THE UNIVERSITY OF MICHIGAN OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR SYSTEM STUDY CONCERNING AN ANTENNA SUITABLE FOR A SPINNING-GONIOMETER DIRECTION-FINDING SYSTEM Technical Report No. 125 3697-3-T Cooley Electronics Laboratory Department of Electrical Engineering By: C. E. Lindahl Approved by: _ _ _ _ B. F. Barton Project 3697 Contract No. DA-18-119 sc-1357 U. S. Army Signal Procurement Agency Ft. George G. Meade, Md. August 1961

ABSTRACT Three antenna arrays suitable for a spinninggoniometer direction-finding system have been analyzed, and their relative advantages and disadvantages with respect to a standard, four-element Adcock system have been studied. An eight-element configuration using two signal goniometers was found to be superior to any of those studied. A double Adcock array consisting of two such eight-element configurations is proposed. It is shown that the bearing error of the double Adcock is on the order of a tenth of a degree, and the sensitivity factor exceeds that of the standard, four-element Adcock by an order of magnitude. ii

TABLE OF CONTENTS Page ABSTBACT~~.*~.. r. ~....*.0...............**..... * r ~ ~ c ii LIST OF ILLUSTRATIONS.................................... iv 1. INTRODUCTION....................*.0*............ 1 2. BASIS FOR COMPARISON.............................. 1 3. CONFIGURATION NUMBER 1............................. 3 4. CONFIGURATION NUMBER 2....................................... 9 5. SENSITIVITY CONSIDERATIONS.................................... 13 6. THE DOUBLE ADCOCK ANTENNA SYSTEM............................ 19 7. SUMIMARY AND CONCLUSIONS....0 0.............. 27 APPENDIND 00 IX........... *. ~.~28 iii

LIST OF ILLUSTRATIONS igurPage 1 Plan view of Adcock antenna 2 2 Octantal error curves for Adcock antenna 4 3 Plan view of Configuration No. 1 5 4 Transformer connection for Configuration No. 1 7 5 rTwo-goniometer connection. The rotors are tied together and phased such that their induced voltages are additive 7 6 Octantal error curves 8 7 Plan view of Configuration No. 2 10 8 Goniometer connections and rotor positions used for Configuration No. 2 10 9 Error curve 12 10 Sensitivity factor; Adcock antenna 14 11 Sensitivity factor; Configuration No. 1 16 12 Sensitivity factor; Configuration No. 2 18 13 Plan view of double-Adcock Antenna System 19 14 Sensitivity factor; standard AN/TRD-4 Adcock; high band 22 15 Sensitivity factor; double Adcock; high band 25 16 Sensitivity factor; standard AN/TRD-4 Adcock; low band 24 17 Sensitivity factor; double Adcock; low band 25 iv

SYSTEM STUDY CONCERNING AN ANTENNA SUITABLE FOR A SPINNING-GONIOMETER DIRECTION-FINDING SYSTEM 1. INTRODUCTION In this study an attempt was made to determine an "optimum" antenna configuration for a spinning-goniometer radio direction-finding system. This system was to be optimized with respect to four quantities: (1) error, (2) sensitivity, (3) bandwidth, and (4) cost. The objective was an antenna configuration which had better sensitivity than the standard, four-element Adcock with a bandwidth of 2 to 30 Me and with bearing errors for all frequencies of less than 2 degrees. The various configurations were studied, first with respect to their inherent bearing errors, and secondly with respect to their relative signal sensitivities. 2. BASIS FOR COMPARISON The first step taken was the investigation of various ways of connecting antenna elements together; that is, would advantages be gained by connecting adjacent elements in series, parallel, or some other type of connection? As a basis for all comparison the four-element Adcock antenna was studied first. The plan view of this system is shown in Fig. 1. Let 0 be the phase center of the array, with a wave impinging in the direction town. Then the voltages induced in each element are: j(Wt + CPN) eN A e j(ct t)PS) eS = Ae

