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JOURNAL CF RESEARCH of the National Bureau of Standards-D. Radio Propagation Vol. 67D, No.5, September-October 1963
VLF Superdirective Loop Arrays Elwin W. Seeley
Contribution from U .S . Naval Ordance Laboratory, Corona, Calif.
(Rec('ived March 15, 1963)
SUjwrclirl'ctil·it.v may be ach iC'vecl with short VLF loop arrays bC'cause thl' bramlyidth d<'pcnds on l ~ ' upon the number of loops and not the length of the array. In a ddition the usual factors limiting slIpel'dircctivity a rc not so prevall'l)\' due to tl1(' (Ircoupling b('( I\'rC' n Vr,F loops, Expressions al'l' deril'ed for thl' bcamwidth, ofl'l'otivl' hl'ight, J'('cl'ption patteI'll, ampli tude and pOsillon of the back lobes and the dfeets of loop voltage phasC' and amplituci!' ciirrrrrncrs brtwl'l'n loop". Thesl' eCj uat ions dl',;cribl' short a rra.vs of any nUl1lbr r of loop';. The most s('rious limitation on thl' directivity of supcrcli rl'ctive loop a rra.vs is t 11(' I'ollagc pha"l' ancl amplitudl' ciiffNe ncl's bC'lwee n loops. These cliffC' rl'nces bl'tw('rn adjacent loops add up to obseul'(' tlw nulls anci dct('riorate tho reception pat terll.
List of Symbols Th ~ elemenLs or receiving a rrays at VLF can very easdy be decoupled from eacll other since they are E
1. Introduction 2. Radiation Pattern Characteristics
T he superdirec tive an tenna with the possibility The impor tant characteristics or the pattern of or infinite gain wi th an infinitesimal sm all an tenna the n-loop array will h e d educed from corresponding has h een discussed hy several au thors [Scbelkunoff , equations or t he t wo- and three-loop arrays. E q ua 1943: Schelkunoff a nd Friis, 1952; Rihlet 1948' Lions 1' 0 1' the position of t he side lobes a nd nu lls, the 'I'ayo I I' , 1948; Y aru, 1951; Stearns, 1961]. ' These , b Nl mwidth , a nd ratios of siele and hack lob e to au thors h ave pointed out that such arrays ar c il1l front lob e will b e presen ted. practica,l due to basic limitation such as nan ow The h orizo n tal p attern or a h orizontal a rray with bandwidth, hig- h losses, and cri Lica I tolerances. Lh e planes of t he loops orien ted in a vertical p lane :YI: oderate superdirectivity h as been achi eved [Ell iott, such as in figure 1 is [Seeley, ] 963] 1054; Spitz, 19.';9] wit h practical arrays. Spitz shows that superdirec tivity can he obtain ed from (1) radi ators t llat nre decoupled rl'O:::! one a nother. 563 for two-loop array and back half of the pattern to reject unwanted signals by varying the delay (0) between adjacent loops. The beamwidth is a measure of the directivity of 2 D IEq, 1= cos cf> [ : (cos cf> -cos cf>o) J"- (2) an antenna. The half-power beamwidth (2cf>A) of I the n-loop array in terms of the null position is for three-loop array, where D<< A and there is I - cos cf> o (o-n-) phase difference between identical adjacent cos cf>A = 0.707 [ cf> cf> In- l (5) loops. By mathematical induction the pattern of cos A-COS 0 n-Ioop array is when D<< A. The narrowest beamwidth occurs when the null position approaches 90°. Then IE q, 1= cos cf> [ ?~: D (cos cf> -cos cf>o) J"-1 • (3) (6)
,*, '900 o When the null position is at 180 0 the beamwidth is ---"'ILOOP 2 LOOP 3 broadest. The beamwidths at these two extremes are plotted in figure 2 as a function of the number of loops in the array. There is a spread of only about 10 0 between the extremes. The beamwidth is not a function of the distance between the loop and herein lies the possirJili ty of obtaining super directivi ty.
90
80 - 200
_-+--=- 1.0 70 Q. 0- o - 160 3 '0'" -60 w I z af- -140 a ~ o f- ~ 50 w -1 20 0 (IJ W 0: 0: ~ 'g40 -100 :2 a cF o ' 180 0 o Q. u f- -80 I ;;i'" 30 I '"W I - 60 w 1 = TOTAL L ENGT H Of ARRAY ~ 20 f- ( b) U he = EFFECTIVE HEIGHT - 40 w "- "- CPo = NULL POSITION w_ 10 -20 .c'" 27°o------=::~"""'::::::.------900 o 7 9 II 13 15 17 NUMBER OF LOOPS
FIGURE 1. Superdirective loop array. FIGU RE 2. Superdirective loop arrays.
