L I B RAHY OF THE U N IVE.RSITY Of ILLINOIS 6ZI.3S5 D?655te no. 2- 14 cop3 Digitized by the Internet Archive in 2013 http://archive.org/details/analyticalstudyo10berr Antenna Laboratory Technical Report No, 10 AN ANALYTICAL STUDY of SPACED LOOP ADF ANTENNA SYSTEMS Contract No AF33(616)-3220 Project No 6(7 4600) Task 40572 WRIGHT AIR DEVELOPMENT CENTER by D G Berry and J B Kreer 10 May 1956 Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois 6M 2^ ABSTRACT The patterns of three multiple loop antenna systems (including a goniometer) are investigated analytically to determine whether these systems offer an improvement in polarization and reradiation error over a single loop when mounted on a long conducting cylinder with a circum- ference of approximately one half wavelength Goniometer patterns are presented and are summarized by plots of indicated versus actual direc- tion of arrival. The four element spaced loop system with all of the loop axes oriented in the direction of the cylinder axis is the most promising, since the polarization error is comparatively small. The four loop system with loop axes in the circumferential direction and a system with loops above and below the cylinder proved to have quite large polarization and reradiation errors The results, and partic- ularly the negative results, may be extended qualitatively to these antenna systems when mounted on missiles and aircraft. 1 1 CONTENTS Page Abstract H Acknowledgment Vl i 1. Introduction 1 2. Spaced Loop Systems Under Consideration 2 3. Methods of Analysis 4 4. Analysis of Results 17 5. Conclusions 32 Bibliography 33 Appendix I 34 Appendix II 38 iii ILLUSTRATIONS Figure Number Page 1. Illustration of a Multiple Loop System Having Opposite Loops Connected Out of Phase and Fed Through a Sine, Cosine Goniometer 3 2. A Spaced Loop Goniometer System Having Loops Mounted Above and Below a Wing, Fuselage, etc, 3 3. The Four Loop System of Figure One Mounted on a Cylinder with the Loop Axes in the Circumferential Direction of the Cylinder 5 4. The Four Loop System of Figure 1 Mounted on a Cylinder with the Loop Axes in the Axial Direction of the Cylinder 5 5. The Loop System of Figure 2 Mounted on a Cylinder 6 6. The Positions of the Loops When Mounted on Cylinders of Diameter X/10 and A/5 6 7. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 0=0° 8 8. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> - 10° 9 9. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> - 20° 10 10. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> - 30° 11 11. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> = 40° 12 12. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> - 50° 13 iv ILLUSTRATIONS (Cont ) Figure Number Page 13. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> = 60° 14 14. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> - 70° 15 15. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from = 30° 20 19. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Axial Direction of the Cylinder and the Exciting Wave Incident from 4> 50° 21 20. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Axial Direction of the Cylinder and the Exciting Wave Incident from - 70° 22 21 Truf; Bearing versus Indicated Bearing for the Four Loop System Mounted on a Long Cylinder (Axes Axial) 23 Goniometer Patterns of the Loop System Mounted Above and IV- low ;i Conducting Cylinder of Diameter A/10 and Exciting Wavf; Lk KJf-rit. from rf> ■ 10° 24 ILLUSTRATIONS (Cont.) Figure Number Page 23. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter A/10 and Exciting Wave Incident from 4> = 30° 25 24. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter A/10 and Exciting Wave Incident from (p - 50° 26 25. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter A/10 and Exciting Wave Incident from 4> = 70° 27 26. True Bearing versus Indicated Bearing for Crossed Loops Mounted Above and Below the Cylinder 28 27. