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Two- and Three-Loop Superdirective Receiving Antennas Elwin W

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Two- and Three-Loop Superdirective Receiving Antennas Elwin W JOURNAL OF RESEARCH of the National Bureau of Standards-D. Radio Propagation Vol. 67D, No. 2, March- April 1963 Two- and Three-Loop Superdirective Receiving Antennas Elwin W. Seeley Contribution from U .S. Naval Ordnance Laboratory, Corona, California (R eceived June 11 , 1962; revised August 2, 1962) The characteristics of two- and three-loop superdirect ive a ntenn a arrays arc presenled . At VLF, t his type of array appears to h ave m any d esirab le qua li t ies, and the usua l dctr'i­ mental ch a racteristics associated wi th superdirectivity are less in evidence. It is shown that the beamwidth is n arrowest, the front-to-back voltage and power ratios m' e greatest, a nd the position of t he back lobes and nulls are most invaria nt whf'n closel.v spaeed loops are used . Inequalities in signals from the individua l loops tend to obscure t he fr ont a nd back lobe' and limit t he proxim ity of the loops. List of Symbols E q,= relalive volLtLge received frorn di rec tion </> compared to voltage Crolll one loop. </> = angle of received sign<Ll in the horizontal plane f!"om the vertical phtll e in ",hiclt th e loops are located. 2'/fD cos rf> 1/; =- A D = distal1ce beLween loops A= free space wavelength - il = phase delay between loop </>0 = null position </>l= position of side lobe maximum R1= ratio of front lobe to side lobe ampliLude Ro= ratio of Jront lobe to back lobe amplitud e Rp=ratio of power collected by the fronL lobe to thaL colleeled b ~- the back lobes I = current in the loop antenna 77 = 1207r, intrinsic impedance of space A = area of loop {3 = 27r/A 8= angle in the plane of the loop r = radius of loop ro = radius of wire used to maIm loop ZL= input impedance of loop VL = loop voltage E,=free space radiation field. 1. Introduction At very low frequencies (VLF) a large land area is required to obtain high resolution and unidirectional antenna patterns with conventional arrays. It appears that the array can be greatly reduced in size by using the principle of superdirectivity. Several authors have discussed the factors that limit superdirectivity in practical antennas [1'aylor, 1948; Wilmotte, 1948; Riblet, 1948; Yaru, 1951 ; di Francis, 1956 ; Stearns, 1961]. It is limited by ohmic losses, narrow bandwidth, and critical tolerances of antenna parameters. Superdirective receiving antennas are realizable at VLF because these limiting factors are less in evidence. There is very little coupling between loops used at VLF so the bandwidth is not 215 narrowed and the ohmic losses are not increased when the loops are used in an al'l'ay. The tolerances of the loop voltages are critical and will be discussed later. In this paper several characteristics of two- and three-loop arrays are derived and presented. 2. Radiation Pattern Characteristics The important chaTacteristics of the patterns of two- and three-loop arrays will be derived. The positions of the side lobes, and nulls, beamwidth, ratio of side and back lobes to front lobe, and ratio of power collected by front lobe to that collected by back lobes will be pre­ sented . Some of the pattern characteristics of the two-loop aITay have been derived and experimentally verified earlier [Friis, 1925]. This paper will be an extension of the work done b.,- Friis. 2.1. Two-Loop Array T he pattern of the two-loop arr a~- is easily derived by pattern multiplication from the IJattern of a loop and that of a two-element isotropic array [Kraus, 1950a]. Consider two identical loops with (0 -'71-) phase shift between them, oriented in line in a vertical plane such as loops No. 1 and No.2 in figure 1. If the loops are receiving verticall~- polarized energy. the radiation pattern in the horizontal plane is given by (18A) in appendix 5.1 which is (1) FIGURE 1. Loop antenna arrab. (a) Physical configuration. ( b) Radiation pattern. ) 0 .707 ( b) 2700--------=:,";;'F~------900 216 a. Null Positions The null positions ill Lbe pattern may be used to exclude unwanted signals. There ~l j' e rOUl' nulls: one each at <7> = 90 ° fwd <7> = 270°, and two between the side lobes and the back lobe as shown in figure 1 b. Oilly the null positions between the back lobe and side lobes will be discussed, since the othel' two arc fixed . Null