j(ut + RPE) e = A e j(Cot -C W) eW = Ae (1) N 9 IMPINGING WAVE d \W< S Fig. 1. Plan view of Adcock antenna. where: txd ON = PS = A 9 cos Trd cpE = cDW = d sin 0 (2) In the goniometer one forms eN - eS = +2j A ejt sin N eN e S -~ N (3) eE - eW = +2j A e sin (4) and the indicated angle of arrival, cp, assuming no errors are introduced by the goniometer, is found from:

eE e sin cpE sin ]sin tan qp = (5) eN -eS sin cpN sin d- cos e:gd If A is small: d sin e tan c= d. c = tan e, and = the true bearing. cos e However, in order to obtain the greatest sensitivity at the lowest frequency received, d should be as large as possible (within practical limits). The array diameter, d, will be limited by the greatest error which can be tolerated at the highest frequency to be received. It is of interest, then, to calculate the error for various spacings, d, and to plot (cp - e) vs. 8 for various relative spacings, d/x. This is shown in Fig. 2. 3. CONFIGURATION NUMBER 1 The first configuration investigated is shown in Fig. 3. Let O be the phase reference point and e be the angle of the incoming plane wave measured clockwise from a center line passing through antennas 1, 5, 6, and 2. The idea here is to add the voltages of antennas 1 and 5, and those of 6 and 2, and subtract (6 + 2) from (1 + 5) to form the northsouth antenna-pair voltage. In like manner, the east-west antenna-pair voltage is formed.

-8 = ERROR CD c,- DEGREES ~.. B~.o o o o o 0 H' 43 6. 4 63 CD O ( CD (D H _!11, U H * I I ~ I ~ + I- IItI II1 1 ( 1 ( c- c - C). H. CDt If (D CD CD CD C - 0 Cj. C-o C. _,D ~ ~ e e+ +H'' " _ CD It o1jL fo 3 - - (3 1 (,, 4'm., v,., O::)n R) H — H'P CD CD C I.) IIt~~~~~~ I I m mO~~~~~~~~0 H."' (D ( C4X) — 4,, cl I C1, D- p' I I II I I II ItT CDf;~ fD 1 D CD CD CD o) 1 PI IfI + I + 0 I CD_ s_ oCDJ+ FJ

56 8 0 72 Fig. 35. Plan view of Configuration No. 1. It can be shown that: jel e~zeg + CP5 C - 5, (e 1+ e) (e2 + e6) = +4jAej~t sin.2 cos (8) and c3 + cp7 c(e3 + e7) - (e4 + e8) = +4jAejit sin c 7 cos 3 (9) The indicated bearing, cp, is: 3 cP 7 F-P7 sn(dli+d 2) cos (dl - d2) sin 2 cos 2 sin L 2A sin cosL 2 sin tan cD 2- -L 2 = sin ____ CP1 CP5 (dI +d.2) (d1-d2) sin 2 2 sin 2 c cos 2 L Letting now d2 = kd., where O. k 5 1: sin [(1 + k) sin ] cos [(1- k) 2 sin () sin [( l+k) cos e cos [(-k) cos ] 5~~~~~~~~~o