Equation (3) i~ expressed in terms of distance T1w amplitudes of the back lobes are an indication between loops m wavelengths, D/A, and the null of the llndirectional properties of the array pattern. I position, cf>o. located between the back lobe and the There are three back lobes, one at cf>= 180 0 and two side lobes (see fig. 1). The null position depends symmetrically located about this one (see fig. 1). upon the delay between adj acent loops and the free The lobe at cf> = 1800 will be called the back lobe and space propagation time between loops the two lobes on either side of the back lobe are call ed the side lobes. The ratio of thE'. front to back lohe derived from (:3 ) is (4) I -cos cf> oJ n-l [ (7) , fo], the n-loop array. It may be moved about the R o= 1 + cos cf> o • I 564 It is obvious from (7) that a large number of loops to deteriorate t.he reception pattern as the number in the array cause the back lobe to be much smaller of loops is increased. The feasibility of a practical than the front lobe. The front-to-side lobe ratio array will depend on how accurately the individual can be derived also from (3) . But first the position loops and delay lines can be matched. This can of the side lobe maximum must be determined by only be determined experimentally. differentiating (3) and equating the results to zero. Having done this the position of the side lobes is 5. Conclusion Superdirectivity may be achieved with short VLF 4>1 = arc cos (~ cos 4>0)- (8) loop arrays because the beam width is not a function of the length of the array but the number of loops ill Equation (8) can now be used in conjunction with the array. Also , the usual superdirective limiting (3) to derive the front-to-side lobe ratio, which is factors such as narrow bandwidth rmd high losses are minute due to the decoupling between loops
l at these long wavelengths. The directivity is limited R1 =(-n ) n[ l -COs4>oJ n- . (9) cos 4>0 I -n by the critical tolerance of adjacent loop voltages. The effective height and all the pattern character From (7) and (9) it is obvious that the loop array istics of the short array such as beamwidth and patterns become very unidirectional as the number back lobe amplitude and position can be expressed of loops is increased. This is important. in applica in terms of the selected null position and the dis tions where the loop array is in a field of multiple tance between loops for a given number of loops. sources such as sferics at VLF. The assumption is made that the distance between loops is much smaller than a wavelength, which is valid at VLF. The directivity is increased by 3. Effective Height the number of loops used ill the array. Equations Quite narrow beams can be achieved with a (6), (7), and (9) bear this out in that the beamwidth moderate number of loops (see fi g. 2), bu t in return and back lobes become smaller for increasing num the effective height is reduced. The effective height ber of loops. (in db's) of an n-loop array derived from (3) is The two limiting factors on the directivit:v are effective height and unequal voltages between l oops. The very small effective heights of the shorter he=20(n-l) 10 g1 {2~D (I -cos 4>0)} (10) arrays could make them impractical unless very large loops are used. The most serious limitation This is the effective height compared to one loop. on directivity of arrays with large number of loops The effective height is plotted versus the number is the voltage amplitude and phase differences of loops for three array lengths in fi gure 2. The between loops. These differences between adjacent area between equal-array-Iength curves indicates loops will obscure the nulls and deteriorate the the range of effective heights resulting from posi reception pattern. Equation (ll) shows that the tioning the null from 90 ° to 180°. differences add up as the number of loops is increased. The very small effective heights of the shorter The feasibility of designing highly directive loop arrays could make them impractical unless very arrays will depend on the accuracy with which the high effective loops are used to make up the array. individual loops and delay lines can be matched. Large ground return inverted loops or perhaps short This can only be determined experimentally. Beverage antennas would be a practical element to use because of their large effective heights. 6. References Elliott, R. S. (Oct. 1954), A five-element super-gain dipole 4. Effect of Phase and Amplitude Errors array, Hughes Aircraft Co. Tech. Memo No. 380. Friis, H. T. (Dec. 1925), A new direcdonal receiving system, Proc. IRE 13. 685-707. The analysis above has assumed signals of equal Riblet, H . J . (May 1948), Note on the maximum directivity amplitude and the proper phase from each loop to of an antenna, Proc. IRE 36, 620- 624. cancel at the null angles. As the number of loops Schelkunoff, S. A. (Jan. 1943), A mathematical theory of linear arrays, Bell System Tech. J . 22, 80- 107. is increased any departure from this assumption will Schelkuno(t', S. A., and H. T , Friis (1952), Antenna t heory reduce the null depths. An analysis of the null and practice (John Wiley & Sons, Inc., New York, N.Y.). voltage with small loop voltage phase and amplitude Seeley, E. W. (Mar.- Apr. 1963), Two- a nd three-loop super inequalities has been made [Seeley, ] 963]. The directive receiving antennas, J. Research NBS 67D (Radio Prop.), No.2, 21 5- 235. resultant null voltage is Spitz, E. (June 15, 1959), Super-gain and volumetric antennas, Compagnie General de Telegraphie Sans Fil, technical final (ll) report, Contract No. AF61(052)- 102 AF CRC- TR- 59- 194. Stearns, C. O. (to be publishcd), Computed performance of moderate size super-gain . Taylor, T . T . (Sept. 1948), A discussion of the maximum which is the sum of the ampli tude differences be directivity of an antenna, Proc. IRE 36, 1135. tween adjacent loops in quadrature with the sum of Yaru, N. (Sept. 1951), A note on super-gain a ntenna arrays, the phase differences between adjacent loops. These Proc. IRE 39, 1081- 1085. voltage differences between adjacent loops will tend (Paper 67D5- 288) 565
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