11218-3-T = RL-2213 011218-3-T THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Redi.ien Lobortory APPLICATION OF THE LARGE GRADIENT VOR ANTENNA By Dipak L. Sengupta and Philip Chan 15 March 1973 Interim Engineering Report No. 3 Contract No. DOT-FA72WA-2882 Project No. WA5R-1-0526/N113-739.0 Contract Monitor: Mr. Sterling R. Anderson, RD 331 Prepared for: FEDERAL AVIATION ADMINISTRATION 800 Independence Avenue, S. W. Washington, D.C. 20591 Space Research Building Hayward Street 2216 2455 Ann Arbor, Michigan 48105 (313) 764-0500

011218-3-T TABLE OF CONTENTS page I INTRODUCTION 1 II FREE SPACE COURSE SCALLOPING AMPLITUDE 3 2.1 Basic Analysis 3 2.2 Free Space Course Scalloping Amplitude 9 2.3 Discussion of Free Space Course Scalloping Amplitude 12 m COURSE SCALLOPING AMPLITUDE IN THE PRESENCE OF A GROUND 15 3.1 Multipath Signal at P 16 3.2 Total Field at the Far Field Point and the Bearing Error 20 3. 3 Course Scalloping Amplitude 23 3.4 Discussion 25 IV RF PHASE RELATIONSHIP BETWEEN CARRIER AND SIDE-BAND SIGNALS 27 V NUMERICAL RESULTS FOR COURSE SCALLOPING AMPLITUDES 33 VI CONCLUSIONS 44 VII REFERENCES 45 i

011218-3-T LIST OF FIGURE CAPTIONS 1. Coordinate system used. 2. VOR system and scatterer above ground. 3. Direct and reflected rays that reach S. 4. a (0), a (e) and a (6) a (0) as functions of 0 for a standard VOR antenna c S S c in free space. kA = 18. 0859, kd = 0.9276, kh = 2. 7755, f = 109 MHz. T T T T 5. ac (), a (9) and a (e) - a (e) as functions of e for a standard VOR antenna c s s c located 15' above ground. f = 109 MHz. 6. a (e), a (e) and a () - a (9) as functions of e for a large gradient parasitic antenna in free space. kA = 18.0859, kd = 0.9276, kh = 2. 7755, kH = 3.6, kB 116.9, kH 2=10.9, kB 2=12.1, f= 109 MHz. T T T T 7. aT (), aT () and a (e) - a (9) as functions of e for a large gradient parasitic c S s c loop counterpoise antenna located 15' above ground. 8. Free space scalloping amplitudes S and S1 Approx as functions of (0-01) with A as the parameter. 9. (a) Course scalloping amplitude S. as a function of (0-0 ) for a standard VOR antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0 = 0. 76~. min 9. (b) Course scalloping amplitude IS2 as a function of (0- 0) for a standard VOR antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0. m76. min 10. (a) Course scalloping amplitude kS as a function of (0 - 01) for a standard VOR antenna located 15' above ground and for different heights of the scatterer. Orbital flight at e = e,83~. max ii

011218-3-T 10. (b) Course scalloping amplitude S2j as a function of (0-0 01) for a standard VOR antenna located 151 above ground and for different heights of the scatterer. Orbital flight at 0 = 83~. max 11. (a) Course scalloping amplitude S as a function of (0-01) for a large gradient parasitic loop counterpoise antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0 = 76~. 11. (b) Course scalloping amplitude S2 as a function of (0-01) for a large gradient parasitic loop counterpoise antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0 = 76~. 12. Course scalloping amplitudes [S21 as functions of (0- 0) for a standard VOR and large gradient parasitic loop counterpoise antennas located 15' above ground for a scatterer located 100' above ground. Orbital flight at 0 = 830. iii

011218-3-T I INTRODUCTION This is the third Interim Report on Contract No. DOT-FA72WA-2882, "Application of the Large Gradient VOR Antenna, " and covers the period 1 December 1972 to 28 February 1973. The present report discusses the theory of course scalloping amplitude in standard VOR bearing indications. Wherever there exists a multipath between a VOR station and a flying aircraft, the multipath signal combines with the desired direct signal to produce various types of siting errors in the VOR indications at the aircraft [11, [2]. Here we discuss only those siting errors which appear as course scalloping in the VOR bearing indications. Detailed discussions on scalloping amplitude and frequency for a standard VOR system in the presence of various types of multipath signals are given in [2], [4]. The references cited give a considerable amount of observed course scalloping amplitude values produced by different types of scattering objects. Approximate theory of scalloping amplitude has been developed [4] for idealized situations. The theory developed, at its best, is able to explain qualitatively some of the general behavior of the scalloping effects [2], [4]. However, because of the basic approximations made, in many instances [2], [5] the existing theory is unable to explain qualitatively the observed behavior of the course scalloping amplitudes in the various VOR indications. In the following sections we at first develop a rigorous theory for the course scalloping amplitudes observed in standard VOR indications in the presence of multipath signals. The antenna characteristics, the scattering properties of the multipath sources, and the ground effects are all built into the theory so that their effects on the scalloping may be investigated quantitatively. The analysis assumes that the VOR receiver characteristics are ideal. It is hoped that the theory discussed here may be found useful to analyze the large amount of experimental data on 1

011218-3-T course scalloping amplitude collected by the FAA over many years [23, [4. The analytical results discussed below are based on an earlier work by one of the authors [5]. The various approximations involved in the theory are then critically examined. Finally numerical results obtained for a few idealized cases are discussed. 2