A few observations can now be made. Consider first antennas 1, 2, 5, and 6. The voltages of antennas 1 and 5 can be added together through the use of transformers, as can those of antennas 6 and 2. The connection is shown in Fig. 4. The subtraction of (e2 + e6) from (el + e5) is obtained when the proper connections are made to the goniometer. Actually the above operation [the subtraction of (e2 + e6) from (el + e5)] can be thought of in a different way. e2 and e6 can be subtracted from eI and e5, respectively, and the resulting voltages can then be added. This operation can be obtained using two goniometers (see Fig. 5). If the transformer scheme were used for implementing Configuration No. 1, eight transformers and one goniometer would be required; whereas only two goniometers would be required for implementing it if the goniometer connection were used. A polyphase goniometer of proper design could also be used. If Eqs. 3 and 4 are compared with (8) and (9), it is seen that the output voltage of the system is doubled. The above analysis does not, however, take into account mutual effects between antennas, but it does appear that something can be gained by using eight antennas instead of four, as far as sensitivity is concerned. If k is set equal to 1 or 0 in Eq. 10, the antenna system degenerates into the ordinary four-element Adcock system. If dl/A is set equal to 2/3, error curves are as plotted in Fig. 6 for k = 1/2, k = 1/3, and k = 2/3. By comparing these curves with those of Fig. 2, it is seen that for a spacing of dl/X = 2/3, the error has been reduced by approximately 5o. Clearly, we appear to be moving in the right direction. If we examine Eq. 10, we see that it can be divided into two multiplicative terms: 6

5 6 2 N S GONIOMETER COIL Fig. 4. Transformer connection for Configuration No. 1. 2 5 6 S N N 101 CO,M 00 S GONIOMETER GONIOMETER NO. I ROTOR NO. 2 w E W E Fig. 5. Two-goniometer connection. The rotors are tiedl together and phased such that their induced voltages are additive.

u~tot4s aq Aou'TTf se GaseD aq. papuT ST sTF s *T'oa, T1q uePq. Do aqq. xoj: 4oajzLoD pTlnox suuaj GATWOTTdTq-Tnm:q; JOa auo auo;B A ~e Rw ns UTl sB uua; -uen aq; aouTcd o; aGq~ssod aq Gce Ru T WW;S s1-sa92hns 4J STV'1TaTAT4flnquI (TT) P3L _ _ _ _ _ _ __ [ -" ~, (_ T)]SOo TSOD - (.i + T)]US.[.. UfS.Z (-. ]..... S (..)] U~SGA@AmD:oIJa T:;:ue 30' 9' 2T. Z3 p?uLaiRq paGDTpu: = oo ( - T)so] [eS (N + T )] US [ S -Z (- [ )SOD] [- S (T + T)] u:S (saax2p uT) Dulaeon Gnx; = S338930 8 06 0o OL e 09 0__og_ 01 0 01 0 g- -em m

We shall use the two-goniometer method for taking the sums and differences of the proper antenna voltages. It appears from the above that if the inner four antennas are rotated by 450 in a counterclockwise direction about the phase reference point,and the rotor of the goniometer associated with these antennas is rotated such that its null position for a signal arriving from North is the same as for the goniometer associated with the North-South, East-West Adcock array, then error cancellation will occur. Actually the inner array diameter should be increased until all antennas lie on a circle with diameter d. This antenna array will now be analyzed and will be called Configuration No. 2 (see Fig. 7). 4. CONFIGURATION NUMBER 2 The induced antenna voltages are: j(Ae t + cpl) j(wt + CP5) el = Aae e Aae j((A t - c2) j(ct - c6) 2 = Aae e6 Aae (Wt + (P 3) j (Ot +'77) e = eA e e A Aae 3 a 7 J (wt - cp4) j (Wt - cP8) e4 = Aae e8 = Aae (12) where: A 3d gd 7c~z = f2 = Ecos e = 6 = cos (e - Cp = =:dsie 7 = =- sin (0 - I) (13) 3 = 4 %- 7I Since the goniometer rotor positions and connections are important, the conventions used in the analysis are shown in Fig. 8. The input voltages to the goniometers are: 9

4'4/' d 4 Fig. 7. Plan view of Configuration No. 2. \ct~ jN9l ~ +94 + Vg2 rV34 3 2 X P V78 A 4-X — V34 V78 1 V12 2 5 V56 6 Goniometer No. 1. Goniometer No. 2. = mt where om rotation rate of goniometers. Fig. 8. Goniometer connections and rotor positions used for Configuration No. 2. 10

