IEEE TRANSACTIONS ON ANTENATAS AND PROPAGATIOM contributions The Characteristicsand Design of the Conical Log-Spiral Antenna JOHN D. DYSON, Abstract-Thebalancedconicallogarithmic-spiral antennais considered as a slow-wave locally periodic structure with a slowly varying period. A study of the near fields and their relationship to identification of the active region or the f a r fields has led to the effective radiating aperture on the antenna and to a clearer understanding of its operating characteristics. Information of the near- and far-field characteristics and on the input impedance for a wide range of parameters is presented in a form suitable for use in the design of practical antennas. INTRODUCTION HE FREQUEKCY-1NDEPEN:DEST antennas havefoundwideapplicationandtheirgeneral characteristicshavebeenoutlinedinnumerous publications [l], [2]. The purpose of the present paper is t o presentthecharacteristicsandpracticaldesign information for one member of this general class, the balancedtwo-armconicallogarithmic-spiralantenna, Dyson [3]. T Manuscript receix-ed March 23, 1964; revised February 25, 1965. This research was sponsored by the Avionics Laboratory, rVrightPatterson AFB, DaJ-ton, Ohio, under Contract AF33(657)-10471. The authoris with the Dept. of Electrical Engineering, University qf Illinois, Urbana, Ill. SENIOR M E D E R , B E E This study has been guided b~7 the concept, introduced by Mayes, Deschamps, and Patton [4], t h a t t h e logarithmic-periodic antennas could be consideredt o be locall>- periodicstructureswhoseperiodvariesslowly with distance from the point of excitation. It extends this basic concept to the conical log-spiral antenna, by initially comparing the propagation constant measured along the surface of the conical antenna toknown propagation constants for the cylindrical bifilar helix. Thestudy of thepropagationconstantandother characteristics of the near and far fields of these antennas has led to the identification of the ‘(active region” or effective radiating apertureon the antenna. Successful design of the antenna depends upon knowledge of the position and size of this active region as a function of antenna parameters. The basic parameters of the conical antenna are defined in Fig. l. The parameter 0 0 determines the cone angle, and a the rate of wrap of the arms. The angular width of the exponentially expanding armsis defined b y theangle 6 which is theprojection of 6’ on a plane perpendicular to the axis of the antenna.-These angles Uyson: ConacaC Log-apzraL Antenna 981u are constantfor any given antenna and the radius vectorsurface of the cylinder. If we consider the difference between these k / P ratios for thehelix and conical spiral, to any point on the arms is given by [3] and express this as a function of the helix ratio, we see P = PO exp b ( 4 - 6) (1) in Fig. 2 that thedifference issimply 1-cos Bo. For cones with an included angle of 20°, for example, this differwhere ence is approximately 1+ percent. For allconicallogspiral antennas which are good unidirectional radiators, this difference is only a few percent. This small difference in k / / 3 ratios for the smaller inThe edges of the first arm are defined by letting 6 = 0, cluded cone angles ~ o u l dlead one to expect that if we and a fixed value between 0 and T. T h e second arm is limit our discussion at any one time toa limited portion obtained by multiplying the defining equations for the of the conical antenna there should be,a t least for these first arm by e-bz. The orientation of the antenna in the smaller cones, a close relationship between this portion associated spherical coordinate system used for radiaof the conical Structure and some corresponding cylintion pattern measurements is also indicated. drical structure. The variationof the propagation constant on periodic antenna structures can beconvenientlydisplayed on the Brillouin or k / p diagram [S]-[S]. One such diagram for the balanced bifilarhelix is shown in Fig. 3. T h e vertical coordinate is given in units of k a l t a n a which, since k = 27r/X, is simply the pitch distancein free-space wavelengths. The horizontal scale is the pitch distance expressedinguidewavelengths on the surface of the antenna. For any one helix, since a and a are constant, the only variable involved is the wavelength of operation. the frequency of operation is increased the propagation constant increases. If we assume a nondispersive wave along the conductors, we find that there is first a region of S ~ O W , closelyboundsurfacewaves. As the propagation constant increases still further, there will be strong coupling to a space wave traveling in the opposite direction. The propagation constant becomes complex [ 9 ] as the structure radiates with aphasing that provides an end-fire beam directed back towardthe Fig. 1. Conical antenna with associated parameters. point of excitation,i.e., a backfirewave. As the frequency of operation is increasedstillmore,energyis THEAYTENNA AS a LOCALLY PERIODIC STRUCTUREradiated a t a nangle from the axisof the structure, and The parameters involved in a comparison of the coni- then in an end-fire beam in the opposite direction [lo]. calspiralandcylindricalhelicalgeometriesareindiFor the conical antenna there are two variables, the cated in Fig. 2, in which is shown one turn or cell of a wavelength and the radius a , which vary linearly with conical antenna with infinitesimally narrow arms, withI distance from the pointof excitation. For a fixed