 60 90 120 150 180 2^ IN DEGREES Figure 20. Coupling Between the n and n+2 Element in a Conical Array 37 t — r i i i i i i i i i i i i r CONTRAWOUND _ a = 60 -Q O 20 30 CL -40 => O O v^. 50 60 70 J I I I L 1 ■ . 1 ♦ IN DEGREES J I 30 60 90 120 150 180 Figure 21. Coupling Between Contrawound Antennas in a Conical Array as a Function of the Array Angle 38 - 10 -50 -60 -70 i ■ i ■ i ■ r C0NTRAW0UND \\t = 29.5 90 80 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 22. Coupling Between Contrawound Antennas in a Conical Array as a Function of Relative Rotation 39 200 Qfl50 100 t — i — i — | — i — i — i — i — | — i — i — i i f J I I L J 1 1 L I .... I -50 X 100 Figure 23. Mutual Impedance as a Function of Spacing in a Parallel Array, a = 60 (The subscript C indi- cates values measured between contrawound antennas) 40 200 Of 150 100 1 1 1 x nc-*== — 1 1 1 1 1 1 1 1 1 . 1 1 1 . 1 1 1 1 1 1 1 50 - --I00 10 8 6 4 2 -2 -4 -6 -8 -10 n 1 1 1 1 1 1 1 i r a = 73' I— s A R|2C-A J I I I I , I I . . ■ . I J I I I- .5 1.0 S/X 1.5 2.0 Figure 24. Mutual Impedance as a Function of Spacing in a Parallel Array, a = 73 41 200 of 150 i ' i i 1 i y A IIC\ R nr. ">., > r i [ i r i i | i i i i | _ 100 ..ill. i i 1 i i i i 1 i i i i 1 +50 x = -50 10 8 6 4 — i — i — i — i — 1 — i — i — i — i — i— a=83° i . . , | . . . . , A A 1 2 / R I2C X J^^^»V 11 x^^C"" *?' R i ' VX I2C ^Z^^ _== U -2 ''N.-S^ -4 - _ b -8 - -10 . ... 1 .... 1 . ... 1 .... 1 .5 1.0 1.5 S/X 2.0 Figure 25. Mutual Impedance as a Function of Spacing in Parallel Array, a = 83 42 10 1 1 ! 1 ' 1 HA — - X*"*N A A - 8 a =60° , f \ \ *r-Q - / \ A ^ _ / \ © 6 - / / \ \ x-€- X |2C 4 ~x / \ * \ / 7 r ^\^ R l2C 2 \ Vi- * ^-R 12 - // \ - -2 '^l, // \ \ \ \ \ - /\ *^r \ -4 - / X / \ — -6 - ft / / / V*I2 - -10 "ill 1 i 1 1 1 - 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 26. Mutual Impedance as a Function of Rotation in a Parallel Array, a = 60 43 200 i 'i I I i i i I = == — — — ^"LTlZTLTirir^-" Z- ._^c x nc — — — ^^— — — — V__ft - 150 100 ^— ■ ST" "lIC i i i i i < i i i i i i i i i i 50,= 100 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 27. Mutual Impedance as a Function of Rotation in a Parallel Array, a = 73 44 200 1 1 1 1 1 1 1 1 _ qT 150 ; ^ ^ x mc ^ ^ - <~R|ir - 100 . 1 i 1 i 1 . 1 i 1 t 1 i 1 i i ■50 10 8 6 4 2 -2 -4 -6 -8 -10 t — i — i — i — i — i — i — i — i — i — i — r a=83 A I2C I2C I . I i I . I . I ■ I ■ I ■ I 90 180 270 360 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 28. Mutual Impedance as a Function of Rotation in Parallel Array, a = 83 45 The real part of the self impedance of the antennas for these cases lies between 110 and 160 ohms. The subscript "c" on all graphs of measured im- pedance indicates measured values between contrawound antennas . 4.2.2 Conical Arrays As shown in Figures 29 through 34, the mutual impedance between ele- ments in a conical array is larger than it is in the parallel array. Figures 29 through 31 should be interpreted in the light of pattern considerations which will limit the normal range of the array angle ty to approximately 60 o o or less . In this range the mutual impedance is quite low for the 73 and 83 antennas . At spacings up to S = \/2 there is little difference in the general level of the mutual impedance between the same or oppositely sensed antennas . 