AFAL-TR-66-57 6633-1-F 6633-1-F = RL-2142 Air Force Avionics Laboratory Research and Technology Division Air Force Systems Command Wright-Patterson Air Force Base, Ohio Derivation of Aerospace Antenna Coupling-Factor Interference Prediction Techniques by J. A. M. LYON, R. M. KALAFUS, Y- K. KWON, C. J. DIGENIS, M. A. H. IBRAHIM and C-C CHEN Technical Report AFAL-TR-66-57 April 1966 Contract AF 33(615)-1761, Project 4357, Task 435709 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory Administered through: OFFICE OF RESEARCH ADMINISTRATION. ANN ARBOR

THE UNIVERSITY OF MICHIGAN - 6633-1-F Derivation of Aerospace Antenna Coupling-Factor Interference Prediction Techniques Final Report June 1964 through September 1965 by J.A.M. Lyon, R.M. Kalafus, Y-K Kwon, C.J. Digenis, M.A.H. Ibrahim and C-C Chen. April 1966 Contract AF 33(615)-1761 Proj.4357, Task 435705 Air Force Avionics Laboratory, AVWC Research and Technology Division, AFSC Wright-Patterson Air Force Base, Ohio 45433

THE UNIVERSITY OF MICHIGAN 6633-1-F When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. Qualified requestors may obtain copies of this report from the Defense Documentation Center (DDC), Cameron Station, Alexandria Va. 22314. Copies of this report should not be returned to the Air Force Avionics Laboratory unless return is required by security regulations, contractual obligations, or notice on a specific document.

THE UNIVERSITY OF MICHIGAN 6633-1-F FOREWORD This report Nr 6633-1-F was prepared by the University of Michigan, Ann Arbor, Michigan, under the direction of Mr. Ralph Hiatt and Professor J. A. M. Lyon and on Air Force Contract AF 33(615)-1761 under Task Nr 435709 of Project 4357 "(U) Electromagnetic Compatibility Control Techniques". The work was administered under the direction of the Air Force Avionics Laboratory, Electronic Warfare Division, Research and Technology Division, Wright-Patterson Air Force Base Ohio. The Task Engineer was Mr. Olin E. Horton; the Project Engineer, Mr. Herbert Bartman. The studies presented herein began 1 Feb 1962 under Contract AF 33 (657)-8178 and were concluded 15 Nov 1965. This report was submitted 3 January 1966., i, tehnical report s been reviewed and is approved. - s,.' -;' -* - *~.;";: -, — - -'', - -"'- h.JOSEPH A. DOMBROWSKI Lt Colonel, USAF Chief, Electronic Warfare Division / ii

THE UNIVERSITY OF MICHIGAN 6633-1-F TABLE OF CONTENTS ABSTRACT v NOMENCLATURE vi INTRODUCTION 1 II FAR FIELD COUPLING FORMULAS 5 2.1 Antennas in Free Space 5 2. 2 Antennas on a Common Surface 7 2. 3 Relation of D(e, 0) to the Usual Definitions of Gain, 9 Directivity, Aperture, and Mutual Impedance 2.4 Effects of Impedance Mismatch and Losses 11 2.5 Effects of Polarization Mismatch for Antennas in Free Space 13 III FIRST ORDER APPROACHES TO ESTIMATING COUPLING WITH NOMOGRAPHS 17 3.1 Thin Monopoles 17 3. 2 Slots 18 3. 3 Approximate Horn Coupling 18 3.4 Approximate Spiral Coupling 22 3. 5 Fresnel Zone Modifications 24 IV DETAILED CONSIDERATIONS OF COUPLING 25 4.1 Monopoles 25 4.2 Slots and Open-ended Waveguides 25 4.3 Spiral Antennas 49 4.4 Conical Horns 68 4. 5 Rectangular Horns 83 4.6 Yagi-Uda Monopole Array 94 4. 7 Coupling of Slot and Horn Type Antennas to Higher Order Modes 100 4.8 Effects of Harmonics on Coupling 102 4. 9 Systems Aspects of Coupling 107 V THE COUPLING BETWEEN TWO ANTENNAS DUE TO THE PRESENCE OF A SCATTERER 109 5.1 Direct Coupling 109 5. 2 Coupling Produced by a Scatterer in Space 110 5. 3 Scatterers on a Common Ground Plane 122 VI EFFECTS OF SURFACE CURVATURE ON COUPLING 125 6.1 Limitations and Applications 125 6. 2 Procedure for Calculating Curvature Effect 125 VII EXPERIMENTAL FACILITIES 133 7.1 Description of Anechoic Chamber 133 7.2 Determination of Chamber Reflections by Swept Frequency Technique 135 7.3 Limitations on Coupling Accuracy 144

THE UNIVERSITY OF MICHIGAN 6633-1-F TABLE OF CONTENTS (continued) VIII COUPLING COMPUTATIONS 148 8.1 Evaluation of Coupling by Nomograph Method 148 8.2 Nomograph Procedure 148 IX REFERENCES 164 X ACKNOWLEDGEMENTS 166 APPENDIX A: ANALYSIS OF PLANAR APERTURE ANTENNA COUPLING 167 A.1 General 167 A. 2 The Coupling of Two Waveguide Connected Aperture Antennas 177 A. 3 The Coupling of Two E-Sectoral Horn-Connected Aperture Antennas (Small Flare Angle Approximation) 197 A. 4 Coupling of H-Sectoral Horn 212 A. 5 Conclusions 220 APPENDIX B: COUPLING DUE TO SCATTERERS 221 B. 1 Coupling Due to An Edge 221 B. 2 Coupling Due to the Leading Edge of a Wing 229 APPENDIX C: RECTANGULAR HORN DATA 262 C. 1 Curves and Tables for Rectangular Horn Coupling Calculation 262 C.2 Experimental Data: E-plane Coupling 310 C.3 Experimental Data: H-plane Coupling 323 iv

THE UNIVERSITY OF MICHIGAN 6633-1 -F ABSTRACT This report emphasizes procedures to determine the power interference coupling from one antenna to another. The two antennas need not be similar. Simplified methods including graphs showing the variation of coupling vs angular orientation, frequency or distance are presented for a number of antenna types. Nomographs have been designed to make possible the rapid calculation of the coupling between two antennas. The last chapter of the report (VIII) shows by example the manner of calculation using the nomographs. In the main body of the report, details are given concerning the methods of obtaining the coupling between two antennas by analysis and also by measurements in the laboratory. Some of the more mathematical aspects of the analysis have been introduced as appendices to the report. For a reader interested primarily in the system interference problem, Chapters I and VIII will be helpful. The simplified formulas for coupling in the far region will prove interesting and the numerous experimental data graphs will be helpful for the various types of antennas represented. The presence of a third antenna or scattering object may occasionally influence substantially the coupling between two antennas, especially in the case of very weak coupling. Some discussion of the influence of scattering objects is given. v

THE UNIVERSITY OF MICHIGAN - 6633-1-F NOMENCLATURE This list applies to symbols used in the main body of this report. Symbols encountered in the appendices are defined therein. a Slot dimension in H-plane; radius of curvature of cylindrical or spherical ground surface. A Effective aperture. b Slot dimension in E -plane. c Aperture radius of conical horn; speed of light. C Coupling. Co Coupling between two parallel isotropic antennas. Cs Coupling due to scatterer. d Diameter of cylindrical waveguide. D Directivity. E Electric field intensity. f Frequency. J Bessel function of the first kind. A A hi Unit vector perpendicular to q- and parallel to the ground plane. A hg Unit vector perpendicular to Q and parallel to the ground plane. H Magnetic field intensity. I Current. k Wave number. (K c) First root of J. (K1 c) First root of J1. L Greatest linear dimension of the largest slot; height of conical horn. p Polarization mismatch factor. P(0) Power density in the p direction. q Impedance mismatch factor. A Unit vector directed from transmitter to scattering center. qi Unit vector directed from trans:mitter to scattering center. vi

THE UNIVERSITY OF MICHIGAN 6633-1-F q s Unit vector directed along the line between scattering center and receiving antenna. r Distance of field point from source point. r As a subscript; refers to the receiver. R First spherical coordinate (field point); with subscript it indicates resistance. Ro Center-to-center spacing of two antennas. s As subscript refers to the scatterer. S Standing wave ratio. S Scattering dyadic. SWR Standing Wave ratio. t As a subscript; refers to the transmitter. Vi Unit vector perpendicular to qi andni. V Voltage. W Power. Y Admittance. f3 Inclination of the major axis of an elliptically polarized antenna with respect to some fixed axis. 7 Second spherical coordinate (aperture point). e Dielectric constant. Third cartesian coordinate (source point). r Second cartesian coordinate (source point). 6 Second spherical coordinate (field point). ea Flare angle of a rectangular horn (H-plane). Ob Flare angle of a rectangular horn (E-plane). ec Flare angle of a conical horn. X Wavelength. u Permeability. Vc Angle between two planes containing the cylinder axis and the two antennas. __ vii

THE UNIVERSITY OF MICHIGAN 6633-1-F Vs Angle between two antennas on a plane containing the center of the sphere and the two antennas. First cartesian coordinate (source point).?:' Hertzian magnetic fector. p First cylindrical coordinate (source point). 7 Second cylindrical coordinate (source point). Third spherical coordinate (field point). Third spherical coordinate (aperture point). w Angular frequency. m viii - ~- -- -- --- -- - -

THE UNIVERSITY OF MICHIGAN 6633-1-F I INTRODUCTION This report presents detailed information on a wide variety of coupling situations involving two antennas. Concentration of effort has been made upon flush-mounted antennas. This has been due to the often specified requirement of flush-mounting for antennas utilized in airborne and aerospace vehicles. Some of the most common types of antennas useful for flush-mounting have been covered. The antennas which have been used in these coupling studies are: a) rectangular slots; b) E-sectoral rectangular horns; c) H-sectoral rectangular horns; d) pyramidal rectangular horns; e) conical horns; f) circular Archimedian spirals; g) square Archimedian spirals. In addition, a monopole antenna, erected perpendicular to the conducting metal plane used as a ground plane often has been used as a probe of the fields of other antennas. Also a Yagi array of monopoles has been used in a brief study of coupling involving this antenna. For those most interested in system interference problems, section VIII illustrates by simple example the use of data from nomographs. These examples show how to compute coupling from one antenna to another like antenna. Also there are examples for the coupling from one antenna to an unlike antenna such as from a circular Archimedian spiral to a rectangular slot antenna. The manner of compensating for curvature of the metal mounting surface is also shown in section VI. For the most part, low- and medium- gain antennas have been used in these coupling studies; all antennas have a directivity of less than 20 db. It is believed that the coupling of antennas having directivities in this designated range presents a relatively severe problem. Antennas with higher directivities can be handled by the same methods. It is necessary to extend the usual definition of directivity 1

THE UNIVERSITY OF MICHIGAN 6633-1-F since in general the main beams of the antennas are not collinear. The directivity function as used in this report is defined as a function of the angles designating direction. In the consideration of antennas of high directivity with main lobe pointed along the metal mounting surface, the directivity function along the radius vector designated could approach the maximum directivity as specified for the main lobe. Such cases are within the scope of this report. The power coupling factor has been studied on the basis of matched antennas. This corresponds to a maximum of power transfer from one antenna to another. All analysis and measurements have been made upon the basis of cw or continuous wave propagation. Modulation of signals can well represent an important factor in the determination of tolerable level of power coupling in the overall interference problem of one system to another. Modulation of signals can be considered as an additional factor which can be dealt with separately since the power coupling based upon the cw assumption applies to most cases of modulated signals. Effects of mismatch, filtering techniques, and connected circuitry beyond the terminal reference planes of antennas are ignored. Such effects can also be accounted for by appropriate additional factors introduced into the interference problem. Similarly, non-linear effects are ignored; these could be introduced although with somewhat greater difficulty into system interference studies. No attempt has been made in these studies to account for the effect on power coupling interference of either mechanical or electronic scanning since this can be treated as a type of modulation. Peak and average power values as well as the details of time sequence in the scanning pattern should be considered ultimately. If the mode of scanning changes under certain operational conditions then it is likely that the interference problem must be solved on a statistical basis. Even with all these possibilities, it is believed that the cw coupling studies contained in this report represent the most basic approach necessary for use in the power coupling interference problem. I 2

THE UNIVERSITY OF MICHIGAN 6633-1-F The coupling problems associated with an antenna having a highly asymmetric pattern such as one with the main beam directed along the metal mounting surface can be treated utilizing information in this report. In such a case, the appropriate directivity corresponding to an angle very nearly 90~ giving a ray along the surface is necessary. Also, in the consideration of such antennas, it is important to have in mind the boundary condition which indicates that the tangential electric field must be zero just off the metal mounting surface. Such antennas with asymmetric patterns will be more highly coupled to another antenna if both of the antennas involved have main beams along the metal surface with the electric field normal to the metal surface. A vertically polarized surface-wave line or area source mounted in a conducting ground plane would be a general class of antennas where coupling could be predicted by the general ideas in this report. Such asymmetric antennas should be considered with pattern compensation made for the location of the antenna within and also flush to the metal surface. The effect of the curvature of the conducting metal surface would be substantial on this type of antenna. The modifications indicated in section VI for curvature of the ground surface would not fully account for the situation involving the vertically polarized surface wave antenna mounted flush on the ground surface, since the surface curvature within the Fresnel zone could be appreciable. These coupling studies have been made with the recognition of dependency upon frequency, spacing, orientation, polarization and scattering objects. In the case of transmitting and receiving antennas mounted in the same surface, the analysis can frequently be made from available coupling formulas of two antennas in free-space. For example, two parallel monopoles operating over a conducting ground surface, have the same coupling as two dipoles having the same spacing and operating in free-space. Likewise, two rectangular slots in a common ground plane show a coupling corresponding to two magnetic dipoles in free-space with recognition that all energy is confined to half-space. 3

THE UNIVERSITY OF MICHIGAN 6633-1 -F In the investigation of the effect of surface curvature it was found that if the radius of curvature is greater than twice the free-space wavelength, its effect can be obtained by the analysis included in this report. In fact, the formulas give reasonably accurate results down to a radius of curvature equal to the freespace wavelength. For frequencies in L-band and higher, such formulas appear to be adequate. For surfaces having a radius of curvature less than one half the free-space wavelength, the surface must be considered an integral part of the radiator, and the problem is more complicated. The low-gain antennas used in these studies were monopoles, rectangular slots, spirals, and, in general, antennas having a lineal dimension less than 1.5 wavelengths. The medium-gain antennas used are characterized by having beams perpendicular to the conducting metal mounting surface. The report includes extended calculations on the effects of scattering from objects other than the two antennas between which coupling is observed. These studies are applicable to a metal edge such as is present on various airfoils and other scatterers. In the presentation of data numerous graphs, charts and nomographs have been utilized. The data considered to be most important or most useful is in the main body of the report. In Appendix A an analysis of the coupling between planar apertures is given. In Appendix B the more detailed information on coupling due to other scatterers is given. Additional experimental data on coupling between rectangular horns is presented in Appendix C. 4

THE UNIVERSITY OF MICHIGAN 6633-1-F II FAR FIELD COUPLING FORMULAS 2. 1 Antennas in Free Space In most textbooks the terms gain, aperture, and directivity are used in claculating the power transferred from one antenna to another. These terms are usually attributed to the antenna itself; for example, it is frequently stated that a horn has a gain of 15 db above isotropic. In order to consider variations with angle, these terms must be generalized. The coupling between two antennas at a given frequency is defined as the ratio of the power received to the power transmitted, or W C r (2.1) c=wt The general form of the expression for free space coupling is C=C (R)D D ( e ( r) P qr qt (2.2) where C (R ) is the coupling between two "parallel" isotropic antennas a distance R apart, given by: (see Fig. 2-1) X 2 23 C (R) ( 47rR ) ' (2.3) D (0, 0) is the free space directivity function, defined by: D (v ) = 2 E 1 sin d (2.4) l I oE( 0) 2 sin 0 dO d0 5

THE UNIVERSITY OF 6633-1-F f(GHz) 10 9 - 8 - 7 6 - 5 4 w R0/x 30 -20 10 9 8 7 6 5 4 3 2 -1. 5 1. 0. 9. 8. 7-. 6-. 5-. 4 C0(db) ~-52 MICHIGAN Ro (cm) — 90 i 80 -50 - -48 - -46 - -44 - -42. -40 -- -38 — 36 — 34 — 32 — 30 — 28 — 26 — 24 — 22 — 20 — 18 — 16 — 14 - 70 -60 -50 -40 0l 0 0 '-4 2-4 3 - 2. 5 -2 - -30 - 25 - 20 -15 - 10 1. 5 1t 1 6

THE UNIVERSITY OF MICHIGAN 6633-1-F This gives the ratio of the power density in a given direction (0, 0) to the average power density. The subscripts t and r refer to transmitter and receiver, respectively. The maximum value of D (e, 0) is called the "directivity" of the antenna (Fig. 2-2). The symbol p is the polarization mismatch, equal to one for a perfect match (Section 2. 5), and q is the impedance mismatch, which accounts for reflection and losses in the antenna (Section 2.4). For two lossless, perfectly matched antennas whose polarizations are matched (e. g., two parallel, linear dipoles), p= 1 and qt=q =1. The advantage of this formulation is that the spacing dependence and angular dependence of each antenna are separated, since for the far-field case D(0, 0) is independent of R. Here the coupling between two antennas is compared with that between two isotropic antennas at the same spacing. Thus if C = -60 db, and neither antenna lies on the main beam of the other, the coupling O will be less than -60 db (assuming no reflections from surrounding objects). 2. 2 Antennas on a Common Surface The major emphasis of this report is on pairs of antennas mounted in a common conducting surface, as the skin of an airplane or missile, or a mounting plate. In this case the tangential electric field is zero (or very small for a good conductor), and the polarization is vertical. Thus the polarizations of the transmitting and receiving antennas are parallel, and p= 1, identically. Furthermore, the coupling between two vertical dipoles in free space is identical with that of two corresponding monopoles in a ground plane at the same spacing. Defining a "semi-isotropic" antenna as one which has a constant pattern factor above the ground plane and zero below, the coupling between two semi-isotropic antennas on the same ground plane is the same as the coupling between two isotropic antennas. m 7

THE UNIVERSITY OF MICHIGAN 6633-1-F z Field point (RO, 0) Source point (e,n,#) x FIG. 2-2a: GEOMETRY OF GENERAL GROUND PLANE APERTURE z (R, 9, e ) a 2 Z'- y Source point -L /4 x FIG. 2-2b: GEOMETRY OF RECTANGULAR APERTURE 8

THE UNIVERSITY OF MICHIGAN 6633-1-F For two antennas in the same flat conducting surface, 0 = 900, and the expression for the coupling becomes: C = CO (Ro) Dt (0t) Dr (Or) qt qr (2.5) where D() =2 / Oi 0J 2r E (7r/2, ) I2 |E(O, 0) 2 sin e d do = D(/2, 0) (2.6) and as before (see Fig. 2-1) C (R) = ( )2 o o 4ir R 0 (2.3) 2. 3 Relation of D(0, () to the Usual Definitions of Gain, Directivity, Aperture, and Mutual Impedance As stated above, the "directivity" of an antenna is the maximum value of D(0, 0). "Gain" usually given in decibels, is equal to the directivity diminished by losses. "Aperture" has usually been associated with the directivity of an antenna, but recently this has been generalized to account for angular variations (Tai, 1961), so that several apertures are defined (for free space): 9

i THE UNIVERSITY OF MICHIGAN 6633-1-F (a) Effective aperture: 2 X2 f A= D (e, ) pq 4ir (b) Effective aperture, polarization matched: 2 X f A = - D (0, )q n 47r (d) Optimum effective aperture: 2 X2 f A = D (2.7) op 47r max where D is the maximum value of D (0, 0). The latter definition is one max usually used for effective aperture, or maximum effective aperture. If both antennas are on the same ground plane, then the apertures must be half those above, e.g. 2 A =r (0, p) pq (2.8) The directivity function D(0, 0) of Section 2.2 will be used throughout this report. 10

THE UNIVERSITY OF MICHIGAN 6633-1-F "Mutual impedance" is a term often used in connection with interference studies. Coupling actually carries less information, unless a phase measurement is also taken. Coupling is related to mutual impedance by: lZ 12 C =412 q qr (2.9) t r where Z12 is the mutual impedance, and Z12 < Rtor R. Rt is the input resistance of the transmitter Rr is the input resistance of the receiver 4S q= (1+ S)2 S is the standing wave ratio (SWR). 2.4 Effects of Impedance Mismatch and Losses Frequently coupling takes place between antennas which operate at different frequencies. Here it is necessary to consider the filtering action of the receiving antenna. The SWR of an antenna at a given frequency is a measure of its rejection of an incident signal. The coupling calculated in the report gives a measure of the power received under matched conditions. Thus from Fig. 2-3 if the SWR of the receiving antenna is 5, the actual coupling will be 2. 5 db below the calculated level. In using the figure, a horizontal line is drawn from one vertical line to the other. Losses are not considered in any detail in this report. Generally they do not affect the shape of the patterns appreciably. Typical results can be taken from the experimental data. These data were obtained by comparing the power I 11

THE UNIVERSITY OF MICHIGAN 6633-1-F SWR 1.1+ 1.2t Transmission Loss (db) Transmission Loss (db) 1.0 1. 3 1. 5 2.0 -2. 5-. SWR 0.05 -4 - 1. 3~. 10.4-.4 -.5 -.6 - 1.0 - 1.5 - 2.0 - 2. 5 - 1.4 T 3. 5 4 - 1. 5 4-. 20 -4 - 1. 6 4 — 4.0 - 4.5 -5.0 - 5.5 -6.0 -6.5 7. 0 - 7.5 - 8. 0 - 8.5... 9.0... 9.5.. 10.0 -12.0 -14.0 -16.0 -18.0... 20.0 -30.0 -40.0 -50.0 - 3.0 + 3. 5 -+ 4. 0.4.. 4. 5 -1-. 1. 74...30 - 5.0 + 1.8 + 5. 5 -6.0 -6. 5 -7.0 7.5 -8.0 9.0 - 10.0 -. 404 - 1. 91 - - (a) (b) FIG. 2-3: NOMOGRAPH RELATING SWR AND TRANSMISSION LOSS