wavethe length of the turn, and a thc radius at any pointin length, wc SCC t h a t as a currcnt wavc progresses from question. Superimposed upon this is one cell of a cylin- the pointof excitation along a structure with an increasdrical helix with the same pitch angle, and with a radius ingradius a , thepropagationconstantmightbe exequal to the geometric mean radius of the conical cell. pected to behavein the manner thatwe attribute to the Forobservationsparalleltotheaxis,theturn-tocylindrical structure with increasing frequency. turn phasing of the helix is determined to a first approxi- Thesolutioninvolvingthepropagationconstant mation by the ratio of the pitch distance p to the turn along the surface of the cylindrical helix applies t o t h e length 2. This assumes a current wave progressing down infinite structure; however,suchsolutionshavebeen the arm at the intrinsic phase velocityof the surround- shown to be useful for interpretation of the characterising medium. On the helix this ratio p / l is equal to the tics of the finite monofilar endfire helix and the finite sine of the pitch angle or cos a. The ratio of the pitch backfire bifilar helix with thin arms. I t is also recognized distance (parallel to the axis) to the turn length on the that extending these concepts to the tapered structure conical structure is equal tocos a cos Bo. is movingevenfartherfromthebasicpremiseupon The ratio ( S ) of pitch to turn length is also the ratio which this diagram is based.I t is, however, a useful tool of the propagation constant k along the arms to the in the interpretation of the characteristics of these anpropagation constant p of a wave propagating along the tennas [11], [12]. SPIRAL s, = shCOS e, FACTOR= s,_sh(IOO%) sh = (I-cos8,) 100% I Fig. 2. 0 Corresponding conical spiral and helicalcells. Y Fig. 4. Fig. 3. 1 2 Brillouindiagram for bifilar helix. 8 - Amplitude of near fields measured with a small shielded loop probe along the surface of one cone and electric far-field radiation patterns corresponding to a truncationa t indicated points (280 = 20", 01 = SO", and 6 = 90"). The curve indicated bya solid line in Fig. 4 is a typical plot of the amplitudeof the fields measured b y moving a smallshieldedloopprobeparallelto,and 0.03 wavelength from, the surface of oneconicalantenna. The vertical axis is the relative magnitude of the probed field in decibels, and the horizontal axis is the distance from the apex along the surface of the cone in wavelengths (p/X) andtheradius of the cone (a,'X). WTe observe that there is a region of closely bound waves near the apex of the cone. &4sthe probe is moved along thesurfaceawayfromtheapex,thiswave becomes more loosely bound, and more energy is coupled to the probe,untilthewavebecomes so loosely bound that energy is rapidly lost through radiation, and the amplitude of the near field decays to a negligible value. As the frequency is changed, this region of rapid decay moves on the antenna so that its location and size in wavelengths remains constant,i.e., the antenna aperture scales with frequency. An arm-to-arm variation in amplitude isvisiblein the region to the right of the peak amplitude. The standing waveon the apex side of the region of rapid decap appears to be due to an interaction between the wave progressing from thetip down theantennaandthespacewaves progressing in the opposite direction. )Then the fields are measured with the probe very close to the antenna (0.004X) the amplitude of thisfirstregion is onlya few d B below the maximum and the amplitude of the standing wave has decreased, e.g., from 10 t o 1 2 d B t o less t h a n 1 d B a t a radius of 0 . 0 3 . \JThen the probe is moved out to only 0.03 wavelength from the surface of the antenna, the relative amplitude in this region drops by approximately 10 t o 15 dB, indicating how tightly the fields are bound to the structure when the conediameterissmallin wavelengths. THEACTIVEREGIOK Since there appears to be a waveguide region and a radiating region on the antenna,we define the radiating or "active region" to be that region which controls t h e primary characteristics of the radiated field. To indicate the extent of the radiating or active regionon thisantenna,the near-fielddistributionand radiation patterns are shown in Fig. 4 as the base diameter becomes smaller in wavelengths. The near-field amplitude indicated by the solid curve and the radiation patterns numbered (1) are representativeof the antenna withoutend effect. As successive turnsareremoved from the base end, thereis a negligible change until the antenna is truncated at a radius such that the original near-field amplitude is approximately 15 d B below the envelope along the peaks of t h e field in the region of rapiddecaywouldbeunchanged. Theactive region recorded maximum. By the time the base is reduced to this size, a definite change in the half-power beamwidth limits were obtained from such a smoothed curve. Figure5 is aBrillouindiagramconstructedfrom and the axial ratio is noted. These data and the nearfield and far-field data for other antennas,as they were amplitude and phase data measured on one narrow arm maintained with a fixed size but with a changing fre- antenna at one frequency of operation. There is indeed of the quency of operation, led to the conclusion that the near a strikingresemblancebetweenthevariation fields more than 15 dB down from the maximum con- propagation constant with position along the structure tributed little to the