'^^/ ; 
 
 LIB R.AFLY 
 
 OF THE 
 
 U N IVER.SITY 
 
 OF ILLI NOIS 
 
 621.365 
 
 I^G55te 
 
 no. 40-49 
 
 cop. 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/newcircularlypol46dyso 
 
ANTENNA LABORATORY 
 
 Technical Report No. 46 
 
 NEW CIRCULARLY POLARIZED FREQUENCY 
 INDEPENDENT ANTENNAS WITH CONICAL BEAM 
 OR OMNIDIRECTIONAL PATTERNS 
 
 by 
 
 John D. Dyson and Paul E. Mayes 
 20 June 1960 
 
 Contract AF33(616)-6079 
 Project No. 9-(13-6278) Task 40572 
 
 Sponsored by: 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
 This report was presented, in condensed form, as a paper at the 
 URSI-IRE Spring Meeting, Washington, D.C., 3 May 1960, under the 
 title, "The Log-Spiral Omnidirectional Circularly Polarized 
 Antenna." 
 
i i 3 \pZ> 
 
 - ENGINEERING LIBRARY 
 
 
 ACKNOWLEDGMENT 
 The authors are pleased to acknowledge the assistance of Professor 
 G. A. Deschamps in formulating the theory of excitation of multi-arm 
 antennas. Discussions with W. T. Patton were helpful. 0. L. McClelland 
 supervised the measurements program. 
 
ABSTRACT 
 
 A conical beam may be obtained from balanced equiangular spiral 
 antennas by constructing an antenna with more than two spiral arms and 
 symmetrically connecting these arms to provide a suppression of the 
 radiated fields on the axis of the antenna. The angle of this conical 
 beam can be controlled and with proper choice of parameters it can be 
 confined to the immediate vicinity of the azimuthal (0 = 90 ) plane. 
 
 An antenna with four symmetrically spaced arms can provide a 
 radiation pattern that is within 3 db of omnidirectional circularly 
 polarized coverage. The standing wave ratio of this antenna referred 
 to a 50 ohm coaxial cable is less than 2 to 1 over the pattern bandwidth, 
 
 This four-arm version retains the wide frequency bandwidths of the 
 basic conical log-spiral antenna, and it provides a coverage which here- 
 tofore has been difficult to obtain even with narrow band antennas. 
 
CONTENTS 
 
 Page 
 
 1. Introduction 1 
 
 2. The Conical Log-Spiral Antenna 3 
 
 2.1 The Basic Structure 3 
 
 2.2 The Radiation Pattern Beamwidth 5 
 
 3. The Conical Beam Antenna 7 
 
 3.1 The Principle 7 
 
 3.2 Radiation Patterns 9 
 
 3.3 Pattern Beamwidth 17 
 
 3.4 The Input Impedance 17 
 
 3.5 Operating Bandwidth as a Function of Antenna Size 17 
 
 4. A Non-Frequency Independent Version 20 
 
 5. Conclusions 24 
 Appendix 25 
 
ILLUSTRATIONS 
 
 Figure 
 
 Number Page 
 
 1. A conical log-spiral antenna with associated coordinate system 4 
 
 2. Variation in electric field pattern of typical balanced 2 arm conical 
 equiangular spiral antenna 6 
 
 3. Possible feeding arrangements for multi-arm structures 8 
 
 4. "infinite balun" feed used on a four arm conical beam antenna 10 
 
 5. Typical electric field patterns and orientation of the conical beam 
 
 as a function of the rate of spiral 11 
 
 6. Antenna C - 15 - 9 etched from copper clad teflon impregnated 
 fiberglass 13 
 
 7. Electric field patterns of a balanced symmetrical 4 arm conical 
 equiangular spiral antenna 14 
 
 8. Electric field patterns of a balanced symmetrical 4 arm conical 
 equiangular spiral antenna 15 
 
 9. Azimuthal coverage of the radiation patterns in Figures 7 & 8 16 
 
 10. VSWR of typical 4 arm conical equiangular spiral 18 
 
 11. Projection of equiangular spiral and Archimedes spiral curves on 
 
 a conical surface 21 
 
 12. Electric field patterns of symmetrical 4 arm conical antennas 23 
 
 1A. Terminal region of a structure having N-fold rotational symmetry 26 
 
 2A. Instataneous electric vectors at 9 = and 6=9 for three values 
 
 of m ° 30 
 
1. INTRODUCTION 
 
 The balanced planar and conical equiangular spiral antennas have been 
 demonstrated to have essentially frequency independent radiation and impedance 
 