4.3 Radiation Patterns 4.3.1 Parasitic Arrays The radiation patterns of the individual antennas are shown in Figures 35-40. Also shown is the distortion in the individual element pattern due to the presence of a parasitic antenna in parallel and conical arrays. For all patterns the feed line to the parasitic element was terminated in its characteristic impedance. There was, however, some mismatch between the line and the antenna. All patterns in this report are electric field pat- terns and those in Figures 35-40 were taken in the plane of the array. The axial ratio is recorded on the axis of the active element unless stated otherwise. The term parasitic array, used to identify the patterns, is used to mean one driven element with one or more parasitic elements present. For parallel antennas the principal charge is the introduction of some squint in the E pattern taken in the plane of the array and a broadening of the beamwidths in this plane at a spacing of one half wavelength. As the separation is increased this pattern modification diminishes until at one wavelength the patterns are very similar to the single element pattern. Pattern modification is less when the parasitic antenna is one with an opposite sense of polarization. As previously pointed out, when the antennas are placed in a conical array the separation between the surface of the elements is reduced and there is consequently considerably stronger coupling. For the smallest array angle used, ^ = 17 , (i.e., an angular separation between the conical surfaces of 2 ) pattern modification is quite severe for antennas with the same sense of polarization. Modification is much less where the parasitic element is of opposite sense although for this close spacing the patterns taken in the plane perpendicular to the array are slightly distorted. In addition the axial ratio on axis may rise sharply as indicated for a = 83 . Patterns taken in the plane perpendicular to the array were essentially unchanged for the same sense antennas . As the array angle is increased the pattern distortion rapidly diminshes and for a separation angle of only 10 the element pattern is modified only slightly by an oppositely sensed parasitic element. As indicated in Figures 41 and 42, the presence of a second parasitic element restores symmetry to the pattern. Although the terminal coupling varies considerably with rotation of the parasitic element, the effect of such rotation on the radiation pattern is usually small compared to the effect due to the physical presence of the element , Radiation patterns of a conical parasitic array as a function of rotation of the parasitic antenna are plotted in Figure 43, Most of the pattern modification in Figures 35-40 can be explained in terms of array theory, For example, a phase delay of less than 7T/2 in the excitation of one element of a two element array scans the beam of the array toward this unit. In Figure 39, the separation between the element phase centers is approximately \/4, for 4* = 17 . The radiation patterns taken in the plane of the array in this case are approximately those of a two element array with S = X./4 and a phase delay of 7T/4 to one element , This would in- clude that a significant portion of the energy is coupled to the parasitic element and reradiated with a phase delay. The excitation of an array with an equivalent phase advance to one element scans the beam in the opposite direction,, In the parallel parasitic array in Figure 36 the energy coupled to the parasitic antenna is delayed 180 by the separation. A further delay in reradiating the energy will appear as a phase