radiation patterns. The radius of and the asymptotic valuewhich we would expect on the the cone a t this point, which we identify as a15+, could bifilar helix with variationof frequency of operation [9]. be considered t o be the lower edge of the active region. At the distancefrom the antenna surfaceat which these measurements were made, a propagation constant beAs the antenna becomes still smallerinwavelengths, comes evident by the tenth turn that has a leading the radiation in the direction of the base rapidly increases. Thepatternsnumbered phase with increasing distance from the apex. This plot ( 3 ) , whichwererecorded for the antenna with a base radius (ala+), such of the variation of the real part of the propagation conthat the original near-field amplitude was 10 d B below stant along the surface lies just below the asymptotic line cos a cos 0 0 . Near turn 18 this propagation constant the recordedmaximum,aretypical of those obtained for a fixed cone size, at a frequency of operation such becomes complex, the real partincreases, and the atten18 t o 23 thattheantenna is toosmall.The effect of these uation per cell rapidly increases. From arms there is more than 25 d B of attenuation. This is the changes in base size, in wavelengths, on the measured “active region” and the phase center of the radiated field distribution on the small end of the antenna is not great,indicatingthat we shouldexpectonlyminor field is consistently located in this region, in this case variations in the input impedance. near arm 20. Comparing this figure t o Figs. 3 and 4, we T h e effect of a similar truncation of the apex of the note that the majorityof this activeregion and the porcone is shown by the curve labeled a3 and pattern (4). tion with greatest amplitude near fields is phased f o r . backfire radiation. The vertical solid line indicates the original truncated When data is recorded at a higher frequency, and the apex a t a radius of approximately 0.03X. Whenthe antenna is truncated ata radius such that the smoothed probe separation maintained fixed in wavelengths, the level of original field distributionis 3 d B below the phasevariationobservedherefromturns 11 t o 22 is maximum,the near-field distributiononthelarger closely repeated at turns 3 through 14. Thus, in conturns is perturbed enough to make majormodifications trast to the uniformly periodic structure, the radius at in the radiation pattern. This radiuswe identify as as-. the region of rapid decay (i.e., the active region),exFurther truncation produced large changes in the fields pressed in wavelengths, remains essentially fixed on the and hence in the aperture distribution in the radiating antenna. If the coupling to the space wave is strong, region. In general agreement with these results, calcuessentially all of the energy is radiatedintheregion which is phased for backfire wave radiation and there lated radiation patterns based upon the measured near fields of the unperturbed antenna, i.e., without end ef- is a negligible amount left to radiate at any angle from fect, as the smaller turns were eliminated, showed little the axis of the structure. change until the antennawas truncated at a point such Having defined the active region of the conical anthat the originalnear-field distributionwasapproxitenna in terms of radii of the cone we note how this mately 3 d B below the maximum. active region depends upon the antenna parameters. In Although changes, and in particular a small decrease Fig. 6, the boundaries of this region are plotted as a in beamwidth, may be noted before this limit is reached, function of the included cone angle 280 and the spiral these results correlated with the other information led angle a. Since both axes of this graph are normalized t o to the adoption of the region from a point 3 d B below the same wavelength these curves give the active region is considered t o be the maximum on the apex side (radius a3-) to a point bounds. If, however, the vertical axis 15 d B below themaximum on thebaseside(radius normalized to the shortest wavelengthof operation and al5+) as these fields were measured, as the effective radithe horizontal axis to the longest wavelength of operaating aperture of the antenna in what wecouldcall tion, these curves give the required radii of the trunnormal frequency-independent operation. Near-field cated apex and base of the conical antenna. data recorded under different conditions would necesA study of the Brillouin diagram in Fig. 5 indicates sarily be subject toa different set of limiting values on that the active region of the or=60° antenna, for exthe active region. ample, should be closer to the apexof the cone than for I t should be noted that if the line along which the the a = 80’ antenna. Figure 6 shows that the onset of fields are measured were to be rotated around the anradiation does occur at a smaller radius, but the loose tenna by 90°, the arms would all be shifted down onespirals are less efficient radiators; the net effect is t h a t half period and hence the arm-to-arm variations in the these antennas radiate over a considerably wider region field structure would be likewise shifted.Theoverall on the antenna. The result is an activeregion that starts 492 TRANSACTIONS IEEE O N ANTENNAS AND PROPAGATION - ACTIVE REGION EANDYIIDTH (6-1 L5 2 R P, I .I3 July a- .R 10 .os 0- .os % .07 .06 .w .04 2T 5 .03 .02 .01 '0 02 m 06 .06 M J2 , I 4 16 20 22 24 .26 28 3 32 3 4 36 a t a smaller radius and, particularly for 