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/5 with the Loop Axes in the Circumferential Direction of the Cylinder 29 28. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/5 with the Loop Axes in the Axial Direction of the Cylinder 30 29. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter A/5 31 30. Spherical Coordinate System 34 31. Locations of the Four Loops 34 32. Definition of Polarization Angle, a 36 33. Definition of the Coordinates 38 34. Definition of (3 42 v i ACKNOWLEDGMENT The spaced loop systems described in this report were originally proposed by Drs E. C Jordan, A D Bailey, and R. C Hansen. Prof, V, H. Rumsey suggested that a theoretical investigation be made of these systems when placed on an infinitely long cylinder. Dr. R. H. DuHamel was the supervisor of the work. v 1 1 1. INTRODUCTION The development of an antenna system for use in radio direction finding in the 3-18 mc frequency range is made necessary by the severe limitations that are inherent in the ordinary rotatable loop antenna in this range, The single loop, whether mounted on an aircraft or in free space, introduces an error in the RDF system whenever the exciting wave is not incident from the horizon This error is due to the voltage in- duced by the horizontally polarized portion of the field. Because ionospherically reflected waves are common in this frequency range, this polarization error is a serious enough defect to make the single loop unusable for airborne RDF, Another error arises from the fact that the longest portions of modern aircraft are long enough to be appreciable portions of a wavelength in the 3-18 mc range. Because of this, large currents are induced in the wings, fuselage, etc., by an incident wave. The fields reradiated from these currents induce voltages in the loop antenna which are not simply related to the direction of arrival of the incident field and, therefore, cause further error to be introduced into the system. The approach taken in an attempt to devise an antenna system which would provide reliable RDF information in the frequency range under consideration was to space a number of fixed, flush mounted, suitably interconnected, loop antennas about the aircraft. Although this idea is not new, recent work with ferrite cores indicates that it is now possible to get sensitivity adequate for RDF work from flush mounted loops, Flush mounting is, of course, essential on high speed aircraft. That a system of spaced loops would give an improvement in polari- zation error is suggested by the fact that two loops closely spaced in free space, connected in phase opposition and oriented so that they are either coaxial or coplaner, are free of polarization error. That is, the pattern of the loops maintains two of its nulls in the same position regardless of the polarization of the incident field. 2. SPACED LOOP SYSTEMS UNDER CONSIDERATION Two specific multiple loop systems have been under investigation. The first, which is illustrated in Figure 1, consists of one pair of coaxial loops and one pair of coplaner loops, closely spaced and mounted so that each loop is located at a corner of a square. Both coaxial and coplaner loops are connected out of phase. The outputs of each pair are fed through a sine, cosine goniometer. The goniometer output from this system, which is derived in Appendix I, is seen to be (for a given polarization and direction of arrival of the incident wave) proportional to sin (<£-y) when the loops are mounted in free space, where is the azimuthal angle of arrival and Y is the goniometer angle. The goniometer output then is a "figure eight" pattern, independent of the polar angle of arrival and the polarization angle, which has its nulls in the direction of arrival (4>) and 180° from that direction. To insure sensitivity in all directions, two of these systems could be mounted at right angles and the one with maximum output used. ! The second system