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L I B R.AR.Y 
 
 OF THE 
 
 UNIVERSITY 
 
 Of ILLINOIS 
 
 6Z1.365 
 I!655te 
 no. 50-57 
 cop. 2 
 
 Liiiijn<Li,mci 
 
Antenna Laboratory 
 Technical Report No. 57 
 
 POLYGONAL SPIRAL ANTENNAS 
 
 by 
 
 C. H. Tang 
 
 and 
 
 0. L. McClelland 
 
 Contract AF33(657)-8460 
 Project No. 6278, Task No. 40572 
 
 June 1962 
 
 Sponsored by: 
 AERONAUTICAL SYSTEMS DIVISION 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
ABSTRACT 
 
 VO. 57 
 
 By relaxing the similarity cnnditii.ii on bhe angular structures, a class 
 )f modified angular structures when used as antennas are shown to have fre- 
 juency independent performances. The class of log-periodic antennas belongs 
 :o the modified conical structure which is a special case of the general 
 ingular structure. The general angular structure can be described by log- 
 spiral curves which lead to the design of log-spiral antennas. In this 
 report, modifications are made on the general angular structures with the 
 riew of obtaining a larger class of frequency independent antenna design, 
 rhe measurements taken on some of the structures indeed give the desirable 
 results. 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/polygonalspirala57tang 
 
ACKNOWLEDGEMENT 
 
 The authors wish to thank Professor J. D. Dyson for reading the 
 manuscript . 
 
CONTENTS 
 
 Page 
 
 1. Introduction 1 
 
 2 , The Design of Frequency Independent Antennas 3 
 
 2d Similar Figures and the Electrodynamical Similitude 3 
 
 2.2 Self Similar Figures 5 
 
 2.3 The Convergence Problem of the Angular Structure as a 
 Frequency Independent Antenna and the Modified Angular 
 Structures 6 
 
 3. Polygonal Spiral Antennas 8 
 
 3.1 A Definition of Spirals 8 
 
 3.2 Polygonal Spirals 9 
 3 .3 Zigzag Structures 11 
 
 3.4 The Scaling Parameter 14 
 
 4. Experimental Method and Measurement Results 19 
 
 4.1 Experimental Method 19 
 
 4.2 Results of Measurement — The Rectangular Spiral Antennas 20 
 
 4.2ol Pattern Comparisons and the Comparison of the 
 
 Axial Ratio 20 
 
 4,2.2 The Standing Wave Ratio and the Input Impedance 35 
 
 4.3 Results of Measurements — The Biangular Spiral Antenna 46 
 
 4.3.1 Radiation Patterns 46 
 
 4.3.2 The Standing Wave Ratio and the Input Impedance 46 
 
 5. Conclusions 56 
 Bibliography 57 
 
ILLUSTRATIONS 
 
 Figure Number Page 
 
 1 A Radial Transformation 3 
 
 2 Self-Similar Figures 5 
 
 3 A Rectilinear Spiral 8 
 
 4 A Log-Spiral Curve and a Polygonal Spiral Structure 10 
 
 5 A Triangular Spiral 12 
 
 6 A Rectangular Spiral 13 
 
 7 A Pentagonal Spiral 13 
 
 8 A Log-Periodic Zigzag Structure 14 
 
 9 A Conical Rectangular Spiral and a Conical 
 
 Log-Spiral 15 
 
 10 The Scaling Parameter of the Rectangular Spiral 17 
 
 11 The Scaling Parameter ot Polygonal Spirals for 
 
 m = 2 } 3 3 4 } 5^ 6o 18 
 
 12 A Conical Rectangular Spiral Antenna 21 
 
 13 A Conical Log-Spiral Antenna 22 
 
 14 A Log-Spiral Zigzag Antenna 23 
 
 15 Pattern Comparison (9 = 7 .5 } a - 73 ) 25 
 
 16 Pattern Comparison (9 = 10 } a = 73 ) 26 
 
 17 Pattern Comparison (9 = 7„5° a = 60°) 27 
 
 o ' 
 
 18 Pattern Comparison (9 = 10° a = 60°) 28 
 
 o ' 
 
 19 Pattern Comparison (9 = 7,5°^ a = 45°) 29 
 
 20 Pattern Comparison (9 = 10°, a = 45°) 30 
 
 21 Comparison of the Axial Ratio - As a Function 
 
 of the Frequency 31 
 
 22 Comparison of the Axial Ratio - As a Function 
 
 of the Frequency 32 
 
Figure Number Page 
 
 23 Comparison of the Axial Ratio - As a Function of 
 
 the Frequency 33 
 
 24 Comparison of the Axial Ratio - As a Function of 
 
 the Frequency 34 
 
 25 Comparison oi the Axial Ratio - As a Function of 
 
 the Elevation Angle 9 36 
 
 26 Comparison of the Axial Ratio - As a Function of 
 
 the Elevation Angle 9 37 
 
 27 Comparison of the Axial Ratio - As a Function of 
 
 the Elevation Angle 9 38 
 
 28 Comparison of the Axial Ratio - As a Function of 
 
 the Elevation Angle 9 39 
 
 29 Comparison of the Standing Wave Ratio 40 
 
 30 Comparison of the Standing Wave Ratio 41 
 
 31 Comparison of the Standing Wave Ratio 42 
 
 32 Comparison of the Standing Wave Ratio 43 
 
 33 Comparison of the Standing Wave Ratio 44 
 
 34 Comparison of the Standing Wave Ratio 45 
 
 35 Comparison of the Input Impedance (9 - 7,5 . 
 
 a = 73°) ° 47 
 
 36 Comparison of the Input Impedance (9 = 10 , 
 
 a = 73°) ° 48 
 
 37 Comparison of the Input Impedance (9 = 7,5 y 
 
 a = 60°) ° 49 
 
 38 Comparison of the Input Impedance (9 = 10 } 
 
 a = 60°) ° 50 
 
 39 Comparison of the Input Impedance (9 = 7,5 } 
 
 a = 45°) ° 51 
 
 40 Comparison of the Input Impedance (9 = 10 3 
 
 a = 45°) ° 52 
 
Figure Number Page 
 
 41 Radiation Patterns of the Log-Periodic Zigzag 
 
 Antenna (6 - 7 .5°, a = 73°) 53 
 
 o ' 
 