advance to that element. The result is a squint away from the parasitic antenna. The fact that a rotation of the parasitic element has only a secondary effect would indicate that the net phase delay, excluding element separation, in the excitation of and reradiation from the parasitic element is approximately the same for any orientation „ 47 200 of 150 1 1 1 1 1 1 1 1 1 /" x n 100 ^ Rue 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 -50 x -100 50 40 30 20 10 -10 -20 -30 -40 -50 i i r a = 60 i i i i i Hsh- g^A >»« *~- I i I i !_i I l .2 .4 .6 .8 1.2 -I S/X .4 # I I I I 1 1 1 I I I I I I I 1 20 60 100 140 180 ♦ IN DEGREES Figure 29. Mutual Impedance Vs Array Angle in a Conical Array, a = 60 48 i i i i — i — i — i — i — r 200 Ctf 150 100 h ' nc IIC I ' I ■ I ■ I I I ■ Jl o x" -50 100 50 40 30 20 10 I o O -10 -20 -30 -40 i r i i i i i r # i . i . i . i . i . i . i & 2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 I ■ I ■ I ■ I 1 . I i I S/X 20 60 100 140 * IN DEGREES 80 Figure 30. Mutual Impedance Vs Array Angle in a Conical Array, a = 73° 200 Of 150 100 i 1 1 1 1 1 1 1 r V X||C IIC 49 + 50 -50 i i i . i . i 50 l ' i ' i ■ i ■ i ' i ■ i 1 1 ■ 1 - 40 - a = 83° s^A - 30 i^r R i2 - 20 - - 10 -10 **<« ^^r x i2 .^- C^ R I2C X 12C - -20 - -30 - -40 cr\ . i . i.i.i.i.i.i . i S/\ - /; 2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 Ltm- i.i.i.i.i.i ,1.1 60 100 140 y\f IN DEGREES 180 Figure 31. Mutual Impedance Vs Array Angle in a Conical Array, a = 83 50 200 of 150 100 I ' I ' I ■ I ■ I ' I ' I ' I ■X- MIC 1 I I I 1 I I I 1 I I I I I 1 -50 x = -100 20 16 1 ' 1 a = i | i | i | i = 60° Hsh _ / "\ s~>f R]Z 12 8 4 \ / " \ s / J\/ R I2C 1 \ ; / / * A- X I2C ' / v -4 -8 -12 J > ■V -\ \ - \ _ \ vv y / / /V-X-z / / / / / S=X/2 -b -/ \ / X^ R I2C -8 ^— • — -10 .1.1.1.1 . 1 i 1 1 1 1 1 90 180 270 360" RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 33. Mutual Impedance Vs Relative Rotation in a Conical Array, a = 73 52 200 1 1 1 1 1 1 1 1 __£" x li - IT 150 ~"^x llc — 100 ^ R " - ^- R i.c i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 + 50 -50 RELATIVE ORIENTATION OF PARASITIC ANTENNA Figure 34. Mutual Impedance Vs Relative Rotation in a o Conical Array, a = 83 a =60° ANTENNA r = 1.13 ^ r -**-^=-n y^ y^^ 3 /ff NA\ / i k / M \ / V Y \ / j) \ 1 ^^- \ yy \ a =60° ANTENNA r = 1.13 yy / "v\ / v \ \ i ^-£ 4^ yy \ WITH PARASITIC ANTENNA WITH C0NTRAW0UND PAR. ANT. r= 1.18 r = 1.14 S = ^iT^-T] =r= ^ =: >^ \\\ / ^ / A / n \ / V \ >v--v - J Cm^Z^ 1 r = l.ll r = l.ll r = Lll ^^^3 \\\ /If n/\ / \ (\ / i\ \ / \ v\ / ^\ \ 1 s ^. \ / // \ L^^-^ - 1 r =l.ll E FIELD PATTERNS IN PLANE OF ARRAY s = ^f y y >X\ A\f \/\ /\ K / "A / tJ \ %*. ys ^*^ i /7f \\ /'X \!/\ / I 1 \ ' J ) \ ^v^. __ \ 'fS^ \ \ / \f f A y\ I vi Jr \ 1 VOX /'/ \ 1 N^ \ /// \ S =5/8X r = 1.13 / X / r = 1.17 r = 1.19 X / / "X A \ Y \ /n \ / xX /fs \ 6-^S 1 r = 1.21 E FIELD PATTERNS IN PLANE OF ARRAY 2VI5' S = 3/4 X 1 s s*'' ■*n\ >v /\ // N \ X / Nv / VX XX /V \ s=x vJa/ 50JL r = 1.16 y^ r ^*" S / ^ s\ X /\ / / M /\ / is! \y \ / v\ /y \ N^Sy r = 1.16 E * Efl r= AXIAL RATIO ON AXIS Figure 36. Electric Field Patterns of Conical Log-Spiral Antenna With and q Without a Parasitic Antenna Present in a Parallel Array, a = 73 a=83 ANTENNA r = 1.17 /^ 1 y^~ ^T\ X^\ A/ a= 83° ANTENNA 55 r = l.l7 WITH PARASITIC ANTENNA WITH CONTRAWOUND PAR. ANT. r=l. r = l.l4 r = 1.12 r = 1.17 — ^>^ X f / /\ // \!