that was investigated, which is illustrated in Fig. 2, also uses four loops and a goniometer. In this case, one loop is mounted on top of a wing, fuselage, etc. and another loop which is coplaner with it is mounted below. These two loops are connected so that voltages induced in them by the incident wave add in phase. At right angles to this pair is another pair of loops similarly connected. As in the system previously described, the loop pairs are fed through a goniometer to some indicating device. This system was chosen for in- vestigation because the balanced loop arrangement has been shown to offer improvement in reradiation error for waves incident from near the horizon . 2- Figure I Illustration of a Multiple Loop System having Opposite Loops Connected Out of Phase and Fed Through a Sine, Cosine Goniometer Pr jr A\ Gon, Figure 2 A Spaced Loop-Goniometer System Having Loops Mounted Above and Below a Wing v Fuselage, etc. 3. METHODS OF ANALYSIS In order to see whether these multiple loop systems held promise for airborne RDF use, the behavior of the goniometer pattern was in- vestigated analytically when the loops were mounted on a conducting body of simple geometry. The idea was that if the multiple loop system showed marked improvement in reradiation and polarization error when mounted near a conducting body of simple geometry (in this case a right circular cylinder with a scattered field which could be easily investigated*), then it would probably prove worthwhile to investigate by experimental methods the behavior of the system when mounted on an aircraft. The method used to calculate the goniometer pattern is as follows; A plane wave, linearly polarized, is assumed to impinge, from an arbitrary direction, on an infinitely long, perfectly conducting cylinder The total field outside of the cylinder is then the sum of the incident wave and the scattered field The voltage induced in a small loop immersed in the total field is proportional to the component of magnetic field that is normal to the plane of the loop. For all of the cases considered the loops were assumed to be mounted at the sur face of this cylinder. The details of the mathematics are summarized in Appendix II. Figures 3, 4, and 5 illustrate the orientation of the conducting cylinder with respect to the coordinate system used in the calculations and the orientation of the loops on the cylinder. The four loop system illustrated in Fig 1 was investigated in two positions with respect to the cylinder Figure 3 illustrates this system oriented with its loop axes in the circumferential direction of the cylinder, whereas Fig. 4 shows this system with its loop axes in the axial direction of the cylinder In each case, loops one and two are connected in phase opposition as are loops three and four The output of each pair is fed into one input of a goniometer The Loop system illustrated in Fig. 2 is shown mounted on the Lndei in Fig 5. Loops one and two are connected so that the vol' i luced in them by the incident field will add and are fed through ;i goniometer as are loops three and four Hir system of 1 1 ■ j I ;. » inc ' h« patterns of small antennas when mounted on a perfectly conducting Infinitely Long, r ik'i' circulai ylindei iraa developed by Carter- It was later ;< ntabla f'.irn Miid r-xt.ended i.o elliptj .1 1 -ylinders l>y Sinclair .3 4 Figure 3. The Four Loop System of Figure One Mounted on a Cylinder with the Loop Axes in the Circumferential Direction of the Cylinder Figure 4 The Four Loop System of Figure One Mounted on a Cylinder with the Loop Axes in the Axial Direction of the Cyl inder -5- Figure 5 The Loop System of Figure 2 Mounted on a Cylinder ± v 40 ± 1,1 k 40 Figure 6 The Positions of the Loops When Mounted on Cylinders of Diameter A/10 and A/5 6 Figure 6 illustrates the cylinder size and the spacing of the loops used in the calculations One set of data was calculated using a cylin- der with