V12 = Re[e1 e2] = -2 Aa sin Cp1 sin wt V34 Re[e3 - e4] -2 Aa sin p3 sin ~t V56 = Re[e5 - e6] = -2 Aa sin (p5 sin Ut V78 n Re[e7 - e8] = -2 Aa sin q)7 sin wt (14) The.complex output voltages of the goniometers are: Vg =Ag2 fJRe[e5 - e6] e + Re[e7 - e4 ] e } (15) where: Agl A A2= Aare constants depending upon the goniometers used. Let VT = output voltage fed to the receiver = Re Vgl + Vg2} It can be shown that: Re V = -2 AgAa sin wt(sin cp1 sin(Z+ sin p3 cos Z] gl ga (16) Re V = -2 A Aa sin wt[sin c5p sin (+) + sin cos )] g2 ga7 cos ((D+ Therefore VT = 2 A Aa sin wt ([sin 1 + sinn 5 sin sin + [ sin p5 + sin 3 + sin7 ] cos (17) The indicated angle of arrival can now be determined from II

sin cD +- sin c)7 + sin cD tan = 1 1 -s in cp sin T7 + sin s pl or sin -- cos(e - ) + sin (- + j2 sin LT sin 0 sin[cos(e )] - sin'd sin (e -i)] + sin cos e] (18) If we let now d/K = 1, the bearing error, cp - 0, can be calculated; this is shown in Fig. 9. It is clearly seen that a large decrease in error is achieved using this scheme, and it appears that a considerable improvement has been achieved. Ui X |d/X I1.U o 0I' __ _ __ Lr -6 0 10 20 30 40 50 60 70 80 90 8 DEGREES -- 0 = true bearing = indicated bearing sin[ %I cos (e - + sin sin ( - + s sin[ sin i] tan m = sin[;. cos (e- - sin[;. sin ( - )] +%Isin[ cos 0] Fig. 9. Error curve. 12

5. SENSITIVITY CONSIDERATIONS One must, however, also consider the sensitivity of the various configurations. A figure of merit of the sensitivity is given by a quantity which we shall call "sensitivity factor." This is a measure of the magnitude of the magnetic field set up inside the goniometer by a signal impinging upon the antenna array. In general it is a function of the bearing of the incoming wave and the antenna array being used. As before, the four-element Adcock antenna was used as a basis for comparison. The output voltage of the signal goniometer of the standard Adcock antenna is given by: VT = -[2 Aa Ag sin cPN sin D+ 2 A Ag sin PE cos()] sin Ot (19) where ( is the instantaneous goniometer position as defined in Fig. 8. The sensitivity factor can be immediately determined from this expression and is defined as the square root of the sum of the squares of the coefficients of the sin D and cos ~ terms; hence, Sensitivity Factor = S.F. = 2 Aa Agsin2 cos o] + sin2 [d sin e] (20) This equation was plotted as a function of 0 in Fig. 10 for values of d/k = 1/5, d/7 = 1/2, and d/j = 2/3. The quantity 2 A A was set equal ag to 1 to normalize the result. It can clearly be seen from this that one must consider the sensitivity of the array as a function of the angle of the incoming electromagnetic wave. For the standard Adcock antenna configuration 13

320 340 0 20 40 VZ`'''',/ I 260~~~~~~~~~~~~~~~~~~~~~~2 \ \~~05 220 200 180 160 140 =-8DEGREES Fig. O. Sensitivity factor; Adcock antenn. * ii \~!