/\ / \r 1 X^v /y 1 E FIELD PATTERNS IN PLANE OF ARRAY 200 = 15° fc) 50 r = 1.17 r = 1.13 r = 1.16 r= 1.20 sL r= AXIAL RATIO ON AXIS Figure 37. Electric Field Patterns of Conical Antenna With and Without a Parasitic Antenna Present in a Parallel Array, a = 83° a =60 ANTENNA r = l.l4 /* / /•K / \ I s / M /ft \ / A. / // \ / / 'Is * = 25° r= 1.67 r = l.05 r = l.08 r=l.09 E FIELD PATTERNS IN PLANE OF ARRAY 20 O =I5° * = 34' / fs \ y\ / X' / v\ /!) \ r=l.07 r = 1.07 r= AXIAL RATIO ON AXIS Figure 39. Electric Field Patterns of Conical Antenna With and Without a Parasitic Antenna Present in a Conical Array, a = 73 a =83 ANTENNA a =83° ANTENNA r = 1.15 s — / V" /X 7 v X \/ \ 1 ^^ 58 r = 1.15 WITH PARASITIC ANTENNA WITH CONTRAWOUND PAR. ANT r = 2.0 r = 1.18 r= 1.17 r = 1.2 E FIELD PATTERNS IN PLANE OF ARRAY * = I7° ^-""■" — ^1 "^^"■^^ *. N ^\ /" /? \ X >^ x // / \ 1 ' / \1 ' // \ / \j 1// \ 1 ^^ ~\ V /\ 1/ \ Jr \ r = 4.8 r = l.22 r = 1.18 r = 1.15 r = AXIAL RATIO ON AXIS Figure 40. Electric Field Patterns of Conical Antenna With and Without o a Parasitic Antenna Present in a Conical Array, a = 83 59 onvy ivixv S33U93CI Nl H1QIMWV39 d H en en O 01 -6- -6- -e- -e- 3- -6- -e- UJ UJ UJ UJ _ 1 1 ! 1 ■ o 1 1 1 1 ! ! - 5 _ Ij o - to v \ • \ \' \i - C^ - M Jr v ^f N ~~^^_l N 1 ■"— «-»^ v k "*^X O _ \ \ \ k. ^ \ \ \ \ \ \ 1 1 1 1 1 1 1 60 uo U CO < t> ouva ivixv S33y93Q Nl HlQIMlMV3a a H o •H II a 84 a • 0) !1J E rH rH G W < CD >> r;i 0I1VU 1VIXV S33U93CI Nl HiaiMWV39 d"H ° / en / If X •«- r l\ A 8 C/) UJ DC od I t- en • h 5 010 00 50 t- z cd UJ 2» UJ H <-3- _J Z UJ UJ If) 2 •HO to P CO h _l ■H t^ co UJ tfl < cd 1! cr !h a > cd d a, h \- ^ o~> < H +-> z CJ CD O ■H S H c a> It iH h O Cd § q: P O fl -H " CD -P -©. CD CO b Gj Ph H «H fl Cfi -H fl P H Cfi CD -P P O 5" Ph CD c -s -H >, <-t a >w Sh ^0 <; ro f- C II 0) S co 0) w • -P c S 0» H E a> tH 0; ■P O a aj od ft C H & C -H O -t-> a •^ aJ -P -P II a c •H CD -& -a -h rf ^ S CrJ O •H 67 ■P c HO pa n i> ouva nvixv S33d93a Nl H1QIMIMV38 £ "H 68 / ^ELEMENT PHASE 1/ CENTER ^ /3d sinif//2 sin0 E = cos n {6 + */2)e + cos n {8-\j//z) e -/3d sin^/2 sin0 Figure 49. Geometry of Two Element Conical Array 69 ~ s N^/ " -^/ / / N. N. / N. \ >Ml \ \ / / MfaC / x \ s^ / ^ S \ /X \ l\ \ \ / / fi\ / S\ s^ X ^ 1 / /It / v^ \^ ~T^L / 1 il / y\ \ \ Tr4(l\s s' X ^^"^ \ ^/^W^^o \I/Mrj^\\^^'^ V^^.. t"""' I 7 " -— jLy*^^ssj« E - CALCULATED EXPERIMENTAL 2C-l£-73-3lw CA 6 yAR. <£ = 0° Figure 50. Radiation Pattern for Two Element Conical In-phase Array, S = \/2 (4* = 34°), a - 73° 70 beamwidth of the measured E pattern is within one degree of the calculated 9 pattern and there is reasonable agreement between the patterns for 0+40 o degrees „ Beyond = 40 the measured patterns are somewhat narrower than calculated o The E array pattern is somewhat narrower than calculated; better agreement could be secured by using a slightly modified element function . Measured patterns for an in-phase conical array with variable spacing are shown in Figure 51, Data on a phased array of 2 elements is shown in Figure 52 and on a three element in-phase array in Figure 53, When interpreting the conical array patterns it should be noted that a change in the array angle ^, which brings about a change in the spacing between the phase centers of the elements is not equivalent to a change in the element spacing in a parallel array. In a parallel array the change in radiation characteristics with frequency can be simulated by maintaining the frequency constant and varying the element spacing,, provided the element