011218-3-T II FREE SPACE COURSE SCALLOPING AMPLITUDE 2.1 Basic Analysis In this section theoretical expressions are developed for the course scalloping amplitudes in the standard VOR bearing indications caused by multipath signals when both the VOR station and the scatterer are located in free space. Although this is a very idealized case, the results may be found useful in studying some of the general behavior of the scalloping phenomena. Let us assume that the VOR station is located at the origin 0 of a rectangular coordinate system x, y, z (Fig. 1). A stationary aircraft is located at the far field z P(r, 6, 0) STATIONARY AIRCRAFT XV A 0 0; S(d1, 9, 1) SCATTERING OBJECT VOR STATION, _________________ y X 0 x 0 1 Fig. 1: Coordinate system used. 3

011218-3-T point P whose coordinates are (r, 0e, 0) expressed in spherical polar coordinates with origin at 0. A scattering object is located at S which is represented by the coordinates (d1, e0, 01). It is assumed that the distance of the scatterer from the VOR station is sufficiently large (kd1>> 1) so that the usual geometrical optics approximation may be used in obtaining the field incident at S. We assume here that the field radiated by the VOR antenna system is horizontally polarized and that the scattering object does not produce any depolarization of the field. The direct signal at P coming from the VOR station may be written as: -i (t-kr) r L Ed(P) X K r ISc() + S (0) cos (wt-), (1) d r S c m ] where, K is a constant, w is the carrier frequency in radians, k = 27 is the propagation constant in free space, Wm is the modulating frequency in radians, S (e) is the free space elevation plane complex far field pattern of c the VOR antenna in the carrier mode, S (e) is the free space elevation plane complex far field pattern of s the VOR antenna in the side-band mode. Notice that in Eq. (1) the modulation index 'm' of the space amplitude modulated wave is not explicitly stated. However, this can be assumed to be included as an appropriate multiplying factor in S (e). Observe that, as indicated by Eq. (1), s the side-band and carrier signals are not in general r. f. phase with each other. The implications of the r. f. phase relationship between the carrier and side-band signals will be discussed later in more detail. Explicit expressions for the standard VOR antenna complex pattern functions have been discussed elsewhere [6], [7]. It will be assumed that S (0) and S (0) are known in the present problem. c s 4

011218-3-T For later reference we write the pattern functions in the following way: ia (0) S (0)= S (0) e (2) c c ia (0) S (e)= Ss(0) e s (3) s s In the absence of any other signal, the VOR receiver at P detects the envelope of Eq. (1) and observes the phase 0 of the detected signal which gives the azimuth of the field point P with respect to the VOR station. However, in the presence of multipath signals, the phase relationship between the carrier and side-band signals are disturbed thereby producing errors in the azimuth information indicated by the receiver. The field incident at the scattering object S, located sufficiently far from the VOR station, may be written as follows: -i (wt - kd1) E.(S)= K-e d Sc(l)+ Ss(0) cos(wmt 01 (4) where the various notations are as explained before., The multipath signals at P will be that portion of field, given by Eq. (4), scattered by the scattering object S in the direction of P. For a finite object the scattered field at P will appear as a spherical wave originating from S. It can be shown that the multipath signal at P may be expressed [5] as follows: -i t-k(rl+dl)] E (P)= Ke r A(01, 01. dl; 0, 0-01) e x 5 r1 x S (8)+S (0+ ()cos(W t-e0l) (5) c 1 s 1 m 1J 5

011218-3-T where, r1 is the distance between the scattering object and the aircraft at P, A(01, 01' d1; 0, 0-01) e6 may be identified with the scattering or reflecting properties of the scatterer. The dependence of the incident and scattering angles as well as the distance dl of the scatterer from the VOR antenna are explicitly written in A to emphasize the fact that in many cases of practical interest the scattered field from the object depends strongly on these parameters For convenience we shall, from now on, suppress the variables 01, 01, d1, 0, 0 from A. In Eq. (5) A and 6 may be identified respectively with the amplitude and phase of the scattering coefficient of the object. By definition both A and 6 are real quantities. Since P is located in the far zones of both the VOR antenna and the multipath object S, we can make the usual far field approximations and can rewrite Eq. (5) in the following form: -i (t-kr) E (P)= K Ae x s r x S c(0)+S () cos (W t-, (6) where, g= kd1(1-cos )+6 (7) cos y = sin 0 sin 01 cos ( - 01) + cos 0 cos 01. (8) The total field received at P is obtained by adding Eqs. (1) and (6) and is given by: -i (wt - kr) S (0l) E(P) K -S (0) l +Ae S1 + r (0) s (O) S (0 ) s c + S () cos(w t-0)+Ae S () cos(wt-). (9) c c 6

011218-3-T After some algebraic manipulations, Eq. (9) may be written in the following form: -i[wt-kr-a (0)-] E(P)= K. e S c(0) x c s (10) r C where, S(6 ) 2 S ) r 2= 1+ 2 C 2 f= +A 1 +2A c () os +a (0l)-c(0), l +A () s() c 1 S (6 ) c 1 A sin +a (6 )-a ( s (() c 1 ( tan L(12) 1+A cod +i (at P ad (() c 1 fe Ncos( t-0+a) (1) f2 2= (0) r cos (wm t-) - A3os(w13) +S ) 5 s 1m '() C C In Eq. (10) f may be identified with the combined carrier signals at P and it contains no azimuth information. Similarly f may be identified with the combined side-band signals at P which contain the azimuth information of the VOR and the scattering object. The detector output at the ideal VOR receiver at P is proportional to the real part of f. Let us denote f = Real part of f given by Eq. (13). It can be shown that f can be written as: f= Ncos(W, t-+a) (14) where, 2 Ss(O)] 2 2 2 'I 1 2 +A oos (es)- a (a) + - s(6) c 7