 characteristics over bandwidths which are at the discretion of the design 
 
 12 * 
 
 engineer ' . These antennas, based upon the equiangular or log-spiral curve, 
 
 have the property that the highest and lowest usable frequencies are independent 
 
 The highest usable frequency is determined by the diameter of the truncated 
 
 region at the origin, which must remain small in terms of the operating 
 
 wavelength, and the lowest usable frequency by the arm length and hence 
 
 the maximum diameter of the antenna. 
 
 The two-arm planar antenna provides circularly polarized, single lobe, 
 bidirectional radiation on the axis of the antenna. An orthogonal projection 
 of the two-arm planar antenna on a conical surface forms an antenna which, 
 over a suitable range of parameters, confines the radiation to a single 
 lobe directed off the apex of the cone. 
 
 It is possible to devise a multitude of frequency independent structures 
 by using the log-spiral arm as a basic element. Many different excitations 
 may be used when several log-spiral arms with a common origin are placed 
 on a cone or plane. Some configurations and excitations produce radiation 
 patterns which are distinctly different from those obtained heretofore. 
 
 * A convenient abbreviation for logarithmic spiral, a synonym for equiangular 
 spiral. 
 
 1. J. D. Dyson, "The Equiangular Spiral Antenna," IRE Trans, on Antennas and 
 Propagation, vol. AP-7, pp. 181-187, April, 1959. Also Technical Report 
 No. 21, University of Illinois Antenna Laboratory, September 15, 1957. 
 
 2. J. D. Dyson, "The Unidirectional Equiangular Spiral Antenna," IRE Trans.. 
 
 on Antennas and Propagation, Vol. AP-7, October 1959* Also Technical Report 
 No. 33, University of Illinois, Antenna Laboratory, July 10, 1958. 
 
It is the purpose of this paper to introduce a simple theory relating 
 the excitation and the radiation fields (insofar as now possible), and to 
 present data showing the performance of the four-arm conical log-spiral 
 antenna with one particular excitation. 
 
2. THE CONICAL LOG-SPIRAL ANTENNA 
 2.1 The Basic Structure 
 
 The conical log-spiral antenna with its associated coordinate system 
 is shown in Figure 1. On a plane surface, the edges of one arm of a 
 
 logarithmic spiral antenna may be defined by 
 
 '■ ' i aff 
 
 P = P e 
 
 and 
 
 / a a(<P - $) 
 
 Since Tan a = 
 
 dPV.d0> . a 
 
 / / ( — - — ) <P 
 P l =P e tan a 
 
 The orthogonal projection of P and P_ on the surface of revolution 
 
 1 ^ 
 
 7T - 9 = G is defined by sin 9 
 
 tan a ' r 
 
 sin f 
 a " d -• (r -2) (<P-h 
 
 where • -sin 6 
 
 2_ 6 . / 
 
 tan a 
 K = e 
 
 P is the radius vector from the origin to the truncation of the spiral 
 at the apex region, and P and p are the radius vectors to the inner and 
 outer edges of the exponentially expanding arm at a given angle <P. The 
 angle ° is a constant and of such value that if the curve traced out by 
 P is rotated about the axis through the angle 6 it will coincide with the 
 curve traced out by p . The angle a, a constant, is the angle between the 
 radius vector and a tangent to the log-spiral curve at the point of inter- 
 section. 
 
Figur* 1 A conical log-spiral antenna with 
 associated coordinate system 
 
5 
 
 The second arm of the balanced structure is defined by rotating curves 
 1 and 2 through 7T radians. Hence the defining parameters are: the included 
 cone angle, 26 ; the arm width determined by °, or K; the rate of spiral, a, 
 the base diameter, D; and the apex diameter, d. 
 2.2 The Radiation Pattern Beamwidth 
 
 The beam width of the two-arm conical antenna can be controlled over 
 
 a limited range by a suitable choice in the rate of spiral. Figure 2 
 
 shows typical electric field radiation patterns for balanced two arm 
 
 .o „„o , ,_o 
 
 ant 
 
 ennas constructed with an a of 73 60 and 45 . Typical half-power 
 
 beamwidths range from 60-70° for a of 82 , 70-80 for a of 73°, and 160-180° 
 
 o o 
 
 for an a of 60 . As the angle a is decreased to 45 the beamwidth increases 
 
 o 
 to 180-200 . Pattern cuts through the axis and perpendicular to the axis 
 
 of the antenna [Figure 2(c)] indicate that this latter case provides essentially 
 
 circularly polarized coverage in one hemisphere and omnidirectional coverage 
 
 on the 6 = 90 plane. 
 