200>10", ex- Fig. 6. Bounds of the active regionin terms of the radius of the cone. Data for the self-complementary antenna (6 = 90"). tends to a larger radius than the equivalent region on the antennas with tighter spirals. ACTIVE REGION BMlWYlDTH [E, These limits on the active region were based upon near-field measurements. In terms of the far-field patterns, they are most accurate for cones with total included angles of approximately 20". They tend to become more conservative as the cone angle decreases and slightly optimistic for greater cone angles. If the activeregion had no width, the operating bandwidth of the antennawould be given by the ratio of the radius of the base to the radius of the truncated apes. Following the convention proposed by Carrel [13]for the log-periodic dipole arrays, this ratiois defined to be the bandwidth of the structureB,. I t is apparent that the operating bandwidth of the antenna is always less than the bandwidth of the structure Fig. 7. Bounds of the active region that may be used for design if appreciable pattern distortion is permissible at lowest operating by a factor defined to be the bandwidth of the active frequency (6 =go"). region B,, CHARACTERISTICS OF THE RADIATION FIELDS In Fig. 8 typicalelectric field radiationpatterns are shown as a functionof t h e included cone angle, and the spiral angle a. T h e well-formed relatively narrow B = BJB,,. beam for small cone angles witha = 80" is indicative of essentially all turns of the active region being phased Figure 6 includes a gridindicatingtheactiveregion for backfire radiation. As the active region broadens, bandwidth B,, as a function of antenna parameters. the radiation pattern broadens and exhibits a tendency For many applications, considerable pattern distortion can be tolerated at the endsof the operating band- to show a multiple beam effect with corresponding irwidth; hence, the antenna can be operated t o a smaller regularities or "ears." It is worth noting that this relasize. For such applications Fig. 7 gives the active region tionship between aperture size and antenna directivity It is brought about bounds as radii a3- and ala+, where ala+ is the radius at is exactly opposite to the usual case. which theunperturbednear-fieldamplitude is 10 d B because the phase distribution across the effective radibelow the recorded maximum, and patterns number (3) ating aperture is such that all parts of the aperture are in Fig. 4 are typical at the lowest frequencies of opera- phased to radiate in progressively different directions. Increaseddirectivity is obtainedfortheperiodic tion. At these frequencies the typical axial ratio on axis structurebyextendingtheradiatingapertureover shouldbe 4.7 d B or less. Foroperationonlytothe is applicable bounds shown in Fig.6, the axial ratio on axis should bemore elements. In principle, this concept to the tapered periodic structures and may be realizable approximately 3 dB orless a t t h elowest frequencies. Hence, the antenna operating bandwidth B is given by Uyson: L'onzcal L o g - 8 p r d Antenna 1965 4Y3 for a slight taper. As the taper is increased to obtain a wide beam. T h e backfire phasing of the 2" cones, for a greater than TO", is clearly evident. practical size antenna that will scale over a wide range The approximate half-power beamwidths of the radiof frequencies, i t is no longer possible t o maintain the reated fields of the conical antennas are plottedin Fig. 11. quired phasing for backfire radiation over a large aperture. The larger turns, or more widely spaced elements,The patterns recorded by orthogonally oriented receivingantennas Eo and E , differ typically 8" t o 10" in are phased to radiate energ17 a t a n angle from the axis beamwidth. The values given in Fig. 11 are the average of the antenna. Hence, for any given ccne angle, the net effect is a n inversion of theusualaperture size- of these beamwidths. I n addition, this mean level of the beamwidth must be evaluated in conjunction with the directivity relationship. This effect is shown in Fig. 9 where the relative mag- indicatedvariationinbeamwidtharoundthislevel nitude and phase of the tangential magnetic field meas- shown in Fig. 12. This indicated variation in beamwidth ured along the surfaceof three log-spiral antennas, each is caused by the fact that the radiated beam, which may with an included cone angle of 15", is shown. When the not be rotationally symmetric, rotates about the axis arms are wound fairly tightly, a = 80", the leading phase of the antenna as the frequency of operation is changed over a relatively narrow aperture concentrates the radi-[14]. This effect, and the fact that the practically conated energy in the backward direction. -4s the rate of structed antennawill not scale exactly a t all frequencies, spiral is relaxed the relative radiating efficiency of the causes a variation in beamwidth in a fixed plane of observation as the antenna is operated over a very wide aperturedecreases,andhencetheaperturesizeincreases. The slope of the phase curve is decreased and range of frequencies. This variation can be minimized the last few turns in the active region radiate apprecia- for operation over a reduced bandwidth and would be ble energy a t a n angle from the axis of the cone. Yi7hen on the order of that for any conventional antenna when of a few percentfor the angle of wrap a is decreased t o 45", a major portion operatedoverthebandwidths which such antennas are designed. of the active region