011218-3-T s (6) s (e ) +2A s() Cs() (0) L a ( i s (e) s (0) - Xos a (00)-5(o+- C] o (O-l) (15) s (16) The bearing indication in the VOR receiver is given by the phase of Eq. (14). Thus we can write the bearing indication B as given by: B= 0-a () 17)s The true bearing of the aircraft with respect to the VOR station is 0. Note that in the absence of multipath signals, A = 0 and Eq. (14) takes the following form: s c tanF ( +f= cos ()cos (t-0). (18)(e s ( s c S (e) s cs- ~c l0 1) L (16) Equation (18) indicates that the detected signal amplitude depends quite ofstrongly on te r. f. phaswriterelationship betweenarrier nd side-band signals. From then by: The true bearin of detection maximum aircraft with respect to the VOR station is a (. Note that in the absence of multipath signals, A = 0 and Eq. (14) takes the following form: f [Sc(0) cos sa ()-ca (e) cos((w t- ). (18) the r. f. phase relationship between the carrier and side-band signals. From the carrier and side-band signals are in phase at all angles 0. In general, a (0) $ a (0). so c It is possible to adjust the phase relationship such that a (6 )-ac(0 ) = 0 in a certain direction 0 =. However, since a (0)-a (0) changes with 0, the phase adjusto S C ments made at 0= 0, will be disturbed at angles 0 0. As we shall see later, O 0 this aspect of the phase relationship will have considerable effects on the scalloping amplitudes in the presence of multipath signals. 8

011218-3-T Thus the second term in Eq. (17) may be considered as the bearing error. Let us write the bearing indication of the VOR receiver at the aircraft as: B=0+A0, (19) where A 0 is the error introduced due to the multipath signal. Complete expression for the error A 0 is given by: -1A c 1os a (O ) -e ( 0) -+ sin(~ ml + A S2) c os (I)- ((0 )++ - sin (0-l ). c1 cL Note that in the absence of multipath signal A = 0 and hence from Eq. (20) it follows that AO = 0. For convenience of reference and later discussion, we rewrite below the explicit expressions for e and i given earlier: = kdl - sin8 sin01 Cos(0-01) - os os 0c1 +, (21) cS(9) r A S (0) sin C+ac ()-ac() = ta l ) c (22) + Af ' cos —73(0)o)+]cos)-%() 2. 2 Free Space Course Scalloping Amplitude In the presence of multipath signal the bearing error in the VOR indications of a stationary aircraft located at (r,, 0) with respect to the VOR station is given by Eq. (20) As the aircraft flies A0 will change due to the variation of the parameters 6 and 0. For a radial flight with respect to the VOR station 0 is variable 9

011218-3-T and 0 is constant. Whereas for an orbital flight at constant height 0 is constant but 0 is variable. In both cases AO changes above and below the true value of 0 with variation of the coordinates 0 and 0 of the flying aircraft. Before investigating the nature of variation of A0 it is instructive to study the variations of the parameters e and 5 (in Eq. (20)) with the aircraft motion. If S (el) S (1) c small quantity (~<<1) and the effects of its variation with the aircraft motion on the final results may be assumed negligible. The parameter I, given by Eq. (21), is large and its variation with 0 is also large. As a result, the terms involving e in Eq. (20) oscillates between +1 and -1 as the aircraft moves. Thus the variation of A0 with the aircraft motion will be a slow variation with 0 (or 0 as the case may be), determined by the terms sin (0 - 0) and cos ( - 01) in Eq. (20) mounted on top an oscillatory variation determined mainly by the parameter ~. It is evident that A0 fluctuates between positive and negative values with the motion of the aircraft. The amplitudes of these positive and negative swings of the error A0 are termed as course scalloping amplitudes. We write the bearing indications of the VOR receiver at the flying aircraft as: B= 0-S 2 ' (23) where, S is the value of AO obtained from Eq. (20) when cos as(0) -a (0)+ - = 1 S is the value of A0 obtained from Eq. (23) when cos a (01)-a (0+?- = -1 S1, S2 may be called the course scalloping amplitudes which give the two bounds of the scalloping errors as the aircraft moves. Note that S1 and S2 are of opposite signs and in general S1 S2j By definition, 10

011218-3-T S1= A0 evaluated at 01' ' ~ 01' (24) where, a C(01)- (0) + 01 - 01= 2n7r, n is a positive integer (25) s 1 c 01 01 Sc() -( A sin 0 +ac (l)- sa tan i6 ' s C) (26) 0+ s (e oo 1+A C 1 cos + + (e )-a (0) 8() Lo 1 c C Equation (26) can be solved for 01 and we obtain: L c Cs (I)- c j1 01= -sin A S 1 sin s(0 a)- c( (27) After using Eqs. (20), (24)-(27) the following is obtained: S (0 )1 A 1s C1 -(-1 0an r(0S ). (28) csO) o ()-(a - +)- o A O (A - S s(C) 01 ( 1 C c Similarly, S2 is obtained from Eq. (20) for values?= s02' ~ = 02 given by a s(6)-a (0) + 02 -02 nr n is a positive odd (29) number s ((0) 7 1 A S ( sin E02+ c(1 )-c() tan 2 (30) 1+A C'1 Cos-a(0ia ( 02+ % (012 )-C 1 (c C 11