THE UNIVERSITY OF MICHIGAN 6633-1-F delivered to a receiver load to that delivered to the transmitting antenna; thus the measurement contains the losses in both the transmitting and receiving antennas. In the case of slots, comparison of theoretical and experimental data indicate about 1 db per antenna; for the horns, from 1 to 3 db per antenna. It should be noted that for convenience brass was used extensively in the experiments whereas better conducting material may be used in practice. 2. 5 Effects of Polarization Mismatch for Antennas in Free Space For the sake of completeness, a brief discussion of the power transfer between elliptically polarized antennas is included here. It should be pointed out that this discussion does not bear on most of the items in this report, due to the presence of the ground plane, which dictates a vertically polarized wave, and matched polarization conditions. The polarization mismatch is expressed by 1+s 2s2 + 2s s Cos (a +a) r t r t r t ( 2 2 (2.10) (1 +s) (1+S ) r t where s and a are defined below. Usually if antennas are elliptically polarized, this ellipticity is expressed by the shape, and the inclination of the major axis; i. e. r and /3 are specified, e e where re is called the axial ratio, given by minor axis re major axis 13

THE UNIVERSITY OF MICHIGAN 6633-1-F 1 is the inclination of the major axis with respect to some fixed axis and the ellipse describes the locus of the electric field vector. The quantity r ranges from -1 to +1, and referring to Fig. 2-4, the following convention is taken with respect to the sign of re: +: transmits or receives a right-handed, or clockwise wave -: transmits or receives a left-handed, or counter-clockwise wave. Note that an antenna transmitting a right-hand wave will receive a left-handed wave, and vice versa. The quantities s and a are related to r and j3 by: 2 2 r + tan I s 2 2 (2.11) l+r tan 13 e 2r e tan a = (2.12) (1-r )sin 2 e Examples: (a) two dipoles - linear polarization: r = 0 for bothe transmitter and e receiver: 2 s =tan3t s = tan=0, andp =cos (t -13 ) r r t t r t t r Then p = 1 if the dipoles are parallel (Bt = ), and p = 0 if the dipoles are perpendicular (3t = 83r 900). t r 14

I THE UNIVERSITY OF 6633-1-F MICHIGAN I I Xr Xt R T FIG. 2-4: GEOMETRY FOR ELLIPTICALLY POLARIZED ANTENNAS 15

THE UNIVERSITY OF MICHIGAN - 6633-1-F (b) Circularly polarized antennas: r = +1 for both transmitter and receiver: e s =s= 1 a = + 90~, a = + 900 r t then 1+ cos (a + a) = (2.13) If both antennas transmit a right-hand wave, then t = 90~, a = -900, and p = 1. If one of the polarizations is reversed, p = 0. For more detail, one should consult Section 5.2, Tai (1961), Deschamps (1951), and Rumsey (1961). 16

THE UNIVERSITY OF MICHIGAN 6633-1-F III FIRST ORDER APPROACHES TO ESTIMATING COUPLING WITH NOMOGRAPHS The knowledge of the coupling between two simple antennas, such as two monopoles, may be used to estimate the coupling level between two arbitrary antennas. In this section simple antennas are first considered and then attention is given to successively more complicated antenna:., providing estimates of the way they differ from the corresponding simpler cases. All antennas considered are flush-mounted, so that the coupling formulas from Section 2.2 are used: C = CO (Ro) Dt(0t) Dr (Or) qt qr (3.1) C=C() ) ~(3.1) This formula holds for spacings greater than L /X i.e. R > L /X, where L is the greatest linear dimension of the largest antenna. The expression may also require modification for angles where D(0) becomes very small, i. e. on a null. These cases will be discussed further in Chapter IV. 3.1 Thin Monopoles If two monopoles are mounted perpendicular to the ground plane, the directivity function D(0) is independent of angle. For short monopoles, i. e., having lengths less than X/10, D(p) = D = 1.5, or 1.76 db. (3.2) max For resonant quarter - wave monopoles D(0) = D = 1.64, or 2.15 db. (3.3) max m 17

THE UNIVERSITY OF MICHIGAN - 6633-1 -F Thus the coupling between two quarter-wave monopoles, each perfectly matched and lossless, is 4.3 db above C (R ). This case is important enough 0 0 to deserve a separate nomograph (Fig. 3-1). 3.2 Slots The directivity function for a short, thin slot is given by D(0) = 1. 5 cos (3.4) except at g = 9o~ (see Fig. 3-2). Near p = 90~, the coupling may be near-field coupling, which cannot be expressed in the far-field coupling form. This is discussed further in Section 4.2.3. By short and thin is meant: for this expression to be accurate within 50/o, b < 0. 2 X and a < 0.35 X. For a thin slot a half-wavelength long, D(O) = 1.64 cos2 (7r sin ) (3.5) Note that the maximum values of D(0) for the two cases above are the same as for the monopoles of equal length. The general slot case is discussed in detail in Section 4.2 along with nomographs for calculations. The values of D(p) above are limiting cases. 3.3 Approximate Horn Coupling The antennas considered thus far have been low-gain, broad beam radiators. The aperture fields of the slots are known with some precision, and the resulting field patterns are relatively unaffected by the inaccuracies. As the aperture is increased in size, this is no longer true. First of all, the aperture fields are more sensitive to small disturbances. Second, the resulting fields are more 1 _18____

v,4 1 —i o-A C co i 3 Y ) I I I I.1 II! CA I I r UI-! I Co i I 1 III Ilil IIII IIIIilll l lll l11 1 11 III I I jI I I 1 I f -- -! I I I I I I I I I I I I F i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ~ W W3 3 C C.4 CA3 CA) CAD CA) t~ i w t & I-A II) 01 4. 0 Co 00 -4 O 0 Cw3 w 0 O 00 I w 0 z Z C< GO 0 C) I i P It 0 z FIG. 3-1: NOMOGRAPH FOR CALCULATING COUPLING BETWEEN TWO QUARTER WAVE MONOPOLES

THE UNIVERSITY OF MICHIGAN 6633-1-F sensitive to small changes in the aperture fields, although on the main beam it remains relatively insensitive. Third, conventional configurations (such as horns) do not give plane waves at the aperture, but rather curved wave fronts. For a common type of horn, as well as many practical antennas, the main beam is perpendicular to the surface, and coupling takes place through the side lobes. A very rough estimate of the coupling between two horns, accurate to about + 10 db, (except at a null), can be obtained without knowledge of the flare angles; in this estimate the following expression is used: 2 D(0) = 6 () (b) cos fv) f(u) (3.6) 7r x 2 / /A 1 2 where the geometry is that of Fig. 3-2, and sin v <7r/2 -- V< 7v/2 V f1(v) I 1. V>7r/2 v I cos u u<ir 4u2 2 f2(u) =(3.7) 1. u> r 4u2 1 - j 20

THE UNIVERSITY OF 6633-1-F MICHIGAN <^I i FIG. 3-2: SLOT GEOMETRY 21 21 I

THE UNIVERSITY OF MICHIGAN 6633-1-F a u = 7T (-) sin V = 7r (-) COS A set of nomographs for finding D(O) by this formula is given in Fig. 3-3. The use of these nomographs is explained in Section 8. 2.3 where an example is given. This expression may be used for obtaining "ball park" coupling levels for aperture type antennas where the axis of the main beam is normal to the surface on which the antennas are mounted. More accurate expressions for the standard horn are given in Section 4.4.2. The curves of Figs. C-1-a through C-2-m demonstrate the comparative coupling levels of the above expression (3.6) and the more detailed and accurate ones. 3.4 Approximate Spiral Coupling The spiral is a low gain radiator which differs from the antennas discussed before in that it is not linearly polarized. Along the beam axis a well designed spiral is very nearly circularly polarized. Due to the presence of the conducting ground plane, the electric field along the ground plane is vertically polarized. Thus the coupling between two spirals is dependent only on Dt(0t) and D ( r), and does not depend on the sense of polarization of each antenna. For example, if two spirals having opposite senses of polarization have the same pattern factor, their couplings with a third antenna will be identical (same spacing, orientation assumed). To a first approximation spirals are omni-directional with a progressive phase shift as the angle 0 is changed. From the experimental data the circular spiral has an average pattern factor given by D(p) = -8 db (axial, or IX mode) (3.8) I 22

A — 6.0o — 5.0 4.0 3.0 2.0 — 1.5 1.0 0. 8 FIG. 3 -3: THE UNIVERSITY 4 6633-1-F b D 0.2 -+10.25-. 3 0 -10.5 -. 6.7- 2.8-:. 9 1.0 -30 -40 1. 5 2.0. -48 NOMOGRAPH FOR CALCULATING A RECTANGULAR HORN MICHIGAN U OwT.5w -. 6w4 V. 7w +. 8w-I 97r 4. 1. 0w -4-..57r<v Ar.4 -.3w -. 28wr -. 26w -. 24wr -. 22wr -. 20wr -. 197w -. 187r.17wr..16wr K. 15wr.14wr 1. 1wr ~ 1. 27r ~ 1.3wT ~ 1. 4w 4. 1. 5wr 4-.137w 1. 67r 1. 7w. 12w 1.8w. 11wr 1. 9w 2. 07. o 2. 1w7 2. 2wr2. 3wr-. 09w 2. 4w 2. 5w. 08w THE DIRECTIVTY FUNCTION OF

I THE UNIVERSITY OF MICHIGAN 6633-1-F The square spiral has a slightly higher value since it has slightly broader beam: D(p) = -7 db (3.9) These are average values taken over the design band. At a particular frequency and angle, the pattern factor may vary significantly from this value, but over a band of frequencies this expression is adequate. Spirals are discussed in more detail in Section 4.3. 3. 5 Fresnel Zone Modifications The far field coupling formulas are usually considered to be valid for the condition: 2L R > (3.10) o A where L is the largest linear dimension of the larger antenna. This criterion was derived by considering a phase difference of 22 1/20 between a plane wave and a spherical wave incident on a flat plate. This apparently has little to do with the criterion desired for coupling, i. e., the spacing at which the coupling deviates from its far field expression by, for example, 1 db. A study by Jacobs (1960) indicates that for the maximum coupling case, i. e. two horns "looking at" each other, a variation of 1 db is encountered at about R = L2/X for several aperture field distributions. For the experimental data taken here, the coupling patterns inside this region change somewhat in shape, particularly at the pattern minima, but the change in level is close to the 6 db per spacing octave which is the fall-off characteristic of the far field. This is discussed further in Section 4.5. a 24

THE UNIVERSITY OF MICHIGAN 6633-1-F IV DETAILED CONSIDERATIONS OF COUPLING In this chapter more accurate calculations and expressions are given for the antennas described in Chapter II, with other related antennas as well. The theoretical results are compared with experiment. Fabrication problems, design criteria, and antenna details are discussed as well as the limitations on the theoretical models. Fresnel zone and near field effects are described and harmonic frequency effects and higher order modes are considered. 4. 1 Monopoles The emphasis in this report is on coupling levels rather than phase information. For monopoles a quarter wavelength long, or less, the directivity function D(0) does not change significantly as the shape of the radiator is changed. An estimate of these changes can be obtained from the slot data of Section 4.2. The coupling of dipoles and monopoles has received much attention in the literature (King, 1946, Blasi, 1954). There the concern has been to obtain high accuracy in calculating phase in applications to arrays. Here, the directivity functions of Section 3. 1 are sufficient. 4.2 Slots and Open-ended Waveguides The general problem of coupling of two rectangular apertures was treated in this study. In particular, solutions were found to the following cases (see Fig. 4-22). 1) Directivity functions of an open-ended waveguide for arbitrary size: TE10 mode, far field. 2) E-plane coupling: TEl0 mode, near and far field. 3) H-plane coupling: TE0 mode, near field only. 25

THE UNIVERSITY OF MICHIGAN 6633-1-F An outline of the derivation of the solutions is given in Appendix A. Only results and curves are presented here, including a nomograph for calculating the coupling between two slots. 4.2. 1 Pattern Factor for Small Slots and Open-ended Waveguides From Appendix A (Sect. A. 2), the normalized pattern factor D(0) for an arbitrary rectangular aperture, assuming a constant phase front at the aperture, is 3 2- 2 Cossiv 2 D(0) 3 2 2 (sin v)2 cos 2 (4. 1) 2F v where a u = 7r () sin 0 (4.2) b v = () cos 0 X and F is a modification factor having the following properties: F 1, for a/X and b/X small, 1- 2 (4.3) F =.2a a/X and b/X large Fig. 4-1 shows F as a function of a/X, with a as a parameter, where a = b/a. For a,< 0. 5, which is the case for slots, and a/X < 0. 8, F is given approximately by: 26

THE UNIVERSITY OF 6633-1-F MICHIGAN o itl II 0 / I I I/ I I, o 4 in - -: - 1 Pk I 0!o Lo Cq I 0 O! FIG. 4-1: MODIFICATION FACTOR "F" VS a/X 27

THE UNIVERSITY OF MICHIGAN 6633-1-F F = 1 - (a )2 (.374+ 1.360a2)+ (-)4 (.130+.365a2 +.556a) - (-)6 (. 154+. 218 a2) (4.4) Figure 4-2 is a nomograph for calculating the directivity function D(0). For convenience the nomograph in Fig. 4-3 is included. It facilitates the calculation of the coupling between a pair of slots or open-ended waveguides. Instructions for using the nomographs, including worked examples, are given in Chapter Vym. There are two assumptions involved in the use of the formulas and nomographs: a) Each slot is in the far field of the other; this is not much of a restriction, since the slots must be practically touching before any significant discrepancy occurs. This is discussed in Section 4.2. 2. b) Neither 0t nor 0 are 90~. When this happens, the far field term approaches zero, and the near field dominates. This is shown in Section 4.2.3. 4.2.2 Broadside Coupling of Open-ended Waveguides When two open-ended waveguides are in the broadside, or E-plane configuration, where 0 = t 0~ or 180~, the problem can be solved for any spacing, including the near field. The problem of two open-ended waveguides having the same dimensions is treated in detail in Appendix A. In order to consider near-field effects it is no longer possible to treat each antenna separately; they must be considered as a pair. 28

THE UNIVERSITY OF 6633-1-F MICHIGAN IOL 11 I I I I 11 I I I I coin.4 -4 R COM R ul 1 I a i I Im I I I 11 O.-4 0 in 0 kn S. C4 CM m en I I I I I I to LM Cl) $ 10 I a aI I I I. In 9 tn 0 1 1 1 1i 4-4 - 0 11 1 1 1!1 1 1 1 i * In 0 I in r i i 0 4 In a I - -.I i i i i i a i i i i 2 i I I i I I I a a i i i i i i i i i i i I i kn qr 0 In LO Ln I 9 In I LO t- a I0 I0 0 0t In 0 Ln C I m In in It I I i i I I I > li i I iI I I a I I I -1 — v — T I * *' I In a inC in C tn 10 10 10 C- C- 1 010 In A o o Qo o< AA ~ U U I I ---- FIG. 4-2: NOMOGRAPH FOR CALCULATING DIRECTIVITY FUNCTION D(0) 29

G GT TF. 025 C(db) 00 * 025.100 * 125 FG.150 * 175 * 200. 225 4. 0 0 0 ci, ri, z e tTI 0 H z. 275 -.300 -. 325 -.350 -. 375-. 250 4 - *- 100.-.150. 175 —.200. 225 - 250. 275 -.300. 325 * 350 * 375.400. 425.450 -. 475 -10 - 15 -20 -25 — 30 — 35 — 40 R r1 1.0 -1.5 2.0 3.0 4.0 5.0: 6.0 -- 7. L - 8.0 -9.0 10.0 H z P-4 Ul) Cw 0.400 4-. 425 -f a/Xk=0. 90 &/),=O. 85 a/X=O. 80 &/XL=0. 75 a/>X=0. 70 a/Xk=0. 65 a/)X=0. 60 &/,X=0. 55.450 + P-4 z. 475 +I ~-45.500 -.L. bR/X -.i-.500 bTA)

THE UNIVERSITY OF MICHIGAN - 6633-1-F The coupling must be modified by the factor N: C = C 0(R0) Dt(o~ D r(Or N qt qr (4.5) where IK11 2 4 2 b 4 sin v4 2r 2 v R a o (4.6) and 4 0+b K -4 o a -b 0 0 2 + (1 +-) si] 4a a 2 dy (b - R - YI) i(1 — ) (a - )) cos ES 0 J 4a a 0 n- _-jk/y2 + 2j n.. / o (4.7) Experimental data were obtained for broadside open-ended waveguides for several spacings, as shown in Fig. 4-4. The straight line represents the 6 db per octave decrease in coupling ass ociated with the far-field formula. The calculated coupling level checks within 0. 5 db of the observed level, which is closer than the expected accuracy of the measuring system. It should be noted that for the extreme left experimental point the waveguides are as close as it is physically possible to place them. Thus for practical purposes, the far-field coupling of Section 4.2. 1 may be used. 31

THE UNIVERSITY OF 6633-1-F MICHIGAN -10 R o -15 k -20 Coupling (db) -25 0 0 0 0 *.\ o Experimental * Theoretical db per octave line -30 1 - -35 R /X (V I I I I I I I I I I I I I I I II -40 I I I L — I L -- - I I - - I I - -M -.. -.-. I.2.5 1. 2..5 10. FIG. 4-4: BROADSIDE COUPLING OF OPEN-END WAVEGUIDES VS CENTER TO CENTER SPACING 32

THE UNIVERSITY OF MICHIGAN 6633-1-F 4.2.3 Generalized Formulas for Slot Coupling When the two open-end waveguides are positioned so that their narrow sides are facing each other (H-plane configuration) the formulas developed in the preceding sections have to be modified. The general formula for coupling can only be obtained through the solution of boundary value problems comprising the whole transmitting and receiving structures. An approach to the problem is outlined in the Appendix A and the complete details were given by Kwon (1965). In the present section, the construction of the general formula will be attempted from the simplified engineering point of view. If the size of the two waveguides is such that the fundamental mode is the only propagating mode, one can view the coupling as a two-port network problem where subscripts "1' and "2" designate the transmitting and receiving port respectively. Since no generators are in the receiving guide and the wave transmitted into the receiving guide from the halfspace does not reflect back (matched conditions), one can set k -((-) 2 y(R) =(R) r (4. Y Y lo=(4.8) 10~ (the subscript 10 refers to dominant mode). Then, the derivation of the following algebraic formula is a purely algebraic matter. 2 2 12 V11 1 V12 i I ) I2 R R' Y i V e 2 2 (R) 2 eLT)*j (R) Y22 2 e (T 10 10 1 +10 (4.9) 33 --

_ THE UNIVERSITY OF MICHIGAN 6633-1-F where * designates complex conjugate and y(T) = G(T) + j B(T) Y' = G +j B (4.10) The meaning of G(T) and B( is obvious. If G >> B(' then W R R e[ V2 I2L WT - R e 1 1 Y 2 (R) G(T) 1 22 22 0 y(R) Y0 (4.11) SinceY =Y12 << Y~ Y or Y,one can have 21 12 11 22 (T) (T) Y12 Y21 G(T)y() = Yl Y2 11 11 22 0 and y(R) = y(R) G 0 22 22 (4.12) (4.13) then C = 4 21 G (true for no admittance mismatch) 4 G11 22 (4.14) which is equivalent to (2. 9). It will be shortly shown that (4.14) reduces to the far-field coupling formula. Equation (4.14) is a generally acceptable formula 34

T THE UNIVERSITY OF MICHIGAN -- 6633-1-F even for the near field case, if the circuit parameters satisfy the assumptions leading to (4.14) Another coupling formula equivalent to (4.14) is WR C-w 2 ^ (R)l2 G(T) IV112 1 1 y(R) (T) 0 12 I2 2 V1 2Y 10 T (4.15) where no approximation is used. The circuit parameters in (4.14) are given by: (Kwon, 1965) 2 G - 11 3 7r at ( —) 6t, M(bt 2 r G22 2 3 b 7r r (4.16) 6 = R(1 (a. b) [see (A. 41c) of Appendix A] b. -- 3T (kb) ( ) F. F. is given by (4.4) (4.17) 3 1 a. i i 1 1 1 Y J (4.18) 12 j - 7j (a b) (a b ) 10, 10 t t r r For 7_, 7r _ (c - ), (see Fig. A-2, Appendix A), F o t 2~(t 2 -r, r 2 35

THE UNIVERSITY OF MICHIGAN 6633-1-F 16 10, 10 R O (7 7r a (f) e 7r -jkR L kb o sin ( t cos t)-C 2 ka ( 7r sin t) where kb ka sin ( r cos ( ) * C ( sin 0 ) 2 r r cos (-sin 0) 1 a -(,ka,2 sin2 7r 1 -(-) sin 2 7r (4.19) (4.20) For t = "10, 10 J -i 10, 10 kat 4 at a jkR cos (2 ) 2( ) (e t 2 2 R k a (1 )7 7T 01 (4.21) where kb r sin(- 2 cosp) kb r 01 6 s p - (kb ) cos2 (r cos ) sinp 1 C os r 2 j r ka C(- sin ) r (see equations A. 36d) and (A. 36e) of Appendix A). Substituting (4. 16) - (4.19) into (4.14), one obtains the following expressions for coupling: 36