 42 The Standing Wave Ratio of the Log-Periodic 
 
 Zigzag Antenna (0 = 7 »5 , a = 73 ) 54 
 
 43 The Input Impedance of the Log-Periodic Zigzag 
 
 Antenna (9 = 7.5°, a = 73 ) 55 
 
INTRODUCTION 
 
 The design of broadband antennas is one of the major tasks for an 
 
 antenna designer* Its theory has been advanced remarkably by the introduction 
 
 1< 
 
 2 
 
 of the angle concept by Rumsey and the principle of logarithmic periodicity 
 
 bv DuHamel 
 
 The angle concept is a consequence deduced from the theory of similtude 
 and sealing,, and it leads to the investigation of angular structures. The 
 idea of making antennas to possess electrical properties which will repeat 
 periodically with the logarithm of the frequency was arrived at in considering 
 the convergence problem of the finite angular structures. Since all angular 
 structures are necessarily of infinite extent, the convergence problem of 
 the electrical properties of the finite structure toward those of the infinite 
 structure must be overcome , Experiments have shown that this can be accom- 
 plished by introducing certain periodic discontinuities in some finite angu- 
 lar structures , These ideas have since stimulated a large amount of activities 
 
 in the antenna field . 
 
 3 4 
 
 Dyson's log-spiral antennas and then Isbell's log-periodic dipole arrays 
 
 were pioneer works in bringing two different types of frequency independent 
 antennas into practice. The term "frequency independent" shall be used to 
 indicate that antennas of this class can be designed in such a way that they 
 will perform with practically constant characteristics — input impedances 
 and radiation patterns, for any desired bandwidth, anywhere in the practical 
 frequency spectrum* It is now well known that log-spiral antennas have a 
 circularly polarized, frequency independent, radiation field while log- 
 periodic dipole antennas radiate linearly polarized, frequency independent, 
 far field patterns, and their impedance characteristics both remain practically 
 constant within the designed bandwidth. Following these discoveries, effort 
 has been directed toward practical applications of these two types of fre- 
 quency independent antennas . The multi-arm log-spiral antennas were investi- 
 
 5 fi 
 
 gated by Mayes and Dyson ' . They demonstrated that an omnidirectional pat- 
 tern could be obtained with log-spiral antennas by using symmetrically spaced 
 
 7 
 multiple arms. Recently Dyson has considered the coupling effect between 
 
 log-spiral antennas arranged in various arrays. Various designs and appli- 
 
 8 9 
 cations of log-periodic antennas have been investigated by Mayes , Carrell 
 
DuHamel et al. ' } } A bibliography of articles on wideband and fre- 
 
 14 
 quency independent antennas has been presented by Dyson. 
 
 The theoretical analysis of these frequency independent antennas has also 
 been pursued by many authors. Mast has solved the static problem of the log- 
 spiral structure . Rumsey, in solving the sheath model of the log-spiral 
 structure, has successfully transformed the exterior vector problem into scalar 
 
 problems by decomposing the field into right-hand and left-hand circularly 
 
 17 18 
 polarized components . Copeland and Lee assumed the current distribution 
 
 along the antenna arm in analyzing the radiation field of non-planar spiral 
 
 9 
 antennas. Carrel analyzed the log-periodic dipole structure based on circuit 
 
 theory „ Generally speaking, work still remains to be done to solve the 
 
 theoretical problem of above mentioned frequency independent antennas. 
 
 The lack of the complete theoretical solution on the problem of fre- 
 quency independent antennas does not prevent an antenna engineer from making 
 progress in the practical designs. Those ideas which were impetus to the 
 birth of the frequency independent antennas can still be used as guiding princi* 
 pies for the design and the experimental method serves as a practical way of 
 verification „ Through an accumulation of workable structures, one may gain 
 a better insight to the frequency independent operation. These outlooks were 
 motivations for the present investigation on the frequency independent an- 
 tennas . 
 
 It is the purpose of this report to point out that there is a group of 
 approximate log-spiral antennas which are also capable of frequency inde- 
 pendent performances, in the above defined sense. The basic consideration 
 which leads to this approximation is the relation of the continual similarity 
 condition and the log-spiral configuration is thus approximated by a set 
 of connected and proportioned segments. They are called polygonal spiral 
 antennas . Several polygonal spiral antennas have been constructed and their 
 characteristics investigated. Experimental results indicate that their 
 behaviors are very similar to that of the corresponding log- spiral antennas. 
 
 Both polygonal spiral antennas and other log- periodic rectilinear struc- 
 tures have only discrete similarity in contrast to the continual similarity for 
 the log-spiral structures „ 
 
2, THE DESIGN OF FREQUENCY INDEPENDENT ANTENNAS 
 
 We shall try to relate some basic concepts and guiding principles 
 useful in making the design of frequency independent antennas so as to 
 give a better insight of the theory. 
 2,1 Similar Figures and the Electrodynamical Similitude 
 
 In the plane geometry, similar polygons are defined as figures having 
 all corresponding angles equal and all corresponding line segments propor- 
 tional. Extending this notion to more general entities, similar surfaces 
 are surfaces which can be made to correspond point to point in such a way 
 that the distance between any two points on one surface is always the same 
 multiple of the distance between two corresponding points on the other. 
 The property of being similar is called similarity. 
 