011218-3-T Equation (30) has the solution: 02 sin S (1) r 02( sin A s (e sin cx (e )-ac(el s () 1 c c (31) Proceeding as before we obtain: -1 S tan 2 (32) 2.3 Discussion of Free Space Course Scalloping Amplitude Approximate course scalloping amplitude expressions are given in VOR Handbook [2]. In this section we discuss in the light of the theory discussed in the previous sections, the scalloping amplitude expression given in [2] for the case of a non-directional multipath source. In particular we discuss Eq. (4-1) on page 22 of [2]. The physical arrangement of the VOR station, the multipath source and the aircraft considered in [2] implies in our present notation: 0= 0=/2 A is a constant, 6=0. Thus we obtain the following from Eqs. (28) and (29): -1 S = -tan 1 -1 2 -1tan A sin (0-01) cos a S(r/2) -ac (r/2)+ s in (Asin(as (r/2 )- c /2) +Acos( - 1) AS C S C -1 - A sin (0-01) (33) (34) 0- - os /2) ) sin A sin 2 -A cos (0 0 Cos as /2) - c(r/2) - sin sin (as(Tr/2> - r/2) -A cos (O - 1) 12

011218-3-T If it is now assumed that a (7r/2) = a (7/2) which implies that the side-band and carrier mode fields are in phase in the direction of 6 7/2, Eq. (33) and (34) reduce to the following: 1 A sin (0-0 ) S= -tan 1+Acos ( — ) (35) A sin (0-01) S2 tan 1-Acos(0-01) (36) Under the assumption of A << 1, Eqs. (35) and (36) reduce to Eq. (4-1) given in [2] which is: -1 S1 +tan Asin(0- 1) 2 (37) + Asin (0 - 01) for A<< 1. If a ac, which is most certainly possible in a real case, Eq. (37) will give values which are too low for scalloping amplitudes even under the assumption of A << 1. To show this we assume ( /2)- c 72)| >>0, (38) A <<1 Under the conditions (38) it can be shown that S, S2 as given by Eqs. (33) and (34) are given by: Asin ( - 01) S +tan. (39) 2 cos [A2) - a( 20 + A cos (0 - 0) Hence unless a ( /2)- aC(r/2) = + nr, the neglect of A cos (- 01) and the assumption of cos Er (/2)- ac (/2) = 1 in the denominator of Eq. (39) are not justified. Thus in general the values predicted by Eqs. (35)-(37) will be lower than those predicted 13

011218-3-T by Eq. (39). On the basis of the free space theory developed and the discussion given here, it appears that the side-band and carrier mode elevation plane complex patterns (in particular, their phases) should be considered in evaluating the course scalloping amplitudes. 14

011218-3-T m COURSE SCALLOPING AMPLITUDE IN THE PRESENCE OF A GROUND In this section we generalize the theory of VOR course scalloping amplitude to the case when the VOR station and the source of multipath signal are located above a perfectly conducting infinite planar ground. This is a more realistic case and is of considerable practical interest. Figure 2 represents geometrically the arrangement of the problem to be studied. The VOR antenna is located at (Z 0, 0), the 0 scatterer at (d., 01, 01) and the ground plane is at z = 0. The free space theory is generalized to the present case by taking into account the image fields produced at P by the perfectly conducting ground. z P(r, 6, 0) STATIONARY / / AIRCRAFT VOR STATION (Z 0, 0, 0) SCATTERING OBJECT y GROUND AT z 0 x Fig. 2: VOR system and scatterer above ground. 15

011218-3-T The field at P due to the VOR station alone may be written as: -i (wt - kr)T T Ed(P)= K. e (O) + T(o) cos (W t-0) (40) r c s m where, -ikZ cos ikZ cos S (8) = S (0) e - S (r - ) e (41) c c c ikZ cos 0 ikZ cos 6 s (0)= S (0)e - S (r -0) e (42) s S s where S (0), S (0) are the free space elevation plane complex far field patterns c s of the VOR antenna in carrier and side-band modes respectively. The derivation of the multipath signal at P is more complicated and is discussed in more detail in the following section. 3.1 Multipath Signal at P At first we need the field incident at S from the VOR antenna. If S is in the far zone of the antenna then it can be obtained from Eq. (40). We assume here that S is not in the far zone but it is sufficiently far away from the VOR station so that kd1>> 1. Under this condition usual geometrical optics approximations can be used To obtain the incident field E.(S) we consider the direct and reflected 1 waves originating from the VOR and reaching the scatterer S (Fig. 3). The incident field at S can be written as: -i (t - kd ) -i (at - kd") 1 E.(S) = K e f (0 0) K e f(7r- " 0), (43) where, f(0, 0) = S (0) + S () cos (w t-0). (44) From Fig. 3 it can be shown that From Fig. 3 it can be shown that 16

011218-3-T z S(dl/ 01' 01) y IMAGE x Fig. 3: Direct and reflected rays that reach S. and, z -1 d' " d, (1 - -- cos 06 Z 1 1 d1 1 1 e sin0 _ 1 l+dcos1 1 sin 0 sin" z 1+ 2 cos 0 1d 1 z for ~ << 1 d1 Z for ~ << 1 d1 (45) (46) 17