THE UNIVERSITY OF MICHIGAN 6633-1-F For 7 2 3t F2 3in- - Cos t sin L(kR0) (kb) (kb ) F F sin( 2 cos ) C r sin t) kb ka 2 2 1 ri L4Roi Ft FR 2 os(L 2 C (kR ) (kb) (kb ) 4R (4.23) L o t r- L FJ t R Li ka J 7T It is easy to show that (4.22) is identical with the far-field coupling formula obtained by using directivity concept. If one uses (4. 15) and the expressions for I2 obtained in the Appendix A, one obtains the other coupling formulas, which are not bound by the assumptions leading to (4.14). The final results for the coupling are: for rtf 32 1 t 1 1[ kb t kat 2 Cf o m = o bt — Lind Cos Cb ( singdrit c nj i2 (kb ) -( — )2 r (kR )2 6 L or t i r ka o r kb ka 2 L sin (-2 cos 0r) C ( sin f)t. (4.24) w 37

THE UNIVERSITY OF MICHIGAN 6633-1-F For 7r ka 2 b cos(-) 2 1 1 t 1 2 2 C =2 1 (4.25) ir (kR ) 1-(-) (kR) o t k a 0 ka o __t I 7T The above coupling formulas for the slots, (4.22) through (4.25) give excellent agreement with experiment for b < a X and R > X as shown in the accompanying figures. The derivation of the coupling formula for R < X is o difficult. However, for a= ar =a, bt =b =b and Ot = Or = 0, the exact 1 1 1 ka 2 1K 2 C 32 2 r 2 '-.6 (4.26) 7r (kb) 1-( —) ka where b -7T b X K =a r - a) dc (1o-uJ - or - X) cos X 22 a a 7r7r R R + g ( -r sin -, a cos + r) (4.27) a a 7r ~ 2 38

THE UNIVERSITY OF MICHIGAN 6633-1-F -10 A Experimental:} Theoretcal -15 -20 -25 -30 -35. * * 0 t2 12e I e1% C (db) t = 0~ -45 -L -5o L -55 L f= 9.20 KMC X = 1,283 in. a = b = 0.9in. b = b 1 0. 4 in. t r -60 A a! a a a I a.. 0.1 0.2 0.3 0.4 0.5 0.6 17 a8 9 1 1.5 2 3 4 5 26 7 i 10 R /X FIG. 4-5: COMPARISON BETWEEN EXPERIMENTAL AND THEORETIC RESULTS 39

~~THE UNIVERSITY OF MICHIGAN 6633-1-F 6 8 I T i I II I L 0CO (LZ=7T/2) I t7/2(a= 0) NO~ DATE i lu i i i wOr — AAX4M 2. I.- 6. — -~20 - Expert..I ment x. Theory'.. 0 -6 0 4$;, > )I 0-Gee0CO'P - Experiment rheory.. C 7 __ i 4- 4- - - n 4. 3:. 6- - 0 -70 - -l. -1 -t -50 i M _ 7;1( _ i — IA+ F9 I I -- - - - Li >- -4 --- -J -2 ---6 -- -6 ---8 it 4!?i -.. I.. I I -7. 7 I I I I I I -60 i w i i 0! lo= 11. 43cm. __r-_ - — ",I,, - - - j Or -''- ---.8-4. ".0. ' I CiO -, I I I I i m a I. -. - i. -, - - 30' 24" 18- 120 60 1 8 1 1 -- -1 ---180 144 1080 720 360 360 1 60 1 I I r r r IR0=11.43cm 12. 1 80 240 720 1 080 144" 30 I I FIG. 4-6a: COMPARISON BETWEEN EXPERIMENT AND EQS (4-24) AND (4-25) f=8. 03 GHz. - 40

THE UNIVERSITY OF MICHIGAN 6633-1-F 2 1 I INO. I DATE 4 6 -20 -00 20 --40 --~50-____ - 1 50 40 30 300 24 0 183 1800 1.440 1080 12" -— I — - - 720 P. 60 — 1 -- -. - 360 10 20 103 40 S 60 120 - 180 240- 310 360' 720 1080 140 0 bO6 -—. - I --. - ___j AND (4-25) f=9. 03 GHz. 41

THE UNIVERSITY OF MICHIGAN6633-1 -F -UNO. 2 -- -~ - -~ - - DATE -10-___ _ -20- - - 07 40. aw -- - - 540320 10 50 -- 4- 3 -- 300 240 180 ~ 120 60 1800 -1440 1 080 720 360 1, % I I I IL 5 60 120 18. 240. 3Q0o ----------------- -I -;- ----— j 360 - 720 1080 1440 1*.ISO I I ~ I I I I -.; 60 FIG. 4-6c: COMPARISON BETWEEN EXPERIMENT AND EQS (4-24) AND (4-25) f=10. 03 GHz. 42

THE UNIVERSITY OF MICHIGAN 6633-1-F I ---I ---- L2 4 6 8 I I I T I I NO DATE - -— 6 — ---—. —.- --------.- - - —. -........ *- -2 0. —. — 2 - - -- ------- -- --— I ---- ------.. -.- - - 1 -- - 20 02. - -- '-1..- —.. —.. —1...-. —...- - I........... -4:,-30 1 - _ --- -.......... --------- - — 0 --- —--- -- _ I ---- ----- ------- ------- --- —. ------ - - - - -40- - 1 - - — I - 50- - — 1 ____I....~.................................+.7.... 1... 2- — j -. - j- - I -4 — 300 240 18~ 180 1 --- - 8 180 1440 108 I I,-, 2' -L - - -.. 72~!. Irl 6- - 36~! - --- -I -- 60 6.. - -,- - rco * Oa 2 - 12" Ir.. 72~ o~ 4 5~ - -- - -- - I 18 24~ 30 - 1 — - - -. I 108~ 1440 oI I I, 'I h m I I I I. - FIG. 4-6d: COMPARISON BETWEEN EXPERIMENT AND EQS (4-24) AND (4-25) f= 11.03 GHz. - - 43 l

THE UNIVERSITY OF MICHIGANI 6633-1-F -0 -2 1 ~NO. DATE l -4 - --- 2 - r I- -. P-O," x x10.- - -f -! K., I I;! I I,,, I I 71i I, 1:6- 1. --- -I -. — - I I 6 V-1 I!;I;I-!I I, - -7 -41, I i iI -- -- I - i I o If. -1 I. 1 i 1, i i m i 1 - I, - A7I - I T: I — t- — r-! I i I ' 7. T i I i I,- 1 i I f, i t I I I I. I I:: I i I I I I. I FIG. 4-6e: COMPARISON BETWEEN EXPERIMENT AND EQS (4-24) AND (4-25) f=12.03 GHz. 44

THE UNIVERSITY OF 6633-1-F MICHIGAN -24 -26 -28 -30 -32 -34 -36 A Experimental o Calculated by Gain formula, eq. (4. 22) ~ Calculated by eq. (4. 24) R =2X o Or (=) -38 -40 R o t o (a T) -42 -44 -46 FIG. 4-7a: COMPARISON OF EQS. (4.22) and (4. 24). f=9. 20 GHz. X=1.283", at=ar=0.9" and bt=br=0.4". 45

THE UNIVERSITY OF 6633-1-F MICHIGAN db -35 r V Experimental o Calculated by gain formula * Calculated by equation 4.24 -36 L -37 - -38 -39 L -40k - 1 - -41k -42 -43 t -44 2X -45 -46 - I I I I I I I I I 2.-% f. 4. - - - e, I -47 U. b 1 1.5 2 2. 5 3 3.5 4 I D X FIG. 4-7b: COMPARISON OF EQS. (4.22) AND (4.24) 46

THE UNIVERSITY OF MICHIGAN 6633-1-F -. FIG. 4-8: SQUARE AND CIRCULAR SPIRAL ANTENNAS

THE UNIVERSITY OF MICHIGAN 6633-1-F Epoxy Glass Board FIG. 4-9: CIRCULAR ARCHIMEDEAN SPIRAL FACE CONFIGURATION 48

THE UNIVERSITY OF MICHIGAN 6633-1-F and ka 2 2 -j -VU +v 7T g(u, v) = e 2 (4.28) 2 U +v For pa = 2 and p = 0, C of (4.26) is compared with experimental results in Fig. 4-5, where C for a = 45~ and a = 80~ is also plotted. Before closing the present section, some comments are necessary about the differences between (4. 22) and (4.24) or (4.23) and (4. 25). The lack of symmetry in (4. 24) and (4. 25) is due mainly to the fact that the assumption of (4. 13) was not used in their derivation. Therefore, as long as the assumption of (4.13) is valid, the coupling expressed by two set of formulae gives similar answers as illustrated in Figs. 4-6 and 4-7. 4.3 Spiral Antennas 4.3. 1 Introduction and Summary of Spiral Properties The widespread use of flush broadband antennas on aircraft and missiles, both singly and in arrays, has motivated a detailed study of coupling patterns. The Archimedean spiral is representative of this class and was thus chosen for further study. It provides a practical way of producing circularly polarized radiation over a wide band of frequencies. It is usually mounted in a conducting ground plane and backed by a cavity. The square and circular spiral configurations are the most common types; samples are shown in Fig. 4-8. These spirals may be wound either clockwise or counterclockwise depending on the sense of circular polarization desired. Spiral antennas are usually fed out of phase in order to effect maximum radiation in the * Equation (4.23) preserves symmetry when o = 7r/2. w 49

THE UNIVERSITY OF MICHIGAN 6633-1-F axial direction (axial, or 1X mode). A balun is used to provide a balanced feed. The operation of the spiral antennas has been explained qualitatively in a number of papers and a summary of the operation is given here. As shown in Fig. 4-9, the antenna can be thought of as a two-wire transmission line which has been wrapped around the origin in such a way as to form a spiral. It is obvious that at any position along this line, the wire lengths differ. Furthermore, this difference in lengths steadily increases for points further removed from the origin. If two out-of-phase waves of current start at the center terminals and travel out along the transmission line, they will eventually become in phase when the difference in the lengths of the line wires becomes X/2. It may be shown that the distance from the origin at which this in-phase condition occurs is given approximately by r = X/2ir =.159X. This value is the radius of a ring one wavelength in circumference. From considerations of this kind, it is generally assumed that most of the radiation from the spiral originates in a region near the one wavelength ring where neighboring current elements have small phase differences. For other regions of the antenna the fields tend to cancel. It is also reasonable to assume that the currents are attenuated in the region of the radiation ring due to radiation. If the ring is not too near the outer extremities of the antenna, the current magnitude may be reduced by radiation enough so that the reflections from the ends of the spiral arms can be neglected. In a rough way, one can approximate a spiral at one frequency by a similarly shaped loop which carries a traveling wave. Thus, for a circular spiral, consider a circular loop with a circumference of one wavelength. The separation of parallel elements is the diameter d = X/ir =. 318X. For a square spiral, this distance is d = X/4 =. 25X. 50

THE UNIVERSITY OF MICHIGAN 6633-1-F If dielectric is used extensively near the spiral (e. g. in the cavity), the radiating region will be reduced in size. The effective dielectric constant is not significantly altered until a substantial amount of dielectric is used, in terms of wavelengths. When this occurs, the effective spacing d decreases, which broadens the beam. Thus the gain of a spiral antenna depends mainly on two factors; 1) the shape of the spiral, and 2) the effective dielectric constant surrounding the elements. A square spiral will have a broader beam than a circular spiral, and thus less gain. Similarly, the effective dielectric constant varies directly with the width of the beam. Thus for maximum gain, a circular spiral should be utilized, using an air-filled cavity, as thin a printed circuit board as possible, and as low a dielectric constant as possible. The above conclusions are born out by the experimental data and a consideration of the resonant circumference condition. Interest to date has centered largely around the near-axis, far-field properties, which can be obtained with rather crude approximations to current distribution. However, the interference effects between antennas in a common ground plane depend upon ground plane patterns which require a more detailed knowledge of current distribution. An interesting factor of spiral ground plane patterns (Fig. 4-10) is their lack of omnidirectionality. However, p-variations are not due to imperfections in spiral construction. These patterns were taken from a circular spiral to a X/4 monopole. Even with a perfectly balanced feed (necessary for the axial mode), and with perfect mechanical symmetry, a coupling variation of several db will occur as the spiral is rotated in the ground plane. With the commercial spirals tested, other effects add to the basic 0-variations. These variations 51

THE UNIVERSITY 6633-1-F 10(a) OF MICHIGAN -30 -36 -30 '0 0 Q -36 -30 10(b) f = 2.2 gc 10(c) f= 2.4gc 10(d) f= 2.6 gc.0- -n actd -36 -30 -36, 180" 360u FIG. 4-10: COUPLING PATTERNS FOR CIRCULAR SPIRAL "A" 52

THE UNIVERSITY OF MICHIGAN 6633-1-F -30 I I -38 -30 -40 v i -30 -~,, -40 -30 -40 -40 -30 -40 f = 2.8gc 10(e) f = 3.0 gc 7 10(h) 00 180~ 360~ FIG. 4-10(Cont.). COUPLING PATTERNS FOR CIRCULAR SPIRAL "A" 53

THE UNIVERSITY OF 6633-1-F MICHIGAN -30 -40 -50 -30 -40 -50 -40 -50 -60 -30 10(i) I. g = l \ / \i=4.o / 10(k) f=. 4. 2g/ /\ 10(1) — 00 1800 3600 FIG. 4-10:(cont.). COUPLING PATTERNS FOR CIRCULAR SPIRAL "A" 54

THE UNIVERSITY OF MICHIGAN 6633-1-F stem from the following secondary causes and apply to all antennas in this class. 1) In-phase feed currents and the subsequent redistribution of current on the spiral. 2) Reflections from the ends of the spiral elements 3) Reflections from the cavity 4) Distributed, induced reflection by neighboring elements 5) Direct radiation from the feed and balun The effects noted above are more critical in coupling properties than in near-axis properties. While it is difficult to isolate the effects of the several secondary causes, additional information thereon would help in the future design of flush antennas. Variations in the ground plane coupling pattern are not completely undesirable for they may make possible appreciable decoupling by proper orientation. However, the observed 0-variations change rapidly with frequency, so the coupling reduction which results from optimum orientation will exist over a narrow frequency band only. For array purposes, it would often be better to have frequency independent 0-variations over the band, even at the expense of higher coupling levels. 4. 3.2 Circular Archimedean Spiral Two similar circular archimedean spirals obtained from AeroGeoAstro Corporation were tested singly and together. Pertinent information is given below. Frequency range 2 - 4 GHz Number of turns 13 Width of conductor 1. 5 cm Width of spacing 1. 5 cm Balun type Roberts, strip-line Diameter of cavity 10 cm. j I 55

THE UNIVERSITY OF MICHIGAN 6633-1-F In order to test each antenna singly, a quarter-wave monopole probe was used as a transmitter and located 11. 3" from the spiral center. A 4' x 4' ground plane surrounded by absorbing material yielded results which agreed to within 0. 5 db at large couplings (-35 db or higher) and to within 2 db at lower levels (-50 db) or the values obtained using a 121 x 12' plane in the anechoic chamber. The general problem of coupling between flush antennas in a ground plane is greatly simplified by the fact that the incident field is linearly polarized, since the tangential E-field must be zero at the conductor. This, plus the fact that in the far field the fields look like an ordinary plane wave, means the great majority of problems can be solved merely knowing the coupling patterns. The implication that the winding sense of the spiral is irrelevant breaks down in the near field case because the receiving spiral in a spiral-to-spiral configuration subtends a large enough angle of the other so that the phase distribution of the incident field over the receiving area is no longer that of a constant-phase plane wave. This effect can be noted experimentally by comparing the couplings at large and small spacings. Figure 4-10 shows typical coupling patterns between the spiral antenna and a tuned monopole probe. These exhibit sharp nulls near the upper end of the design band for both A and B antennas, due to cavity, balun radiation and secondmode effects. (The second mode is excited by in-phase currents at the feed, and has a null on the axis.) Figure 4-11 shows coupling patterns taken at three spacings for two similar circular spirals A and B at one frequency. At other frequencies the behavior was similar. (These data were taken before the experimental setup allowed continuous coupling patterns.) The patterns are fairly consistent as the spiral-to-monopole spacing is varied even at the extremely close spacings where the probe is 1. 5" from the periphery. Even more consistent j 56

THE UNIVERSITY OF MICHIGAN 6633-1-F -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 0 FIG. 4-11: 60 120 180 240 300 360 Relative Orientation Angle, 0 (degrees) COUPLING PATTERNS AT SEVERAL SPACINGS, CENTERTO-CENTER 57

THE UNIVERSITY OF MICHIGAN 6633-1-F is the coupling level, which at the pattern maxima follows within 0. 5 db of the ordinary 6 db per (distance) octave variation at distances of 1 wavelength or more. At closer spacings, the level tends to rise somewhat above the 6 db line by up to 1.5 db at 1/2 wavelength (Fig. 4-10). Extensive tests were taken on the coupling patterns from 1. 7 - 4. 2 GHz in 50 MHz increments on spirals A and B. Rather than present all the curves involved, typical patterns are given in Fig. 4-10. Using data from Fig. 4-10 and the expression R C = C ( ) Dt (0t)r o X t t' r R where R = X and C (-) =constant o oX D (0t' f) = Circular Spiral Directivity Function, and t t' D = Monopole Directivity Function (independent of 0, f), the maximum and minimum values of Dt (0) for a particular frequency were calculated. These values are plotted as a function of the frequency in Fig. 4-12. The coupling decreases noticeably near 3. 8 GHz, where simultaneously the radiation pattern is becoming narrower. This increase in gain means more power is radiated along the axis, and thus less along the ground plane. The variations are small at lower frequencies and increase to extreme values at the higher frequencies, where sharp nulls occur. Just above the rated frequency band, the coupling increases sharply with small variations. The reason for this is that the second mode, which has a null on the axis of the spiral, becomes predominant. This mode occurs when the feed currents differ appreciably from the out-of-phase condition. There is a large in-phase component when the balun is I 58

0 -5 -10 -15 I H z rm r-4 H 0 — CA co CD I IItJ -20 -25 -30 0 0

THE UNIVERSITY OF MICHIGAN 6633-1-F operated outside its designed band. Furthermore, the second mode has more energy concentrated along the ground plane, which accounts for the rise in the coupling level. Figure 4-13 shows the angular position of the maxima as a function of frequency. An expanded angular scale is used for clarity; one should remember that 0 = 0 + 2n 7r in the figures. At the lower end of the band, the maxima and minima shift quite rapidly. This is due to standing waves arising from reflections at the ends of the spiral elements. These standing waves are more predominant at lower frequencies because the current traveling wave is not sufficiently attenuated by radiation damping. The curves of Fig. 4-13 demonstrate that nulls of the pattern, which move in much the same manner as the maxima, can only be used for decoupling over a narrow frequency band. 4. 3. 3 Square Archimedean Spiral Right-hand and left-hand wound square spirals were experimentally tested on the 12' x 12' ground plane in the anechoic chamber. Pertinent information is given below. Type Cavity-backed, two-arm, 20 turns Frequency range 2 - 4 GHz Balun type Strip-line, Roberts, located in cavity Manufacturer Advanced Development Laboratories Model No. 1L (left-hand wound) 3R (right-hand wound) Overall Dimensions Overal Ds 5 cm x 5 cm x 2.5 cm (antenna and cavity) II 60

THE UNIVERSITY OF MICHIGAN 6633-1-F 1500 8-r 6ir 1000 47r 500 27r 1500 1000 500 0 4.0 Frequency (gc) 4-13a 2.0 2.5 3.0 3.5 4.0 Frequency (gc) 4.5 4-13b FIG. 4-13: ANGLES OF MAXIMA OF COUPLING VS. FREQUENCY, SQUARE SPIRAL "IL" (a) FOR CIRCULAR SPIRAL "A" (b) FOR CIRCULAR SPIRAL "B" 61

THE UNIVERSITY OF MICHIGAN 6633-1-F Coupling measurements were taken between the square spiral antenna and a quarter-wave monopole. The monopole was perpendicular to the ground plane, and located 11.3" from the center of the square spiral antenna. Coupling patterns were recorded for frequencies from 2 - 4 GHz at 100 MHz intervals. Typical curves for spiral 1L are shown in Fig. 4-14. Further study of these coupling patterns reveals a preponderance of maxima at locations radially out from the antenna corners (0 = 00, 90~, 180~ and 2700). Figure 4-15 indicates the maximum and minimum of the directivity function of the spiral antenna calculated in the same manner as for the circular spiral. Due to the relationship between coupling and directivity function (see Eq. 4. 29), these curves indicate: a) a relatively small variation in coupling as a function of frequency, b) a relatively constant difference between maximum and minimum levels over the frequency range, and c) a trend toward increased average coupling as the frequency is increased. Comparison with the circular spiral indicates a) a greater beamwidth than that observed for the circular spiral, and b) a relatively constant gain over the frequency range of the antenna. A number of measurements of coupling between the two square spirals 1L and 3R, were made for center-to-center spacing at a minimum. One side or face of the square housing of one spiral was placed in contact with a similar side of the second spiral. Over a frequency range of 2. 4 - 4. 0 GHz the coupling was approximately -33 db. The observed variation of coupling over this frequency range was 1 db. 62

THE UNIVERSITY OF 6633-1 —F MICHIGAN -30 I "0 %-"4 0 QF. -38 -30 "0 P-"4 0 Q -38 -30 '111-f f 2.220go (b) (a) f = 2. 6 go (d) la bfl.-"4 0 Q -38 -30 "0 brI~ V-"4 0 Q -38 1800 3600 FIG. 4-14: COUPLING PATTERNS FOR SQUARE SPIRAL "IL" 63