 It has been pointed out that a modified version of the balanced conical 
 
 antenna is obtained when the width of the arm is made constant rather than 
 
 2 
 
 tapered . This form of the antenna is readily constructed of wire or cable. 
 
 However, it is only an approximation to the true equiangular spiral structure. 
 The approximation is good for relatively tightly spiraled antennas, i.e., a 
 greater than 60 , and results in only minor pattern changes. As the angle 
 a is decreased to the neighborhood of 45 to 50 marked pattern changes occur 
 for the wire approximation, including a multilobing of the main beam and 
 large radiation off the base of the cone. Thus to realize the patterns such 
 as shown in Figure 2(c) the decrease in the rate of spiral must be accompanied 
 
 by an increase in the arm width, i.e. increase in o (or K) . The patterns shown 
 
 o 
 in Figure 2(c) were for an antenna constructed with the parameters; 29 = 20 , 
 
 o 
 
 a = 45° K = .75 ( 6 = 94°). 
 
<f> =0°, 9 VAR 
 
 (1=73' 
 
 <£VAR, e=90 < 
 
 ^.=60* 
 
 b. 
 
 «=45 ( 
 
 Figure 2 Variation in electric field pattern 
 of typical balanced 2 arm conical 
 equiangular spiral antenna 
 
 E^, Eg polarization. 
 
 9 = 10 . 
 o 
 
3. THE CONICAL BEAM ANTENNA 
 
 3.1 The Principle 
 
 When using multiple-arm structures the number of choices of feeding 
 
 3 
 systems increases . There are basic excitations of multiterminal antennas 
 
 which are simply related to the azimuthal variations of fields of the 
 
 - -jirfp 
 form e J associated with solutions of Maxwell's equations. The parameter 
 
 m must be an integer to make the field single-valued. 
 
 Excitations of the spiral arms which correspond to each of these 
 
 radiation "modes" are readily apparent. The customary excitation of the 
 
 two-arm spirals, as shown in Figure 3(a), corresponds to m = + 1 and would 
 be expected to produce a field which varies primarily as e . For antennas 
 which are not large compared to the wavelength the lower order terms will 
 be predominant. 
 
 With four-arm structures the number of basic excitations increases. 
 An excitation corresponding to m = 1 is shown in Figure 3(b) . Hence in 
 order to obtain the e fields with a four-arm spiral it is necessary to 
 excite the two pairs of arms with a 90 degree phase shift between them. 
 
 With the four-arm structures it is possible to produce fields which 
 correspond to higher values of m . For example, the excitation shown in 
 
 Figure 3(c) corresponds to m = +2 and should produce fields which vary 
 
 -i2fP 
 primarily as e . This concept may be generalized to any number of arms, 
 
 N, and a discussion of the more general case is given in the Appendix. 
 
 3. G. A. Deschamps, "impedance Properties of Complementary Multiterminal 
 
 Planar Structures," Trans. IRE, Special Supplement, Vol. AP-7, Dec, 1959, 
 p. S371. Also Technical Report No. 43, University of Illinois, Antenna 
 Laboratory, Nov. 11, 1959. 
 
 * First pointed out in Quarterly Report No. 5, Contract AF33(616)-6079, 
 Antenna Laboratory, University of Illinois, 31 December 1959, pp. 9-11. 
 
+ CT7 (a) 
 
 
 
 +j 
 
 -I + IC=3 (b) 
 
 "J 
 
 fl 
 
 D 
 
 
 fl 
 
 Figure 3 Possible feeding arrangements 
 for multi-arm structures 
 
9 
 
 Examination of the solutions of Maxwell's equations in spherical 
 coordinates shows that values of m different from unity are always accompanied 
 by a null in the 9 - function (associated Legendre polynomials) along the 
 polar or 8 = axis. Therefore we expect an excitation which corresponds 
 to any in / + 1 to produce a conical beam. The excitation of four arms 
 corresponding to m = 2 as shown in Figure 3(c) , is the simplest case. 
 