pattern does not change with frequency. In a conical array this is not true since a change in 4* changes the element pattern with respect to the array geometry „ The element pattern can be approximated by a function such n 4* as cos (0 + — ) o Thus a change in S^ and hence 4J<, effectively changes this function as well as changing the element spacing. The array patterns show i ° a sharp decrease in beamwidth with increasing 4^ up to approximately 70 the maximum angle investigated^ and incidentally increasing S „ That the conical array geometry does preserve the beamwidth of the array with changing frequency of operation is shown in Figure 54, The conical array half -power beamwidth remains essentially constant (within the limits of the pattern range) over a two to one change in frequency, The half-power beamwidth of a linear array of the same elements decreased 40 percent. In addition a four element array^ a "conical quad-spiral array" 22 has been constructed to cover a 10 to 1 range of frequencies , This par- ticular array has a nominal directivity of 12 db over a circularly polarized isotropic source. 71 i i i o o CD CT> o o y- -9- -O- -e- -e- -©- CD o UJ UJ f- / / II / / II o r / II < // oiivh nvixv S33y93Q Nl H1QIM1AIV39 d H Sh CD cd C > aJ i-H - a, a o u u n < „ c >> o ca U u u +J a a tu u 6 a 0) ci> iH a w •H i— i o 5 HO ro !h r^ O M-l li 01 ri £ a> CM -p «* 4-> m n a, rH s -H- 72 ouva ivixv HIQIMWV3a d'H > ctf 03 - C >> -H Sj rH fH ^ U <0 ro rH O at u ij •H C d o o - CM ■P \ ci ^: 0) a ii iH CO W +> =S CI H 0) s ^ cu rH «h ca m -•^ \ . ' T , c--^ fc "«* \ - \ \ \ \ \ \ \ \ ] ) i o- i i i I \ \ \ \ \ Jj m ^ o II -©. -^^7^ \ o s a, \4 u? -e- w^- , y " ^ 'o " -~ -e-u, O " © i i -e- lu i I I 1 1 ft oi±va ivixv S33a933Q Nl H1QIMWV39 d 'H CO © CO S l> W 74 70 60 * = o 500 600 700 FREQ. IN Mc. 800 900 1000 Figure 54. Half -power Beamwidth of Two Element Parallel and Conical Arrays 75 5 > CONCLUSIONS A consideration of the conical log-spiral antenna as a locally periodic slow wave structure has led to somewhat better understanding of its charac- teristics and operation. Under this concept an approximate expression for the location of the phase center of the antenna has been obtained. An experimental investigation of the coupling between conical log- spiral antennas indicates (1) that coupling is low; on the order of -30 db or greater for element to element spacing of X/2 or more in a parallel array, (2) This coupling varies with rotation,, being a minimum for a 90 rotation between elements . (3) Coupling is on the order of 10 db greater in the o conical array . Minimum coupling for 15 cones occurs at an array angle of approximately 35 to 50 depending upon the spiral angle a, (4) Changes in the basic element pattern caused by the presence of other elements are minor for an element to element spacing of at least one-half wavelength, and consist mainly of a broadening of the element pattern beamwidth in the plane of the array . (5) A good approximation to the array pattern of small arrays may be gained from the use of the isolated element pattern function. 