011218-3-T If the scatterer S is located at a distance D from the z-axis and at a height H above the ground plane then it can be shown that: D tan = 1 H dlcos 01 H (47) dlcos 1= D After using Eqs. (45) and (46), Eq. (43) can be written as follows: Z ikdL (1- cos l 1 d 1 ikd1(1+T1 cose) -iwt e -K'e " f(7r-0^^1). (48) Z d (1 + - cos ) As before we assume that the scattered field at P is related to E.(S) by: ikr1 i6 e E'(P) = E.(S) A (01 01; O, 0) e (49) s 1 1 r In obtaining the scattered field at P we must take into account the effect of the ground. In this case we can assume that P is located in the far zone of S and the usual far zone approximations may be made. The total scattered or multipath signal at P is then obtained as in the following: 18

011218-3-T -ikZ cos O -i (wt - kr) o 1 E (P)= K.e A Z f('f 01) S r 1 1-Z cos1 i where, cos z f sin 0 sn- 01) + cos cos (-5 After some algebraic manipulations Eq. (50) can be rearranged in the following form: -i(wtt-kr) E (P)= K.- e 2Asin(kHcos0)e - sin(kHcos0) x s r ikZ cosO0 ikZ cos 0 d 10 1 COS0d 1+-cos 0 L 11 - e 1, (50) where, cos kd sin sin 1 cos (0- ) + 6-2 * cos53) z z In many practical situations - < 1 and we can neglect the terms After some alg nder these conditions Eq. (52) simplifies to the following: E (P)= K e 2Asin(kHcos)el sin(kH cos) x ikZ cos e ikZ cos 1 o 1 o 1 f =)-s e.......sf(- + ) (52) Z Z In many practical situations 1 and we can neglect the terms in the denominators of the terms within parenthesis in Eq. (12) and also can write ';" "-' eIl. Under these conditions Eq. (52) simplifies to the following: 19

011218-3-T -i (.t-kr) ig' E (P) K* e 2A e sin(kH cos 0) x s r x T() + ST(Ol) cos (mt-01, (54) T T c s ml where ST(0) and S (e) are as defined by Eqs. (41) and (42). C S We shall use the simplified expression given by Eq. (54) for the scattered field at P instead of the more accurate expression given by Eq. (52) which is given here in case it is needed for more accurate results. 3.2 Total Field at the Far Field Point and the Bearing Error The total field at P is obtained by combining Eqs (40) and (54) and is given by: -i (wt - kr) E(P)=K SC () x r c T x 1+2Ae sin(kHcos 0) T (e) s(e) -i(wt-kr-a( )(56) + 2Ae 1 sin(kHcos 0) cos (w t - 0. (54) S(0)m (1 c After some algebraic manipulations Eq. (55) may be rearranged as follows: T C,, E(P) K*- S e) f (56) r c c 20

0112 18-3-T where., T 2 2 ~Sc (61 ) (f ) 14 T sin (kflcos60) + Sc(i +4A T 1sin kCos) CosI aT aT(k S T(6) 1 C (57) ST( 2A - sin (kHlcos60) sin 6 a( )-aT(6 ST~ T S 6 C tan IfI -- (58) I T+2 sinl (kH cos60) cos f +a.(6) -la (6J ST~ ff=S e -rtCos (w t -0) + ST(6) m C ST( + 2A 1sin kCos6) ST (6) T IT = T Ss(=Ss(6) ei jf v Cs w t0 (59) E iT(0 aT(0 ia(6 51 0 (60) E As before, let f ' Real part of f I. From Eq. (59) we obtain the following express sion for f ': 21

0112 18-3-T ST( f s T T Cos a (0) -a (0) os-Wt') 5 + T co TO)aTO+?j csw-1 S 0 (61) We now express f'I in the following term:f'-Nlcos(w t-0+a9, (62) where., T 2 2,(0) 2 T 0 T~e (N) Cos a. ()a (oT 2 2 2 55(1) 2 FT T + 4A sin (kH cos 0) T Cos a (0 )-a (0)+~?~I S (0)1 C3 C 5T(0) 5T(0) + 4A T T, sin (kH cos 0) Cos aT0)- a -O x 5T(0) 5 (0) S, C C C x Cos [a T (o)-a T(O1) -i(63) S (0 siT 2A T sin (ldHICos0) Co [a O1 a (0jo+ sn( ISc (0)I tana,' = C 5S(0) T T SS,( 0) T T CScs a0 ce -a )- 2A T sin kCos 0 a (01)-.a (0)+V'- VCS( ST(0) Ls -J 5(0) 1 C C (64) 22

011218-3-T The bearing indication in the VOR receiver is B= 0-oC= =0+A0. (65) The bearing error A0 in Eq. (65) is then given by: 2A T sin(kH cos 0) cos a(0)-a(0)+ ' ('sin(0-0) -s1) I_ ______ _ S 6) s c c =-tan ST( ST () T s c + 2A s c - sin (kH 0) c) cos a(1) -a(0)+~'- cos(-0 + T sk / [Ts ( c0 c (66) where, P'=kd l-sin0sin01coso(-0 ) + 6 - /2 (67) 2A c 1 sin(kHcos sin Q +a T(e)-a ( -t1 _________ c 1c?tan ' (68) 1+ 2A j) sin (kH cos 6) cos?' + a (01) - T(0O sT(0) c 1 c 3. 3 Course Scalloping Amplitude Proceedings in a manner discussed in Section 2.2, we obtain the following expressions for the course scalloping amplitudes: 23