is phased for broadside radiation. The approximate directivity with respect to a circuTo maintain good backfire patterns the leading edge of 11 was the effective aperturemusthavean excess of phase larlypolarizedisotropicsourceplottedinFig. obtainedfromthe use of theseaveragehalf-power shift for backfire radiation, and this aperture must be beamwidths in the expression [IS] an efficient radiator. If n7e recall from Figs. 3 and 5 that the ratio of the 32 600 D = 10 loglo dB, velocity of propagation of the surface wave to a freespacewavewasapproximately cos a! cos Bo, and solve for the intersectionof this asymptotic line and the where q51 and O1 are the average half-power beamwidths region boundary given by in two orthogonal planes. Thefront-to-backratios of theradiated fields are ka pa -- - I - plotted in Fig. 13. This is the typical minimum value tan a tan a t h a t should be expected for a well-constructed antenna operated at a frequency such that there is no effect due we obtain the expression to the truncated base or tip. The decrease in front-toa 1 sin Q cos Bo back ratio with increasing cone angle is to be expected --x 2 1 C O S Q cos00 . since the planar structure is bidirectional. The antenna is elliptically, and essentially circularly, This and similar expressions for the intersection of the polarized in any direction in which there is substantial asymptote and thelines pa = t a n a! (broadside radiation) radiation. Typical valuesof axial ratio recorded a t angles and from the axisof a 20" antenna are shown in Fig.14. T h e Pa ka lon7er curve is indicative of the increased beamwidth and -- - l + increased energy radiated in directions approaching that tan Q tan CY perpendicular t o t h e axis of the antenna as the rate of (end-fireradiation)havebeenplottedasthedashed wrap a! is decreased. curves in Fig. 10. The conical log-spiral antennas do not havea unique The radii ax- and a15+that outline the active region center of phase,howeveroveraportion of the main have also been plotted for 2" and 20" cones. Although beam an "apparent phase center" may be defined [16]. this is only an approximate indication of the relative As shown in Fig. 15, the apparent phase centers of the phasing of the radiating aperture, it clearly indicates the smallerconeantennas lie wellbelow theslow-wave phenomena observed in Fig. 9. Recalling the amplitude region boundary on the Brillouin diagram. For 200 5 20" distribution from Figs. 4 and 9, we see t h a t for the 20" the position of the apparent phase center can be given cones the major part of the radiating aperture of the by the straight-line approximation in Fig. 15. In terms tightly wrapped antennas is phased for backfire radiaof the cone radius this m a g be expressed as: tion. From these we get relatively narrow well-formed 1 1.2 sin a beams.However, as theangle a decreases,thenet alx r result of the1radiation from the wider active region is a 2n 1.4 cos 01 ~ + + IEEE TRANSACTIOA'S OAT dATTENAiAS AND PROPAGATION 494 July .x) - - 3.5 x.,,; .I5 - 10- .os I I M 0 SPIRAL , 5007 > t- a 100; I .IO CONSTANT . ......... . . .I5 b ACTIVE REGlMU , - . . . .<pH*= ..................... . . ..... .............................. a=45' . . 1 0 DISTANCE ALONG CONICAL SURFACE L"/A) , 3 0 4 0 SPIRAL Fig. 9. Relative amplitude and phase of magnetic fields measured along the surface of conical antennas (260=15, 6=90°), and corresponding far-field radiation patterns. Fig. 1I. 5 0 MIOLE m m m 3 0 a Averagehalf-powerbeamwidth andapproximate tivity of the conical log-spiral antennas (6=90°). direc- Theheavyweighting of theportion of theaperture nearest the apex of the cone, due to the high amplitude fields present there, tends t o keep the phase center in thisgeneralregioneventhoughtheradiationregion, albeit a t lower amplitudes, extends considerably farther down the antenna. THEANTENKA FEEDSYSTEM SPIRAL ANGLE a Fig. 12. .Approximate variationinhalf-powerbeamwidthtobe expectedwhen antenna is operatedoverwidebandwidths. 28, = 15" 28. = 20° 29, = 300 30 25 20 I5 dB IO 5 0 90 8 0 180'0 90 1880 6 90 l80"O 90 6 180" 6 ANGULAR ARM WIDTH Fig. 13. a Minimumfront-to-backratio of radiationpatternsas function of ea, a,and 6. 2.0, 6 ANGLE e IN DEGREES Fig. 14. Typicalvariation of axial ratio off the axis of antenna. Measured for one antennawith 200=20", 6 =go". .4 - .3- \ \ 0 .I .2 .3 .4 .5 .6 .7 .8 0 9 .9 Bo tan D Fig. 15. Measured "apparent phasecenters"forseveralconical antennas ( 200 = 20"). 1 LO The two-arm conical antenna is a symmetrical structure, and when excited in a balanced manner it radiates a beam on the axis of the cone without squint or tilt. I n all cases a tilt in the radiation pattern can be traced t o t h ephysical construction or to imbalance in the feed. The antenna may be excited b y bringing a balanced transmission line along the axis, to the apex of the cone, and connecting one wire to each arm. I t is preferable, although not necessary, that this be a shielded line. T h e presence of a metallic shield or cylinder o n axis has a minimum of effect on the antenna characteristics if the diameter of this cylinder (or metal cone) is no more than one third the diameter of the antenna at any point on the axis. The transition from an unbalanced coaxial line to a balanced line may be made by placing a balun inside the antenna if the balun is nonradiating and not affected by the fields inside the cone. Conventional baluns may fall into this class, but are very limited in bandwidth. The tapered line