 The general transformation of similarity is the product of the general 
 Euclidean transformation composed possibly of a translation and a rotation, 
 and a radial transformation. A two dimensional radial transformation is a 
 transformation expressed by the equations 
 
 T y 
 
 (i) 
 
 Figure 1. A Radial Transformation 
 
For an example,, P and P in Figure 1 are related by the radial trans- 
 formation, The real number t is the ratio of similitude and the origin 
 is called the center of similitude . 
 
 Analytically,, the general transformation of similarity can therefore 
 be expressed by the equations 
 
 x = t(x cos G - y sin + a) 
 
 y' = t(x sin + y cos 4- b) 
 
 (2) 
 
 for two-dimensional figures . The similarity transformation of figures in the 
 space can also be expressed analytically. 
 
 With the above definition of similar figures, one can develop a theory 
 of linear modeling for electromagnnetic problems, usually referred to as 
 electrodynamical similitude. To be specific, the term "similar electro- 
 magnetic boundary value problems" will be used to imply the relation between 
 two antenna problems in which the electrical properties of the modeled 
 antenna are the same as that of the original except for a change in scale. 
 
 Through dimensional analysis of Maxwell's equations, it is readily 
 
 19 
 
 shown that the similitude conditions imposed on the similar problems are 
 
 f x /T (3) 
 
 Vi (4) 
 
 \l O = u. (J /t (5) 
 
 T T 1 Y 
 
 where f^ } H e r and 0" t are the operating frequency, the permeability, the 
 permitivity and the conductivity of the modeled antenna system respectively, 
 while f , H. e and C are those of the original antenna system. 
 
 It is seen that if H 1 and e are kept constant, doubling of the size of 
 the antenna requires that the operating frequency and the conductivity of the 
 
antenna structure both be halved, while shrinking of the size of the antenna 
 into a half requires that the operating frequency and the conductivity of the 
 antenna structure both be doubled. Scaling of the conductivity however, 
 causes a great inconvenience in practical designs. Fortunately, in most 
 cases, if the conductivity is high, it can be kept unchanged and the linear 
 modeling of antennas will still hold approximately. This may thus be termed 
 as the approximate linear modeling of antenna problems, and its theory in- 
 volves the finding of similar figures. 
 
 20 
 Belyea_, Low and Segel reported to have met some difficulties in scaling 
 
 a certain aircraft for an x-band radar cross-section experiment with the 
 
 linear modeling theory. They have developed a non-linear modeling theory for 
 
 the electromagnetic problem and others . 
 
 2.2 Self Similar Figures 
 
 Figures with the whole similar to its part are called self-similar 
 
 figures . Usually, many self-similar figures may be drawn for a given figure. 
 
 For example, the following shaded triangles are similar to their original 
 
 triangles . 
 
 Figure 2. Self-Similar Figures 
 For every pair of similar figures there is a ratio of similitude or 
 the scaling parameter t„ For a self-similar figure, the set of similar figures 
 can be transformed into each other by a similarity transformation with an 
 associated scaling parameter t Since a general similarity transformation, 
 
6 
 
 as indicated in last section^ is the product of the general rigid motion 
 and a radial transformation^ it follows that for the self-similar figures^ 
 the similarity transformation contains only the product of the rotation 
 and a radial transformation. 
 
 It is required by the theory of similitude that in order to achieve the. 
 true frequency independent operation,, the configuration of the antenna must 
 be a self-similar figure with < t < oo This implies a continuous similarity. 
 An infinite conical configuration is an example of these structures. The 
 general structure of this type was found by Rumsey to be that described by 
 the log-spiral curves. 
 
 r = r e a ^ (6) 
 
 o 
 
 where a and r are constants. For the conical structures, the similarity 
 o ' 
 
 transformation involves only a radial transformation while for the log-spiral 
 structures,, the similarity transformation contains both the expansion and the 
 rotation and they are related by 
 
 9 = In T/a (7) 
 
 where 9 is the angle of rotation. 
 
 When the admissible values of t for the similarity transformation form 
 
 a discrete set (t ) } then the self-similar figure possesses only a discrete 
 
 similarity. The log-periodic structures belong to this class and have T = 
 
 t } n = integers . We shall show that the polygonal spiral structures are 
 
 also self-similar figures with discrete similarity and a construction method 
 
 will be given. Both the log-periodic structures and the polygonal spiral 
 
 structures are modified angular structures. 
 
 2,3 The Convergence Problem of the Angular Structure as a Frequency Independent 
 Antenna and the Modified Angular Structures . 
 
 All angular structures are infinite extent. To be used as antennas^ they 
 
 must be truncated. The amount of end effects due to the truncation serves 
 
 as another criterion for being a frequency independent antenna besides the 
 
 similarity condition. It is known that not all angular structures lead to the 
 
frequency independent antenna design; the log-spiral structures were shown 
 experimentally to have only negligible end effects under suitable design, 
 while the radiation of the biconical structures can be attributed to the end 
 effect. This implies that when the similarity condition is insisted, the design 
 of the frequency independent antennas is restricted to a special angular 
 structure — log-spiral antennas . 
 
 It has also been shown experimentally that some angular structures 
 which can not be used for the frequency independent antenna design under the 
 strict similarity condition become useful structures when the similarity 
 condition is relaxed in a certain way = The log-periodic structures are examples 
 of this situation. They all belong to modified angular structures — angu- 
 lar structures with periodic discontinuities. 
 
 Based on the principle of scaling and in view of above experimental re- 
 sults, one realizes that much flexibility could be gained in the design of 
 frequency independent antennas (or rigorously speaking, approximate frequency 
 independent antennas) by relaxing the similarity condition. As the log-periodic 
 structures are derived from the conical angular structures, we shall demon- 
 strate a way of relaxing the similarity condition for the log-spiral structures. 
 