a diameter of A/10 and a loop spacing (between members of a pair) of A/40. Another set was taken for a cylinder with a diameter twice as large but with the same loop spacing The data were calculated in the following manner The incident wave was assumed to be polarized in the 9 direction and arriving from the direction defined by 6 = 80°, 4> = 0° With the incident field as described and the goniometer connected to the output of the two sets of loops under consideration the goniometer was rotated (mathematically in ten degree increments) through 360° If the loops were mounted in free space this pattern of the goniometer would be sin(<£ y)> Y being the goniometer angle Next, the incident wave was set to arrive from 9 - 80°, 4> - 10° and the goniometer process repeated This was re- peated in 10° increments through 4> = 90° These results are illus- trated in the (a) parts of Figs 7 through 15. Next, a wave having the same polarization was assumed to come from the direction defined by 9 - 45°, 4> ^ 0° Goniometer patterns were again calculated in 10° (0) increments These results are shown in the (b) parts of Figs 7 through 15* This entire set of patterns was then calculated for an incident wave having 45° polarization, i e , polarization that has equal 9 and 4> components These results are shown in the (c) and (d) parts of Figs 7 through 15. The complete set of patterns is shown for the system illustrated in Fig, 3. The patterns of the other systems are shown only in part. Figures 16, 21, and 26 summarize the information obtained from the goniometer patterns Each figure is a plot of indicated bearing (position of minimum goniometer reading) versus actual bearing (azimuthal direction of wave arrival) for all sets of goniometer patterns calculated These curves indicate the susceptibility of these systems to error due to the presence of horizontally polarized and reradiated fields -7. uces No Signal 80° 4b° Pol , i| 5 o 45° Pol. Figure 7. Goniometer Patternr of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 0=0° 80° l*5°Po 45° »+5° Pol Figure 8, Goniometer Patterns of the Four Lood System Mounted on a Conducting Cylinder of Diameter X/10 with the Lood Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> '0° = 30° Pol h « 60 45° Pol -■ 45° 45° Pol Figure 9 Goniometer Patterns of the Four Loop System Mounted >n -> Conducting Cylinder of Diameter A/10 with the Loon \xes i. the Circumferential Direction of the Cylinder an Exciting Wave Incident from 4> ■ 20° 10 = 80° h5° Pol = 45° 45° Pol. Figure 10 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> = 30° 11 wwvawnY of U imn ■ 80° 45° Pol 6 » 45° 45° Pol. Figure II. Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> = 10° 12 e - 80° 45° Pol 4-5° 15° Pol Figure 12 Goniometer Patterns of the Four Lood System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from -- 50° 13 h « 80° hb° Pol 45° 45° Pol Figure 13 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from ■ 60° 14 Figure 14- Goniometer Patterns of the Four Loon System Mounted on a Conducting Cylinder of Diameter X/IO with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from = 70° 15- e = 80° 45° Pol = 1+5° 15° Pol Figure 15 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Circumferential Direction of the Cylinder and the Exciting Wave Incident from 4> ■ 80° 16 4. ANALYSIS OF RESULTS Figures 7 through 15 show the goniometer patterns of the spaced loop system of Fig, 3 mounted on a cylinder of diameter equal to A/10. These were calculated for values of in 10° steps from 0° to 80° and for 9 = 80° and 6 - 45° for both 0° and 45° polarization.. It can be seen that the shape of these patterns (and hence the information de- rived from them) is not dependent on 4> alone as is desired but varies greatly with elevation and polarization angles of the incident field. Figure 16, which is a plot of null position versus the azimuthal angle of incidence, points this up