balun [I71 with its extremely wide bandwidth wou!d seem to be ideally suited for use with this antenna.When placedinside theantenna,however, there appears to be an interaction or interference between the fields around the balun and those of t h e a n tenna, with the result that a truly balanced feed is seld realized. For some purposes, however, the degradation of thepatternduetothe use of thisbaluncanbe tolerated. T h e excellent coaxial hybrids now on the market make satisfactory baluns [lS]. These units provide equal amplitude out-of-phase signalsat the side arms. Fifty-ohm coaxial cables connected t o these arms may be carried along the axisof the cone, and the two center conductors of the cables thus become a shielded balanced 100-ohm transmission line. For maximum symmetry, and to preventthetwooutersheathsfrombecomingatransmission line for any unbalanced currents, these cables should be bonded together or placed insidea metal tube orcylinder.Most of thesehybridsarelimitedtoan octave bandwidth and provision must be made for replacing or switching of hybrids. In the VHF and UHF range,however,low-powerhybridshaverecentlybecome available foruse over ten-to-one and greater bandwidths. To overcome the limiting bandwidths of the baluns, the method of feeding the antenna by carrying the coaxial cable along one arm was devised[14]. T o maintain phqFsica1 symmetry a similar cable must be placed on the other arm. This method, whichhasbeentermedthe "infinitebalun"feed,remainsthemostsatisfactory 496 IEEE TRANSACTIONS ONPROPAGATION ANTENNAS AND method if the physical presence of the cable does not limit the sizeof the truncated apex region, and if the loss in the relatively long length of cable can be tolerated. I t is a truly balanced feed under the condition that the antenna currents have decayed to a negligible value a t the truncated base of the cone and a symmetrical connection is maintained a t t h e feed point. INPUTIMPEDANCE OF THE lhTEXNd Ju2y ance levelsof these antennas have not yet been precisely defined. Figure 1 7 does indicate approximate levels for design andthetrend in thislevelwithangulararm width. The presence of cable on the arms tends to give the narro\v arms near the apex an effectively greater cross section and, hence, shifts the impedance level. An approximate curve of the measured input impedance for an antenna with280 = 20", a = 60, and a tip truncated at The geometry of the truncated tip, including thepossible presence of the feed cable on the antenna arms, has a markedeffectuponthecharacter of the input impedance of the antenna. Since there is an unlimited combination of cable diameter-truncated tip combinations, it appears to be meaningful to present only the impedance of the basic antenna when fed on axis, and indicatethetrend as youdepartfromthisbasic structure. Figure 16 indicates the effect that the feed geometry can exert on the measured impedance. SlTmmetry, balance, and tapered leads from the axis to the surface of the cone are required. The use of the coaxial hybrids as baluns makes possi- Fig. 16. Effect of the feed region geometry on the input impedance. ble a very convenient method of making balanced im500 pedance measurements, if identical slotted lines are inserted in the coaxial lines between the antenna and the hybrid [191. T o determine the impedance level of the antennas, measurements referred to a reference plane at the tips of the tapered leads on the truncated apex of the cone were made over a five-to-one bandwidth. The normalized impedances plotted on the Smith Chart were enclosed by a circle. The "center" of this circle, chosen t o make the hyperbolic distance toall parts of the circle a constant [20], was considered to be the characteristic impedance of the antenna. The maximum VSWR referred to this characteristic impedance was typically 1.3 t o 1.4 t o 1. The variation of the characteristic impedance with arm width is shown in Fig. 17. The input impedance of ANGULAR ARM WIDTH ( 8 ) the antenna is primarily controlled by the arm width, 17. Approximate characteristic impedance level of the conical varying from around 320 ohms, for the very narrow armFig.log-spiral antenna as a function of the angular arm width. antennas, to around80 ohms, for the corresponding very wide arm antennas. The impedance increases as the includedconeangleincreases,with thatforthe selfOr-----7 complementaryantennas (6 = 90") approachingthe theoretically predicted 60a or 189 ohms for the planar antenna [21]. T h e measured impedance variation with changeinthespiralangle a wassmall,althougha tendency for this impedance to increase with increasing a was observed. The apparently clear trend in impedance level with cone angle shown in Fig. 1 7 was not observed for all angles, and the impedance levels for cone angles less than 15" were not lower than those shown for 15". For example,for 280 = 5", a = 80", the characteristic impedance measured for 6 = 16, 90, and 164" u-as approxi- Fig. 18. Approximateratio of the radius at the baseend of the active region for very narron- or very wide arm widths to that for mately 300, 1S5, and 72 ohms, respectively. The impedthe self-complementary antenna. I I 6.16' t Q - Q .. Q Q Fig. 19. Typical electric field radiation patterns indicating general change in shape with cone angle eo, spiral angle a,and angular arm width 6. , 0.75 inch, fed with RG141jU cable of diameter 0.141 inch is shown. This curve indicates the general impedance level forthisparticularcable-truncationcombination