01 1218-3-'T am1 S -tan 2A si1 sin (kH cosO0) sin(0.- 0 ) S T(0) ST( 5(9)Co aT )-aT0 V + sT(e).. 0 T +2A -S1sin(kHcos 6)cos(0.-0) ST(OT LA SO1 sin (kH cos&e) sin J(e1)-.aT(O )~ (69) 0 sin (70) and, -1 S =2tan ST( S (6 ci 5T(0) 5T(6) -'c02 STO ST((0) Cc (71) C=sin FT 1 2A T 1DInL(kH cos 0) sin SO) L ST(&) K (72) iaT(6) =~ a(0), i. e., if the phase of the carrier mode complex elevation S C 24

011218-3-T plane pattern is equal to the phase of the side-band complex elevation plane pattern, then 1' = 0 = 0 and Eq. (69) and (71) reduce to the following: 01 02 2A SsT(e ) sin(kHcos )sin(0-0l) S -tan, (73) tan T(0) +2AS (0 sin (kH cos0)cos(0-0 (73) Ls + 2A S T( 1) sin (kH cos 0) sin (0 - 0 S tan S (74) S (0) - 2A T(0e1) sin(kHcos0)cos(0-0J s s Equations (73) and (74) indicate that for orbital flights S1 evaluated at r + ( - 0 = S2 evaluated at ( -01) ' S2 evaluated at r + (0 - 01) S1 evaluated at (0-0). The above relations imply that it is sufficient to calculate S1 and S2 in the range 0 _ (0-01) < 7r in order to study the course scalloping amplitude variations during an orbital flight. It is also interesting to observe from Eqs. (73) and (74) that the course scalloping amplitude reduces to zero whenever the following relation holds: Hcos 0 = n= 1, 2, 3,... (75) 3.4 Discussion The general expressions for the course scalloping amplitudes in the standard VOR receiver bearing indications are given by Eqs. (69) and (71). The theory clearly indicates that the course scalloping amplitudes depend on the following parameters: (a) the amplitude A(01, 01 dl; 0, 0) of the scattering coefficient of the multipath source located at (d1, 01, 01), (b) the amplitude and phase of the carrier and side-band mode elevation plane complex patterns of the standard VOR antenna, (c) the location of the multipath source with respect to the VOR station. 25

011218-3-T After studying the scalloping amplitude expressions, the following observations can be made with regard to the general behavior of the course scalloping amplitudes in the VOR bearing indications observed by an aircraft in orbital flight around the VOR station: (i) The scalloping amplitude is zero in directions 0 = 01 and 0 = T+ 01 If the multipath source scatters energy isotropically (i. e., A = constant) then the amplitude of scattering assumes maximum value in directions 0 = /2 + 01 and 0 37/2 + 01, only if A <<~ 1. (ii) The two envelopes of the bearing error (S., S ) are opposite in sign but in general are not equal in amplitude. (iii) In the case of a non-isotropic scattering behavior of the multipath source, the directions of the scalloping amplitude maxima will be determined mainly by the parameter A. (iv) The side-band mode pattern S (0) is quite critical in determining the scalloping amplitudes. The scalloping amplitude increases in the directions of minima in the pattern S (0). S (0) does not seem to have strong influence s c on S1and S2. However, the relative phase difference between the side-band and T T carrier mode fields (a (0) - a (0) influences considerably the amount of scalloping s c amplitudes. (v) S I and S 2decreases with increase of d., where (d., 01 0) are the coordinates of the multipath source. (vi) S and S2 increases with increase of height H of the multipath source when the aircraft is flying at low elevation angle from the horizon. All of the above observations are in qualitative agreement with the observed course scalloping amplitude data discussed in [2] and [4]. 26

011218-3-T IV RF PHASE RELATIONSHIP BETWEEN CARRIER AND SIDE-BAND SIGNALS It was mentioned earlier that the r. f. phase relationship between the received side-band and carrier signals plays an important role in the detection efficiency of the VOR receiver at the aircraft. From the scalloping expressions developed in the previous sections it is also found that the same phase relationship has considerable effects on the observed course scalloping amplitudes. For an antenna located in free space this phase difference may be represented by a (0) -a a(0) where a (0) and a (0) are the phase angles of the side-band and carrier complex patterns respectively, as defined by Eqs. (3) and (2). For an antenna located above infinite planar ground the phase difference is represented by a (0) - a (0) where T T s c a (0) and a (0) are as defined by Eq. (60). The results discussed in the previous sections indicate that in general if a (0) - a (0) or a T(0) = aT(0) for all 0, then the detection efficiency is maxic s c s mum and also the observed course scalloping would be minimum. It is therefore important to study the variation of the r. f. phase difference between the two signals as a function of 0 for a given antenna. Complete expressions for the carrier and side-band mode complex patterns of various standard VOR antennas have been given in [6]- [8]. Here we discuss only the variations of the phases of the complex patterns as functions of 0. Figure 4 shows the variations of a (0), a (0) and a (0) - a (0) as functions c s s c of 0 for a standard VOR antenna located in free space. The corresponding results for a standard VOR antenna located 15' above an infinite planar ground are shown in Fig. 5. Figures 6 and 7 show the corresponding results for a large gradient parasitic loop counterpoise antenna under similar conditions. For the standard VOR antenna it is found that the phase difference is almost constant for all 0 in both cases. For the parasitic antenna case the phase difference is found to be reasonably constant in the range 550 < 0 < 95~ for the free space case; the phase difference fluctuates with an average value near 1000 for 500 < 0 < 85 for the 27