 This lowest order conical beam excitation is readily achieved by 
 connecting opposite arms together and feeding one pair against the other, 
 i.e. 180 degrees out of phase. It is apparent also from the symmetry of 
 the input currents in this case that there will be zero field along the 
 antenna axis. The antenna can be fed by a balanced feed line, or a coaxial 
 line and balun, placed on the axis of symmetry. It may also be fed by 
 
 carrying the feed cable along one of the arms as outlined in the previous 
 
 2 
 paper m Details of this latter method are shown in Figure 4. The balance 
 
 and symmetry of the feed is important if symmetrical patterns are desired. 
 
 3.2 Radiation Patterns 
 
 Figure 5 shows typical radiation patterns of symmetrical four arm 
 
 antennas fed in the manner shown in Figure 3(c). As indicated, the rate 
 
 of spiral, (the parameter a), which was the primary factor in controlling 
 
 the beamwidth of the balanced two arm antenna, determines the orientation 
 
 of the conical beam of the balanced four arm antenna. Conical antennas may 
 
 be constructed to provide a conical beam with any angle of orientation 
 
 from around 40 to more than 90 off the axis of the antenna. The case 
 
 where the beam maximum is located at 9 = 90 is of particular interest 
 
 since it fills a need for a simple, very broad band, circularly polarized, 
 
 omnidirectional source. 
 
10 
 
 \ 
 
 \ 
 
 TOP 
 
 
 \ 
 
 V 
 
 SIDE 
 
 Figure 4 'infinite balun" feed used 
 
 on a four arm conical 
 
 beam antenna 
 
<*=73° 
 
 oC = 60° 
 
 11 
 
 Q^=45° 
 
 uj 
 
 UJ 
 
 o 
 
 CD 
 
 40 r 
 
 50 - 
 
 60 
 
 70 - 
 
 - 80 
 
 90 
 
 /' 
 
 •J- 
 
 40 50 60 70 
 
 o< IN DEGREES 
 
 80 90 
 
 Figure 5 Typical electric field patterns and 
 orientation of the conical beam as a 
 
 function of the rate of spiral (7.5 ^ < 10 ) 
 
 • ■ 
 
 . 
 
Figure 6 shows a typical four arm balanced equiangular spiral antenna 
 constructed on a 15 cone. This antenna was etched from a flexible, copper- 
 clad, teflon-impregnated, fiberglass material and then formed into a cone. 
 The feed cable is rg 141/U. The energized cable is carried along one arm; 
 dummy cables are placed on the other arms to maintain structural symmetry. 
 To obtain the desired bandwidth, the arms on this particular antenna were 
 later extended to a cone base diameter of 31 centimeters. Radiation 
 patterns of this antenna are shown in Figures 7 and 8 from 550 mc where the 
 base is .57 wavelengths in diameter up to 4000 mc where the diameter of 
 truncated apex is approximately 0.2 wavelengths. The patterns are for E~ 
 and E.~ polarized fields. The first two columns are pattern cuts through 
 the axis of the antenna and the third column is for a cut perpendicular to 
 the axis, on the U = 90 plane. 
 
 The azimuthal coverage shown in these patterns may be examined in greater 
 detail in Figure 9 where the total deviation in decibels from omnidirectional 
 coverage is plotted for the orthogonal polarizations. The axial ratio 
 on the 6 = 90 plane varies somewhat with the angle i P. It is shown for 
 one particular angle, which is a representative angle of orientation and 
 not an optimum case. Over a considerable bandwidth the total amplitude 
 deviation is less than 3 db and the axial ratio is 3 db or less. 
 