THE UNIVERSITY OF MICHIGAN 6633-1-F -30 I 0 QF. I f = 2. 8 gc -38 -30 bfJ 10 Q.. -38 -30 (e) f = 3.02 gc (g) (h) btb 0 Qr. -38 -30 la bD 0 U -38 1800 3600 0-'FIG. 4-14: (Cont.) COUPLING PATTERNS FOR SQUARE SPIRAL "iL"1 64

THE UNIVERSITY OF MICHIGAN 6633-1-F -30 04 Q -40 -30 bf) 0 Qr. f = 3. 6 gc L(jI f = 4. 0 gc -40 -30 ~0 0 Q -40 I - 1800 (k) 3600 FIG. 4-14: (cont.) COUPLING PATTERNS FOR SQUARE SPIRAL "iL"1 65

0 0 0 Maximum --- —--— Directivity ----, --- —-------- Ci -5 > -10 o -15 f (GHz) -20 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 z rri m cn 0 -To -4 0 C) &. FIG. 4-15: VARIATION OF DIRECTIVITY FUNCTION VS FREQUENCY (SQUARE SPIRAL TO MONOPOLE)

t1 -30 - -34 1 Coupling (db) < -4 a) S; Crl C3 CI3 0 0 -38 k c0 — J -42 1 -46 I I I I 1 1 I 1 1 I ler r t A -.1A?l E A 2. 2 FIG. 4-16: 2. 4 2. b 2. 8 j. U j. 2 3. 4 35. b.5. 0 4. U P4 Z-4 C) z f(GHz) NEAR FIELD COUPLING OF SQUARE SPIRAL TO SQUARE SPIRAL NORMALIZED TO ONE WAVELENGTH SPACING

THE UNIVERSITY OF MICHIGAN 6633-1-F 4. 4 Conical Horns 4. 4. 1 Theoretical Derivation A theoretical derivation of the far field coupling for the conical horn can be made. By the postulate of a tangential E-field, E and Ep in the aperture of the conical horn the Hertzian magnetic vector can be derived as _ - e *(R)= Jj [ kE( ])xn - dS (4.29) The factor in front of the integral already includes the imaging effect of the ground plane. E x nA, the tangential E-field, vanishes in the ground plane. Therefore, the only contribution to the integral comes from the horn aperture. From the Hertzian magnetic vector, the fields can be obtained as E = -jwp VxT7 (4.30) H = V(V- ') + k2 (4.31) Using a spherical coordinate system as indicated in Fig. 4-17, only Eo is non-vanishing in the ground plane (0 = /2). The field component E is given by (Silver, 1949) jke-jkR [ 0] e =27R 2 (N cos + N sin 0) (4.32) e 27rR x y where N and N represent the integrals in f and V, and x y x y 68

THE UNIVERSITY OF MICHIGAN 6633-1-F z FIG. 4-17: COORDINATE SYSTEM FOR CONICAL HORN FIG. 4-18: COORDINATE SYSTEM IN GROUND PLANE FOR CONICAL HORNS 69

THE UNIVERSITY OF MICHIGAN 6633-1-F N =fj(Et) ejk(c~ + sin)dS (4.33) and N t=/j(Et) ejk( cos + sin 0)S (4.34) y ty Using the relations e = p cos, ri = p sin T (see Fig. 4-18), one gets N =ff(Et) e pdpdT (4.35) and N J(Et) e jp pdpdT. (4.36) y jy y In order to evaluate the far field pattern, the tangential aperture field Et has to be derived or postulated. For the present case, it was assumed that in the aperture the tangential E-field could be represented as a combination of circular waveguide modes TE1 and TMl, both of them having their phase centers at the point where the extended walls of the conical horn would intersect (Fig. 4-19). There is experimental evidence available (Potter, 1963) that a mixture of these two modes actually exists in the aperture of even moderate flare angle conical horns. The aperture fields assumed to exist are given by: TEl 11 jw o F sinJ1 (K11 p) E = (4.37) P P 70

THE UNIVERSITY OF MICHIGAN 6633-1-F i- L I d FIG. 4-19: CROSS SECTIONAL VIEW OF CONICAL HORN c=4.6cm, d=2.54cm, L=26.4cm, 0 =9.9~. c 71

THE UNIVERSITY OF MICHIGAN 6633-1-F E = -jWc K11 F cos J1 (K1 p) T ~11 ** 1 (4.38) where F = exp -j - L + p L j J (4.39) I I (for the definition of L, see Fig. 4-19) and where (Kl c) is the first root of J1; I t J1 (Kllc) = 0. (4.40) TM11 E = 2jw K11 F sin J1 (K11 p) 1 2j ip K11 F cosTJ1 (K11 p) E = T711P (4.41) (4.42) where (K 1c) is defined as the first zero of J1; J1 ( 11 = 0 (4.43) and F is defined above. In order to evaluate N and N (4.35) and (4. 36), it is x y convenient to decompose the aperture fields into rectangular components, giving: TE11 11 E =1 j K F (Kl p) sin 27] f 2 juK11 FL2 11 (4.44) --- 72

THE UNIVERSITY OF MICHIGAN 6633-1-F 1 E = 2jwK1 F J (Kl P) -J2(K p) cos2T] (4. 45) TM11 E = -u Kl FJ2 (K1 p) sin2T Er = j K 1 [F o (K11P) + J2 (K11p) cos 2] For the TE11 mode there results N = A F sin T ejkpC -T) (K p) pdpd x 2 11 (4.46) (4. 47) (4.48) and N= A/ F y VW [ (K1 P) - J2(K p) cos 27] o 11 2 11 ejkp cos (0 -T)pd pd e' pdpdT (4.49) where 1. 2 j=o/I K11. Using the expansion eJX Co = J (x + n = 1 2j J (x) cos ny, n (4.50) these become N = -2r A sin 20 2 (K 11) Ny= 2A [o(K 1) + F(Kll) cos 2] y I 1 2 1 (4.51) (4.52) and 73

41 N r 4 I THE UNIVERS iITY OF MICHIGAN - 6633-1-F where a r0 () = O a r2(7) = rFM( ) Joy FP Jo(p) Jo(kp)dp (4.53) and Fp J2(7p) J2(kp)dp. (4.54) For the TM1 mode one obtains: 11 N -2A F sin 27 eikp cs(0-7) J2(K11p) pdpdT (4.55) and = 2AJ F[J (K P) + J(K ) cos 2].eJk cos(T - )pdpdT (4.i 56 (4.56) I which become N = 47r A sin 2 r (Kll) (4.57) (4.58) and N =47A[r (K )- r2(K1 )cos 20. 0 11 1 ' Combining the contributions due to the x and y polarizations according to (4.32), one gets: a E= R- [r (K 11) (K11) sin 0 E0 jkA r (K )+ (K 1)]sin 0 R 11 11 for TE mode for TM11 mode 1 1e (4.59) (4.60) ind 74

THE UNIVERSITY OF MICHIGAN 6633-1-F or equivalently, WK k r E = 12R1 w K 1. k 0 R [r11+ 2 (Kll) sin 0 r2 (K 1) sin 0 I;(K11] sin ^ for TE mode (4.61) for TM mode. (4. 62) 11 and The integrals [- and P2 (4. 53 and 4. 54) cannot be evaluated in closed form because of the factor F. However, in the limiting case of an aperture fed by a circular waveguide of the same size, this factor becomes equal to one. In this case r and 72 can be evaluated exactly using the relationship (Silver, 1949); YJn (y) Jn (3y) dy = 2Jn(ac)3J (1) - J (c) J (ac)]. 0 - (4.63)! I Now by definition J (K lc) = 0 for the TE1l mode, and J1 (Kl c) = 0 for the TM mode, and the following evaluations can be made: I '2 (K 1 c 2 J(K11 c)[K cJ (kc) -kJ (kc) '0 11 K2 -k 11 2 11 t2 2 c)Jl(kc) - cJ (kc) F2(K kc J (K J(kc) 0 11 '2 k2 o 11 1 (4.64) (4.65) (4.66) 75

I THE UNIVERSITY OF MICHIGAN 6633-1-F 1 and kc r(K) = J (K c)J (kc) 211 '2 2 0 11 11 (4.67) Therefore, the particular combinations needed for deriving the normal E-field equations (4. 61) and (4. 62) are: ( 11) - r(K11) = J (K C) Jl(kc) O1 +2 11) J(k 1 2ke r(K )+r2(Kll) = - 2k J(K c) Jl(kc). o 2 11 2 2 0o 11 K -k 11 (4.68) (4.69) and WK1 c K E0 R J(K C) J (kc) sin 0, 2wuK k o 1 =k J (K c) J (kc) sin 0 e8 (K 2 k2)R o 11 (K11 for TE mode 11 (4.70) and for TM lmode. (4.71) The directivity of the antenna is given as =P(0)2 rR2 Wt~2R (4.72) where P(O) is the power density in the 0-direction and W is the total radiated power by the transmitting antenna. Then, 1 E 2 2R D =w2 E e (4.73) MI 76

THE UNIVERSITY OF MICHIGAN 6633-1-F 1aperture, giving: To calculate W, evaluate - Re(E xH ) over the aperture, giving: 2 t t W=2wH 27r c (IEj2 + Ey2) pdpdT'. 0 0 (4.74) There results for the TE11 and TMl1 modes, respectively i13K~oi^s 2c 2 ' ( I 'O w 11 J2(K p)+J2(K p) pdp = 1 (KL E= 4 1 2 11 4 11 0 for the TE mode 11 2 -1)J (K c) (4.75) where 2 1 '2 11 11 (4.76) and 2K2 11 2 WM =16r IA Ji (K 11C) BVII, for the TM1 mode 11 (4.77) (4.78) where P12 2 011= K -K 11 11a The total radiated power by the conical horn will then be the sum of that contained in the TE1 and TM1 modes, i.e. W 2 +R2 total 1 E 2 M ' (4.79) 77

THE UNIVERSITY OF MICHIGAN 6633-1-F where R1 and R2 are the amplitudes with which the TEl1 and TMl modes are excited, respectively. Using the directivity of the conical horn one can derive the coupling by the formula: C = Dt(t)Dr(r)C(R) (4.80) where Dt = directivity of the transmitting horn in the direction of the receiving horn D = directivity of the receiving horn in the direction of the transmitting horn C = X2/47rR2 o o R = distance of separation of the ponical horns from center to o center measured in the common mounting plane. Now, for identical transmitting and receiving horns each with circular polarization D(0) = Dt(t)= Dr(0r) (4.81) and the coupling becomes D = D ()C (4. 82) While one can derive a closed form expression for F = 1 for r and F 0 2 (equations (4.53) and (4.54) ), these equations can be evaluated very easily numerically for any F. A computer program was written for this purpose, evaluating the integrals by using Simpson's formula. Figure 4-20 shows the theoretical maximum coupling value obtained as a function of frequency with a I 78

THE UNIVERSITY 6633-1-F OF MICHIGAN -50 o-o:3 bb P-d 0 r g *a r) x -60 -70 -80 FIG. 4-20: COMPUTED COUPLING LEVELS FOR CONICAL HORNS AS A FUNCTION OF FREQUENCY. R1=. 9, R2=. 1, a=4.6cm, Ro= center-to-center spacing=36. 6cm. 79

THE UNIVERSITY OF MICHIGAN 6633-1-F I. B I M) a -60 (db) -70 -80 -90 -70 (db) ~ w l4 1 -80 -90 g -1: - -100 -180 FIG. 4-21: CONICAL HORN COUPLING PATTERNS (-) E-PLANE ( —) H-PLANE. (LINEARLY POLARIZED) 80

THE UNIVERSITY OF MICHIGAN 6633-1-F (a) (b) E H XI H E E-plane Coupling H-plane Coupling FIG. 4-22: DIFFERENTIATION BETWEEN E- AND H-PLANE COUPLING 81

THE UNIVERSITY OF 6633-1-F MICHIGAN -50 -60 F f= 8Gc -70 - -80 - I I I,,,aI -I -181 U -9o O 90 1 FIG. 4-23: CIRCULARLY POLARIZED CONICAL HORN COUPLING PATTERN

THE UNIVERSITY OF MICHIGAN 6633-1 -F pair of horns of radius a = 4. 6 cm, center to center spacing 36. 6 cm and length L = 26.4 cm. R =. 9, R =. 1. Only a plot of the maximum coupling is needed in order to deduce the coupling for any orientation of the conical horns, since the directivity in the ground plane varies as sin 0. 4. 4. 2 Experimental Results Experimental data were gathered for a pair of conical horns mounted in a ground plane with L = 26. 4 cm, c = 4. 6, 6 = 9. 9 (Fig. 4-19). The gain of each of 0 these horns is 18 db at 9 GHz. The data were taken for the linearly polarized case at 8, 10 and 12 GHz at a center to center spacing of 36. 6 cm. The two curves shown in Figs. 4-21a to 4-21c, labelled E- and H-plane coupling, are described by the geometry of Fig. 4-22. It is seen from these curves that, indeed, as predicted by the analysis the coupling levels vary as sin 0. The maxima of the H-plane coupling are down 6 to 16 db from the maxima obtained for the E-plane coupling. At this spacing of 36. 6 cm, the lowest coupling level obtainable for the linearly polarized pairs of conical horns is about -90 db. Figure 4-23 shows the E-plane coupling pattern as a function of rotation angle p (of the receiving horn) at 8 GHz with both horns circularly polarized. Because of the phase quadrature relationship between the two orthogonal components existing in the horns, the nulls of the coupling patterns disappear. The slight variations in coupling are due to the aperture fields being somewhat elliptically polarized. 4.5 Rectangular Horns To completely specify a two-horn coupling situation, it is necessary to know twelve parameters: the two dimensions of each horn aperture, the two flare angles of each, the spacing, the orientation angles and the frequency. For very crude approximations, one can reduce this to eight (see Section 3.3). 83

THE UNIVERSITY OF MICHIGAN 6633-1-F 4.5. 1 Directivity Function To get an expression for far-field coupling, a knowledge of the aperture fields is necessary. The assumption is made here that the waveguide fields (TE10 mode) are reproduced over a curved phase front at the mouth of the horn. Thus higher order modes are ignored in representing the aperture field. This gives surprisingly accurate results, even for H-plane, or near field, coupling. This will be discussed later. The electric field of an aperture-type antenna can be expressed in terms of an integral over the aperture: (see Fig. 2-2): E (R) = -Vx aperture e-jkr — E () xrdS 27r (See for example, Harrington, 1961) where r = R-p. Along the ground plane, the vertical component in the far field is: -jkR E (R ) - eR E cos ga(u) gb(v) where g (u) = 27 1= gb(v) = a/2 X (e) exp ( a u) d~ -a/ 2 b/2 (r) exp ( v) d -b/2 84

THE UNIVERSITY OF MICHIGAN - 6633-1-F a u = 7r() sin | b v = 7r(b) cos x and the assumed aperture distributions are given by 77e f () =cose cos (- ) exp ( csc ) a 2 )p a fb(r)=cos0 exp ( csc 0) 0- = tan1 (I 2 tan e ) a a e = tan- (271 tan Ob) e0 =tan-l tanb). The "length" of the horn can be used as well, where a 1 = cot 0 a 2 a and 1 = cot b 2 b These quantities are shown in Fig. 4-24. Note that as a and 0b approach zero zero 85

THE UNIVERSITY 6633-1-F OF MICHIGAN - a/2 A FIG. 4-24: RECTANGULAR HORN GEOMETRY. H-PLANE 86

- THE UNIVERSITY OF MICHIGAN 6633-1-F f () = cos a a and cos u ga(U).__ 4u2 1 2 2 7r sin v as in the case of open-ended waveguides. The integrals for ga(u) and gb(v) cannot be evaluated for all angles by ordinary means. For this reason, discussion in the literature has been limited to small flare-angle approximations. Here the IBM-7090 computer was used to evaluate these integrals for flare half-angles up to 45. The results are tabulated in Table C-1. A set of design curves is also included in Figs. C-la to C-lm. The formulation here is for far-field only; section 4.5.2 which follows, considers near-field terms as well. The two formulations match in the far-field, and give results which match well with the experimental results. There are slight differences in the shapes of the coupling curves, in that the maxima and minima differ somewhat (on the order of 1 -3 db), but the angular dependence is preserved very well. The levels of experimental and theoretical coupling generally match within 5 db. In most cases the theoretical coupling is above the experimental coupling; this is thought to be due to losses. The accuracy is 87

THE UNIVERSITY OF MICHIGAN 6633-1-F better than anticipated, considering that higher order modes were ignored, and noting that the sensitivity of coupling to changes in the aperture distribution is high. 4.5.2 E-Sectoral Horn Coupling The coupling of the two E-sectoral horns is to be considered in this section. The transmitting horn is assumed to be excited by a waveguide supporting a TE10 mode and the receiving horn is also connected to a similar waveguide (which supports only TE10 mode), in which a measuring device is located. The details of derivation for coupling formulas are given in the Appendix A. The results are: For ft 2 a 1 (t) ka ka a I 1 2 (t)P 1 1 rk2 t 2 1 at ('y a)2 (1 R) 7r W MO1 (qb) kat 2 2 M* I I-[ —C(- - sin Ot] cos 2tcos J v ('Y bt cos t)J "cH lbr o 72 r 0c t * jHCC( lb cos 0) C ( sin ). 1r r I r r 88

THE UNIVERSITY OF MICHIGAN 6633-1-F For t t (t) a 12 p (ka ka C=-647r(-)(-.( (t)2 1 1 1 ka 647r(- (-) —) (7y P. -- 1 2( ir2_ ~2 C 6 b k )(a )Y1P 2 2a ( R )2 (kR )2 1 M01(y1b0) 1 a 2 2 in (y b obts 1 r rrt r, 1 'r w 2 k C r r c (y r sin )r r ka 2 C( r sin 0 ) r rj Numerical examples are shown in Figs. 4-25 to 4-26, together with + 4 (.1 br5. Cs 0P ( l) - y (3 b -Seor 0) cos r sCinr ka 2.C( r sinr)j TheThe formulas for the coupling ofare very inconvenient for numericalns calculations. waveguides suppornota ti ngons, referthe TE mode is tohe Appendix A. The formulas for the coupling are very inconvenient for numerical calculations. 89

THE UNIVERSITY OF MICHIGAN 6633-1-F 1 I I I f 2 4 6 8 Ot=0 -40 db - I NO DATE (a) f=8. 03 GHz I 0 t -- lu -----— 11 An- I -f -----, - -2..4 - 6 - -8 - - -8-,tN rS XX x.5C,. X.. X. - x '60 0 0 I/Nb \ I - - \J 0 O N\. zu i of i Y 1 -.. -- i i v ii P 4 - Ld z 0 — 2 --4 -6 --8 - A — — X. 0 0 0 - e~ -.... -. *. - - I.0 -LJ: d 2 -4 --6 - -8 — -AN A. -1 -.7..+ i..... -- - —.- —........................-. -.. ----.-. -.-.. —t. —............ -8Q.......... ZU- I, i -2 - - - - - 2- -- — 4 -- -6 8 - - - 8 - 4 5o I 1 -- I 5o { - 30" 1 80 4 24~ 144' I -2 -4 --6 8 - 3, 18 108 -1 o Theory.. I..,........ Experimental _ - -- -..- -.-. ---. --- I.-. - 0 1 1 I... -- '0 0 6' r r- -- 36" 12 72' i. -.. -. - - 6' 36~ 12' 72 3o 18" I - - 108~ I I. 144' I 5o 30" 180~ I ii. I flu 1 ' i, i FIG. 4-25a: COMPARISON OF EXPERIMENT AND THEORY E-SECTORAL HORN. R =22. 86cm, ar=at=0. 9", br=bt=3 22", a=0.9, b =0.4", p(t)=p(r)=3.78". O 90

THE UNIVERSITY OF MICHIGAN 6633-1-F I 0 1 1 2 4 6.8 0 t0 -40 au NO. DATE (b) f=9. 03 GHz Al-, I,-xK<-Xt50 dbI 2 ------ xx X 0 -20 — z- 8 - 0 0-7 -40 -- -50- ___ I — - 300 I - 1800' 40 30 64 I-1 I - 8 -1J1440 1080 120 720..6 jOi. I - - - - - L — ---- -4 — - 60 0 0 60 4 --- r r 360 36 --- - 1 — * -- 2) 120 - — A ---.720 — +- -j -- -, -— H 30 4D 50 — 1 ----— 1 —. --- -I — - - --- 1080 - 144 1800IB II I bU I FIG. 4-25b: COMPARISON OF EXPERIMENT AND THEORY E-SECTORAL HORN. Horn data same as in Fig. 4-25a. 91 - m

THE UNIVERSITY OF MICHIGAN 6633-1 -F 1 Cl 4 6 8 -10o2 -.4 —. — 6-. 8 - I I I =0 Ot I -40-db' -50 db - I I v NO. DATE (C) f=10. 03 GHz 0t=7r/ 2 i m i Al 0,0 0 I> OQ f 44u- i -i V4".-4 + "" A r. i ii of s H i A6 LaJ z 0.2- -.4 -6 —.8 - I - 01. -.-0 xv' —.-. __j_(X -,An - I i I I.L I n -.3v i i -- 9 %j 9 m i A % Iw 310 I-0 -a. -2 -4.6 — 8-8 - — 8C 0 - 0 1 40 1 4 - 4i -i 4 i u - - 2 --— 4 --- -- 6-.8 I. - -F-' O- 4 --- —-- - -4 4 — 4 -4 4 3Q0 180 24: 1440D -2.4 6 -8 - 3" 1 80 1 08' -11 720 I ' 6' 36" 66 1 I I -- 360 2 0 I - 1 20 -- I1 720 3-. 180 I - - 1080 ii40 - -- I - 24" 144' - I 30' - I 18C bU FIG.- 4-25c: COMPARISON OF EXPERIMENT AND THEORY E-SECTORAL HORN Horn data same as in Fig. 4-25a. 92