3. POLYGONAL SPIRAL ANTENNAS 
 
 The polygonal spiral antenna was developed by relaxing the continuous 
 similarity condition of the log-spiral structures. First a generalized 
 definition of spiral structure will be given and then certain similarity 
 condition imposed so as to give a polygonal spiral structure which inscribes 
 a log-spiral curve, 
 3 = l A Definition of Spirals 
 
 A spiral curve can be defined as a line which, starting from a point 
 of origin, which is called the pole, continually diminishes in curvature as 
 it recedes from that point, in other words, the radius of curvature continually 
 increases, A log-spiral is such a curve that the arc intercepted between 
 the pole and any other point whatsoever on the curve is always similar to 
 itself , Therefore a log-spiral curve is self-similar with continuous simi- 
 larity , 
 
 To modify the structure by replacing the curved structure with straight 
 elements, the rectilinear spiral curve is constructed as follows: from some 
 starting point, or the pole, a straight is drawn in a given direction to a 
 definite point P at a fixed distance 1 and changing the direction, a second 
 straight line with length 1 is drawn to a point P , From P a third line is 
 started in a different direction from that of the second line. With length 
 1 , a point P is reached, The process can be continued indefinitely as shown 
 in the following figure. 
 
 Figure 3. A Rectilinear Spiral 
 
9 
 
 Using this definition of rectilinear spiral structures as a construction 
 principle in relaxing the continuous similarity condition of the log-spiral 
 structure, a discrete similarity property can be obtained for these structures 
 by requiring 
 
 Vi n \i> 
 
 x 1 , 
 
 (8) 
 
 P x = P 2 = P 3 = . • • = P (9) 
 
 It is seen that if t = 1, the structure is a regular polygon. In which case, 
 the angle (3 is determined by the number of sides of the polygon m 
 
 p=ii^i (10) 
 
 r m 
 
 If t < l, it becomes a converging polygonal spiral while t > 1 gives a di- 
 verging polygonal spiral. It is shown that each of these polygonal spiral 
 structures, a log-spiral curve can be found to circumscribe it. 
 3.2 Polygonal Spirals 
 
 The construction of polygonal spirals was given in the preceeding section. 
 Its relation to log-spiral structures shall be pointed out presently, 
 
 A proposition on the characteristics of log-spiral curves can be stated 
 as follows . 
 
 Proposition : In a log-spiral, the polar radii of points whose vectorial 
 angles are in arithmetic progression are themselves in geometric progression. 
 
 If the above said points are connected, then a polygonal spiral structure 
 is obtained, as shown in Figure 4. A corollary follows from the above 
 proposition. The sides of the polygonal spiral are also in geometric pro- 
 gression . 
 
 In Figure 4 
 
 ?.?-?-?-. . . = <p 
 
10 
 
 and 
 
 Figure 4. A Log-Spiral Curve and a Polygonal Spiral Structure 
 
 a<fb afPo 
 
 r i = r o e ' r 2 ?c r i e ' 
 
 Therefore 
 
 r r 
 
 a<£b 
 
1 1 
 
 The side of the polygonal spiral can be expressed as 
 
 2a<p a(p 
 
 1., = r A /l + e - 2e cos <P = r t 
 1 o V o o 
 
 l 2 = *1 * 
 
 etc 
 
 hence we also have 
 
 Thus from these we obtain 
 
 12 o 
 
 a<p 
 t = e ° (13) 
 
 and 
 
 (3 = 7T - 0? 
 
 Several particular polygonal spiral structures, a triangular spiral, a rec- 
 tangular spiral and a pentagonal spiral are shown respectively in Figures 5, 
 6, and 7, 
 3,3 Zigzag Structures 
 
 Under the general definition, the rectilinear spiral structures also 
 include the biangular configurations as a special case. To satisfy the self- 
 similarity condition we impose the following conditions for these structures. 
 
 
0--T- 
 
 t = e 3 
 
 12 
 
 Figure 5 . A Triangular Spiral 
 
13 
 
 Figure 6. A Rectangular Spiral 
 
 Figure 7. A Pentagonal Spiral 
 
14 
 
 \ = h =h = ?4 
 
 (15) 
 
 where 
 
 P 2 = 27T - p 2 
 
 P 4 = 277 - P< 
 
 hence they become the log periodic Zigzag configurations 
 
 Figure 8. A Log-Periodic Zigzag Structure 
 
 3,4 The Scaling Parameter 
 
 It is seen that a polygonal spiral is uniquely determined by specifying 
 the parameters (3 and t . Their relations to the circumscribing log-spiral 
 
15 
 
 curve is given by 
 
 2TJ 
 
 (77-(3)a m 
 
 T = e r = e 
 
 (16) 
 
 for a planar log-spiral 
 
 a = - 1/tan a 
 
 (17) 
 
 where a is the spiral angle, i.e, ; the angle between the inward radius vector 
 and the tangential vector, For a conical log-spiral structure 
 
 a ~ - sin /tan a 
 o 
 
 (18) 
 
 where 9 is the half-cone angle, A conical rectangular spiral is shown 
 o 
 
 in Figure 9 in comparison to a conical log-spiral structure. Note that in 
 
 the case of polygonal spirals the spiral angle a is a variable. 
 
 r= const. 
 
 J 3 
 
 Figure 9, A Conical Rectangular Spiral and a Conical Log-Spiral 
 
16 
 
 From Equations (16) and (18), the equivalent conical polygonal spiral 
 can be determined for a given conical log spiral, i.e, for a given set of 
 cone angle and spiral angle a, the scaling parameter T is fixed for the 
 polygonal spiral, Figure 10 gives the scaling parameters for a rectangular 
 spiral related to its corresponding log-spiral structures. This shows the 
 relationship between a conical rectangular spiral and a conical log-spiral. 
 Similar relationships can be obtained for other polygonal spiral and their 
 corresponding log-spirals . Figure 11 shows the scaling parameter of poly- 
 gonal spirals of m = 2, 3, 4, 5, 6, for the spiral angle of the log-spiral 
 
 o o o 
 
 a = 35 , 70 and 80 . 
 