the greatest overall sensitivity occurs when d/A = 1/2. When d/A > 1/2, it is true that the sensitivity is increased for some directions, but in others it is decreased below that at d/k = 1/2. The output voltage of the goniometer for Configuration No. 1 is: VT =-4AA sin(l + k) sin ] cos [(- k) 2 sin (] cos CD T a 2e 2J aid Tcd (21) + sin[(1 + k) 2 - cos e] cos [(1 - k) cos e] sin sin wt From this the sensitivity factor is given by: S.F. = 4AaAg A +B2 (22) where: Al = sin[(l + k) 21 sin ] cos [(1- k) 2 sin ] and B1 = sin[(1 + k) 21 cos ] cos [(1 - k) 2T1 cos ] For k = 1/2 and dl/X = 2/3, Fig. 11 shows the variation of sensitivity factor as a function of arrival angle, e; again, 2 AaAg was set equal to 1 for comparison purposes. Here we see the same type of behavior as in the case of the standard Adcock. The sensitivity factor is approximately twice that given by Eq. 20. It must be kept in mind, however, that the mutual interaction of the antennas was not taken into account. Configuration No. 2 has the following output voltage: VT =2 AaAg{A2 cos + B2 sin } (25) 15

320 340 0 20 40 300 220 200 180 160 140 *- 0 8DEGREES Fig. 11. Sensitivity factor; Configuration No. 1. 16

where: A =.sin coS + sin[( sin (e - )] + sin[r sin e] and B = -sin- co (s - )] L -si sin (e - ) + sin[ cos ]. The sensitivity factor is obtained from this equation and is given by: 2 2(24) S.F. 2 AaA +B2 (24) where A2 and B2 are defined as above. Figure 12 shows a plot of Eq. 24 as a function of e with d/7 = 1 and 2 AaAg = 1. From this figure it is seen that not only is the sensitivity greater than that of the standard Adcock, but it is more uniform as a function of the angle of arrival, e, of the electromagnetic wave. In summary, three antenna systems have been studied with respect to sensitivity and inherent errors. Of the three, Configuration No. 2 has the greatest merit and shows the most improvement over the 4element Adcock. Incidentally, it is interesting to note that this antenna system is currently being used in the U. S. Navy GRD-6 directionfinding system; however, as far as the author knows, no detailed analysis of the system has ever been published. Indeed, it has many features which were not initially obvious when the study was first begun. The remaining part of this report will deal with further details concerning Configuration No. 2. 17

HO 340 20 40 v\ /~ ~~~~~~~~.520 280 80~~80GRE 240Fig 120niiiy atr oigainN.2 220 20 180 60 14 DEGREES Fig 1. enitviy acor CnfguatonNo 2

6. THE DOUBLE-ADCOCK ANTENNA SYSTEM The frequency band will be split into two parts: (1) from 2 to 8 Mc, and (2) from 8 to 30 Mc. The antenna configuration being considered for coverage of the entire frequency range consists of two rings of eight antennas each. Figure 13 shows a plan view of the system envisioned. Fig. 13. Plan view of double-Adcock Antenna System. The inner ring is to be used for the higher frequency range (8 - 3O Mc); the outer, for the lower frequency range (2 - 8 Mc). Let the diameter of' the outside ring be 100 feet and that of the inside ring 27 feet. As usual, the spacing is a compromise between the greatest 19

bearing errors at the higher end of the frequency band and the minimum sensitivity of the array which occurs at the lower end. It will now be of interest to determine the relative increase of sensitivity of this double Adcock configuration with respect to a four-element Adcock which has a maximum error of only one degree in the band being considered. Through the use of Eq. 5 it can be shown that a spacing of d/? = 0.204 gives approximately a one-degree maximum error for the fourelement Adcock. Hence, for the low band at 8 Mc: dlo = (0.204)(37.5) = 7.65 meters; and for the high band at 30 Mc: dhi = (0.204)(10.0) = 2.04 meters. Obviously, these values for the diameter of the high and low bands are unrealistic. It was decided, therefore, to compare the double-Adcock array shown in Fig. 13 with that of the AN/TRD-4A directionfinding set. For this array dlo - 33' and dhi = 18.83'. For these spacings = 0.574 for 30 Mc and = 0.268 for 8 Mc From the literature" one can obtain the values of maximum error for the above antenna spacings. They are: (1) for the low band at 8 Mc,Error = max (P - 0) 1.90; and (2) for the high band at 30 Mc, Error = - 0) max - 10 We shall now calculate the maximum errors which occur in the proposed double-Adcock array. From Fig. 9 it is seen that the maximum error for this array type occurs at approximately 11.25~ and at odd, integral multiples thereof. With e = 11.25, dlo = 100', dhi = 27', and through the use of hq. 18, it can be shown that for the high band (8 - 30 Mc) at 30 Mc: Errormax = (Co - e)max' 0.12~; and for the low band p. G. Redgment, W. Struszynski, and G. J. Phillips, "An Analysis of the Performance of Multi-Aerial Adcock Direction-Finding Systems," J. Inst. Elec. Engrs. (London), Vol. 94, Part III A, No. 15, 1947, pp. 751-761. 20