THE UNIVERSITY OF MICHIGAN 6633-1-F 1 i I r -;U-r "IU I K 2 4 6.8 t 0 0 t /2 NO DATE f=8. 23 GHz x x r- fl - 1V 3 1 rU - __ -Z 20- — 4060 - ------------- —........... 1 ---~- t~- C ---- ~ — — ~~ ------— I - - 68 --- - - - - --- --.... - -. -— ~ ----.- 0 --- —--— t-C+ — - 40 -0X - ~30 —50 -701 -- - - - t- f — -.-.-. —.. I —0.-.c..-+~-t — ~tt-f ct — 7 t LA - -- -. ~ I, ~I-- -- --- 6 -- -~. ---.-~-t - -+ t --- ~ ~~0 ~t ---- - 1 —~Theory - "7-r-7- - - - --—. —. --- I- c --..- - W-0 AO -60 -80 EXperiment _ _ _ _ _ _ _ _1 I I 71171 —2711 71 1ff — - I-j 7- — 1 71 7 1 71 77 t. - 3 20 0 0 10 20 30o 40 5 I --- —-J --- —-tt-~ —f —t — -— 1 - -1 I --- —— I --- —- I --- —-— [ ---l --- —j- ------ — I 30" 240 180 120 60 6 120 180 240 0 1800 1440 108" 720 360 360 720 1080 1440 18C fi(. - FIG. 4-26: COMPARISON OF THEORY AND EXPERIMENT. H-SECTORAL HORN 93

THE UNIVERSITY OF MICHIGAN 6633-1-F It has been learned that, for the purpose of computation, further refinements, particularly with respect to the approximations used in the analysis, must be made. The formulas in the present form yields satisfactory results, if one picks two or three modes. Shown in the Fig. 4-26 is the comparison between experimental and theoretical result. The complete results of numerical calculations are contained in the separate memorandum by Kwon (1965). 4.5.4 Pyramidal Horn Coupling A pair of pyramidal horns mounted in a ground plane was also tested at the same center-to-center spacing of 36. 6 cm. There horns were fed from X-band waveguide (1.01x 2. 28 cm), which was flared out over a length of 10.1 cm to an opening of 4. 77 x 6. 04cm. The horns have a gain of about 12 db at 9 GHz. Figure 4-27 shows both the E- and H-plane coupling curves. The coupling pattern for the E-plane case is more directional than for the conical horns because there is more tapering of the E-field in the pyramidal horn aperture. It is to be noted that this more selective coupling occurs even though the conical horns have a much larger aperture, and correspondingly, much higher gain. Due to the lower level of the sidelobes in the ground plane for the pyramidal horns, an antenna designer might want to consider using a pyramidal horn instead of conical to achieve lower coupling levels between adjacent horns. The general level of coupling observed for pyramidal horns is believed to be dependable. However, there was some scatter from the other objects in the anchoic chamber which entered into the observed coupling values, particularly as nulls were approached. 4.6 Yagi-Uda Monopole Array The medium gain antennas discussed so far in this report have not had the main beam along the ground plane. The Yagi-Uda monopole array was chosen for study since it has its main beam along the ground plane, and polarization 94

THE UNIVERSITY OF 6633-1-F -5(i m '|||||,|||! i - (db t;i; i-:*: F-,t - i i...!|t -60 -70 -60 X 211 ~~TF F UL7Fz MICHIGAN -70 (db) GHz 80 il: -90 i [IIIIIII:I:I ( m-c+ Lff- | -ou q I -90 FIG. 4-27: PYRAMIDAL HORN COUPLING PATTERNS. (-)E-PLANE, ( —) H-PLANE. 95

THE UNIVERSITY OF MICHIGAN 6633-1-F vertical to it. It also allows an investigation of the Fresnel zone region in the region of the main beam as well as the sidelobes. An eight-element array was chosen, plus a taper section of two elements (Fig. 4-28). The reflector length was chosen to give maximum front-to-back ratio. The antenna was designed to give maximum gain in accordance with known design criteria (Jasik, 1961, Ehrenspeck and Poehler, 1958). Further adjustment of the element lengths gave little improvement, indicating the accuracy of the design procedures. A monopole probe was used as a receiving antenna with mounts as indicated in Fig. 4-28, allowing sampling of the field at a broad range of spacings at 30 intervals. The measured gain of the yagi was 12 db with a front-to-back ratio of 19. 5 db. In Fig. 4-29, the results are plotted in actual distance, R0, from the center of the circular plate housing the antenna. The center of the antenna was roughly at the third director from the feed. In particular, the curves for 0 and 300, both of which are on the main beam, show very smooth behavior, and adhere to the 6 db/octave line, this indicates the phase center of the main beam is near the center of the antenna. It was anticipated that the main beam coupling would fall away from the 6 db/octave line at close spacings - an effect observed with large aperture antennas. Evidently this effect is not encountered to any significant degree with gains of 12 db or less. In the sidelobes 60 and 90 0, the coupling falls below the 6 db/octave line, and with the backlobes, rises well above. If a new phase center is defined for each angle of approach, the curves are more regular (Fig. 4-39). The phase center here is defined by taking the distance from the far hole to that point which gives coupling 12 db higher, as being 3/4 of the way to the phase center. This procedure may imply that the data is being forced to fit pre-conceived notions, but data, so presented, does appear to be more regular and reasonable. I 96

THE UNIVERSITY OF 6633-1-F + A MICHIGAN '-4 cq~ 0) '-4 z 0 z 0 0 z 0 CtzJ 0 EU 97

THE UNIVERSITY OF MICHIGAN 6633-1-F -15 -20 -25 '-30 -35 -40 -45 -50 20 I -.a, W R /X 180 FIG. 4-29: COUPLING BETWEEN A YAGI-UDA ARRAY AND A MONOPOLE VS. SPACING FOR DIFFERENT ORIENTATIONS 98

THE UNIVERSITY OF 6633-1-F MI CHIGAN o 0 o o CM LO -- -. C I t T 0 0 z 0 0 0 O 0 O Cq --.01 bf I - 4 o - I, FIG. 4-30: COUPLING BETWEEN A YAGI-UDA ARRAY AND A MONOPOLE VS. SPACING FOR DIFFERENT ORIENTATIONS nn

THE UNIVERSITY OF MICHIGAN 6633-1-F Another representation which is meaningful is shown in Fig. 4-31. Here the lines of constant coupling amplitude are plotted. This shows the concentration of power along the elements, and the end-fire radiation. Some of the lesser deviations from regularity can be explained by reflections, but the gross effects are greater than experimental error. Repeatability was within 1/2 db, and few of the rapid deviations were greater than this. 4. 7 Coupling of Slot and Horn Type Antennas to Higher Order Modes A study of coupling, in which the individual modes present in the apertures are considered, complements other coupling studies discussed earlier in this report. In considering the influence of one system upon another it is possible that some of the modes actually produced through the incidence of a wave upon an antenna aperture are not present in the interior of the receiving system. These higher order modes can be excited at frequencies below the frequency level necessary for propagation in various elements of a system. Changing the frequency of operation can change the prevalence and strength of various modes. In waveguide systems some modes can be created that will not be transmitted if the operating frequency of the system is within the usual prescribed limits with respect to the cutoff frequency of the waveguide used. Usually the operating frequency is so chosen that only the principal or dominant mode, such as the TE 1 mode in a rectangular waveguide, can exist. However, if a transmitting system is operating at frequencies well above the nominal frequency of a receiving system, which is subject to interference, signals of a higher order mode can be accepted. In discussing frequency and harmonic frequencies as related to systems which can interfere with each other it is appropriate also to consider the modal aspects of the interference. In analyzing and computing the level of modal coupling and later measuring modal coupling the values obtained may turn out to be astoundingly high. However, such modal coupling values must always be 100

THE UNIVERSITY 6633-1-F OF MICHIGAN 225~ 195~ 180~ 165~ 1 'O 240~ 255~ 270~ 285~ 300~ 315~ 0o FIG. 4-31: COUPLING BETWEEN A YAGI-UDA ARRAY AND A MONOPOLE: CONSTANT COUPLING CONTOUR LINES 101

THE UNIVERSITY OF MICHIGAN 6633-1-F considered with the total filtering effects of each system brought to bear on the entire interference problem. In considering the applicability of filtering to eliminate higher order modes it is therefore important to consider the design features of each system, particularly those of the receiving system. Interference caused by coupling determined on a modal basis can be very pronounced when the receiving system is designed to operate with more than one mode. Special attention must be given where one of the systems is of the dual or multi-mode variety. An example of higher order mode filtering is shown in Fig. 4-32; some of the modes whose cutoff frequency is influenced by the waveguide height may be eliminated through the reduction in height. 4. 8 Effects of Harmonics on Coupling 4.8. 1 Harmonic Content of Signal Sources The harmonic content of signal sources is of great importance in the evaluation of interference caused by the coupling of one system to another. The presence of substantial harmonic content is a given transmitting system operating at a design frequency below that of a second system which is sufficiently coupled may result in a high interference level in the second system. The second system designed for a higher frequency than the fundamental frequency of the offending system does not have the advantage of frequency filtering under the critical condition where, say, the third harmonic of the offending system coincides with the fundamental of the receiving system. It is in such instances as this that the harmonic content of the signal source becomes a major factor in interference. An example would be the third harmonic of an S-band transmitting system interfering with an X-band receiving system. i,... I. 'l...: ~r '' ~r -.. '. r r ' i. r 7 -~;J-~` ~ i f~-..:. 1; -~lt.- r.)~ ~ -` ~ ~ "' 102.

~~THE UNIVE-RSITY OF MICHIGAN 6633-1 -F TM12 TEOZ TE12 v TM11 TM21 TM31 TE1O TE 20 TEOI TEI1 TE 21 TE 30 TE31 (T a i A - i.- R R -i I-.iM - - 0 I 27 18 Fundamental operating freq. Second Harmonic 7 I Frequen~ GHz i icy 1 a=2. 840"1 = 7. 214 cm b = 1. 340"1= 3. 404 cm Third Harmonic (a) TE 10 TE 20 TE30 a1 TE40 0.445"1 E 11 6 i! I i i Frequency in GHz (b) FIG. 4-32: THE EFFECT OF WAVE GUIDE SIZE ON MODE FILTERING. DATA BASED UPON MET (1959). 103 I

I THE UNIVER SITY OF MICHIGAN 6633-1-F Or -- Designed fundamental signal and accompanying harmonics. Spurious signals. -10o I --.E -20 E-C c20 D a) H | -30 $ 3-40 O o 0 -50 = I I I ii,........ I...., Fundamental 2nd 2nd 3rd f 4th Harmonic Frequencies FIG. 4-33a: TYPICAL POWER - FREQUENCY OF MAGNETRON Data based upon Matthaei (1964) 104 - -

THE UNIVERSITY OF 6633-1-F MICHIGAN H-' z 0-! I8-4 1-. 10 20 10 ------.. —I --- —-------------- 30 40........... 50.... rO —6 70 80 90 nn 1 U- * I -. I 4 4 I I I U- 9 a _ _ _ a a - -m a & _ _ _ t 2nd 3rd 4th Fundamental 1800 MHz 5th 6th Harmonic Frequencies FIG. 4-33b: TYPICAL RELATIVE POWER - FREQUENCY SPECTRUM OF A KLYSTRON Matthaei (1964) - m 105

I THE UNIVERSITY OF MICHIGAN 6633-1-F The harmonic content of signal sources varies among the different types of signal sources. Figure 4-33 indicates the harmonic content of a number of frequently used signal sources. This information is submitted only as being reasonably typical. In this report, the ultimate influence of harmonic content is not incorporated in any interference problem. However, it is easy enough to use such information: it is convenient to speak of a given harmonic as being so many db below the power level of the central operating frequency of the source. Harmonic contents will then be considered at power levels which are from 10 - 20 db below the incident power levels at the central operation frequency. 4.8. 2 Spiral Antenna Data Taken at Harmonic Frequencies Data were taken on both square and circular spirals at harmonics of the normal operating frequencies. All the measurements were taken in an anechoic chamber with the spirals mounted in a 12' x12' ground plane. Coupling patterns in the ground plane were taken for both the square spiral 1L and the circular spiral A, at frequencies between 8 - 10 GHz. These patterns are not included here but will be described briefly. The coupling patterns for the circular spiral have 3- 5 irregularly spaced maxima. The ratio of absolute maximum to absolute minimum coupling varies from 15 - 25 db. The coupling patterns for the square spiral have 5 - 7 irregularly spaced maxima. The ratio of absolute maximum to absolute minimum is greater than for the circular spiral since there are either one, or three, very deep nulls in many of the patterns. 106

- THE UNIVERSITY OF MICHIGAN 6633-1-F 4. 9 Systems Aspects of Coupling In considering the coupling effect due to higher order modes in the system interference problem, attention must be given ultimately to the filtering action of the detection system used. As an example of higher order mode coupling, consider an X-band system transmitting and an S-band system receiving. The presence of higher modes in the receiving system is not dependent on their existence in the transmitting system. Fortunately, in many cases a substantial amount of filtering is provided by the detection system. This means that a given receiving system may be much more sensitive to the desired mode, than to any of the higher order modes. The filtering action of the detection system can be of two kinds; 1) the associated frequency sensitive circuits of the detection system can afford frequency filtering, or 2) the placement of detection probes can result in modal filtering. The frequency filtering action of a detection system of a receiver would be a factor in the overall frequency filtering formula for the entire receiving system. If the receiving system had for example, 75 db filtering against an undesired signal at a specific frequency perhaps 10 db of this would be due to the frequency filtering action of the detection system. If the local oscillator of a receiver and its function is included is the ultimate detection process, then an additional factor which must be considered in the presence of harmonic content in the local oscillator signal. The presence of these harmonics which may mix with undesired harmonics from an offending transmitter may very well result in a substantial decrease in the frequency filtering action of a given receiving system. In the further consideration of filtering due to detection systems the sensitivity of the detection system to various types of modulation is important. The sensitivity of a detection system for one type of modulation as compared with another type of modulation represents a substantial item for study. The modulation aspects and the differences in filtering action due to sensitivity or lack of sensitivity to various types of offending modulation are not covered in this report. r I 107

THE UNIVERSITY OF MICHIGAN 6633-1-F The earlier discussion has shown that various higher order modes may exist somewhere between the radiator of a transmission system and the signal display or output of the receiving system. It is expected that quite often these modes will not exist at the output port of the receiving system. This may be due to the modal filtering which occurs due to the design of the receiving system elements including of course the cut off characteristic of any waveguide used. Detection systems and directional coupler devices are really modal filters and modal converters. All of the energy occurring at harmonic frequencies should be eliminated or absorbed in some manner before it become part of the receiver output. Mode separating devices with absorbers for harmonic energy would then become justified. Studies of this type have not been undertaken in this project, but might very well be worthwhile in the overall interference problem. The two port network representation for two systems where one interferes with another is a useful one to consider. Whether or not higher modes are available from the port which constitutes the output of the receiving system, depends upon the details of the receiving network. Somewhere in this network, perhaps in the design of directional couplers, detectors or modal fitters, provisions can be made so that an unwanted mode can be eliminated. 108

THE UNIVERSITY OF MICHIGAN 6633-1-F V THE COUPLING BETWEEN TWO ANTENNAS DUE TO THE PRESENCE OF A SCATTERER The concern herein will be with flush mounted antennas on a common ground plane, and the effect of a scattering object upon the coupling between the two antennas. When the two antennas are loosely coupled (i. e. the direct coupling is negligible) the presence of a scattering object may or may not produce significant coupling depending upon the parameters involved. It is the purpose of this section to determine the coupling due to the presence of a scatterer, with special emphasis on scatterers that would be of concern for aerospace vehicles. 5.1 Direct Coupling First the expressions for the direct coupling (no scatterer present) between two flush mounted antennas on a common ground plane, will be briefly considered. The coupling is given by Eq. (2. 5) for qt = qr = 1 by assumption: C (R,) Dr) = C(r) Dt (0t) (5.1) where the directivity functions D(0) are given by |E(f, 0)12 D(p)= I - )2 2 | E (0, /) )2 sin 0 dO d0 as in equation (2.6). as in equation (2.6). 109

THE UNIVERSITY OF MICHIGAN 6633-1-F R represents the distance between the centers of the two antennas, and Pt and Pr denote the orientation of the antennas with respect to the line joining the centers of the two antennas. For convenience the angle 0 will be the angle between the E plane of the antenna and the line joining the centers. (See Figure 5-1.) 5. 2 Coupling Produced by a Scatterer in Space Restrictions (some of which will be later removed) are now placed on the scatterer, transmitter and receiver geometry. Namely, the scatterer must be in the far field of both antennas, and in addition the receiving antenna must be in the far field of the scatterer (treated as a reradiating object). If R and Rt are the distances from the transmitting antenna and receiving antenna, respectively, to the center of the scatter, then not only must R ~X and R ~X, but if L and r t r L are the maximum cross sectional dimensions, respectively, of the aperture of the transmitting and receiving antennas, then 2L2 2L2 t r < 1, and <1 RX RX t r must hold. This is the usual condition that the scattering object be in the far field (see Section 3.5). In addition if L is the maximum cross-sectional s dimension of the scatterer as observed by the receiving antenna, then 2L2 i7 -<<1 R X r should also hold. However, this latter condition limiting the size of the scattering object may be removed under certain conditions, and certain large scatterers may be treated when the receiving antenna is in the near field. If 110

THE UNIVERSITY OF 6633-1-F MICHIGAN R 0 r I FIG. 5-1: SLOT GEOMETRY 111 11..

THE UNIVERSITY OF MICHIGAN 6633-1 -F the dominant scattered return from the large object comes from one or more finite scattering centers, then the scattered field can be treated as arising from these scattering centers. In this case R and R are the distances to the scattering r t center. As an example, the dominant return for high frequency scattering from a sphere (wavelength much less than the radius of the sphere) arises from a specular point (a single scattering center). Thus, the scattered field is treated as arising from a small region around the specular point, and the restriction that 2L / (R X) < 1, where L is the diameter of the sphere, is not required. On the s r s ~~ other hand for a flat disc of diameter L this restriction is required for the case of specular return since the scattered field arises from the whole surface of the disc. However, when the scattered return from the disc is not specular, but arises from the neighborhood of two points, one on the near edge and the other from the far edge, there are two distinct scattering centers. The restriction 2L / (R X) < 1 may be removed, but the return must be treated as arising from two s r distinct scattering centers. Summing up, when the whole scattering surface gives rise to a return at the receiving antenna the restriction 2L / (R X) < 1 must be used. When the s r scattered return arises from one or more finite scattering centers, then this restriction may be removed. Thus with this preamble one can employ the concepts of radar cross-section and scattering matrices to obtain the coupling. Let the far field of the transmitting antenna be given by -jkRt Et Rt e t (et., 0t (5.3) t where (Rt, 01 0t) are spherical polar coordinates with the origin at the transmitting antenna. The azimuth angle 0t is measured from the intersection of the 112

THE UNIVERSITY OF MICHIGAN 6633-1-F E field vector of the antenna with the ground plane. (See Figure 5-2). The directivity function is given by 'tB (, 8p,t (etr' t)2 2n i t ( t, 0 2 sin dt dot J o 0 (5.4) and the power per unit area incident at a point S designated by (Rt, 7t, ot) is w p = - D2 t O t) 2 DRt(Yt 't) 27r R t (5.5) The vector et (Yt' d O) can be written in the form t t c et (t'i = et (t' t) Pt (5.6) where Dt is a complex vector such that t.), ( pt' t) = 1 (5.7) A - A where pt is the complex conjugate of p. The vector p is a function of t and t t t t Ot and has the form Pt = at t + bt (t where Yt and Ot are unit vectors in the spherical polar coordinates. From (5.7), we have I atI2 +lbI2 1 (5.8) 113

H a m Vs v Ak s Vt Rt z P)4 0" Pd1 Or T fit /o R /It - FR4 0 FIG. 5-2: TRANSMITTING AND RECEIVING ANTENNA CONFIGURATIONS > Z

THE UNIVERSITY OF MICHIGAN - 6633-1-F The vector p can be written in a more specific form, depending upon polarization as follows: Linear polarization, t =cos 3t +sint.t (5.9) Left-hand circular polarization, tr= j('Yt-jt) (5.10) Right-hand circular polarization, 1 A A Pt+= 7t+ j.t) (5.11) For the receiving antenna there are similar expressions. However, the coordinate system (Rr, 'yr, Or) for the receiving antenna is centered at the receiving antenna and oriented with the E plane of the receiving antenna as is shown in Figure 5-2. For convenience some auxiliary unit vectors are now introduced; they are indicated in Fig. 5-2. Define. to be the unit vector directed along the line from i the transmitter to scattering center; h. is defined as parallel to the ground plane tA A* and perpendicular to q; v is taken to point away from the ground plane and be toA A perpendicular to qi and h., so that the set (q., vi, h.) forms a right-handed A triad. Let q be the unit vector directed along the line between scattering A AA A center and receiving antenna; h and v are defined as h and v. above, so that s i i AA A (q, v, h ) also forms a triad. Thus the following relations hold: 5 5 5 115

THE UNIVERSITY OF MICHIGAN 6633-1-F A - AA A tA A S vi -tti h,v,r (5 12) and t = -at b h (5.13) With this, the concept of the scattering matrix, or dyadic, may be used. The scattered radiation in direction q produced by an incident field E (q.) A 1 1 (propagating in direction qi) is given by (s) = A(s A i A A (5.14) S S S I l l where S is the scattering dyadic, which can be written in the form A A A A v v.S +v h. S \ s 1 vv s i vh A A (5.15) v. S +h h. S s Vi hv s hi h Here it should be noted that Ei (ai) is the transmitted electric field to be evaluated at the scattering center; also, the dyadic S is a function of the distance R from the scattering center to the receiver. The dyadic corresponds to the r more common term "scattering matrix" in the obvious way, i. e. the corresponding matrix is Svh [s]= (5. 15a) hv w A- A If the incident field is horizontally polarized, E (q) = h. E, then the scattered 1 1 1 O field is given by 116