positions between the side lobes were derived in ftppendix 5.1 and rrom (23A) (2) Equation (2) is plotted in figure 2 which can be used to position the nulls in a loop <H'l'ay. b. Back Lobe Positions A picture or the radifttion pattern would not be complete without a knowledge of the side Jobes fwd the back lobe. The side lobes are the two lobes whose peaks are situated at equal an gles on either side of ¢ = 180° and the back lobe is the lobe whose peak: is at <7> = 180°. If (22A) ilnd (2:3A) are s ubstit uted into (1) a volLage pattel'll equaLion results in te rms or Lhe null position selected, E q, = 2 cos <7> sin [71'~ (cos <7> - COS <7>0 ) ] j-(tp)- (3) Only t he amplitude is needed, IE",I = 2 cos ¢ sin [71'~ (cos <7> - COS <7>0) J (4) The positio ns of the maxima of all back lobes can be derived from (4) , by difl'eren tinJing E q, \\'ith respect to <7> <wei setting the results equal to zero. Two equalions are obLftined from which tllJ'ee b ack lobes Cftn be asc~rta in ed. 6 cos tpo = 2nD/>- ~ 1.0 "e; o~GOIS • .... N «> - ..J V> '"o 0a. B g .... z ~ ~ 0 .5 "'z ....;< - '" ...o CD Q ~ .... ..J « '" '" 0 120 150 18 0 210 240 NULL POSITION, CPo (DEGREES) FIGUHE 2. N ull po.,ition as. a function of delay. 217 sin </> = 0, (5) find (6) Equation (5) yields one back lobe maximum at 180°, and (6) yields two more side lobe max­ ima, since there is symmetry about </> = 0°. A graphical solution of (6) for three different. null positions shows that the side lobe positions change very little with D/ A.. These solutions are plotted in figure 3. For small loop spacings (D/ A. < O.l ) (6) reduces to 2 cos ¢ 1= cos ¢o. (7) vVhen the null position has been selected, the corresponding back lobe position may be com­ puted to a fail' degree of accuracy especially at VLF frequencies where the wavelength is very long. c. BeaIDwidth A measure of the directivity of an antenna is its beamwidth. The half-power bearnwidth of tbe two-loop anay is determined by setting (8) and solving for </>.4, wbich is half the half-power beamwidtb. (9) When (9) is solved and plotted (see fig . 4) the first evidence of superdirectivity is revealed, for the narrowest front lobe occurs with tbe sJnallest loop spacing. The beam width is only 175,-------------------, 175 <jIo ::: Null Position ' 0 = Null Poa itlon .D 150 tan [T(.D C08 <P I - cos ~o )J = -T C06 <J> j 150 2+ A. :'!" H alf-Power Beamwldth 125 125 ¢ oOI BOo ,po: 150" w" 100 " 0 ~ 100 <Po: 120 ..,:- 2- ~ w 75 75 ~=======- m ~ w a ~ 50 50 25 25 o o 0.10 020 0 ' 0 040 0 50 °0~--~OL.I --~0~. 2---Q~3---~0.4---~0. 5 DISTANCE BETWEEN lOOPS ( WA VELENGT H ) DISTANCE BETWEEN LOOPS ( WAVELENGTH ) FIGU U E 3. Two-loop array: side-lobe positions. F IGUUE 4. T wo-l(j)0p a1Tay: beamwidth. 218 plotted up to a value of loop spacing where lobe splitLin g begins; at this point the front lobe splits into two lobes. This loop spacing depends upon the null positions, and occurs where .. dIE.. I · " L Imit --" lS posltlve. </>-)0 clA/> Lobe splitting begins at tbe following loop spacings : Djt-. = 0.364 for a null at 180° Djt-. = 0.385 for a null at 150° Djt-. = 0.461 for a null at 120°, d. Front-te-Back Lobe Ratios The front-to-back: r atio )llay well be the most important charll,cteristic of the two-loop array, for it shows to what degree signals from the real' (</> = 90° to </> = 270 °) are exclud ed. If (4) wi th </> = 0° is divided by the same equation with </> = </> 1, the position of the back lobe maxima, the lowest fron t-to-back voltfLge ratio is obtained 1'0[, the two lobes on either side of </> = 180°, (10) and for the sin gle bac k: lobe at c/> = 180°, E </>= o sin [7r~ (I -cos c/> o) ] --=R o= . (ll) E </>= 180o . s ill[7r~ (l + cosc/>o) ] The front-Lo-back ratios of back lobes c/> = c/>l alld </> = 180° nre ploUed ,lS a function of Djt-. for severfLl null positions and shown in fi gures 5a and 5b, The l \VO back lobes off </> = ] 80° are called the side lobes to distinguish them from the back: lobe at </> = 180°, These curves represent the lowest front-to-bac k ratios. At all other angles </> , the £ront-to-back ratio is greater. It is interesting to note that the solution fo[, H I and Ro ,,-hen Djt-. is VClT small is a fair approxim ation for a good range of Djt-. values, I -cos </>0 (12) , R I -cos </> 0. I1m '0- (13) .Q-)o 1 +cos CP o A When loop spacing is small there is an optimwn null position for t he greatest front-to-back ratio.
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