(2 - 8 Mc) at 8 Mc: Error max - e)max 0.11~. Clearly the results here are far superior to these achieved by the reference AN/TRD-4A antennas. A comparison must now be made of the relative sensitivities of the AN/TRD-4A antennas and the double Adcock antennas. This was accomplished by calculating the sensitivity factors at 2 and 8 Me for the low band and at 8 and 30 Mc for the high band of both antenna systems. The results are shown in Figs. 14, 15, 16, and 17. From these curves it is easy to compare the two systems. Table I summarizes the results with 0 = 0, since at this angle of arrival the sensitivity factor is minimum. This table gives the ratios of the sensitivity factor of the double Adcock to that of the TRD-4A antenna for the limiting frequencies of the two ranges of coverage. In all cases 2 A aAg was set equal to 1 in Eqs. 20 and 24. Frequency Double Adcock TRD-4A Ratio Low 2 Mc 1.213 0.209 5.8:1 Band 8 Mc 1.930 0.746 2.6:1 High 8 Mc 1.298 0.462 2.8:1 Band 30 Mc 1.896 0.973 1.9:1 Table I. Sensitivity factor comparison. Although Table I shows considerable improvement in sensitivity with the double Adcock system, this is not the whole story. One might well ask what value of dj/X would give the greatest minimum sensitivity. For the standard Adcock this occurs at d/X = 0.5, which can be seen in Fig. 10; however, the bearing error is too great for the system to be of any use 21

320 340 0 20 40 V"~ II i~~~~~~~i,'~~f"Sc' 30MC';'Z260 X I00 ~40 ~~~~~~~~~~~~~~~~~~~~~120 220 200 180 160 140 * —- 8 DEGREES Fig. 14. Sensitivity factor; standard AN/TRD-4 Adcock; high band. ~~~~~~~'?2

320 340 0 20 40 I;~~.'.~~'.::.0 ~ ~.~' %~ ~.. 230 0 60 Fig. 15. ensitlvlty factor, double Adcock" high band. 07 280 80 f=f8MC t 21.35MC 260 100 240,120 220 200 180 160 140 DEGREES Fig. 15. Sensitivity factor; double Adcock; high band.

320 340 0 20 40 300 60 O~~~~~ 60 I,;i~~~~~~~~~~~~~! 2 80 ( r_3 —--—`~il:~-~"~ —-"K\)~\ —nm)~3~Y - ~~~~Y~~~~I — ~180 f=2MC.... — f = 8MC 260 100 9 DEGREES Fig. 16. Sensitivity factor; standard AN/TRD-4 Adcock; low band. 24