THE UNIVERSITY OF MICHIGAN 6633-1-F E () =(S hh s + Svh ^) E (5.16) s s hh s vh s o and if the incident field is vertically polarized, E. (q.) = v. E, and 1 1 10 E (q ) =(Shv + S v ) E (5.17) s s hv s w s o The coefficients Svh and Sh of the scattering matrix represent cross-polarization effects. If the scatterer produces no cross-polarization then Svh = Sh = 0. Following the treatment for the radiation patterns of the transmitting antenna, the radiation pattern of the receiving antenna viewed as a transmitter is represented in the form -jkR E = e e ( (5.18) r R r r Pr r where A A/ ' p r P =1. r r From equations (5. 3), (5.6) and (5. 7) for the field incident on the receiving A antenna, where Ei (q) = Et (Rt), one obtains -jkRt 0 e A A A s Rs) et (t' I t) s' i)t (5. 19) The component of this which is coupled into the receiving antenna is E (q ) r; hence the effective power per unit area incident upon the receiving antenna is 117

THE UNIVERSITY OF MICHIGAN -- 6633-1-F Teui2nd ^ th rP E | 'p =nglet( eot)l I2 Pt & (5. 20) The coupling due to the scatterer is thus given by the relation Cs = (Rt)Dt (yt' ) D (yr, Dr r) |P s a pt (5. 21) The coupling is expressed in terms of the function of the two antennas, and the scattering matrix. Reduced expressions for rS (q s qi) t p are derived below from the special cases of linear and circular polarization. When the far field radiation pattern of the transmitting antenna is linearly polarized pt is a real unit vector of the form A A A Pt = cos t ot + sin t or Pt = cos t vi - sin t h. Similarly when the receiving antenna relation holds (5.22) (5.23) pattern is linearly polarized, the following s A A p = cos 7 r+ sin r r/ r r r r A A A p = -cos 3 v +sin 3 h r r s r s (5.24) (5.25) or This gives, on using relation (5. 15, or (5. 16) and (5. 17), the following expression: 118

THE UNIVERSITY OF MICHIGAN 6633-1-F ^( S~, qi). i t =(cos(s s 3tS -sin 3 tSh) Pr st = cohr vv -sin (sin3 Shh-cos 3t Svh). (5. 26) For circular polarization the far field of the transmitting antenna has the form -jkRt - Re et At) (5.27) with t =r (yt t) = - B( i~ j (5.28) where the plus sign is for right-hand circular polarization and the negative sign for left-hand circular polarization. A similar relationship holds for a circularly polarized receiving antenna, viewed as a transmitting antenna. -jkR e E e (yr ~r)^ (5.29) r R r r r r r with 1 + 1 a A p J (yr qr) - s- -jhs) (5.30) When both antennas are polarized in the same sense as transmitters, then the following relation holds r (qs qi)p -Pt 4 (Svv+ Shh +j(Shv -S 119

THE UNIVERSITY OF MICHIGAN 6633-1-F when the upper sign is for left-hand circular polarization, and the bottom sign is for right-hand polarization. When the antennas are polarized in the opposite sense s ) t= 4 (S + (Svh+ hv (5.32) where the upper sign is for the receiving antenna being left-hand polarized, and the lower sign for it being right-hand polarized. The special case where S = - Shh and S = S = 0 will now be investwvv hh vh hv igated. Before considering the particular scattering objects which possess these properties, the effect on the coupling of these relationships between the scattering dyadic elements should be discussed. For linear polarization, from equation (5. 26) one obtains Pr S (qs' i)' P t = Sv s cos (f -t) ' (5.34) When both antennas are circularly polarized in the same sense, it follows from equation (5.3) that PIr S s. )* P2t 0. (5.35) In this case the coupling is zero. The factor 1S = Shh2 can be related to the radar cross section of the scattering object. The bistatic radar cross section a is given by Crispin, et al (1959, pg. 155): 120

THE UNIVERSITY OF MICHIGAN 6633-1-F r = lim 47rR I s pr r.. (5. 36) r E2 where E is the scattered field produced by an incident field E. In the case where S = -Shh and Svh = S = 0, expression (5.36) can be given by the relationship o = lim 47r R2 IS 2 r v r v (5.37) R - ao r The expression for the coupling due to the scattering object can be represented in the form 4va s2 C = (R)C (Rt)Dt (yt, t)D (yr' )cos (1 -t) s 2 o r r r r t (5.38) when both antennas are linearly polarized. When the scattering object is far enough into the far field so that the vectors. and qsare anti-parallel, l reduces to the monostatic, or backscattering, cross-section. For back-scattering and nose-on incidence to bodies of revolution, the following relations hold among the cross-polarization terms: S = -Shh and S = Sh w hh vh hv Explicitly, the direction of incidence qi is parallel to the axis of revolution of the body. For back-scattering s is parallel to the same axis. 5 121

THE UNIVERSITY OF MICHIGAN 6633-1-F For large convex bodies where the radius of curvature everywhere is greater than the wavelength X, the scattered field arises from two contributions, the specular field associated with the geometric optics approximation and the diffracted field. For very large radii of curvature the specular contribution dominates. In this case a = 7TR1 R2 where R1 and R2 are radii of curvature of the scattering body at the specular point. For back-scattering the specular point is a point on the surface where q. is normal to the surface. For large flat plates with dimensions much greater than X and the incidence vector q. normal to the surface, the main contribution to the back-scattered field is from the specular reflection which arises from the whole surface (edge contributions are negligible). The back-scattering cross section for a flat plate of area A is A2 C= T5.3 Scatterers on a Common Ground Plane The problem is now restricted to the case when the scatterer is on the ground plane or else, if the object is large, to the case when a dominant scattering center lies on the ground plane. In this case Pyt = r = 7r/2. Due to the presence of the perfectly conducting ground plane the field incident upon the scatterer has a vertical component only and the scattered field at a point on the ground plane has a vertical component only. This yields the relationship A'/ A A |2 |_v|2 giving the expression (5. 39) giving the expression for the coupling I 122

-THE UNIVERSITY OF MICHIGAN 6633-1-F C =Co(R)Dt (' S (540) 5 ) t 2 t r2r vv The direct coupling between two antennas separated by a distance of R is given by C(R C (R, r t ) = C ( Dt(0t) Dr() (5.41) Thus equation (5. 40) can be expressed by C (R ) o t s C( R ) w C(RS C(R ~ ) C(Ro) (t rS vv 2 s= CoiRo v o t r r vv (5.42) The importance of this result is that it has two separate factors, one depending upon the properties of the scattering object, and the other, on the direct coupling between the two antennas for specific orientation and distance apart. Thus previously known values of the direct coupling can be inserted. As an example, suppose that the E-plane of the receiving antenna is directed toward the scatterer, and the E-plane of the transmitter is directed away from the scatterer in which case ~r = 0, it = rt. If the antennas are H-sectoral horns, we r have from Appendix C, that the coupling C (R 0, O, 0,) is 39. 3 db down, when R = 45. 3 cm and the frequency is 10 GHz. For a scattering object on an infinite ground plane, S v is the value associated with the free space scattering by the object and its image. For example, (S ) for a hemisphere on a ground plane is the value obtained from vv the free space scattering by a sphere. Values of S or the related bistatic cross section, (5. 37), are given by Crispin, et al (1959) for many scattering shapes. In addition, techniques used to compute Svv are prescribed here. m 123

THE UNIVERSITY OF MICHIGAN 6633-1-F A few results are quoted in the following table where reference is made to the equivalent scatterer (the object and its image). The dipole results are obtained from Harrington (1964). Equivalent Scatterer Sphere (radius a where a>>X) Small Resonant Dipole, (Reactive Loading, losses neglected). Active Loaded Dipole at Resonant Length: (Load Resistance/Dipole Resistance = R /R ) (a) R/R = 0 (b) R/R = -.4 (c) R/R = -.8 1 Bistatic Cross-Section a 2 wa (not valid in forward direction)..716X2.8X2 2.5X2 20X2 m 124 -I

THE UNIVERSITY OF MICHIGAN 6633-1-F VI EFFECTS OF SURFACE CURVATURE ON COUPLING 6.1 Limitations and Applications Under certain conditions which are met in a number of aerospace configurations, the effect of a curved conducting skin on the coupling between two antennas along the skin can be claculated knowing the flat-plane coupling between the same antennas and the geometry of the skin. These conditions are: 1) The diameter of the skin is greater than about two wavelengths. 2) The curvature in the region of the antenna is small. 3) The Fresnel boundary is smaller than the radius of curvature. These restrictions can be relaxed somewhat and still yield good results. The problem was formulated in cylindrical coordinates but is valid for spherical geometries, or any geometry where the radius of curvature along the ray path between the two antennas is constant. The chief reference on this work is by Hasserjian and Ishimaru (1962). Their approach can be extended to treat other than thin resonant slots. 6.2 Procedure for Calculating Curvature Effect Under the conditions listed above, the coupling between antennas T and R on a cylinder of radius a (see Fig. 6-1) or a sphere of radius a can be calculated by multiplying the flat-plate coupling by a factor f(y), where - - 125

H ri T z axis of cylinder rn 0-4 m - w H I-A M av c! 0 lid -4 0 FIG. 6-1: GEOMETRY OF TWO SLOTS ON A CYLINDER 0 z

THE UNIVERSITY OF 6633-1-F 20 19 18 17 16 15 14 13 12 11 10 alX.0 -I 9 -8 - 7 - rs/X 1.5 - 2 3 4 -5 6 — 7 — 8 — 9 10 -15 -20 3 4 - 5m 60 — 70 — 80 — 90 — 10 m a 0-. 0..m.. - 0 v r lib m BIB I I lB m 500 -400 300 -200 -100 - 80 - 60 -40 - 30 - 20 6 -- 5 4 _ 3. 5 - 3 - 2.5 - 2 - MICHIGAN 00 180~ -160 -140 — 120 — 100 Em - 10 - 8 - 6 -5 - 4 - 3 - 2 1.8.6 -.4 -.3 I" mm 90, 80. 70 - 60 m - 50 - 40 30 28 - 26 24 22 - 20 l 18 I. 1.08.06 FIG. 6-2: Y FACTOR ON A CYLINDER 4 MM

THE U NIVERSITY OF 6633-1-F MICHIGAN a/X 20 19 18 17 16 15 14 13 12 11 10 9 y 60 - 50 - 40 - 30 - 0 20 18 16 14 - 180, - 1600 - 1400 - 1200 - 1000 - 900 - 800 - 700 - 600 - 500 - 400 12 + 8 -7 -6 -5 -4 -3.5 -3 - 2. 5 - 2 - 1 I L V 9 -8 - 7 -6 -5 -4 - w P-4 V) P4 0 4 -0 I CD 2 -1-. 9-. 8 -. 7 - - 300 - 280 - 260 - 240 - 220 - 200 - 180

- THE UNIVERSITY OF MICHIGAN 6633-1-F 1 f( w -. IC -Iv _ MAGNITUDE 60 -6- 4C -7 -8- E X -3( -9 -10- 2( -II -12 ASYMPTOTIC -10 E, EXPRESSION -13 - -, O ) (A w w 0 w cn I CO aJ 0 D(.I.2.3.5.7 I y 2 3 5 7 10 FIG. (-4(a): CURVATURE FACTOR f(y) VS y (LOW RANGE) 2 3 5 7 10 20 30 50 70 100 y - FIG. 6-4(b): CURVATURE FACTOR f(y) VS y (IIGH RANGE) 129

THE UNIVERSITY OF MICHIGAN 6633-1-F For a Cylinder: 2 2 ka v (a/X) V C -4 c y =- (v in radians) = 7.64x10 (V in degrees) V/kr7 c c c / c — 2 ~ 222 av 2 r = z +a v (v in radians) = +( 53) (v in degrees) c c c 57. 3 c = distance from T to R along the surface. v is the angle between two planes containing the cylinder axis and the two c antennas For a Sphere: --- 3/2 -3 f1 3/2 y = a v3/2 ( in radians) = 5.78x10 / v / (v in degrees). S S S S v is the angle between the antennas on a plane defined by the two antennas and S the center of the sphere. The flat surface coupling is the coupling of T and R separated by r or r, c s whichever is applicable, along a conducting plane. Nomographs to calculate y are given in Figs. 6-2 and 6-3. The function f(y) is given below, and in Figs. 6-4. Detailed instructions on the use of Figs. 5-2 through 6-4 are shown in Chapter VIII. The analytical expression for f(y) is 2 3 1 -e / 4~+ 7y el / y.... small 21 - 2 4 120 1024 f(y) = o n 2/3 t, r/4 exp -] j ) (M t e i e ' [I not small m=I m m=l 130

THE UNIVERSITY OF MICHIGAN 6633-1-F where t 1.019-j7r/3 t = 3. 248 e-jT/3 2 t = 4.820 ejT/3 3 t = 6. 163 ej/3 t = 7.372 e-j r/3 5 For large values of y, only the first term need be considered, and the asymptotic approximation holds: f(y)J 1. 550 y1/3 e- 7995 h2/3 f(y) db= 8.686[1 log y+.4380 -.7005 2/3 d 20 log y +.1902 -.3042 y/3] db and the phase lag over the flat plate value is given by 2/3 Phase f(y) =.4045 y2/3 -.2818 radians = 23. 18 y2/3 - 15. degrees The curves for f(y) are shown in Figs. 6-4(a) and 6-4(b). 131

THE UNIVERSITY OF MICHIGAN 6633-1-F The error introduced by using this expression is less than 0. 5 percent in magnitude for y 2, and in phase for y > 4. The latter terms of the series rapidly become more negligible as y increases. To a first approximation, then, it is necessary to reduce the coupling found by "flattening" the cylinder into a plane by the curvature factor j f(y). As the curvature goes to zero, y approaches zero and jf(y)j approaches unity. Whenever more than one ray path may contribute significantly to the coupling, modification of the above procedure is necessary. While in theory there are an infinite number of paths corresponding to multiple encirclements of the cylinder, for the region of interest (a > 2 X) all except the two paths where 0 > 3600 yield terms on the order of 70 db below the latter. Thus the modification need only be extended to treat two ray paths. The waves will interfere or reinforce according to the relative phase. In no case will the reinforcement be greater than 6 db over one path alone. k 132

THE UNIVERSITY OF MICHIGAN 6633-1-F VII EXPERIMENTAL FACILITIES 7.1 Description of Anechoic Chamber The dimensions of the anechoic chamber are 50' x 30'x 15', with a 12' square, 1/8" aluminum ground plane mounted in the center of one end wall (Fig. 7-1). The nine individual sections of the ground plane are joined by 2" aluminum tape to form an electrically continuous surface at microwave frequencies. The center section has a removable 2' x 3' section for the test antennas. The ceiling and floor are covered with B. F. Goodrich HV-4 vinyl-covered, four-inch hairflex microwave absorber. The rear and wide walls are covered with VHP-18, 18" pyramidal absorber. The center of the rear wall is covered with VHP-26 absorber since it is normal to, and centered on, the mean axis of propagation for most measurements. The chamber is lined with interlocking aluminum foil sheets to provide a known uniform termination for the absorbing material and to shield the room from external signals. When the chamber is used for azimuth antenna patterns or radar scattering experiments, the ground plane is covered with HV-4 absorber. 2 All absorber used is in the form of 2 ft sections, secured to the walls and ceiling by Velcro micro-hook fastener strips glued to the back of the absorbers, and velvet strips glued to the walls and ceiling. This allows rearrangement of the absorber to suit a particular measurement problem. The frequency range of the chamber is 500 MHz - 50 GHz and has a nominal reflection coefficient, for normal incidence, of -50 db for X-band and above, tapering exponentially to -20 db for the lower frequencies. Typical return levels are much lower depending on the particular experiment, since normal incidence is usually avoided. 133

THE UNIVERSITY OF MICHIGAN 6633-1-F t;T- ' ' ~d — :_il 1 ---17-1LII:- L — " 'ibM` —~ *~.I?~,~7 ~- " - c~. ~i -e n ;-c'i 2:: — I 4r -"".;I;" c.;; L.C.:;~t: I' i -t 1 k~x -~~- ~: ~*~ t.*ic2" f, / r I C7Y Jr:t~\U Z i,,-^ ^-i 'i 4 I. i~i '.'' * I1 - *4^ A 'w'f^ I i I f: FIG. 7-1 ANECHOIC CHAMBER GROUND PLANE 134

THE UNIVERSITY OF MICHIGAN 6633-1-F 7.2 Determination of Chamber Reflections by Swept Frequency Technique In order to analyze the possibility of errors in the measured low-level coupling between widely separated conical horns (the repeatability of these data was very poor), it was decided that a swept frequency technique would be helpful. The first results indicated that there was more than one path for the coupling energy between the horns, due to the presence of strong periodic perturbations of the coupling level as frequency was varied. Figure 7-2 shows the situation which can lead to errors. For maxima on the interference pattern, 2R-R = nX =(n+l)X =(n+2)X... =(n+m)X +1(7.1) o 1 2 3 m+1 For m cycles of interference, Af = fm+ - f m+1 1 2R -R o nc 1-~ or f = 1 n 1 2R —R 2R -R (n + m)c X U or f..i+m)c m+1 n + m or m+1 2R -R o = f 2R -R m (7.2) = m+l- 1 2R - R o Af 0 or I ] for R>R (7. 135

r Ground plane Transmitter Horn R 0 Target (undesired scatterer at range R) c! Pd ~( CO 1-4 I-q co C3 00 00 1 I - Chamber t -4 O FIG. 7-2: GROUND PLANE AND SCATTERER GEOMETRY z

THE UNIVERSITY OF MICHIGAN 6633-1-F where: R is the range of the interfering scatterer (target) m is the number of interfering cycles Af is the range of frequency sweep R is the distance between antennas o c is the speed of light Equation (7. 3) relates the periodicity of the perturbations of the differential path length between the direct and unwanted coupling paths. To prove the equation, a baffle of absorbing material, which provided a weak reflecting area and isolated many reflecting obstacles in the chamber, was placed normal to and six feet in front of the ground plane containing the conical horns. A swept-frequency coupling pattern was then recorded under these conditions. The measurement setup is shown in Fig. 7-3, and the expected result is obtained in the example below. Example (Actual Results for Baffle at Six Feet): For Af = 5 x 10 Hz R = 36.6 cm o m =5 c =3 x101 cm/sec R = 15 x 10 + 36.6 = (336. 6) = 168. 3 cm = 5.5 ft. (7.4) - 5 x 108.. In addition to the fundamental five-cycle interference pattern from the baffle, higher-order interference was present, indicating the presence of other reflecting objects. A systematic procedure was followed from this point on, whereby the unwanted reflections were eliminated, by recording swept 137

THE UNIV Scientific-Atlanta Model 402P Wide-Range Receiving System 1000 Hz Output R.F. A.F.C. L.O. Input Output Input. A ERSITY OF MICHIGAN 6633-1-F "':Use HP Model X920A shorting plunger three turns from "nearest crystal" position to obtain flattest mixer frequency response. Or Alfred Model 632DB S-Band Sweep Oscillator R.F. F.M. Trigger Output Input Input m "Start Sweep" Pulse Generator - I LII t ~ j I., Mon II0 I RCVR HORN XMTR HORN J MRI Ferrite Isolator S-A No. M8.2 Tuned Mixer,: Alfred Model 645K X-Band Sweep Oscillator F. M R. F Tripoo-r I ____j These flanges mate Input Output for reference level. _ HP No. X382A \ Precision Atter \ Airtron "Twistaguide" <rnainr P^ ^- Ona Input muator - JI. v. AJ. LU ILUlLtL 1. S-band oscillator unleveled at +10 dbm. 2. X-band oscillator unleveled at maximum output. 3. Adjust f ("start" freq.) and f2 ("stop" freq.) and "sweep speed" on each oscillator for near-constant 65 MHz i. f. for receiver; a. f. c. will correct minor tracking errors. 4. Set receiver sweep width (1000 Hz) to 3 MHz for best compromise between sensitivity and stability. FIG. 7-3: CIRCUIT BLOCK DIAGRAM FOR SWEPT FREQUENCY MEASUREMENTS 138

THE UNIVERSITY OF MICHIGAN 6633-1-F frequency patterns with the baffle at increasing ranges to progressively expose more and more of the chamber. These data indicated that severe reflections from the VHP-18 pyramidal absorber on the floor and ceiling, and from over half the light fixtures in the ceiling (Fig. 7-4) were interfering with the coupling measurements. The pyramids were placed with HV-4 vinyl covered Hairflex and the problem lights were covered with the same. This reduced the reflections from the room from -70 db down to -90 db, thereby increasing measurement accuracy and repeatability of the low-level coupling between conical horns. A similar problem reoccurred when it became necessary to measure the coupling between two H-Sectoral Horns. Because these antennas have fan-shaped beams in the E-plane, the E-plane coupling is very large (-30 db) and the H-plane coupling very small (-70 db). Since the antennas were located along a horizontal line on the ground plane, strong reflections resulted when they were oriented for H-plane coupling; this was due to the fan beams 'looking' at the floor and ceiling only seven feet away, and at the edges of the 4" Hairflex absorber and the top and bottom bearings for the swinging boom at the perimeter of the ground plane (Fig. (Fig. 7-5). These reflections caused the main beam coupling to dominate the direct H-plane coupling which was to be measured. To reduce the reflection from the boom bearings and the sides of the 4" Hairflex, a 12' diameter masonite ring, 6" wide, was mounted on the ground plane. The interior of this ring was covered with VHP-5 pyramidal absorber (-45 db reflection for normal incidence at X-band) facing the center of the ground plane (see Fig. 7-1). To reduce reflections from the floor and ceiling in the E-plane of the horns, the 2' x3' sub-ground plane was rotated 90 so that the fan beams 'look' at the side walls which are 15' from the antennas. Then the swept frequency technique was applied to the problem. Measurements still indicated the presence of a strong scatterer at a range of 10'. This was traced to the portable baffle used in the previous measurements, I w 139