 Two pattern characteristics should be noted. For large a, the conical 
 beam patterns are smooth and well formed and, if desired, the arms may be 
 approximated by wire or cable. As the angle a decreases beyond 60° the 
 
 on patterns exhibit minor irregularities and are not as symmetrical. 
 In ;i ■« loosely spiraled antennas require wider exponentially 
 
 expanding arms. Th< ••-.<■ ' h. istics correspond to those noted for the 
 
13 
 
 Figure 6 Antenna C - 15 - 9 etched from copper clad 
 teflon impregnated fiberglass 
 a = 45° D ■ 20.5 cm 
 
14 
 
 8°, OVcuu. 
 F=550Mc 
 
 700 Mc 
 
 CJ>-90° e vcuo. 
 
 (j)Vojo Q = 9Q < 
 
 900 Mc 
 
 
 N \ 
 
 
 S \ 
 
 / 
 r 
 
 
 ^ 
 
 
 
 
 1200 Mc 
 
 Figure 7 Electric field patterns of a 
 balanced symmetrical 4 arm conical 
 ilangular spiral antenna. 
 
 .", a 1 .", K = .925, D = 31 cm, d = 1.5 cm 
 
0s 0° e "z^t/. 
 1600 Mc 
 
 ^=90° e vcuu. 
 
 (£ VoMj. ^ Q r 9QO 15 
 
 3000 Mc 
 
 
 ^ 
 
 "h 
 
 'y.c. .^'^^s-:^ 
 
 V-< ' -v 
 
 7 ^?r 
 
 4000 Mc 
 
 
 y 
 
 
 s; 
 
 
 ^r=— 
 
 
 
 \ 
 
 n 
 
 I : 
 
 \ ' 
 
 / 
 I / 
 
 -^~ 
 
 e = 
 
 Figure 8 Electric field patterns of a 
 
 balanced symmetrical 4 arm conical 
 
 equiangular spiral antenna. 
 
 7.5°, a = 45°, K = .925, D = 31 cm, d = 1. 5 cm 
 
a 
 
 o 
 
 •H 
 +-> 
 
 cS 
 
 •H 
 
 •o 
 
 <D 
 
 X! 00 
 -P 
 
 eg 
 «H 
 
 O «> 
 
 ID (0 
 bD <D 
 
 Q) U> 
 > 'H 
 
 O Ek 
 
 o 
 c 
 
 •H 
 (0 
 
 c 
 
 Fh 
 
 0) 
 
 -p 
 +-> 
 a 
 a 
 
 X! 
 ■p 
 
 S 
 •H 
 N 
 
 a) 
 h 
 
 •H 
 
 o08l = 4> 
 
 o06=e 
 
 OllVti 1VIXV 
 
 qp Nl NfcGlJMd 
 
 Nonviavd o06=e jo 
 
 N0I1VIA3Q 3anilldkNV 
 
17 
 two arm axial beam antennas. 
 
 3.3 Pattern Beamwidth 
 
 The beamwidth in a <P = constant plane is relatively insensitive to a 
 change in antenna parameters. Antennas constructed with both 15 and 20 
 
 included cone angles (2« ) with 45 < a < 73 and with cable arms or with 
 
 o - - 
 
 exponentially expanding arms had half-power beamwidths ranging from 35 to 55 
 degrees, with an average value of 45 degrees. 
 
 3.4 The Input Impedance 
 
 The input impedance of the four arm antenna, fed in the manner of 
 
 Figure 3(c), rapidly converges to a characteristic value. Antennas constructed 
 
 o o 
 with 15 or 20 included cone angles with RG 8/U arms or exponentially 
 
 expanding arms fed with RG 141/U, typically have an input impedance of 
 
 from 45 to 55 ohms for a ranging from 45 to 60 degrees. As a is increased 
 
 to 73 degrees the impedance rises to the neighborhood of 70 ohms. These 
 
 values are approximately one half those noted for similar two arm antennas. 
 
 The input voltage standing wave ratio of the antenna referred to in 
 
 Figures 7 and 8 is plotted in Figure 10. Note that it is less than 1.5 
 
 to 1 referred to 50 ohms over most of the usable pattern bandwidth. 
 