 These graphs serve as a mean in relating the operating characteristics 
 of continuous self-similar structures to that of discrete ones . The design 
 criterion expressed in terms of the scaling parameter T for the log periodic 
 structures can therefore be related to those criterions expressed in terms of 
 the half-cone angle 9 and the spiral angle a for the log-spiral structures, 
 
17 
 
 a=85° 
 
 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° C 
 HALF CONE ANGLE IN DEGREES 
 
 Figure 10. The Scaling Parameter of the Rectangular Spiral 
 
18 
 
 >a=85° 
 
 Va = 70< 
 
 ka = 35' 
 
 10° 20° 30° 40° 50° 60° 70° 80° 90° 8o 
 HALF CONE ANGLE IN DEGREES 
 
 Figure 11. The Scaling Parameter of Polygonal Spirals for m = 2, 3, 4, 5. 
 
19 
 
 4. EXPERIMENTAL METHOD AND MEASUREMENT RESULTS 
 
 4.1 Experimental Method 
 
 The polygonal spiral structures, discussed in Chapter 3, when used 
 as antennas, will be referred to as the polygonal spiral antennas. For 
 convenience of the feeding, m is restricted to be an even integer. When 
 m = 2, it becomes a bifilar log-periodic zigzag antenna. 
 
 Based on the property of self-similarity, we infer that the polygonal 
 spiral antennas can be designed to give frequency independent performances. 
 Evidently, for large enough m its behavior will approach that of the corres- 
 ponding log-spiral antenna. The practical problem is, therefore, to deter- 
 mine the number of sides of the polygonal spiral antenna which behaves prac- 
 tically like a log-spiral antenna a This was our endeavor and we have shown 
 experimentally, that a rectangular spiral antenna is already a good approxi- 
 mation to its log-spiral counterpart . 
 
 In order to compare the operation characteristics between that of the 
 rectangular spiral antenna and that of the corresponding log-spiral antenna, 
 six different rectangular spiral antennas were constructed and tested,, The 
 parameters of these structures were chosen to give a comparison with some 
 
 typical characteristics of the log-spiral structures , 
 
 21 
 It has been shown experimentally by Dyson that the pattern beam width 
 
 of the radiation field of a two arm conical log-spiral antenna, although 
 frequency independent, is a function of the spiral angle a - In general, 
 it may be stated that the beam width increases as the spiral angle a de- 
 creases „ The numerical results, due to Dyson, can be shown as follows: 
 
 BW 
 
 where BW is the half power beam width of the radiation pattern. He also 
 observed that the log-spiral structure can be approximated by the wire version 
 only when the antenna is relatively tightly spiraled, say a > 60 , As the 
 
20 
 
 o o 
 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. 
 
 For these reasons, the following parameters for the testing antennas 
 were chosen, They are listed together with the corresponding scaling para- 
 meter t , 
 
 (a) 9 - 7*5° 
 
 a 45° 60° 73° 
 
 T 0,814 0,892 0=94 
 
 (b) e - 10° 
 
 o 
 
 a 45° 60° 73° 
 
 T 0,76 0,858 0,92 
 
 o o 
 One of the structures, 9 =7,5 a - 7 3 is shown in Figure 12 ana 
 o 
 
 its corresponding wire version log-spiral structure is shown in Figure 13. 
 RG8/U cables were used for these antenna structures, 
 
 The measured results on the radiation patterns and the input impedances of 
 these antennas show a great resemblance to that of the wire version log-spiral 
 structures , 
 
 The radiation patterns and the input impedances of a biangualr spiral 
 structure were recorded. They show very small variation, both in the radia- 
 tion pattern and the input impedances, for a very wide range of frequencies, 
 The structure, 9 = 7,5 , a = 73 is shown in Figure 14, 
 
 Results of the measurements taken on these antennas are shown in the 
 following sections . 
 4,2 Results of Measurement — The Rectangular Spiral Antennas 
 
 4,2,1 Pattern Comparisons and the Comparison of the Axial Ratio 
 
 In order to show the similarity between the operation characteristics 
 of the conical rectangular spiral antennas and that of the corresponding log- 
 spiral antennas, the radiation patterns of these two types of antennas are 
 
2] 
 
 Figure 12. A Conical Rectangular Spiral Antenna 
 
22 
 
 Figure 13. A Conical Log-Spiral Antenna 
 
23 
 
 / 
 
 Figure 14. A Log-Spiral Zigzag Antenm 
 
a 
 
 - 
 
 73° 
 
 a 
 
 = 
 
 73° 
 
 a 
 
 = 
 
 60° 
 
 a 
 
 -s 
 
 o 
 60 
 
 a 
 
 = 
 
 45° 
 
 a 
 
 = 
 
 45° 
 
 e 
 
 = 7,5° 
 
 o 
 
 
 
 o 
 
 e 
 
 = 10 
 
 o 
 
 
 
 o 
 
 e 
 
 = 7,5 
 
 o 
 
 
 e 
 
 = 10° 
 
 
 o 
 
 e 
 
 = 7,5 
 
 
 o 
 
 e 
 
 = 10 
 
 24 
 
 first compared for a range of frequencies <> In the subsequent figures^ the 
 radiation patterns for the conical log-spiral antennas of the wire version are 
 shown by solid lines while those for the conical rectangular spiral are shown 
 by dotted lines. The following list gives the correspondence between the an- 
 tenna patterns and the parameter of the antenna structures , 
 
 Figure 15 
 Figure 16 
 Figure 17 
 Figure 18 
 Figure 19 
 Figure 20 
 
 o 
 In each figure the patterns are shown both for dj = plane and for 
 
 it = 90 plane^ where cb angle is referred to the line determined by the 
 feeding points . 
 