only. The maximum VSM'R referred to this impedance level was approximately 1.75 t o 1. The tapered leads at the truncation are still required if the width of the arm on the surface of the cone a t this truncation is substantially wider than the diameter of the cable. The impedance bandwidth is consistently greater than the radiation pattern bandwidth and hence the active region was defined in terms of the latter. EFFECTOF ARM IVIDTH The angular width of the expanding antenna arms is defined in terms of 6 where 0 <6 <T. In nearlyall cases, thecharacteristicsof these antennas have been investigated for very narrow arm structures (6= 16"), for very wide arm structures (6= 164"), and for 6 =90". This latter structure is defined t o be selfcomplementary in the sense that the geometry of the arms and the space between armsis identical except for a rotation of 90" about theaxis of the structure. 'Lt:ith the exception of the impedance whichis directly related to the arm width, nearlyall d a t a presented thus farhave beenfortheself-complementaryantenna. Changes in the other characteristics do occur as this width is changed. For example, the rate of decay of the near fields becomes less as 6 departs from 90". Thus uI5+ becomeslarger. I n Fig. 18 a factor 114 indicating the ratio of the maximum radius required for veq- narrow or very wide arms to that required for arms with 6 = 90" has been plotted. For values of 6 between 90" and 16" or 16-1" linear interpolation from :71=1.0 to the indicated value should be sufficient. Since us- changes little compared to the change in uljf, this modification factor M can also be taken to be the ratio of the active region bandwidths B,,, for 6 = 16" and 164", to that for6 = 90". For 5 20" the variation with6 is less for a 2 55", and hence, for these small-cone fairly tightly wrapped antennas it is possible to use thin arms and the constant width wire or cable arms version of the antenna [3], with only a small change in characteristics of the radiation pattern. The change in the electric field radiation patterns with change in arm width is indicated in Fig. 19. T h e principalchange is a n increase in beamwidth (So t o 8" for a = SO") brought about by the increase in active region width. These patterns are typical of those to be expected when the antenna is operated a t frequencies such t h a t there is no distortion due to the truncation of the base or tip. As the frequency of operation is decreased and the lower edge of the activeregion becomes distorted by the truncated base, the amplitude of the back lobe will increase rather rapidly.As the leading edgeof the active region moves to the truncated tip, the pattern beamwidth may at firstbecome narrower and the pattern become rough. 4 n y lack of precision in construction a t t h e apes region will cause pattern tilt or distortion.A further increase in frequency may cause the pattern to broaden with a tendency to break into lobes. Radiation patterns for the very loosely wrapped antennas,a = 45", with self-complementar). width arms only are shown. As the arm width deviates from 6 = 90" for antennas with 2 0 0 2 15' and aS45", the pattern breaks into many lobes with a majorportion of theenergy radiated in the direction of the base. A decrease in the front-to-back ratio of the radiation patterns, as indicated in Fig. 13, may be noted as 6 departs from 90". AXTEXXA DESIGNAND COKSTRUCTION The antenna engineer is usually interested in designing an antennawhich has a given directivity or radiation pattern beamwidth and input impedance over a given frequencyband. A satisfactorydesignshouldbeobtained from the previously plotted data. The data was obtained for antennas constructedof 2-mil thick copper, etched on 10-mil teflon-impregnated fiber glass, with a balanced two-coaxial line feed carried along the axis of the cone. Hence, the data should be representative of the basic antenna structure. The use of a n infinite balun feed or a cable approximation to the expanding arms should modify the antenna characteristics in a predictable manner, since the cable gives an effectively greater width to narrow arms. Half-power beamwidths on the order of 40" t o 50" can beobtainedwiththesmallconeantennas.However, whenverywidebandwidthsare t o becoveredthese antennas may become quite long. Physical requirements, such as allowablelengthor height, may dictate thechoice of the wider cone angles and a compromise of antenna characteristics. The antenna length or height in terms of the longest wavelength of operation X L may be expressed as k _ --~ 1 XL 2 tan& ; --__ 3 . In constructing these antennas it must be realized that they are potentially extremely wide-band antennas, andareexpected to performwithconstantcharacteristics w e r these bandwidths. To do this, the active region must be able to scalet o a constant geometry and toconstantphysicalparametersexpressedinwavelengths at all frequencies. Hence, all such parameters should scale in size with distance from the apex. Symmetry, balance, and detail and precisionof construction are important. Violation of any of these principles can only lead to degradation of the antenca characteristics as the frequency of operation is varied. CONCLUSIONS T h e conical log-spiral antenna canbe considered to be a locally periodic structure, with slowly varying period. 