THE UNIVERSITY OF 6633-1-F MICHIGAN w z A pl rz 0 -c 9 c P m Ec ff T-i 5 1= 1%:2:1 c c c:1 c r-:2! 14 1 EC PL I 140

THE UNIVERSITY OF MICHIGAN 6633-1-F I FIG. 7-5: ANECHOIC CHAMBER SET-UP FOR FAR-FIELD PATTERN MEASUREMENTS 141

THE UNIVERSITY OF 6633-1-F MICHIGAN Light Fixtures Light Fixtures Ceiling and Floor C3 Baffle HV-4 Ground Plane Receiver Transmitter Horn Horn FIG. 7-6: CHAMBER BEFORE ABSORBER REARRANGEMENT 142

THE UNIVERSITY OF 6633-1-F MICHIGAN FIG. 7-7: CHAMBER AFTER ABSORBER REARRANGEMENT 143

THE UNIVERSITY OF MICHIGAN 6633-1-F which had been stored in a corner of the chamber at the junction of a side wall with the ground plane wall (Fig. 7-6). Again the problem was due to severe reflections from the sides of the VHP-18 pyramids mounted on the baffle. Removal of the baffle from the chamber reduced reflections to the point whereby a 15 db improvement resulted from these three corrections. For these H-sectoral horns, room return was then approximately -85 db, thus permitting accurate measurements of the coupling for any of the horn antennas. The final layout for the absorber is shown in Fig. 7-7. 7.3 Limitations on Coupling Accuracy The measurement equipment is shown in Fig. 7-8. The limitations on coupling accuracy are described below. 1. Chamber reflections can cause a serious error in the measured value of antenna coupling, as shown in Fig. 7-9, e.g. a reflection 20 db below the coupling to be measured can cause an error of ~ 1 db, depending on the relative phasing. 2. Minimum coupling level recordable with a tolerable noise is limited to approximately -90 db by a maximum receiver sensitivity of -85 dbm, by a maximum transmitter power of 20 dbm, and by losses (undesired losses plus deliberate padding) of 15 db. 3. Accuracy of coupling is limited by mechanical accuracy of the horns (and wedges), i. e., if a horn is mechanically (physically) asymmetrical, it cannot be expected to yield a symmetrical coupling pattern. The degree of flatness and continuity of the conducting ground plane also affects the accuracy of the coupling; the ground plane consists of nine pieces joined by metal tape and is flat to within one-eighth inch overall, and much less locally. 4. Coupling accuracy is also dependent on system linearity, which at best is approximately 0.25 db over a 60 db dynamic range. m 144

-<I co 00 *C Ot ~- r tTI Pn r 0 c I > M a H co tM O ti C) Varian HP No. XE No. X-13 Slide Scr< Klystron Tuner in Narda No. 990 Tube Mount FXR No. Z815B Universal Klystron Power Supply S/A No. 402P wide-range receiving system Monitor Scope Input - S/A No.APR-23 Rectangular Antenna Pattern Recorder - -- 370A ew MRI HP No. X750D HP No. X750D HP No. X375A PRD No. 140-A Ferrite Directional Directional Variable 3db Fixed Isolator Coupler Coupler Attenuator Attenuator... - - HP No. X382A Precision Variable Attenuator PRD No. 140-A 3db Fixed Attenuator, I Receiving Horn Antenna Transmitting Horn Antenna Aluminum F Ground Plane -- I I I I I I -iiHP No. X870A Slide Screw Tuner PRD No. 140-A 3db Fixed Attenuator,,, - - -wr, \/ I HP No. X870A Slide Screw Tuner.\.tro N'. a) Cl c03 CO I t —a! IIT H z rl C 0 z) (3 O > Z S/A No. M8.2 Tuned Mixer. i These flanges mate to determine reference level

THE UNIVERSITY OF MICHIGAN 6633-1-F +6 +4 +2 0 -2 -4;' -6 0 - -8 TD O -0 -12 -14 -16 -18 -20 0 -24 Interference Level, db (relative to coupling level) FIG. 7-9: RANGE OF POSSIBLE ERROR IN MEASURED COUPLING BETWEEN ANTENNAS DUE TO REFLECTIONS FROM UNDESIRED SCATTERING OBJECTS (FREQUENCYCOHERENT INTERFERENCE). 146

THE UNIVERSITY OF MICHIGAN 6633-1-F 5. The precision attenuator used to establish a reference level coupling has an accuracy of + 1 db at its 50 db setting although the probable error is less; this further limits measurement accuracy. 6. Another source of error is the drift of the klystron oscillator output power on the receiver gain, which may amount to ~ 0. 5 db over the time required to make a series of measurements before recalibrating the system. 7. At best, i. e., with no noise (moderate coupling level) with no interference due to chamber reflections, and with the system carefully calibrated, the maximum overall accuracy of a given measurement is estimated at ~ 1 db. For low-level coupling (-75 db or less) where chamber reflections and/or noise can be present, the typical accuracy is probably ~ 3 db. 8. The separation between centers of the two antennas is accurate to within ~ 1 millimeter, an insignificant error. 9. The frequency accuracy is + 0.01 per cent when the Dymec DY2650A Synchronizer is employed (e.g. 8.03 - 9.03 GHz) also an insignificant error. 10. The accuracy of orientation angles indicated is limited to + 3~ because of the compact scale of the recording paper, but is typically within + 1~. 147

THE UNIVERSITY OF MICHIGAN -- 6633-1-F vm VIIT COUPLING COMPUTATIONS 8.1 Evaluation of Coupling by Nomograph Method It is observed that all measured far field coupling falls off as 6db per spacing octave. Therefore, for estimating the coupling between any combination of flush-mounted antennas, it is only necessary to know the directivity of each antenna and the spacing between them. The far field coupling formula is given by C = Co(R) Dt(0) D ()q (8.1) 8.2 Nomograph Procedure The value of coupling C can be determined quickly by the nomograph method in the following three steps: 1. The dimensions and orientation of both transmitting and receiving antennas must be known or postulated. D(p) can be obtained from the directivity nomographs in the previous sections of this report. 2. Enter (Fig. 8-1) the values of spacing R/X. Look up C (R) from the C scale. o 3. The reduction of coupling due to mismatched impedance q factor can be found in Section 2.4 (Fig. 2-3) using VSWR. The coupling is given by C(db) = C db + Dtdb+D db + qdb (8.2) The accuracy of the coupling obtained is within + 2 db except for the horn antennas. I 148

THE UNIVERSITY OF MICHIGAN 6633-1-F R /X C (db) o o 1.0- -22 — 23 -24 - -25 1.5 -— 26 - -27 2.0- -28 -29 2.5 -30 -31 3. - -32 -33 4.0 — -34 -35 5.0 — -36 — 37 6. -38 7.0 ---39 8.0- -40 9.0 -41 10.0 — -42 C (R) vs R /k o o0 FIG. 8-1: COUPLING BETWEEN PARRALLEL ISOTROPIC ANTENNAS VS SEPARATION DISTANCE

THE UNIVERSIT' OF MICHIGAN 6633 1-F Figure 8-2 shows the coupling between some simple antennas. The scales indicate the coupling between antennas of the same type. For a combination of different antennas, the coupling is the average of the two values read. The coupling between other antennas is illustrated in the following examples. 8.2.1 Example 1 Evaluate the coupling between a rectangluar slot antenna and a circular spiral antenna under the following conditions (see Fig. 8-3): Slot dimensions, a = 0. 8X, b = 0.4X Orientation of the slot antenna with respect to the spiral antenna, = 450 Spacing between the slot and spiral, R = 2X SWR of the transmitting antenna, 2. From the coupling formula (2.2), u = - 7Tr sin 0 = 0. 5656, v = - 7r cos 0 = 0.2828. X X Enter into Fig. 8-4 the values of a and b and obtain point(. Draw a horizontal line from ( to line F and obtain point (~). Locate points () and (5with the values of v and u. Draw a straight line between ( and ( to obtain (~). Enter the value of 0 at () and draw a straight line between ( and ) intersecting the D scale at point ( which gives D = -2.9 db. From Fig. 4-12 (Section 4. 3), the average directivity of a circular Archimedean spiral is -8 db on a conducting plane surface. 150

THE UNIVERSITY OF MICH 6633-1-F R o X short — spacing -monopoles x 4 monopoles 2 db 24 db 26 -2.5 26 -3 - 28 -28 -3. 5 — 30 -4 30 32 5 32 6 34 34 7 15 -m 2 36 -8 36 9 38 10 38 11 12 -- 40 13 -- 40 14 - 15 -- - 42 42 44 20 — 44 44 -22 22 -, IGAN semiisotropic db 30 -32 34 -36 38 40 -42 - 44 - 46 48 - m i FIG. 8-2: COUPLING BETWEEN SIMPLE 151 ANTENNAS

THE a FIG. 8-3: UNIVERSITY OF 6633-1-F MICHIGAN Spiral Antenna Slot Antenna H b 1 - COUPLING BETWEEN SLOT AND SPIRAL ANTENNA 152

THE UNIVERSITY 6633-1-F OF MICHIGAN CO oOn C O 0 -4 C1J c~j cn, Ln 0 LO IV I LO Ln 0 to 0 co co t,0 T 40 LO 0 IJC o 0 Lo 0 LoOo 0) co O t- t- co co Lo ii U1 It II II II II FIG. 8-4: NOMOGRAPH FOR CALCULATING DIRECTIVITY FUNCTION D(o) FOR A SLOT ANTENNA 153

THE UNIVERSITY OF MICHIGAN 6633-1-F Enter Fig. 8-1 at Ro/X = 2. Find C is -28 db. From Fig. 2-3, q = -0. 52 db for a 2:1 SWR. The result yields a coupling of C(db) = - 28 -2.9 -8 -0.52 = -39.42 db. 8.2.2 Example 2 If both antennas in Example 1 are mounted on a cylindrical surface with a radius equal to 4X, and angle ic = 20. Then the curvature effect factor, f(y), can be found by using Fig. 6-4. Enter Fig. 8-5 at a /X = 4, r /X = 2 and I/ = 20. Follow the progression c c c of the circled numbers through the figure. There y = 0. 86. From Fig. 6-4, f(y) = -1.7 db. Thus the coupling between these two antennas on a cylindrical surface is -39.42 -1.7 or - 41.1 db. 8.2.3 Example 3 Evaluate coupling between a E-sectoral horn antenna and a quarter wavelength monopole. Assume the following conditions. Horn dimensions: a = 0.8X, b = 2.5X Orientation of horn and monopole: = 20~ Spacing between horn and monopole: R = 10X SWR = 1. Then a b u = - sin = 0.274 r, v=-T cos = 2.35 r. X X 154

THE UNIVERSITY 6633-1-F OF MICHIGAN a /X 20 19 18 17 16 15 -14 13 12 11 10 rs/ y C 9 7 - c u - 4 Q 3. 5 3 - 2. 5 2 - 0 1 - 1.5 2 - 3/ 6 -7 -\ 8 9 10 -15 -20 30 40 50 60 70 80 90 100 - -I -500 400 -300 -200 - 100 - 80 - 60 - 40 - 30 - 20 - 10 8 1 6 - 5 - 4 - 3 - 2 180~ 160~ 1400 120~ 100~ 90~ - 80~ - 70~ - 60~ - 500 - 40~ - 30~ - 28~ - 26~ - 24~ - 22~ - 20~ - 18~ - 1.8.6. 4 -.3 -.2 - 1.08 0E -.06 FIG. 8-5: Y FACTOR ON A CYLINDER 155

THE UNIVERSITY OF MICHIGAN 6633-1-F Enter Fig. 8-6 the directivity nomograph of a horn antenna with the values of a, b, u, v. Follow the order of progression as indicated by the circled numbers in the figure. On scale D, -7.3 db is found. The directivity of a quarter wave monopole perpendicular to a conducting surface is 2.15 db. Enter Fig. 8-1 at R /X = 10 and find that C = -42 db. With the SWR = 1, q is 0 db. The resulting coupling is C(db) = -42 -7. 3 + 2 + 0 = -47.3 db. 8.2.4 Example 4 Assume there is a quarter wave monopole located 4X away from the center of the square archimedean spiral 1L at the frequency of 3.3 GHz. From Fig. 4-15, Section 4.4, the directivities of the square spiral 1L at 3.3 GHz are: D =-5.4db, D =-7.0db, and D. =-9.0db max avg mm Enter Fig. 8-1 at R /X = 4 and one finds that C = -34 db The resulting couplings are: C (db) = -34 -5.4 + 2 = -37.4 db max C (db) = -34 -7.0 + 2 = -39 db avg C. (db) =-34 -9.0 + 2 =-41 db mln 8.2.5 Example 5 The problem is to evaluate the coupling between two parallel slots given the size of the slots, the distance between the slots and the frequency of operation. Assume the following conditions: 156

THE UNIVERSITY OF MICHIGAN 6633-1-F a/X _1. 5 - 1. 4 -.1. 3 - 1. 2 "-1. 1 1. 0 -.9 -0. 7 - 0. 6 FIG. 8-6: b/X D(db) u v.6 olir. 57r v.70.2ir A4r.8.9.4-r 37r 1.0. 287T -10.67r.227r 1. 5 2.77r 17 -15 2.0.177r.167r.157r 0 87r 2. 5 -20 l147r.137r 3.0 l127r 3.5 -25.9ir.117r 4.0 1 07r 4. 5 -30 5.0.0O97r 5. 5 1.Or 6.0. 087r NOMOGRAPH FOR CALCULATING DIRE CTIVITY FUNCTION OF VRECTANGULTTAR HORNSM j

THE UNIVERSITY OF MICHIGAN 6633-1-F f = 9.20 Kmc; X = 3.26 cm = 1.283", R = 2. 566" = 2 Transmitting antenna: at = 0. 9" = 0. 7X bt = 0. 4" = 0. 312X Receiving antenna: a = 0. 77" = 0.6X, b = 0. 32" = 0. 25X r r Enter Fig. 8-7 with the transmitter parameters at and bt and obtain point (). Draw a horizontal line from Q to the line Ft, obtaining point (. In a similar manner, use the receiver parameters a and b to obtain point) and ( draw a straight line between ) and ) so that line F is intersected at. Enter the values of bt and b on the Dt and D axes, respectively, and obtain t r t r points ( and. Connect) and ( with a straight line intersecting the D axis at point. A straight line between () and( gives ( on the central axis. Enter the value of R on the R axis ) and draw a straight line between ( and (1) intersecting the C axis at ). The coupling is given by the value at ) on the C axis. In this case the coupling is -24. 5 db. If Ro exceeds 10X the nomograph can still be used provided -20 db is added to the coupling obtained from the nomograph. Suppose R = 20X. Enter the value of R on the R axis as 2X which will give a coupling of -24.5 db as before; the total coupling will consequently be -24.5 -20. 0 = -44. 5 db. In case the receiving antenna is rotated on an angle of r =600 with respect to the axis between receiver and transmitter a modification is necessary. On the nomograph of Fig. 8-8 mark a point on the angle 0 bar at the 600 mark, call this point (. Mark a point on the b/X bar corresponding to the value 0. 25 which is the bt dimension. Call this point(). Draw a line between ( an ( and let this line extend over to the bar marked G. Call this intersection point(. Reading off this bar find a value 0. 125 at(. Returning to the main nomograph 1 158

D D C(db) R (r/X) 0 r t.025. ~0250 00 -10 1.1007.-100..125-. T.25 T.150,.. T 175 150 2 -2 2.0 _ t.200 F 2F T R.225 225 1.5 h::l.250.27 '; -/*- ~:. 2'. ' S /,,<..? -2 0 / 275.27 ^0.0 FR F T 0 cO ~ 30. -/ ~ / ^ ^ ^ 300 '. r. a/X=0. 90,6.0 a/X=0. 85. - > -"..40 7.0 a/2. -0 75 >8 A0.7.25 0 05 ~:'.*j/.iX 0'1 50. vo 3.0 000005-2 —. 375.400 —.400.3 o ~425 -.425' a/L=0. 90 -35._ 5.0 a/ X;k=0.85 ~ 450 0. 4 50 6.0 a/,,=O. -4 40 7.0 ' a/X=O. 75.475 - -. 475. a/k —O. 70 8.,-' l.., I8W.500. 0.O,&;-. lo,:'O.. 20* 0.,2,5 '0,.3.0:0. 3 0.4 0.4 -. 5":../) ~, ~ - -t o 0) CO y iCD I o z

THE UNIVERSITY OF MICHIGAN 6633-1-F of Fig. 8-7 and using the center stage of the nomograph enter the value of bt (0. 312X) on bar Dt. Also enter the corrected Dr (0. 125) obtained as point( on the nomograph of Fig. 8-8. This latter value of 0. 125 is to be marked as point ( on bar D in the center of the nomograph of Fig. 8-7. The value of b which r t has been entered on Dt is to be called(. Draw a line from() to (. The intersection of this latter line with bar D is to be designated as (). Draw a line from the previously established pointG to). The intersection of this line is to be designated as (on central bar. On the extreme right-hand bar of the nomograph of Fig. 8-8 indicate the spacing of two wavelengths at 2.0 and call this ). Draw a line from () to. The intersection of this line with bar C is to be called (. Point @( designates a value of coupling of 23. 8 db. This value of coupling is not the true coupling for the sample problem but is an intermediate one which must be corrected in the manner described below. On the nomograph of Fig. 8-9 enter the value of a (0. 6) on the bottom r horizontal bar. Call this. Enter the angle 600 on the slant bar toward the bottom of the nomograph of Fig. 8-9 and call this(). Draw a line between( and ) letting it intersect the horizontal line immediately above and call this (. Draw a vertical line from ( upward to intersect the curve. Call this intersection ). Draw a horizontal line from() to the right and let this line intersect the first vertical line. Call this intersection. On the extreme right hand bar of the nomograph of Fig. 8-9 put the value of 60~ and call this (. Draw a line from to( and designate the intersecti othislineon bar K as. The value at ()is -8.2 db, a coupling correction. Ctt= C db + K db = -23.8 db -8.2 db = -32 db where Ctot is the total resulting coupling. When both transmitting and receiving antennas are rotated, the same procedure on the supplementary nomographs 160

T THE UNIVERSITY OF 6633-1-F MICHIGAN G 90~ 1. 50.45 80~ 70~.40.35.30 25 20 50O 15 40O N\ 30O S0 05 20~ 0 0 NOMOGRAPH (B) FIG. 8-8: ANGULAR CORRECTION FOR WIDTH 161

THE UNIVERSITY 6633-1-F OF MICHIGAN K(db) -22 -21' -20 -1 9 -1 8 -1 7 -16 -I.- 4 -13 -12 -I I -I0 -9.-8 -7 -6 -5 -4 - 3 -2 - I 0 80~ 75~ 70~ 65~ 60~ 5.5~ 50~ 45~ 40~ 35~ 30~ 20~ 100 0 I I I I I NOMOGRAPH (C) 0.8 0.9 FIG. 8-9: ANGULAR CORRECTION FOR HEIGHTH 112

THE UNIVERSITY OF MICHIGAN 6633-1-F Figs. 8-8 and 8-09 must be made for each slot. The total coupling then would be; Ctot = Cdb+ K db + K db tot r t K db and K db are obtained from Fig. 8-9. t r 8.2.6 Comparison of Calculations on Coupling Some other typical combinations of coupling are listed in the following Table VI-1 with spacing normalized to one wavelength and assuming a perfect impedance match. X/4 monopole Circular Spiral Rectangular Slot E-Sectoral d = 10 cm a=0.8X, b=0.4X a=0.75X, b=2.5X -17.7 -28 -17.8 -27 X/4 monopole Circular Spiral Rectangula Slot E-Sectoral Horn r -28 -17.8 -38 -28 -37 -28 -18 -27 -37 -27 -36 -27 163

THE UNIVERSITY OF MICHIGAN - 6633-1-F IX REFERENCES Blasi, E. A. (1954), "The Theory and Application of the Radiation Mutual-Coupling Factor," Proc. IRE, 42, No.7, pp. 1179-1183. Crispin, J.W. Jr., R.F. Goodrich and K.M. Siegel (1959), "A Theoretical Method for the Calculation of the Radar Cross Sections of Aircraft and Missiles," The University of Michigan Radiation Laboratory Report No. 2591-1-H, AD 227 695 UNCLASSIFIED. 425 pgs. Deschamps, J.G. (1951), "Geometrical Representation of the Polarization of a Plane Electromagnetic Wave," Proc. IRE, 39, pp. 540-544. Ehrenspeck, H. W. and H. Poehler (1958), "A New Method for Obtaining Maximum Gain from Yagi Antennas," Air Force Cambridge Research Center Technical Report AFCRC-TR-58-355. UNCLASSIFIED: Harrington, R. F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill p. 111. Harrington, R. F. (1964), "Theory of Loaded Scatterers," Proc. IEE, (London) 111, No.4, pp. 617-623. Hasserjian, G. and A. Ishimaru (1962), "Excitation of a Conducting Cylindrical Surface of Large Radius of Curvature," Trans. IRE, AP-10, No. 3, pp.264-273. Jacobs, E. (1960), "Maximum Power Transfer Between Large Aperture Antennas in the Fresnel Region," The University of Pennsylvania Report IS-60-UR-6, pp. 99-112. Jasik, H. (1961), Antenna Engineering Handbook, McGraw-Hill, New York. King, R.W. P. (1965), The Theory of Linear Antennas, Harvard University Press. Marcuvitz, N. (1951) Waveguide Handbook, McGraw-Hill, New York. Matthaei, G. L., L. Young and E. M. T. Jones (1964) Microwave Filter, ImpedanceMatching Networks, McGraw-Hill, New York. Met, V. (1959), "Absorptive Filters for Microwave Harmonic Power," Proc. IRE, 47, pp. 1862-1769. Oberhettinger, F. (1954), "Diffraction of Waves by a Wedge," Comm. Pure and Appl. Math., 7, 551-563. 164