76 REFERENCES 1. Dyson, J, B., "A Survey of the Very Wide Band and Frequency Independent Antennas - 1945 to the Present", Journal of Research of the N.B.S, Vol, 66D, No, 1, pp, 1-6, Jan, -Feb, 1962, 2. Dyson, J, D., "The Conical Log-Spiral Antenna in Simple Arrays", Abstracts of the 11th Annual Symposium on the USAF Antenna Res, and Dev , Program, sponsored by Aeronautical Systems Div., Air Force Systems Command, Oct, 16, 1961, 3. DuHamel, R, H. and Berry, D„ G,, "Logarithmically Periodic Antenna Arrays," IRE WES CON Conv , Record , Part I, 1958, pp , 161-177, 4. Webster, R, and Lyles, J, F,, "Application of Frequency Independent Feeds to Automatic Tracking Antennas," Abstracts of the 11th Annual Symposium on the USAF Antenna Res. and Dev, Program, sponsored by Aeronautical Systems Div,, Air Force Systems Command, October 16, 1961. 5. Bales, B, W„ and Voklenburg, E., "Antenna Feed Has Ten-to-One Band- width," Electronics , Vol. 34, No, 48, pp. 56-58, Dec. 1, 1961. 6. Bresler, A, D., Jasik, H. and Kampensky, A,, "A Wide Band Conical Scan Antenna Feed System," The Microwave Journal , Vol, 4, pp. 97-101, Dec. 1961. 7. King, R. W, P,, "Coupled Antennas and Transmission Lines," Proc, IRE , Vol, 31, November, 1943, pp , 626-640, 8. Blasi, E. A,, "The Theory and Application of the Radiation Mutual- Coupling Factor," Proc. IRE , Vol. 42, July, 1954, pp . 1179-1183, 9. King, R. W, P,, "Theory of Linear Antennas," Harvard University Press, Cambridge, Mass,, 1956, p. 349. 10, Stratoti, A. R, and Wilkinson, E, J., "An Investigation of the Complex Mutual Impedance between Short Helical Array Elements," Trans. IRE , Vol. AP-7, July, 1959, pp . 279-280. 11, Rupp, W. E,, "Coupled Energy as a Controlling Factor in the Radiation Patterns oi Broadside Arrays," Abstracts of the 11th Annual Symp , on the USAF Antenna Res, and Dev, Program, Sponsored by Aeronautical Systems Div,, AFSC, Oct, 1961 „ 12, Mayes, P, E., Des champs, G, A,, and Patton, W, T., "Backward-Wave Radiation from Periodic Structures and Applications to the Design of Frequency-Independent Antennas," Proc. IRE , Vol. 49, No, 5, pp. 962- 963, Mav 1961, 77 13 o Oliner, A, A,, and Hessel, A,, "Guided Waves on Sinusoidally-Modulated Reactance Surfaces/' IRE Trans ., Vol. AP-7, Dec. 1959, pp . 8201-8208, 14. Ishimaru, A., and Tuan, Ho, "Frequency Scanning Atnennas", 1961 I RE International Conv . Record , Part I, pp, 101-109 „ 15. Sensiper, So, "Electromagnetic Wave Propagation on Helical Conductors," Tech, Report No, 194, Res. Lab. of Elect,, MIT, May 16, 1951, 16. Marston, A, E„, and Adcock, M. D,, "Radiation from Helices," Report 3634, Naval Research Laboratory, March 8, 1950. 17. Maclean, T. S. M., and Kouyoumjian, R. G., "The Bandwidth of Helical Antennas," IRE Trans . , Vol, AP-7, Dec, 1959, pp , S379-S386. 18. Patton, W. T., University of Illinois, unpublished work. 19. Mayes, P. E., "'Coupled Mode Analysis of Stop-Band Characteristics of Modulated Reactance Surfaces," Internal TM 62-2, Antenna Lab., University of Illinois, February 1962 (Unpublished), 20. Dyson, J. D. and Mayes, P. E,, "New Circularly Polarized Frequency Independent Antennas with Conical Beam or Omnidirectional Patterns," IRE Trans ., Vol . AP-9, July 1961, pp. 334-352, 21. McClelland, 0. L'„, "An Investigation of the Near Fields on the Conical Equiangular Spiral Antenna," M.S. Thesis, Dept „ of Electrical Engineering, University of Illinois, 1962. Also TR No. 55, Antenna Laboratory, Uni- versity of Illinois, Urbana, Illinois, May 1962, 22. Dyson, J, D., "An Antenna to Cover the 220 through 2400 Mc Telemetering Bands," Proceedings 1962 National Telemetering Conference, Vol. I, May 1962. 