quite clearly. The same can be said of Figs 17 through 20 although the effect of polarization is much less pronounced. These figures are the goni- ometer patterns of the spaced loop system of Fig, 4 mounted on a cylinder of diameter equal to A/10. These patterns were calculated for the same values of 0, and polarization angles as the previous set, but only the patterns calculated for fields incident from <£ = 10°, 30°, 50°, and 70° are shown here. Figure 21 is the plot of true versus indicated bearing for this antenna system. Figures 22 through 25 are the goniometer patterns of the antenna system illustrated in Fig. 5. These were calculated for the same parameters as the previous systems but again only the patterns calcu- lated for fields incident from 4> = 10°, 30°, 50°, and 70° are pre- sented- The patterns of this antenna system are also affected by elevation angle and, to a very great degree, by polarization angle, as shown in Fig, 26. The above comments are, in general, applicable to Figs. 27, 28, and 29 which are the goniometer patterns of the same antenna systems as above when mounted on a cylinder of diameter equal to A/5. 17 6 Polarization 9^80° . Polarization 9 45° _ 45° Polarization 6=80° 45° Polarization 9 = 45* (A & ■o O O ■6 c |**U / / ' cV l 1 I / inn. 1 1 \ \ i 1 / \ \ ^v_ i / ol_r 1 I /// 1 «"> 1 I / <. -7 1 / / / , / ACi 1 \ > / \ \ ( ■s*. > > / / 1 ' ^"< >>*--/<. r*"" ■""" ( • f* / / / / C\J ,< f ^ / > \Tmr < >- »-"* 20 40 60 80 True Bearing (degrees) Figure 16 True Bearing versus Indicated Bearing for the Four Loop System Mounted on a Long Cylinder (Axes Circumferential) 18 Figure 17 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/IO with the Loop Axes in the Axial Direction of the Cylinder and the Exciting Wave Incident from 4> - 10° 19 80° i+5° Pol 9 ■ 45° 45° Pol Figure 18 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Axial Direction of the Cylinder and the Exciting Wave Incident from <$> - 30° 20 e = 80° 45° Pol 45° 45° Pol Figure 19 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter X/10 with the Loop Axes in the Axial Direction of the Cylinder and the Exciting Wave Incident from = 50° 21 e ■ 80° 45° Pol - 115° 45° Pol Figure 20 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/10 with the Loop Axes in the Axial Direction of the Cylinder and the Exciting Wave Incident from 4> ■ 70° 22 c o c o o o °o poo "o is sz o ii m" OO (DO ^o *-o >> O C o (saaifcp) 6uuDag pajoojpui 23- e = 80° e Pol. 80 45° Pol --■■ 15° 45° Pol Figure 22 Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter of X/10 and Exciting Wave Incident from ■ 10° 2\ 45< Pol ■ 80° i+5 Pol 45° 45° Pol Figure 23. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter of X/10 and Excitini Wave Incident from - 30° 25- e = 80° e Pol 9 - 80° 15° Pol 9 * 15° i+5° Pol Figure 24. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter of A/10 and Exciting Wave Incident from ■ 50° 26 - 80° 45° Pol. 1+5° Pol Figure 25. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter of A/10 and Exciting Wave Incident from = 70° 27 \ -V c c o o o o o o o N o N N c ■^ Do oo ft S.CO ii ii CLOO O n m II CD CD ©CD *ta> ^■(D I ! I A -Q- t \ \ \ \ \ t I I «- c £1 CD ^ C O „, t- Ji - \ \ \ \ \ \ X \ \ X \ / I I \ \ \ \ f \ \ / ^*^ ^ g V- ^ 8 (sa3j6ap) 6uuoeg pajDoipui o to o CM CN CI L ! 20 6 = 80° = 10° 1*5° Pol 9 = 80° = 50° 45° Pol 9 = i+5°