 3.5 Operating Bandwidth as a Function of Antenna Size 
 
 The usable antenna bandwidth is fundamentally determined by the diameter 
 of the truncated apex and the antenna arm length. As with the two arm 
 antennas, the radiation patterns tend to deteriorate as the apex region 
 approaches 1/4 wavelength in diameter. It was previously noted that the 
 balanced two arm antenna constructed on a 15 or 20 degree (total apex angle) 
 cone, with an a of 73 could be operated to a frequency such that the cone 
 base diameter is on the order of 1/3 wavelength. As the rate of spiral 
 
 
ed 
 U 
 
 •H 
 C 
 
 o 
 o 
 
 E 
 u 
 
 a 
 •<# u 
 
 •H 
 
 h a 
 
 O 
 
 •H U 
 
 a eS 
 
 >> rH 
 
 tuD 
 
 <H C 
 
 rt 
 
 •H 
 
 ^ cr 
 to <D 
 > 
 
 s-t 
 
 bO 
 
 
 « 
 
 e 
 
 o 
 
 ro 
 
 
 CD 
 
 in 
 
 <* 
 
 ll 
 
 U~ OS 01 Q3dd3J3d tiMSA 
 
19 
 
 is decreased, i.e. a decreased to 45 there is not sufficient radiation 
 surface on this size cone to dissipate the energy without back radiation 
 and hence the size of the cone must be increased to the order of .6 
 wavelength at the lowest frequency of operation. The four arm structures 
 exhibit very similar characteristics and hence omnidirectional coverage on 
 the = 90 plane requires an antenna whose base diameter is on the order 
 of .6 to 2/3 wavelength at the lowest operating frequency. 
 
20 
 
 4. A NON-FREQUENCY INDEPENDENT VERSION 
 
 Thus far we have considered only antennas constructed from the 
 equiangular spiral curve. These antennas are frequency independent 
 
 in the sense that, within the limits imposed by the physical size, the 
 
 4 
 scaling principle is fulfilled . The pattern characteristics of these 
 
 log-spiral antennas (such as the beamwidth of the two arm antennas and the 
 
 angle of orientation of the conical beam of the four arm antennas) are 
 
 constant for a change in the frequency of operation. These characteristics 
 
 are directly related to the constant parameter a, which indicates the rate 
 
 of spiraling of the arms. 
 
 It is possible, as shown in Figure 11, to construct conical antennas 
 
 from other curves, such as the Archimedes spiral. Although these antennas 
 
 may be operated over wide frequency bands, they are not frequency independent 
 
 since the parameter a at any point on the curve is directly related to the 
 
 angle <P at that point. As the frequency of operation is changed the active 
 
 aperture of the antenna is composed of a structure with a changing rate 
 
 of spiral. This shows up as a definite widening of the beamwidth of the 
 
 two arm conical Archimedes spiral antenna as the operating frequency is 
 
 increased. There is also a variation in the angle of orientation of the 
 
 conical beam o! the four arm conical Archimedes spiral antenna with a 
 
 change in frequency. 
 
 A. V. H. Kurmey, Frequency Independent Antennas," 1957, IRE National 
 
 "1, P'. 1, pp. 114-118. Also Technical Report No. 20, 
 University '-f tlllnoie, Antenna Laboratory, October 25, 1957. 
 
p =ke 
 
 Q(j> 
 
 21 
 
 a = arctan tt 
 
 (a) 
 
 p = ka<£ 
 
 << = arctan <£ 
 
 (b) 
 
 Figure 11 Projection of equiangular spiral 
 
 and Archimedes spiral curves on 
 
 a conical surface. 
 
22 
 
 Radiation patterns for a four arm conical Archimedes spiral antenna 
 
 are shown in Figure 12(a) . This antenna was constructed to provide a range 
 
 o o 
 
 of a from approximately 45 at the apex region to 85 at the base. As 
 
 indicated in Figure 12 the complete range of beam orientation from 
 
 o o 
 approximately 45 to 90 off axis is swept out as the frequency is varied 
 
 from 1000 to 2000 mc . For comparison, patterns for an equiangular spiral 
 
 antenna wound on the same cone are shown in Figure 12(b) . 
 
ARCHIMEDES SPIRAL 
 
 P =1.026* 
 
 f = 1000 Mc 
 E * 
 
 B 
 
 EQUIANGULAR SPIRAL 
 
 ( 
 
 />= e 
 
 sin 10° 
 tan45° 
 
 )4> 
 
 23 
 
 f = 1400 Mc 
 
 f = 2000 Mc 
 
 Figure 12 Electric field patterns of 
 symmetrical 4 arm conical 
 antennas 
 
 © = 10°, D = 29.5 cm, d = 4.5 cm 
 
 o ' ' 
 
 (<P = 90°, Q var pattern) 
 
24 
 5. CONCLUSIONS 
 
 It is possible to obtain a conical beam from the balanced equiangular 
 spiral antenna by constructing the antenna with more than two arms and 
 connecting these arms to provide a higher order <P - variation which is 
 accompanied by an axial null. The angle of this conical beam can be 
 
 controlled; in particular, it can be placed in the immediate 
 
 o 
 
 vicinity of the 6 = 90 plane to provide an omnidirectional pattern. 
 