4 study of the antenna in this context has led to ready identification of many of its characteristics. This study indicates that the fairly tightly spiraled self-complementary antenna tends to be the “optimum antenna,” in the sense t h a t it has the narroxest active region and, hence, the greatest operating bandwidth for a given physical size, and the most compact symmetrical radiated beam with the least energy radiated in back lobes. The input impedance, however, depends primarily upon arm width, with typical valuesof 140 t o 165 ohms for these self-complementary structures. -4CKSOWLEDGhiEXT The author gratefully acknowledges the assistance of A. L. Davidson, G. R. Fruehling, 31. D. Johnson, V. P. Rash, R. J. Kopczyk, and other members of the University of Illinois Antenna Laboratory,x7ho helped with themeasurementprogramcarriedonduringthisinvestigation. REFERENCES [l] Dyson, J. D., Frequency-independentantennas-survey of development, Electronics, 1-0135,Apr 20,1962, pp 39-44 (includes extensive list of early references). [2] Jordan, E. C., G. A. Deschamps, J. D. Dyson, and P. E. Mayes, Developments in broadband antennas, ZEEE Spectrum, vol 1, Apr 1964, pp 58-71. [3] Dyson, J . D.,The unidirectionalequiangularspiral antenna, I R E Trans. on Antenms and Propagation, vol AP-7, Oct 1959, pp 329-334. [4] Mayes, P. E., G. A. Deschamps, and \V. T. Patton, Backwardwave radiation from periodic structures and application to the design of frequency-independentantennas, Proc. I R E (Correspondence),vol 49, May 1961, pp 962-963. [5] Oliner, A. .I., Leaky waves in electromagnetic phenomena, in Electron~ugneticTheory and Antennas, E. C. Jordan, Ed. Iiew York: Pergamon, 1963, pp 837-856. Rumsey, V. H., Propagationover a sheet of sinusoidal mires .ibid., and its application to frequencyindependentantennas, pp 1011-1030. glittra, R., and E;. E. Jones, Theoretical Brillouin ( k - 0 ) diagram for monopole and dipole arrays and their application to logperiodic antennas, 1963 ZEEE Iizternat’l C o m . Rec., pt 1, pp 118-128. Hudock, E., and P. E. Mayes, Propagation along periodic monopole arrays. 1963 G-AP Internat‘l S p z p . Digest, pp 127-132. Klock, P. \V.,and R. Mittra, On the solution of the Brillouin ( k - p ) diagram of the helix and its application tohelical antennas, 1963 G-AP Internut’l Symp. Digest, pp 99-103. Ishimaru, A., and H. S. Tuan, Theory of frequency scanning of antennas, I R E T a n s . on Antennas and Propagation, vol 4P-10, Mar 1962, pp 144-150. Dyson, J. D.,The coupling andmutual impedancebetween conical log-spiralantennas in simple arrays, 1962 I R E Znlewut’l Corn. Rec., pt 1, pp 165-182. Dyson, J. D., and G. L. Duff, Kear field measurements on the conical log-spiral antenna, 1963 G-AP Ifzternat’l Synzp. Digest, pp 137-142. Carrel,R.,The design of log-periodicdipole antennas, 1961 I R E I n t e n d l Conv. Rec., pt 1, pp 61-75. Dyson, J. D., The equiangular spiral antenna, IRE Trans. on Antennas and Propagation, uol AP-7, .lpr 1959, pp 181-187. Stegen,R.J.,Thegain-beamwidthproduct of anantenna, I E E E Trans. o?z Antennus and Propagation (Correspndence), vol XP-12, Jul 1964, pp 505-506. Dyson, J. D., and R. E. Griswold, Measurement of the phase centers of antennas, Tech Rept 66, Antenna Lab., University of Illinois, Urbana, Dec1963. and 1‘. P. Minema. 1OO:l bandwidthbalun Duncan, J. ij:., transformer, Proc. I R E , vol 48, Feb 1960, pp 156-164. Alford, A,, and C. B.\4:atts, Jr., -4 wide band coaxial hybrid, 1956 I R E COHV. Rm., pt 1, pp 171-179. Dyson, J. D., Balanced impedance measurements with a coaxial of Illinois, Ursystem, Tech Rept,AntennaLab.,University bana, to be published. TeleDeschamps, G. A., X hyperbolic protractor,Federal commuaication Labs., h-utley? X . J.; 1953. 1957 I R E Rumsep, V. H., Frequencyindependentantennas, ITat’l Cono. Rec., pt 1, pp 114-118. Antenna and Wave Theories of Infinite Yagi-Uda Arrays R. J. ?dL$ILLOUX,SENIOR MEXBER, IEEE , A b s t ~ ~ ~ t - Aintegral n equationforthepropagationconstant along an M t e l y long Yagi structure is derived by expanding the vector potential function for such an array in terms of the spatial be harmonic solutions of wavetheory. This equation is shown to on the basis of array identical with the integral equation derived theory and transformed by the Poisson summation formula. this newwavetheory Withtheidentity of arraytheoryand formation established, the wave theory is used to discuss allowed wave solutions and the physical characteristics required of dipoles in order that they support a wave solution. The fundamental integral equation is solvedby means of the array theory of King and Sandler; the numerical results are found to agree quite well with previously published data. Finally, the problem of two parallel nonstaggered Yagi arrays is considered,andit is shown that the propagationconstant of the composite structure either decreases or increases over that of the isolated array depending upon whether the symmetric or the antisymmetricmodeisexcited.Somepeculiareffects are notedwith IiIanuscript received September 14, 1961; revised March 1, 1965. respect tothis antisymmetric solution, and these lead the to existence The research in this paperwas supported in part by National Science of conditions under which no unattenuated wave solution is possible. Foundation Grant GP85l and in part by joint Services Contract This is referred to as the“cutoff condition.’! Nonr 1866(32). Numerical results are acbiaved which agree very well with exof A3pplied The author is with the Gordon McKay Laboratory perimental data obtained a s part of this research. 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