 The resemblance of the radiation patterns of these two types of antennas 
 is apparant as seen from the figures . In many cases they are actually iden- 
 tical. For a - 73 , the patterns are unidirectional. The difference in the 
 patterns of these two type of antennas is very small . As the spiral angle a 
 
 decreases the beam width of the pattern increases. Although the difference 
 
 o o 
 
 for these patterns is somewhat larger in the cases for a - 60 and a - 45 3 
 
 there is no difficulty in pointing out their similarities, In the case of 
 
 o 
 a = 45 _, the wire version log-spiral structures fail to approximate the exact 
 
 log-spiral antennas. The similar effect of multilobing exists in both the 
 pattern of the wire log spiral structure and that of the corresponding rec- 
 tangular spiral antenna. 
 
 The axial ratio of the polarization ellipsis are determined a) as a 
 function of frequency for the polarization ellipsis of the electric fields 
 measured on the antenna axis and b) as a function of the angle 0, where 
 is the angle between the axis and a radial vector,, drawn in cb = plane and 
 the operating frequency in these cases is 700 Mc , 
 
 The axial ratios as a function of frequency are shown in Figure 21 
 through Figure 24, The following list gives the corresponding parameters of 
 the structures , 
 
 
25 
 
26 
 
21 
 
28 
 
29 
 
 en 
 
 u 
 a 
 a, 
 
 6 
 o 
 
 < 
 
 ft s 
 
 O U <D 
 J DC Sh 
 
 3 
 
30 
 
 2 3 
 
 Q. Z 
 
 CO < 
 
 CD O 
 
 O UJ 
 
 -i en 
 
31 
 
 o 
 
 
 o 
 
 o 
 
 
 +-> 
 
 en 
 
 
 c 
 
 D 
 fa 
 
 d 
 
 
 o 
 
 M 
 
 
 2 
 
 < 
 
 o 
 
 
 I 
 
 o 
 
 z 
 
 
 
 a) 
 
 
 •H 
 
 
 >- 
 
 +-> 
 
 
 o 
 
 OS 
 
 
 z 
 
 ,_, 
 
 
 UJ 
 
 rt 
 
 
 z> 
 
 ■H 
 
 X 
 
 O 
 
 o 
 
 < 
 
 O 
 
 UJ 
 
 
 I s - 
 
 (T 
 
 X! 
 
 o 
 
 •r-t 
 
 o 
 
 ^ 
 
 CO 
 
 a, 
 
 
 s 
 
 
 
 
 
 u 
 
 
 H 
 
 O 
 
 N 
 
 O 
 
 
 
 in 
 
 U 
 
 
 3 
 
 
 M 
 
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 Figure 21 a = 73 = 7 .5 
 
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 Figure 23 a = 60° = 7.5° 
 
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 The axial ratios as a function of the angle are shown in Figures 25 
 through 28. The corresponding parameters of the structures are 
 
 Figure 25 a = 73° = 7.5° 
 
 Figure 26 a = 73° = 10° 
 
 Figure 27 a = 60° =7.5° 
 
 Figure 28 a = 60° = 10° 
 
 In each of these figures, the axial ratios of the corresponding log- 
 spiral structure are accompanied in dotted lines. These experimental re- 
 sults again give the following two indications . 
 
 (a) The frequency independent operation of the conical rectangular 
 spiral antenna 
 
 (b) The closeness in the performances of the rectangular spiral antenna 
 and of the log-spiral antennas , 
 
 4.2.2 The Standing Wave Ratio and The Input Impedance 
 
 The standing wave ratios of these rectangular spiral structures are shown 
 in Figures 29 through 34 in comparison with that of the log-spiral structures 
 (dotted lines) . The parameters of the antenna structures for the set of data 
 are indicated in the following list. 
 
 Figure 29 
 Figure 30 
 Figure 31 
 Figure 32 
 Figure 33 
 Figure 34 
 
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 The input impedances of these antennas have also been determined, They 
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 structures and their figures are in the same sequence as before, namely; 
 
 Figure 35 a = 73° 9 ^7.5° 
 
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 These experimental results again indicate the practibility of the rec- 
 tangular spiral structure used as a frequency independent antenna. 
 4,3 Results of Measurements — The Biangular Spiral Antenna 
 
 Since a log periodic Zigzag antenna is a biangular spiral antenna which 
 is just a special case of the polygonal spiral antenna, it is expected that 
 it should also possess the frequency independent characteristics which were 
 defined earlier, To confirm this fact, a log-periodic zigzag antenna was 
 built (Figure 14) and its characteristics determined, The results are shown 
 below , 
 
 4.3.1 Radiation Patterns 
 
 The radiation patterns are taken for a range of frequencies (600 ~ 1500 Mc), 
 both for the E-plane and the H-plane , The variations, as shown in Figure 41, 
 are small within this frequency range, 
 
 Note that for this type of structure, the radiation pattern is linearly 
 polarized, while those of other polygonal spiral antennas are expected to be 
 circularly polarized, 
 
 4.3.2 The Standing Wave Ratio and the Input Impedance 
 
 The standing wave ratio measured for the antenna is shown in Figure 42 r 
 while the input impedance is given in Figure 43. It is seen that the variations 
 which existed in these results over a wide range of frequencies (500 ~ 1400 Mc) 
 are small » It thus shows the applicability of this structure for the frequency 
 independent operation, 
 
 
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 ^S 
 
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 E- PLANE— ^ = 90°(E H -Eg) 
 H- PLANE— ^0(E v -E^) 
 
 Figure 41. Radiation Patterns of the Log-Periodic Zigzag Antenna 
 
 (0 = 7.5°, a = 73°). 
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56 
 5 . CONCLUSIONS 
 