300 r t v) r3z 200 H Fig. 4: a (0), as(0) and a s()- c(0) as functions of 0 for a standard VOR antenna in free space. kA = 18.0859, kd = 0. 9276, kh x 2.7755, f= 109 MHz. a (0) a s(0)- ac(0) 5 C,.100 -60 -60 0 IN DEGREES -- 90 100 110 120 28

T a (e) C T) aT (0) s T T a (0)- a (0) S C \ 100 C! u;4 gQ 1 0 60 70 80 0 IN DEGREES — 90 - 100 T T T T Fig. 5: a (0), a (8) and aT ()-a (0) as functions C S S C of 0 for a standard VOR antenna located 15' above ground. f = 109 MHz. 29 -200

3C00 - Fig. 6: a (0), a (0) anda (0)- a (0) as functions of 0 for a C S S C large gradient parasitic antenna in free space. kA x 18. 0859, kd=0. 9276, kh=2.7755, kH1= 3. 6, kB a16. kH =10. 9, kB =12. 1 f 109 MHz. 2 2 9/ I 200K A 04;4 ioo H as (0) - ac(0 / a (8) I I 0 I 40 50 60 70 80 0 IN DEGREES-P. I - -— 4 100 110 120 -Omloo 0 30

300 a (0) 8 C 200 0 L; rC o07 100 / t \a (s a() ioo a (9) (, T9e) 0 30 40 50 60 70 80 90 IN DEGREES — T T T T -100 Fig. 7: a (0), a(0) and a (0)-a (0) as functions of C s s C 0 for a large gradient parasitic loop counterpoise antenna located 15' above ground. 31

011218-3-T same antenna located above ground. On the basis of the results given here it can be concluded that it is sufficient to adjust the r. f. phase between the carrier and side-band signals radiated by a standard VOR antenna at a certain convenient angle (say, at the monitor). With a parasitic loop counterpoise antenna it appears that the r. f. phase difference between the two signals cannot be maintained at zero for all angles by such simple adjustment. 32

011218-3-T V NUMERICAL RESULTS FOR COURSE SCALLOPING AMPLITUDES In this section we discuss the numerical results obtained for the observed course scalloping amplitudes in a standard VOR indications for some selected cases. The results have been obtained numerically by using the expressions developed in sections 2 and 3. In the free space case the observed course scalloping amplitudes are given by Eqs. (35) and (36). For convenience we rewrite them in the following form: - A sin (0 -0(1) S -, (75) 1 l+Acos(0-0 ) 2 where the upper and the lower sign on the right hand side corresponds respectively to the subscripts 1 and 2 on the left hand side. The various approximations involved in obtaining Eq. (75) have been discussed earlier. In reference [2], further approximation A << 1 is made in order to obtain the following from Eq. (71) SApprox tan1 Asin(0-0) (76) 2 Figure 8 shows S1 and S| SP vs. (0-01) in the range 0 < 0-01 < 180~ for two values of A. It is evident from Fig. 8 that for A < 0. 1 the use of the approximate expressions give by Eq. (76) is quite justified. For A > 0. 1 the use of Eq. (76) will produce erroneous results. For example, Fig. 8 indicates that with A = 0. 2 the maximum scalloping amplitude does not occur at 0 - 01 7/2. In fact for large values of A the symmetry of the scalloping amplitude curves around 0 - 01 = r/2 are lost and the maxima appear at angles larger than 7/2. In the following sections we discuss the numerical results obtained for the observed course scalloping amplitudes for a standard VOR system located above an infinite planar ground. The scalloping amplitudes have been obtained numerically 33

11 10 co 9 p4 8 0 Q 7 x 6 0; a. 5 "" 4 - 3 2 1 0 0 20 40 60 80 100 120 140 160 180 Fig. 8: Free space scalloping amplitudes S1 and S pprox as functions of (0-01) with A as the parameter.

011218-3-T from Eqs. (73) and (74) which we write below in the following form: 1 2A ST(O1) sin(kHcos 0)sin(0-0) S X + tan — (77) 2 ST(o) +2A S (0) sin (kH cos 0) cos (0-0 Notice that Eq. (77) indicates explicitly the dependence of the scalloping amplitude on the side-band mode pattern of the antenna and the height of the scatterer above ground. In obtaining Eq. (77) it has been assumed that the r. f. phase of the carrier mode complex pattern is equal to that of the side-band mode complex pattern. For the present discussion we consider the cases when the standard VOR system uses a standard VOR antenna and a large gradient ( 23dB/60 gradient) parasitic loop counterpoise antenna. In both cases the antenna is assumed to be located 15' above the ground and the operating frequency is 109 MHz. In our first quarterly report [9] it has been shown that the side-band mode T pattern S (0) for a standard VOR antenna located 15' above ground contains a local s minimum at 0 =0 76~ and a local maximum at 0 =0 83. Forthe large mmn max gradient antenna at the same height S (0) does not have any appreciable maximum of minimum in this region. We compare the observed scalloping amplitudes with both antennas for orbital flights at these two angles. It is assumed that the scattering source is isotropic and that A = 0. 1. Figures 9 (a) and 9(b) show S1 and S as functions of (0- 0) for the standard VOR antenna located 15' above ground for the scatterer located at different heights above ground. These results apply to an orbital flight around the station at 0 = 0mi 76. Corresponding results for the same antenna for an orbital flight at 0 =0 83 are shown in Figs. 10(a)and 10(b). In general it is found that the max scalloping amplitudes are less for 0 = 0 max Corresponding results for the large gradient parasitic loop counterpoise antenna are shown in Figs 11 (a) and 11 (b) for orbital flights at 0 = 0min. In comparing Figs. 11 and 9 it is found that the large gradient system produces smaller 35