= 50° i|5° Pol. = H5° = 50° Figure 28 Goniometer Patterns of the Four Loop System Mounted on a Conducting Cylinder of Diameter A/5 with the Loop Axes in the Axial Direction of the Cylinder 30 = 80° 0=10° i+5° Pol = 80° = 50° 45° Pol ■ 15° 4> = 10° 45° Pol = 45° * 50° 45° Pol Figure 29. Goniometer Patterns of the Loop System Mounted Above and Below a Conducting Cylinder of Diameter X/5 31 5. CONCLUSIONS The only encouraging results are those for the spaced loop system with the loop axes oriented in the axial direction of the cylinder Although the indicated bearings are in considerable error(see Fig. 21), it is seen that the variation of the bearing with polarization is only on the order of ten degrees. This comparatively small polarization error is probably due to the fact that the loops are not coupled to the axial current flowing on the cylinder Thus, with calibration, this spaced loop system might prove to be fairly accurate when placed on a long narrow body. Since the loop patterns have a null in the axial direction, the system is best suited for signals arriving from direc- tions within about 60° of broadside For the spaced loop system with the loop axes circumferential, the large variation of the indicated bearing with polarization (see Fig 16) makes the system useless for direction finding For this case, the loops are strongly coupled to the axial current on the cylinder. It is seen that even for 9 polarization, the bearing error due to reradiation is quite large.. The large variation of the indi- cated bearing with polarization is also due to the reradiation. It is possible that by using four loops above and four loops below the cylinder (in an attempt to decouple the antenna system from the axial current on the cylinder) that the bearing errors could be reduced However, the improvement would have to be considerable in order to warrant the added complexity of the antenna system. Unfortunately, time was not available to investigate the eight loop system. Although this investigation was concerned with loops mounted on conducting cylinders, the results (particularly negative results) can be extended to long, narrow bodies of any shape with circumferences on the order of a wavelength, such as an aircraft wing or fuselage.. 32 BIBLIOGRAPHY 1. "High-Frequency Airborne Direction Finding," Technical Report No. 30, Project No. 591, Stanford Research Institute, December 1952. 2. Carter, P. S. "Antenna Arrays Around Cylinders," Proc. I.R.E. Vol. 31, 671-693, December 1943. 3. Sinclair, G. "The Patterns of Antennas Located Near Cylinders of Elliptical Cross Section," Proc, I.R.E.., Vol. 39, 660-668, June 1951. 33- APPENDIX I A plane wave from the direction described by 4> and 9 with the magnetic field in the "9 direction (Fig. 30) has rectangular components where Hq = Hq e^ ^ r . H z = -H e sin Hy = Hq cos 9 sin H^ = Hq cos 9 cos 4> (1) (2) (3) (4) Figure 30^ Spherical Coordinate System The voltage induced by this wave in four loops with axes in the x direc- tion, located at x - 0, y s ±d/2, and y = 0, x = ±d/2, (Fig. 31) are, for the loop at y = +d/2, V 0i ■ ^ (0, d/2, 0) = Hq cos 9 cos 4> e jP(r - d/2 sin 4>) (5) — X 3 -i 2 1 , i ' 1 -d 7 ■ — i — ■ i i i -d 4 7 Figure 31 Location of the Four Loops 34 for the loop located at y = -d/2 V 02 = H x (0, -d/2, 0) = Hq cos 9 cos cp e for the loop located at x = +d/2 Vq 3 = H x (d/2, 0, 0) = Hq cos 9 cos 4> e and for the loop located at x = -d/2 V 04 = H x (-d/2, 0, 0) = Hq cos 9 cos 4> e j(3(r +d/2 sin 0) j3(r ~ d/2 cos ) (6) (7) (8) To get the voltage of loops one and two when connected out of phase, Eq. 6 is subtracted from Eq» 5. '1-2 Hq cos 9 tos 4> j3(r - d/2 sin 4>) j 3(r + d/2 sin 4>) e -e jPr, ■2jHq cos 9 cos 4> e sin (3 d/2 sin direction the rectangular components are H z = H y = H^ cos 4> Hx = H^ sin 4> where H^ - H 0o eJ & (11) (12) (13) (14) In a manner similar to the foregoing, the output voltages of loops one and two connected out of phase and of loops three and four connected out ■35- of phase that are induced by this 4> polarized wave are V^-2 = -2J^ sin eJ r ° sin ((3 d/2 sin ^ Va = -2jfU sin e sin (3 d/2 cos 0) . If a is the angle of polarization (Fig 32) then Hg = Ho cos a H^ = Ho sin a Figure 32. Definition of Polarization Angle, a Substituting Eqs. 17 and 18 into Eqs„ 9, . 10, 15, and 16 yields Vq = -2jH cos a cos 9 cos 4> e sin (3 d/2 sin 4>) Vq = -2jHo cos a cos cos e sin (3 d/2 cos 0) -2jH sin a sin 4> e sin (3 d/2 sin <£) i 3r /<£ = -2jH sin a sin e sin ((3 d/2 cos 0) '*!