 In this report, limited literature on the problem of the frequency 
 independent antennas is reviewed, It is felt that although the complete 
 theoretical solution of the problem is not yet available, the antenna en- 
 gineer can make progress in the practical design, based on the basic principle 
 of scaling, The main idea suggested was the relaxation of the similarity 
 condition on the rigorous angular structures, This has been one of the 
 design methods in the development of the log-periodic antennas. The method 
 was used to modify the log-spiral antennas, It is hoped that through an 
 accumulation of workable structures theoreticans may have a better idea in 
 making assumptions for the solution of the problem, 
 
 The construction of the polygonal spiral antennas was given in terms 
 of the construction of the polygonal spiral curve, Through the construction 
 the relation between the structure with the continuous similarity condition 
 and that with discrete similarity condition was demonstrated, The experi- 
 mental method was used in determining the applicability of the polygonal 
 spiral structure for the use of the frequency independent antenna. 
 
 The measurement results showed that the rectangular spiral antenna is 
 a good approximation to the log-spiral antenna, for two equivalent antennas 
 of these types behave practically the same, as far as the frequency indepen- 
 dent operation is concerned, 
 
 The measurements for a bifilar log-periodic Zigzag antenna, a special 
 case of the polygonal spiral antenna, were performed for a range of frequen- 
 cies and results given. The frequency independent operation of this type of 
 antennas was also indicated, 
 
 
57 
 
 BIBLIOGRAPHY 
 
 Rumsey, V, H», "Frequency Independent Antennas/' U.I. TR20, October, 1957. 
 
 DuHamel, R, H,, and Isbell, D, E., "Broadband logarithmically periodic 
 antenna structures/ 1 1957 IRE National Convention Record , U.I. TR19, p. 119 
 
 Dyson, J, D a , "The equiangular spiral antenna/' IRE Trans. Vol AP-7, 
 April 1959, pp. 181-189, also T.R. No. 21 Contract AF 33(616)-3220, 
 Dept o of EE, University of Illinois, September 1957. 
 
 Isbell, D, E. r "Log-periodic dipole arrays/' U.I. TR 39. 
 
 Dyson, J, D., and Mayes, P. E., "New circularly polarized frequency inde- 
 pendent antennas with conical beam or omnidirectional patterns," IRE Trans. 
 AP-9, July 1961, pp. 334-352. 
 
 Mayes, P. E., and Dyson, J. D., ''Multi-arms logarithmic spiral antennas," 
 10th Annual Sym. on the USAF Ant. Res. and Develop. Program, sponsored 
 by Aeronautical Systems Div„, AF Syst . Command, October 1960. 
 
 Dyson, J. D,, "The conical log-spiral antenna in simple arrays," 11th 
 Annual Sym. on the USAF Ant. Res. and Develop. Program, October 1961. 
 
 Mayes, P. E., and Carrel, R. L., "Logarithmically periodic resonant-V 
 arrays," WESCON, 1961 also TR No. 47, Contract AF 33(616)6079, EE Dept., 
 University of Illinois, June 1960. 
 
 Carrel, R „ L., "Analysis and Design of the log-periodic dipole antenna," 
 U.I. TR52, September 1961. 
 
 DuHamel, R. H., and Berry, D. G.,, "A new concept in high frequency antenna 
 design," IRE National Con. Rec , Vol. 7, p. 43, 1959. 
 
 DuHamel, R. H., and Ore, F. R., "Log-periodic feeds for lens and reflectors, 
 IRE National Con. Rec , p. 128, 1959. 
 
 DuHamel, R, H., and Berry, D. G., "Log-periodic antennas arrays," IRE 
 WESCON Con. Rec , part 1, August, 1958. 
 
 DuHamel, R. H., and Ore, F, E., "Log-periodic antenna designs," IRE 
 National Con. Rec , p. 139, 1958. 
 
 Dyson, J. D 7 "A survey of very wideband and frequency-independent antennas 
 — 1945 to the present," Journal of Research N.B.S. Vol. 66D, No. 1, 
 January-February 1962, pp. 1-6. 
 
 Mast, P, Eo, "A Theoretical Study of the Equangular Spiral Antenna," 
 U,I, TR35, September 1958, 
 
58 
 
 16, Rumsey, V. H,, "A new way of solving Maxwell's Equations," IRE Trans . , 
 AP-9, No, 5, p. 461, September 1961= 
 
 17, Copeland, J. R,, "Radiation from the balanced conical equiangular spiral 
 antenna/' O.S.U. Report, 903-12, September 1960. 
 
 18= Lee, W, C, '"Analysis of non-planar spiral antennas, O.S.U. Rep. 903-15, 
 November 1960. 
 
 19. Stratton, J. A,, Electromagnetic Theory , McGraw-Hill Book Company, p. 488, 
 1941. 
 
 20. Belyea, L. E., Low, R, D,, and Siegel, K, M., 'Non-linear Modeling of 
 Maxwell's Equations," Contract AF19(604)-4993, No. 2871-4-Y, Radiation 
 Lab,-. The University of Michigan, December, 1959. 
 
 21. Dyson, J. D., "The unidirectional equiangular spiral antenna, ' IRE 
 Trans . , AP-7, No, 4, p. 329, October, 1959. 
 
 

 ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 Contract AF33(616)-310 
 
 "Synthesis of Aperture Antennas," Technical Report No. 1 , C.T.A. Johnk, 
 October, 1954 .* 
 
 "A Synthesis Method for Broad-band Antenna Impedance Matching Networks," 
 Technical Report No, 2, Nicholas Yaru,, 1 February 1955.* AD 61049. 
 