20 18 16 C) w -4 Cl)z 14 12 10 8 6 4 2 0 20' 0 20 40 60 80 100 120 140 160 180 (0- 1) IN DEGREES Fig. 9 (a): Course scalloping amplitude Sl as a function of (- 01) for a standard VOR antenna located 15t above ground and for different heights of the scatterer. Orbital flight at 0 = 0 Ai 760~ min

r i CO - O,1 03 0 2 — 0 20 Fig. 9(b): 40 60 80 100 120 140 160 180 (0-10) IN DEGREES Course scalloping amplitude S2 as a function of (0-01) for a standard VOR antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0 x m 760. mmin

12 H = 100' 0 wCA) 20 40 60 80 100 1: (0-01 ) IN DEGREES 20 140 160 180 Fig. 10(a): Course scalloping amplitude S 1 as a function of (0-01) antenna located 15' above ground and for different heights Orbital flight at 0 = 0 " 83~. max for a standard VOR of the scatterer.

12 r2 ( C X cQ C.0 0-4 tf 0 20 40 60 80 100 120 140 160 180 (0 - ) IN DEGREES Fig. 10(b): Course scalloping amplitude S2 as a function of antenna located 151 above ground and for different Orbital flight at 0= 0 ^ 83~. max (0 - 1) for a standard VOR heights of the scatterer.

8 (0 ) 50' ~70' Fig. 11(a): Course scalloping amplitude ISl as a function of (0-01) for a large gradient parasitic loop counterpoise antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0 = 76~. I

I 8 Cs, 6 2 0 H =100' 20 40 60 80 100 12Q 140 160 18( (- 01) IN DEGREES Fig. 11(b): Course scalloping amplitude S as a function of (0- 0) for a large gradient parasitic loop counterpoise antenna located 15' above ground and for different heights of the scatterer. Orbital flight at 0 = 760~.

011218-3-T course scalloping amplitudes. This is attributed to the large gradient possessed by the vertical plane side-band mode pattern of the parasitic antenna. Figure 12 shows S2 vs. (0-0 ) for standard VOR and large gradient antennas located 15' above ground and for orbital flights at 0 = 0 ~ 83. In max both cases the scatterer is located 100' above ground. Observe that in this case the difference between the two scalloping amplitudes may be assumed to be negligible except near the maxima where it is about 0. 5. 42

'I 12 10 'ANDARD VOR ANTENNA, LARGE GRADIENT PARASITIC LOOP COUNTERPOISE ANTENNA U) 4O 0 o [. 0 20 40 60 80 100 120 140 160 180 (0-0 1) IN DEGREES Fig. 12: Course scalloping amplitudes IS21 as functions of (0-01) for a standard VOR and large gradient parasitic loop counterpoise antennas located 15' above ground for a scatterer located 100' above ground. Orbital flight at 0 = 830.

011218-3-T VII CONCLUSIONS A rigorous theory has been developed for the course scalloping amplitudes in the bearing indications of a standard VOR receiver located at a flying aircraft. The receiver characteristics have been assumed to be ideal. The general behavior of the available observed scalloping amplitude amplitude results appears to be in agreement with the theory. Numerical results have been obtained for the idealized case of an isotropic scatterer located at various heights above ground. The results indicate that in general the use of a large gradient antenna reduces the observed scalloping amplitudes. During the coming period numerical results will be obtained for various heights of the VOR antennas and the scattering object above ground. Thoughts also will be given to the case of more realistic scatterers, i. e., nonisotropic multipath sources and also to the influence of the detailed scattering properties of the multipath source on the scalloping amplitudes. 44

011218-3-T VII REFERENCES [1] Anderson, S. R., VHF OmniRange Accuracy Improvements, IEEE Trans. on Aerospace and Navigational Electronics, Vol. AME-12, No. 1, pp. 26-35, 1965. [2] Handbook, VOR/VORTAC Siting Criteria, No. 6700.11, Department of Transportation, Federal Aviation Administration, Washington, D. C., August 7, 1968. [3] Anderson, S. R. and H. F. Keary, VHF Omnirange Wave Reflections from Wires, Technical Development Report No. 126, Civil Aeronautics Administration Technical Development and Evaluation Center, Indianapolis, Indiana, 1952. [4] Electro Technical Analysis Corporation, Analysis of Vector Analogue Computer, Final Report No. RD-67-37, P. 0. Box 5741, Tucson, Arizona 85703. [5] Sengupta, D. L., Theory of Course Scalloping Amplitude in Standard VOR Bearing Indications —Part I-Analysis., Radiation Laboratory Memorandum 011218-501-M, September 1972. [6] Sengupta, D. L. and V. H. Weston, Investigation of the Parasitic Loop Counterpoise Antenna, IEEE Trans., AP-17, No. 2, pp. 180-191, 1969. [7] Sengupta, D. L., Theory of VOR Antenna Radiation Patterns, IEE (London) Electronics Letters. Vol. 7, No. 5, pp. 418-420, 1971. [8] Sengupta, D. L., Theory of Double Parasitic Loop Counterpoise Antenna Radiation Patterns, IEEE Trans., AP-21, No. 1, pp. 94-97, 1973. [9] Sengupta, D. L. and P. Chan, Application of the Large Gradient VOR Antenna, Interim Engineering Report I, 011218-1-T, University of Michigan Radiation Laboratory, September 1972. 45