-. (15) (16) (17) (18) (19) (20) (21) (22) In order to get the total voltage induced in the loop pairs by a wave of o. polarization, Eqs 19 and 21 are added as are Eqs. 20 and 22. i Sr ^total = 2jH e sin (3 d/2 sin<£)(cos a cos 9 cos + sin a sin 4>) -K sin(3 d/2 sin 4>) (cos a cos 9 cos + sin a sin 4>) j3r . (23) totals " "2jHo e sin(3 d/2 cos 4>) (cos a cos 9 cos + sin a sin 4>) K sin(3 d/2 cos ) . (24) If a goniometer is used that multiplies Eq. 23 by cos Y and Eq. 24 by sin y (where y is the goniometer angle) and then subtracts the two, the result is Vgonionift.f-r K(sin(3 d/2 sin ) cos Y sin(3 d/2 cos ) sin y] Xfcos ot cos cos + sin a sin ] (25) 36 which for small (3 d/2 (closely spaced loops) becomes V goniometer * K (sin ^ cos Y - cos 4> sin y) = X (cos a cos 9 cos 4> +sin a sin \ , is desired. The equation! that will determine this response are in terms of the primed coordinate! Hence it is necessary to transform the given 9, and IBtO the COrreaponding values of 9, and 0. It can he shown, using 38 spherical trigonometry, that the equations of transformation are 0i ■ tan (tan 0! cos 0i ) (27) 0i = cos" (sin 0i sin 4>i) ■ (28) Applying these values of 0i and X to the equations yields the desired response in terms of the voltages induced in the antenna by the 0'and components of incident field. The voltages induced in the antenna can then be obtained in terms of the and

cos a = cos : — —, — . (31) where sin The equations for Y^y and VQ/of a small loop near a long conducting cylinder as taken from Sinclair 2 are: For a loop the axis of which is in the axial direction of the cylinder V ^= -e j3z Sin 9 V e n ( J ) n [J n (w) + C n H n 2) (w)] cosn(<*>'- 4>o) (32) ^ n = Vg > 0. (33) For a loop the axis of which is in the circumferential direction of the cylinder • j (3z cos v— 7 / y e /= 21 ) e n (j) n [J n (w) +C^Hi 2) (w)] cosn(0'- <&) (34) n - U ' ' 6 0' °° -J cos eJ z co y (j) n n[J (w) +Cn H n 2) (w)]sxn n<*'- 4>o) ^w L^ n n n n = U where 4>'o is the location of the loop around the cylinder w = 3bsin -39- J n (3a sin 9') Si r# 2) (3a sin 0') J n ((3a sin 9') Si H,i 2) Oa sin 0') a is the radius of the cylinder b is the radial distance of the loop from the cylinder axis e n = Neumann's number = * ^ or n = n 2 for n ? J n , H^ , J n , Hj~ 2 ^ are the appropriate solutions of Bessel's equation and their derivatives with respect to w. The patterns of the multiple loop configuration illustrated in Fig. 3 can be obtained from Eqs. 34 and 35. Allowing b = a = A/20 (flush mounted loops), the patterns of loops one and two mounted at z :; 0, 4> - ± 14.3° (making the circumferential distance between the two equal to A/40) and connected out of phase are described by oo _4 V (J) n [cos n(0'- 14.3°) - cos n(tf>' + 14.3)] Vq' = —*- ) (36) m " n^i HJ2>( W ) v / - -4 cos 8 ' Y (j) n Uin n(j>'- 14.3°) - sin nQ * 14.3°)] The patterns of loops three and four mounted at 4> ~ 0, z = ±A/80 and connected out of phase are described by _ 2 sin K C° s / A V 9 ;. 4 ® > e n (j) n [Jn(w) +C h H | ( 1 2) (w)]cos n ^ (38) n « 1 4 8in * cos 9 ' « V. 4 — / (j)"n[J n (w) + C^H I i 2) (w)]sin n <*/. (39) ^ i^ Sin ^ " " 1 I I pure 4 i I I list. rates the same loop system as Fig. 3 as far as spar, id,- Location, and connection are concerned The difference is the loop-. Iiavf now been rotated by 90°. The patterns of the pairs 40 of loops oriented in this manner can be obtained from Eqs. 32 and 33 and are described by D l- 2 00 V1-2 = " 4 Si " ^ Y (J) n [J n < w > + C n H^ 2) (w)]sin n 0'sin n 14.3° (40) n n = 1 ^3-4 00 V.--4 = ~ 2j Sin ^^f^ 75 - 1 ^- 1 ' Z £ n ( J )n[J n^) + C n H(2)( w )] cos n ^.(41) n = The patterns of the loop system illustrated in Fig. 5 can be obtained from Eqs. 32, 33, 34, and 35. The loops are flush mounted at z = 0, 0=0 and Tt„ The equations describing these patterns are V 0i- 2 = T H e n^) n ^n^) + CnHn 2) ( w )^ os * ' 1 n odd -4i cos \ V,- 2 = i . , L e n(j) n ^n^)