78 APPENDIX A 1. To determine Z and Z from Z' and Z 8 : Given measured quantities z ii and t L± at some reference plane B on the transmission line. We see from Figure 2a that for IJ.O (A.l) Z A 2 OC " "J Z o COt ^2 (A.2) Z Al 0C = Z ll " Z 22 + Z (A, 3) Z Al 0C + J Z o tan P'l Z ll =Z Bi OC- Z o Z Q + j Z ;an £>£ ± (A, 4) and Z' = Z' 12 11 (?) (A, 5) where Z , Z } and Z n are the impedances to the right at planes A > A and B with I' = 0. z \-,> v ' and v -! are measured at reference planes B and B , From (A „4) Z ii " J Z o tan •»! (A, 6) j Z^ tan (3i x 79 From (A, 3) Z ll Z 22 + Z ll Z A 2 OC " Z 22 Z Al 0C " \oC \ Now consider the equivalent circuit of Figure 2b which includes i and i For V ' = \ = Z B lSC = z n-4- and A 2 SC also From (A .10) (A. 9) Z 12 (Z 22 " Z 12 + Z A SC ) z a sr = (z n ~ z i9 } + 7 TJ (A ° 10) A 1 SC U 12 Z 22 + Z A SC Z' (Z' - Z ? ) Z BSC ■ ^ - Z l2>^ " Z- 9 ^ W.ll) 1 2, 1 1 A Substitution of Equation (A»2) and (A .6) leads to Equation (8) 2, To determine Z ' , Z I ^ and Z no from tne terminal quantities and feed \Z 2i2t i Z line lengths: From Figure 2 and V = Z I + Z I 1 11 1 12 2 V - Z I + Z I 2 21 1 22 2 V s = Z' I 1 + Z' I» 1 11 1 12 2 V ' = Z ' I ' + Z ? I ! 2 21 1 22 2 (A. 16) (A. 17) For Z no = Z on and I» = 12 21 2 Z A OC = " j Z o COt ^2 (A " 18) 2 Now with V • = With I« = *J V C U Z 22 + Z A 2 OC Z A 00 + J Z o tan f"l A 1 SC » Z 22 + Z A 2 S0 «...-« 1 Z A SO + J Z o tan ^1 HI (A, 19) = Z (A, 20) 11 BW o Z o + j Z^ tan (^ V Z' 2 ^4 = Z_ „„ = Z*. - -i2_ (A. 21) i; Bl sc ii z 22 Z A 2 SC = J Z o tan P' 2 W L Weeks >> 5 J une 1959- "Log Periodic Dipole Arrays," Technical Report No, 39 , Do E. Isbell, 1 June 1959, AD 220651 "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory/' Techni cal Report Nov 40, S Dasgupta and Y. T, Lo, 17 July 1959. "The Radiation Pattern of a Dipole on a Finite Dielectric Sheet/' Technical Report No. 41;, K G. Balmain. 1 August 1959. "The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem in a Waveguide/' Te c ; hn ic a 1 Re port No, 42, Raj Mittra, 1 October 1959. "impedance Properties of Complementary Multiterminal Planar Structures," Technical Report No 43, G, A. Deschamps, 11 November 1959, ,8 0n the Synthesis of Strip Sources/ 1 Technical Report No, 44, Raj Mittra, 4 December 1959 ""Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides," Technical Report No. 45 JL C. H. Tang and Y, T, Lo, 11 March 1960, "New Circularly Polarized Frequency Independent Antennas with Conical Beam or Omnidirectional Patterns/" Tech nical Report No. 46^ J. D. Dyson and P, E. Mayes, 20 June 1960 AD 241321 "Logarithmically Periodic Resonant-V Arrays," T echnical Repor t No. 47, P. E. Mayes ; and R L Carrel, 15 July 1960 AD 246302 "A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Technical Report No 48 , Y To Lo s June 1960 "Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems," Technical Report No 49 , R„ Ho MacPhie, 25 January 1961 "Synthesis Of Antenna Product Patterns Obtained from a Single Array," Technical Report No 50 „ R H, MacPhie, 25 January 1961 , "On the Solution of a Class of Dual Integral Equations," Technical Report No. 51 , Ro Mittra, 1 October 1961 . AD 264557 "Analysis and Design of the Log-Periodic Dipole Antenna," Technical Report No, 52 , Robert L. Carrel, 1 October 1961 »* AD 264558 "A Study of the Non-Uniform Convergence of the Inverse of a Doubly-Infinite Matrix Associated with a Boundary Value Problem in a Waveguide," Technical Report No* 53 , Ro Mittra, 1 October 1961. AD 264556 * Copies available for a three-week loan period , ** Copies no longer available . 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