 "The Assymmetrically Excited Spherical Antenna," Technical Report No, 3 , 
 Robert C. Hansen, 30 April 1955 . * 
 
 "Analysis of an Airborne Homing System," Technical Report No, 4 , Paul E. 
 Mayes, 1 June 1955 (CONFIDENTIAL). 
 
 "Coupling of Antenna Elements to a Circular Surface Waveguide," Technical 
 Report No. 5, H, E. King and R. H. DuHamel, 30 June 1955.* 
 
 "Axially Excited Surface Wave Antennas," Technical Report No. 7 , D. E. Royal, 
 10 October 1955.* 
 
 "Homing Antennas for the F-86F Aircraft (450-2500 mc)V ' Technical Report No. 8, 
 P. Ec Mayes, R. F. Hyneman, and R. C. Becker, 20 February 1957, (CONFIDENTIAL), 
 
 "Ground Screen Pattern Range," Technical Memorandum No. 1 , Roger R. Trapp, 
 10 July 1955.* 
 
 Contract AF33(616)-3220 
 
 "Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. 
 Scott and R. H. DuHamel, 16 April 1956, 
 
 "An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report 
 No. 10 , Do G. Berry and Jo B. Kreer, 10 May 1956. AD 98615 
 
 "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," 
 Technical Report No. 11, J. W. Duncan and R. H. DuHamel, 15 July 1956.* 
 
 "Directional Characteristics of a U-Shaped Slot Antenna," Technical Report 
 I No. 12 , Richard C. Becker, 30 September 1956.** 
 
 "Impedance of Ferrite Loop Antennas," Technical Report No. 13 , V. H. Rumsey 
 I and W. L. Weeks, 15 October 1956. AD 119780 
 
 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
 I No. 14, Richard F. Hyneman, 20 December 1956. 
 
Distributed Coupling to Surface Wave Antennas/' Technical Report No. 15, 
 Ralph Richard Hodges ,, Jr., 5 January 1957. 
 
 "The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical 
 Report No, 16 , Robert L. Carrel, 15 January 1957.* 
 
 "On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17 , 
 Walter L. Weeks, 10 April 1957.* AD 143989 
 
 "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source 
 Antenna," Technical Report No. 18 , Walter L. Weeks, 20 April 1957. 
 
 "Broadband Logarithmically Periodic Antenna Structures," Technical Report No. 19 , 
 R. H. DuHamel and Do E. Isbell, 1 May 1957. AD 140734 
 
 "Frequency Independent Antennas," Technical Report No. 20 , V. H. Rumsey, 25 
 October 1957 . 
 
 "The Equiangular Spiral Antenna," Technical Report No, 21, J. D. Dyson, 15 
 September 1957. AD 145019 
 
 "Experimental Investigation of the Conical Spiral Antenna," Technical Report 
 No. 22, R. Lo Carrel, 25 May 1957.** AD 144021 
 
 "Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical 
 Report No, 23 , E. J. Scott, 10 August 1957. 
 
 "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," 
 Technical Report No. 24 , J. W. Duncan and R. H. DuHamel, August 1957. 
 
 "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
 Cross Section," Technical Report No. 25 , Robert L. Carrel, August 1957. 
 
 "Cavity-Backed Slot Antennas," Technical Report No. 26 , R. J. Tector, 30 
 October 1957. 
 
 "Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical 
 Report No. 27, W. L. Weeks, 1 December 1957. 
 
 "Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," 
 Technical Report No. 28 , W. L. Weeks, 20 December 1957. 
 
 "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary 
 Cross Section," Technical Report No. 29 , J. D. Dyson, 10 January 1958. 
 
 "Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. 30 , 
 D. E. Isbell, 20 February 1958. AD 156203 
 
 "Electromagnetic Fields in Rectangular Slots," Technical Report No. 31 , N. J. 
 Kuhn and P. E. Mast, 10 March 1958. 
 
"The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder,' 
 Technical Report No, 32 „ J. W„ Duncan, 25 May 1958 , 
 
 "A Unidirectional Equiangular Spiral Antenna/' Technical Report No, 33 , 
 J. D, Dyson, 10 July 1958 . AD 201138 
 
 "Dielectric Coated Spheriodal Radiators," Technical Report No. 34 , W. L. 
 Weeks, 12 September 1958 . AD 204547 
 
 "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report 
 No. 35, Po E. Masts, 12 September 1958. AD 204548 
 
 Contract AF33 (616) -6079 
 
 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Technical 
 Report No, 36, R. H. MacPhie, 30 April 1959. AD 215558 
 
 "On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and 
 Infinite Ranges/ Technical Report No, 37 , Raj Mittra, 15 May 1959, 
 
 "Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical 
 Report. No, 38 , W, L. Weeks, 5 June 1959, 
 
 "Log Periodic Dipole Arrays," Technical Report No, 39 , D. E, Isbell, 1 June 1959. 
 AD 220651 
 
 "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical 
 Report No. 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," Technical Report No, 42, Raj Mittra, 1 October 1959. 
 
 "impedance Properties of Complementary Multiterminal Planar Structures,? 
 Technical Report No, 43, G, A, Deschamps, 11 November 1959. 
 
 "On the Synthesis of Strip Sources," Technical Report No. 44 , Raj Mittra, 
 4 December 1959 , 
 
 "Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides," 
 Technical Report No, 45, C. H. Tang and Y, T, Lo, 11 March 1960. 
 
 "New Circularly Polarized Frequency Independent Antennas with Conical Beam or 
 Omnidirectional Patterns/' Technical Report No. 46, J. D. Dyson and P. E. Mayes, 
 20 June 1960. AD 241321 
 
 "Logarithmically Periodic Resonant -V Arrays," Technical Report 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, 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. Ho 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, R, Mittra, 1 October 1961. AD 264556 
 
 * Copies available for a three-week loan period. 
 ** Copies no longer available. 
 
AF35(657 )-8460 
 
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