320 340 0 20 40 300 ~~~~~~~~~~~~~~~~~60 280 ~~~~~~~~~~~~~~~~~80 240 ~~~~~~~~~~~~~~~~~~~~120 220 200 180 160 140 = 8DEGREES Fig. 17. Sensitivity factor; double Adcock; low band. 25 ~ iiI/,'~ J~X

to us. For the eight-element antenna, on the other hand, a spacing of d/x = 0.586 gives the greatest minimum sensitivity (see Appendix). This is useful in our case because this spacing occurs within the frequency ranges of interest. Figures 15 and 17 show plots of the sensitivity factors at the frequencies where the greatest minimum sensitivity occurs. For this type of antenna, then, the sensitivity increases from the lower frequency limit of the band covered, reaches a maximum at approximately the mid-frequency, and then decreases again when the highest frequency limit is reached. In contrast to this the sensitivity of the four-element Adcock increases monotonically from that at its lowest operating frequency to its greatest sensitivity at the high end of the band covered. The effect of sensitivitjy peaking near midband in each frequency range is another strong point in favor of the double-Adcock configuration. A few more points must be brought out with regard to the antenna configuration proposed. In all of the above, mutual coupling between antennas was not taken into account. In reality, of course, there will be mutual coupling taking place. For this reason it is imperative that symmetry of the array be maintained. Unless the antennas are equally placed on the circumference of a circle, bearing errors caused by mutual antenna coupling between elements will occur. Another problem which must be solved appears when the inner array is being used. Since the outer system will probably approach its inherent resonance at some frequency in the high band, something must be done to reduce the radiation coupling between the outer and inner antenna rings. It is believed that some sort of passive filter could be placed at the bases of the antennas in the outer ring which would absorb energy at their inherent 26

resonance points and which would not affect them in their own region of operation. Evidence to the fact that this can be done occurs in a report of a similar double-Adcock system manufactured by the Telefunken Co. in West Germany. It is true that such a technique will cause a shielding effect on the inner antenna ring, but one never gets something for nothing. All these effects will have to be evaluated experimentally to determine the actual, overall quality of the proposed system. 7. SUMMARY AND CONCLUSIONS Three antenna systems have been analyzed, and their relative advantages and disadvantages have been studied with respect to a standard, four-element Adcock system. An eight-element configuration using two signal goniometers has been found superior to any of those studied. It is proposed that two such eight-element configurations be erected in two concentric rings with diameters of 27 and 100 feet, respectively. The inner ring will be used for the higher frequency band (8-0o Mc); and the outer ring will be used for the lower frequency band (2-8 Mc). At the highest frequency in each band the bearing error is on the order of a tenth of a degree, and the sensitivity factor exceeds that of the standard four-element Adcock by an order of magnitude. Practical problems remain which will have to be worked out before the system configuration becomes wholly adequate, but it is believed that they can be solved. 27

APPENDIX Determination of the Value of d/X Which Gives the Greatest rMinimum Sensitivity for Configuration No. 2 From Eq. 24: S.F. = 2AA /A +B a g where: A = sin[ - cos( - )] + sin sin(8 - A) +sin sin 0] (26) A =""'L +sin in B = sinr cos( - )] 1le - +sin sin(A s ] (27) The minimum sensitivity factor occurs when e = 0. Therefore, (26) and (27) become: A O and B 2 sin[ ] + sin A (28) Consequently, the sensitivity factor becomes S.F. = 2 AgAa [ sin( )+ sin a (29) where: nd a = -- Taking the derivative of (29) with respect to a and setting the result equal to zero gives: Solving now for values of a which satisfy (h0) results in: 28

a 2 cos a + cos a = 0 2 (a + ) = 0w acos 2( ) 0 whe(n ~(+ )1 = + n m where n = 0,1,2,... and cos a 1 = 0 when -) = + n n where n = 0,1,2, Therefore a = (1 + 2 n) where n = 0, 1, 2,... (s1) 1 1 and a.n(l + 2 n) where n = 0, 1, 2,... (32) 12 From (51) it can be shown that a maximum occurs when n = 0; this results in d/) = 1 = 0.5858. All other maxima obtained from (31) and 1 + -- >12 (32) give values of d/A which are greater than one and, hence, are unuseable because the bearing errors of the antenna system would be intolerable. 29

UNIVERSITY OF MICHIGAN 3 9015 03483 1837111111 3 9015 03483 1837