 An antenna with four symmetrically spaced arms can provide a radiation 
 pattern that is within 3 db of omnidirectional coverage. The standing 
 wave ratio of this antenna, referred to a 50 ohm coaxial cable, is usually 
 less than 1.5 to 1 over the pattern bandwidth. 
 
 This form of log-spiral retains the extremely wide frequency band- 
 width and circular polarization properties of the basic conical log- 
 spiral antenna and it provides a coverage which heretofore has been difficult 
 to obtain even with narrow band antennas. 
 
 Since the Archimedian spiral approximates a log-spiral with changing 
 parameters, this non-frequency-independent version of the antenna may be 
 constructed such that the conical beam may be frequency scanned in the 
 polar angle. 
 
25 
 
 APPENDIX 
 
 Consider a radiating system composed of a number of identical conductors 
 
 on the surface of a cone. Assume that the structure has N-fold rotational 
 
 2ir 
 symmetry, i.e., a rotation about the cone axis through the angle will 
 
 leave the structure unchanged. Figures 3(b) and 3(c) of the text show 
 
 examples of four-fold symmetry. 
 
 The excitation of the conductors will be accomplished at the apex of 
 
 the cone where the dimensions are small compared to the wavelength. There 
 
 will be N terminals available in the excitation region, symmetrically 
 
 spaced about a circle of small radius. Denoting the input current at the 
 
 n terminal by I , the excitation can be described by the current vector 
 
 I = ( V V v v w 
 
 Any index is defined modulo N. To satisfy the requirement of conservation 
 of current, we note that 
 
 N 
 
 £ 1=0 (1A) 
 
 T n 
 
 n = 1 
 
 This viewpoint that each possible excitation is represented by a 
 vector, I leads to the examination of the possible basis for the vectors 
 of this space. Because of the n-fold symmetry the choice of "symmetrical 
 components" as base vectors proves more convenient than others. 
 
 Rotation of the excitation by one step 
 
 : a = ( v v v V (2A) 
 
26 
 
 ® 
 
 / 
 
 © 
 
 © 
 
 © 
 
 © 
 
 Figure 1A Terminal region of a 
 structure having N - fold 
 rotational symmetry 
 
27 
 
 2 IT 
 
 would simply produce a field which is rotated in space by the angle " 
 Hence the transformation of excitation 
 
 I 1 as PI (3A) 
 
 where P is th*~ n x n permutation matrix 
 
 1 ... 
 
 1 ... 
 
 P= 0001. ..00 (4A) 
 1 
 
 1 
 
 produces a simple change in the field. The matrix P is a special case 
 
 . 27Tk x 
 of circulant matrices. The eigenvalues of P are exp. ( — ) and the 
 
 eigenvectors are 
 
 27TR . 27Tk(N-2) . 27Tk(N-l) 
 
 -1/2. J N~ J N J N . 
 
 A k = N (1, e , , e , e ) 
 
 (5A) 
 
 From these eigenvectors we can obtain the basis of our vector space. 
 The eigenvectors of P for the case N = 4 are 
 A 1 = 1/2 (1, j, -1, -j) 
 
 A 2 = 1/2 (1, -1 ; 1, -1) 
 A 3 = 1/2 (1, -j- -1, j) 
 
 (6A) 
 
 A 4 = 1/2 (1, 1, 1, 1) 
 
 Note that A A and A satisfy the condition of Equation (1A) whereas 
 A does not. The former three vectors provide an orthonormal basis which 
 spans the vector space of all possible excitations when N = 4. 
 
28 
 
 A . A * = 1/4 (1 + 1 + 1 + 1) = 1 
 
 A 2 ' A 2 = 1/4 (1 + 1 + 1 + 1) = 1 
 
 A 3 . A * = 1/4 (1 + 1 + 1 + 1) = 1 
 
 A . A * = 1/4 (1 - j - 1 + j) « (7A) 
 
 A . A * = 1/4 (1 - 1 + 1 - 1) = 
 
 A 2 . A 3 * = 1/4 (1 - j - 1 + j) = 
 
 Let us now examine the properties of these basic excitations. 
 Excitation I = A produces field in which a rotation by 77/2 is equivalent 
 to a 7T/2 change in phase. Solutions of Maxwell's equations in spherical 
 coordinates can be written with azimuthal variation of the form e (m integer) 
 Of these solutions only those with m = 4k + 1 (k = 0, + 1, + 2, +3...) 
 will satisfy the above relationship between rotation and change in phase. 
 Hence the excitation I = A will produce fields which can be expressed 
 in the following way 
 
 F, = L a (P 
 
 k 4k + 1 (8A) 
 
 where <Pm = g (r 9) e J 
 
 in ^ 
 
 nfP 
 
 Similarly, I = A yields m = 4 k + 2 and I = A yields m = 4 k + 3, 
 ^ 3 
 
 and these each will produce fields 
 
 F = E a <P 
 P k 4k + p (9A) 
 
 A Bimil.ir ftrgtUMnl Cftl i <«d through for arbitrary N. 
 
 Ari satisfying (1A) can be expressed as a linear combination 
 
 A ] " A 2' A 3 '' Produced thereby as a linear combination of 
 
 F ji y y T, " 15i '><>>> de«( ribed by E will produce the fields 
 
29 
 
 F = E (A * E*) F (10A) 
 
 J J 
 
 If the coefficients a above were known, the fields produced by any 
 excitation would be completely determined. Unf ortunately, this problem 
 has not yet been solved for log-spiral elements. 
 
 Some useful observations can be made for the log-spiral however, 
 by interpreting the above results in terms of experimental data. 
 Consider the case N = 2 which has been extensively investigated. The 
 eigenvectors for the permutation matrix in this case are 
 
 \ " \ (1 > ^ 
 
 A 2 = 1(1,D 
 
 (11A) 
 
 and only A satisfies Equation (1A) . This corresponds to the excitation 
 of the two-arm spiral sketched in Figure 3(a) . The field F for the case 
 N = 2 would be of the form 
 
 F = £ a (P 
 Ik T 2k + 1 (12A) 
 
 and the possible ^-variations are 
 
 e j(2k + 1) 9 
 It has been observed that the radiation produced by 2-arm log-spiral 
 antennas is very nearly circularly-polarized over the major portion of 
 the beam. Circular polarization with non-zero field on the axis requires 
 that E^ and Eg vary as cos(<»>t + <f) and sin (<*>t + <P) respectively, the m = 1 
 case. Functions with higher values of m would contribute to the off-axis 
 fields. Figure 2A shows a sketch of how the circularly polarized fields 
 
= 0, 
 
 m=2 
 
 m = 3 
 
 7.h Instantaneous electric vectors at 
 ■ and ■ for three values of m. 
 
31 
 would appear instantaneously around the polar axis for a few of the small 
 values of m. The values m = 1 and 3 would predominate for the two-arm 
 case. Note that these add alternately in and out of phase in orthogonal 
 cross-sections. Hence the rotational symmetry of the beam should provide 
 an indication of the relative magnitude of these two fields, neglecting all 
 others. A tightly- wound spiral produces more nearly rotationally-symmetric 
 beams, hence predominantly the m = 1 case. 
 
 The excitation of Figure 3(c) used to produce the conical beams is one 
 of the eigenvectors of (6A), namely A . This excitation produces fields 
 
 F = £ a, <P 
 
 2 k 4k + 2 
 
 and the experimental results indicate that, for certain parameter choices, 
 the lowest order terms of this series predominate 
 
 F ~ a <P n + a <P „ 
 2 ~ o 2 -1-2 
 
 These two cases seem equally likely on the basis of the order of the 
 
 functions involved. However these two cases correspond to circular 
 
 polarized fields of opposite sense. One sense of polarization is favored 
 
 over the other by the direction of the spiral winding. Hence it is reasonable 
 
 that one of the coefficients a . a be small, a conjecture which is again 
 
 o' -1 
 
 confirmed experimentally.