THE UNIVERSITY OF MICHIGAN 6633-1-F Rumsey, V.H. (1951), "Transmission Between Elliptically Polarized Antennas," Proc. IRE, 39. 535-540. Kwon, Y. K. (1965), "Details of Mathematical Analysis of Rectangular Horn Apertures," The University of Michigan Radiation Laboratory Internal Memorandum. Potter, P.D. (1963), "A New Horn Antenna with Suppressed Sidelobes and Equal Beamwidths," Microwave J., pp. 71-78. June. Tai, C. T. (1961), "On the Definition of the Effective Aperture of Antennas," Trans. IRE, AP-9. No. 2, p. 224. Vainshtein, L.A. (1959), "Waves of Current in a Thin Cylindrical Conductor-I, II" Sov. Phys. Tech. Phys., 4, pp. 601-626. Vainshtein, L.A. (1961), "Current Waves in a Thin Cylindrical Conductor-II, IV" Sov. Phys. Tech. Phys., 6, pp. 19-38. Silver, S. (1949). Microwave Antenna Theory and Design, McGraw-Hill, pp. 336-337. 165

THE UNIVERSITY OF MICHIGAN 6633-1-F X ACKNOWLEDGEMENTS Dr. Vaughan H. Weston and Dr. A. I. Simanyi of the Radiation Laboratory were major contributors of material in this report; Dr. Weston was responsible for the analysis of scattering as a factor in coupling as shown in Appendix B; Dr. Simanyi analyzed modal coupling and arranged the experiments on this subject; Mr. Richard B. Harris of The University of Michigan Institute of Science and Technology, was responsible for most of the experimental data included in this report. 166

THE UNIVERSITY OF MICHIGAN 6633-1-F APPENDIX A ANALYSIS OF PLANAR APERTURE ANTENNA COUPLING A. 1 General The present section is an excerpt from thesis work of Y-K Kwon to be published separately. Therefore frequent references are made to a complete internal memorandum (Kwon 1965). The purpose of the present study is the theoretical investigation of: (i) Coupling of waveguide connected aperture antennas, and (ii) Coupling of waveguide - horn connected aperture antennas. To keep uniformity of the theory, (i) is treated as the degenerate case of (ii). The two horn connected aperture antennas are shown in Fig. A-1 and Fig. A-2. The transmitting horn T2 is excited by the waveguide T1 and a measuring device is located in the waveguide R1 connected to the receiving horn R. The coupling C between the two antennas is defined as the ratio of the power 2 W received by the measuring device at R1 to the power Wt transmitted from the aperture of transmitting antenna: W C = (1) Wt The calculation of W and Wt requires the correct knowledge of fields in R - and T2- regions, which can only be evaluated through solving boundary value problems comprising the whole transmitting and receiving structures. This task is, no doubt, a staggering one, in view of the fact that the rigorous solution of a flush mounted antenna of the simplest geometry is yet to be obtained. However, 167

THE UNIVERSITY OF 6633-1-F MICHIGAN r a or b r r r (r) P 2 z r z z =0 r S T2 at or bt z t= t FIG. A-1: ANTENNA CONFIGURATION (SIDE VIEW) 168

THE UNIVERSITY OF MICHIGAN 6633-1-F 77 I I I bt L FIG. A-2: ANTENNA CONFIGURATION (FRONT VIEW) 169

THE UNIVERSITY OF MICHIGAN 6633-1-F under some reasonable assumptions, one can get formulas which predict experimental results with sufficient accuracy. To determine the fields inside and outside of antenna systems, it is necessary and sufficient to specify the fields at the junctions, namely zt = 0, z = 0 and z = 0. More specifically 1), the tangential components of Electrical fields at the apertures (z = 0) and junctions between waveguides and horns (zt = 0 and z = 0) uniquely determine the fields everywhere. For regions T1 and R1, the tangential components of Electric field, E and tangential components o E magnetic field Ht are given by: it Et (rT)= Vi (zt) ei (Pt) + i=l It t 1 i i (At) l r (R '= V (z ) i. (r) 1 R r i r i=l j 00 =t+ 1 00 t -=i +1 r.( (z ' fl t) Ij (zt hj (Pt (i ( (A. 1) and Vj (Z) (Pr) j tI 00 r t (r R1) = I. (Z ) i r) + I. (z (hr ) i=1 j= I+1 jir (A. 2) 170

THE UNIVERSITY OF MICHIGAN 6633-1-F In the above equations i and j are to be considered double summation indices and the first St or i modes are propagating ones and the remaining modes are t r decaying. The orthonormal transverse vector functions e. and h. are listed in 1 1 Marcuvitz (1951). The amplitude factors V. and I. can be uniquely specified in 1 1 terms of tangential electric fields at zt = 0 or at z = 0 and the incident fields r (Kwon 1965). The position vectors in the waveguides are decomposed into longitudinal components and transverse components: rT t z zt R= p + z z:Z unit vector in z-direction (A. 3) For region S, the magnetic field is given by H(R)= s Pt)MT(Pt z=O)dST-/Y(R:pr) R(r z=O)dSR l (A. 4) where the integrations are performed over the aperture ST and SB respectively, and -ikjR -p | si 2 e (s:P) =+2r. (k I+VV) (A.5) (V operates on R ) MT =E( z = 0) x (A. 6a) b 171

THE UNIVERSITY OF MICHIGAN 6633-1-F Mr = E (, z =0) x I = unit dyad (A. 6b) (A. 6c) Next, the fields in the regions T2 and R2 should be specified. For an E-sectoral horn, the transverse components of the magnetic fields are given by (Kwon 1965 Section III): oo 00 m2rx1 Hx = 7 Z (y P) sin 1 cos =x m p7 m a 2 m=l p=O 29 0 0 (A. 7a) 00 H = m=y m = 1 > ( —) i - Z (Y p) cos ^-Ja p dp p m 20 p = 0 20 0 p 20 0 1 m7rx P (7mP) sin 2 Cos p 20 a 2e 0 +joE~ I:y 1 W (( np) cos np p 2qr 2 n=0 q=l 2i 0 O u d ndrx - u W ( p) sin. Cos 1 p dp n 20 a 20 o (A. 7b) I 172

THE UNIVERSITY OF MICHIGAN 6633-1-F where p, p, 0, a, etc., are pictorially shown in Fig. A. 3 and 1 ~ a x x+ 2 0 =0+0 o p= /y + u= p t(or zr) I t r 22 2 2 m r m - 2 a (A. 8) (A. 9a) (2) Z (Y p) = A H (y p) + B H (y p) p m mp p m mp P m 20 20 20 0 o o w (Y P) (r n) 20 20 o and A, B, Cnq mp mp nq conditions. = Cnq Hq (nP) +Dnq Hr (np) (A. 9b) 20 20 o o and D are complex constants to be determined by boundary nq As one can see from the above horn mode-functions the absence of orthogonality with respect to y-coordinate makes the determination of constants, A, Bmp, Cn and D extremely difficult. Of course, with respect to a cylindrical mp nq nq coordinate system with x as the axial coordinate, one can construct orthonormal vector functions of the E-sectoral horn. Then one encounters ever more serious difficulties in the waveguide regions and half-space region. In other words, it is impossible to find one coordinate system which preserves orthogonality of vector 173

THE UNIVERSITY 6633-1-F OF MICHIGAN x () kO) a y (r1) u z zt or z t r y (rn) FIG. A-3: COORDINATE OF E-SECTORAL HORN 174

THE UNIVERSITY OF MICHIGAN 6633-1-F mode functions in all regions. For E-sectoral horn, this difficulty is less serious, because it still keeps the orthogonality in x-direction. This indicates that if one considers only TE -modes in the E-sectoral horn, the preservation mo of orthogonality is achieved. One of the major difficulties in the analysis of the present problem is the lack of orthogonality in the horn region with respect to a rectangular coordinate system in which the vector mode-functions of waveguide and the total field in the half-space beyond the metal plane are specified. To overcome this difficulty, a formalism to express the fields in the horn region in terms of uniform waveguide modes is developed (Kwon 1965). With this formalism, the tangential components of fields can be expressed in forms exactly the same as equation (A. 1) or equation (A. 2) with the same forms of transverse vector mode functions e. and h.. The 1 1 amplitudes of the modes V. and I. are determined by the differential equations: 1 12 12 dvi t t i A. V.+jw- I=. E-.. V- M. V (A. lOa) dI t t t t 1 — A. I.+j e V. = - H.. I. N. I (A.lOb) dz i 1 1 j( i) ij j P p pl The summations over the double index j exclude j = i. These equations are for TM modes. 175

THE UNIVERSITY OF MICHIGAN - 6633-1-F Ml ii f l 7 +1 I I? it dz- qIq + jc.,c-2V q= -% M.Ij- )H pIp dz qk ci p(p Lq) qp dV it,1it+ 1? 11 +AV + jwiuI N.V. E V dz q q q i j p(p ~=q) qp~ P (A. l a) (A. l ib) The summation over the double index p excludes the case p = q. These equations for TE-modes. The coefficients are functions of z and are defined as A 2tan 0 111 ds 2 -~ tan e h1 ds A.- tanG j0 J ds = tan 0 I12ds JJ dz j dz 3i ~ir dh Ffde H h - dS= 'E dS =E HqpJ q dz jjdz q pq (A. 12a) (A. 12b) (A. 13a) (A. 13b) M. MJ dz =11 ii" - dSI UdS = pdz pit p Z p (A. 13c) 176

THE UNIVERSITY OF MICHIGAN 6633-1-F dh TdeN p- h P E d= ~.e dS- N (A. 13d) ip i dz dz i pi S is any transverse cross-section of the horn and s is the contour of S. o(0) is half of the flare angle. For rectangular horns, these coefficients are calculated and tabulated (Kwon 1965, Section V). When 0 becomes zero, the A's, de. dh. d and - go to zero, hence equations (A. 11) are reduced to the ordinary wavedz dz guide transmission line equations. A. 2 The Coupling of Two Waveguide Connected Aperture Antennas The present problem (Kwon 1965, Sections III and VII) is the degenerate case of the general situation in which the regions T and R are eliminated and 2 2 the regions T! and R1 are directly connected to the half-space S. Now, (A. 1) and (A. 4) are connected by invoking the continuity of tangential magnetic field on aperture T. Equating z x H on the aperture T, one obtains the following integral equation: t O IiT) xh.(Pt) + 7 I) xh (t) =~x I (:Pt)zxi(pt) dSTA 1 It t j t t T i=1 j=it+1 + zxJ (Pt Pr)' [xE(pj)] dS (A. 14a) The same operation on the aperture R yields: 177

THE UNIVERSITY OF MICHIGAN 6633-1-F o jI(R)+ ) XhdS i 1 r j jr i=l j=A +l + x (r:P )Z x E (P)dSR (A. 14b) r L rJ R Since zt and z in eqs. (A. 1) and (A. 2) are replaced by z throughout the present t r section, the subscripts "T" and "R" are used to indicate modal amplitudes of two waveguides. An exact solution of the integral equations (A. 14) is extremely difficult. Thus, it is desirable to employ some approximations. The usual approach to the approximation is setting up a variational formulation which is not sensitive to the exact nature of the aperture fields (Kwon 1965). The limited space for this section does not permit going into this formulation. As an alternate approach, eq. (A. 14b) is multiplied in dot product fashion by 2 x h. (r ) and the result is integrated over the aperture. Then, the following equations are obtained due to orthonormality of h. (Marcuvitz 1951). 1 r +sRasR h (hr):P~ rr),x ~(P r (A. 15) Similar expressions for I(i) can be obtained by interchanging Pr and P-t in the above equation. The accurate knowledge of the aperture fields E(Pjt) and E(;Pr) determines modal amplitudes in the guide regions, hence the fields. Since 178

THE UNIVERSITY OF MICHIGAN 6633-1-F the determination of (p) and E(p ) is difficult, one usually resorts to some assumptions compatible with boundary conditions and physical intuition. It is also to be noted that E(p') on the transmitting aperture and E(p ) on the receiving aperture are mutually related. To clarify the inter-relation between E(pt) and E(p ) and introduce some assumptions, the following observations are made. 1. On the receiving aperture, the tangential magnetic fields satisfy: A A - A - zxH. =zxH +zxH (A. 16a) in s o or zx!H -H ] =zxH, (A. 16b) where H is field produced by the source (i. e., magnetic current in the transmitting aperture) when the receiving aperture is closed by perfect conductor. H. and H are the scattered fields inside the guides and in the half-space when in s the conductor is removed. 2. Equation (A. 16a) is exactly equivalent to eq. (A. 14b). Since z x Ho never vanishes and both of the scattered fields H. and H are smaller, in in s magnitude, than H, then one can write: z xH =-c z xH. (A. 17a) where c is complex number and, zx-. = z xxH (A. 17b) in 1+c o 3. If the waveguide is removed, one has a symmetrically located groundplane with a hole in it. In this case, as in diffraction theory, C equals to 1 and 179

THE UNIVERSITY OF MICHIGAN 6633-1-F x H. = 2 z xH (A. 18) in 2 o 4. Therefore, one can conclude that -- < 1 (A. 19) In the subsequent analysis, c is assumed to be 1. In view of the above approximation and (A. 14b), eq. (A. 15) becomes (R) ^ =f<). y (-P -. E(p5 (A. 20) Ii 2 dSR JJdSTir rt) 0 Next, (A. 20) is used to compute the TE 0 mode coupling of two waveguide terminated antennas (Kwon, 1965). It is, now, assumed that the transmitting aperture field is a linear combination of TE MO modes of transmitting guide, i. e., co - e(T)m'0 E(PA) E VmO eMI (Pt) (A. 21) m'= 1 Substituting (A. 21) into (A. 20), one gets: oo (R) = f1 (t o 2 m'0 JJ rtM t m'=l (A. 22) where the relation z x m',0 = hm is used. Since only TE modes are considered furtherm' mo sidered further, double primes are omitted henceforth. Now 180

~~THE UNIVERSITY OF 6633-1-F MICHIGAN h MO ab mirxI a sin -a,where xi =X 2 (A. 23) and using eq. (A. 5)., the eq. (A. 22) becomes I(H) = 1 1 MO 2irj wIA.s~(a bt)(arbr) OD zt~ at b a br (T) t~dl x d 1io d 1fd i 1xid 1 m'O0 v where m I7Tf 1.m7rx 1 A 2=T, A,,sin sin xf(k +V7).g G(Q.,rl: x, y)., at ar -jk 2 G(Q, iv:x y) = -e (x _ ) + (y-_ ) X-)2 + yr)2 (A. 24) (A. 25) For other notations, refer to Fig. (A-2). After some algebraic manipulations one gets: (R) 1 1 'MO 2 7irj (a b)( b) OD I: v(T) M= 1 (A. 26) 181

THE UNIVERSITY OF MICHIGAN 6633-1-F where rat m r F 22 mO, m' de sin - cos (k - 2 ) m0 a2 Jo r \ t r br m7rx dr1 dxi dyl sin a G(x,y:g, r) az cos dn dy[(-l)mG(x= ) -G(x=- c-) + sin 3 ( ) drj dxo cos G(y= )( - G(= - -) a 2 2 (A. 27) where a a G(x=-) = G(x=- y:E,r) etc. (A.28) 2 2 The exact evaluation of J m m'0 is extremely difficult. However, for special cases, J O,0 can be reduced to more manageable form. Before evaluating the integrals in (A. 27) approximately, the special cases will be studied. Case 1) = 0, at = ar =a, bt = br = b. For this case (A. 27) becomes: 182

I THE 1 mO, m'O UNIVERSITY OF 6633-1-F b - 7 ka)2 a d (b 2 7r -l a MICHIGAN --- 7r a1) dX O 6 (I -,) [ { 2 2.L k a 2 2 1 m 7r (n -X) cos mX + -(1 + ) k a sin m Xl 2 2 COs (m+m ') sin (m-m') mm 2 2C m - m L. k a L (m-m'). (m+m')Xcos 2 sin 2 2 n + m',,, (m7T)2 1 sin m ka mn R O. h a L 7rcos a+X, R r sin a+a) a R + h(- 7rsina-X, a R - rsin +a) a (A. 29a) (m and m' are either both even or both odd) where j K t2 + s 7T h(t,s) = t + s R is the center-to-center distance of two apertures. 0 183

THE UNIVERSITY OF MICHIGAN 6633-1-F b -7T 7 Jka2 2a bdc U) dX mOm'O a b a 2 2 sin(m-m9)X (m+m)X. (m+mr)X (m-m)X1 m22 2 k 2 a2 ' sin o 2 - " - cos s c 2 2 m+ m' m-rm' a a a a (A. 29b) (for m = odd, m' = even or m = even, mt = odd) The following theorems can be obtained from eqs. (A. 29a) and (A. 29b). R R o em 2: If cos a>, sin af 0, m = even m' = odd, J is, = od -a mOm'n, of the order of s becomes large. (Up to theorem 4, is assumed). 0 R em 3: If - cos >1 sinf 0 m = evenand m' = eveneve, is odd) or (m = odd, st, oftheorderof - as H becomes large. a sven), Jm0. O becomes proportional to cos aR/R as R becomes large. o b em 4: If sina and -1>- J o 0 is, at most, of the order of - aa m0 ' 0O R:era 3 Theor order Theor at moi Theor = ( Theor 184

THE UNIVERSITY OF MICHIGAN 6633-1-F Nomenclature of these theorems is indicated in the Fig. A-4(a) and A-4(b). Before going into the next case, the exact coupling formula for m = 1 and m' = 1 is given for the purpose of comparison with experiment. The formula has the following expression C- 1 ) ka 1 2 (A. 30) 732 2 (lb) ) R b -~r r7T a b 2 2 K = (- 7r -[a)dda dX(- (1T -X) cosX+(l+k2) sinX] (a k2 a2 k2 a2 Jb 0 -— r o a R R R R r o 0 0 [h (- 7rcosa+X, -7rsin+a)+ sin+) a a a a J (A. 31) =k ( )2 If =0 If ~ Cos a> I Law 0 m 185

THE UNIVERSITY OF MICHIGAN - 6633-1-F.T Ia. —a -l — h - T - - u * - - 11 I -- a — ai (a) 1 (1) J ^m'0 (far-field coupling) for m = odd, m' = odd MnO, m' R 0 (2) J < 1 ( ) mO, m' O - 2 0 (near-field coupling) for m = even, m' = even (3) J. os for m = odd, m' = even mO, m'0 R O or m = even, m =odd R=0 R If 1 a sina=0, JmOm 2 (near-field coupling) I M, oO R T T — 1 R bT ---- IU I I. a. t i1 - I i | a — I tL___I (b) FIG. A-4: PICTORIAL ILLUSTRATION OF THEOREMS FOR = 0 186

THE UNIVERSITY OF MICHIGAN 6633-1-F R is given by eq. (A. 43). C is plotted in the graph of Fig. A-5 for a = 00, a= 10, a= 450 and a= 90. For a=O0 and a= 90, the experimental values are also given. A computer program is available for similar cases as shown in the graph, but for any size slots the transmitting and receiving slots need not be the same size. Case 2),=For this case (A. 27) becomes: bt at ar 2 2 2 mO, mO a 2 )J a a 2 2 2 r Tm m1'r mwrx m'7rg dx (cos )(cos-2)cos — cos a L2 2 ar a mr. m'. m7rx mrT m7r m1' mirx m'tr -(sin-) (si in in- cos a + (cos) (sin) cos- cos 22 a 2 2 a at r t r t r t m mr, tM a. mxrx. m'brrg -(sin-)(cos2 —) sin —sin g R cosa. ---,R sinm at+ x.-r) b -g(R cos a+ -, R sina+x- r) 0 2 0 - (A. 32) where ejk/u2 + g(u, v) = e u2 u +v 187

I THE UNIVERSITY OF MICHIGAN 6633-1-F Further reduction of the above triple integral seems to be impossible unless a = bt. Some physical significance is extracted from (A. 32) when put in the form r t of theorems. 7r Theorem 1: If a = -, the odd modes of the transmitting aperture field do not 2 contribute to the coupling, i. e., J m, ' = 0, if m' = 0. mOm'O Theorem 2: If a = 0, no odd modes are generated in the receiving guide, i.e., JM = ~ 0 if m = odd mOmtO 7r Theorem 3: If a = - m' = even, and 2 (bt +a) a + bt R- > t then o 2 2 J is, at most, of the order of as R becomes large. m,m R0 0 o Theorem 4: If a = 0, m = even and (at +b) at +bt R t r ___ o 2 2 J is, at most, of the order of - as R becomes large. The above four 0 theorems are summarized in Fig. A-5. I! 188

THE UNIVERSITY OF MICHIGAN ------ 6633-1-F A; T r r 0 — IT,B= (or -) If -Ar.sina= 1 (1)J 0 to =0 for m' = odd mO, m' (no coupling) (2)J ' R2 for m' = even 0 (near-field coupling) I I4 ii -- ~ _ _ __ _ _ — atbr (a) "Ii' I 1I T,B= 2-(or - -) If, 2 sina= 1 (1) JM0 '0 = 0 for m = odd ( mO, m'O (2) Jm m < 2 for m=even (near fied coupling) 2 0 (near field coupling) 1 -I. | I I I -a^ '.. a t L iO tat - bri* R -—... 0 a r I (b) FIG. A-5: PICTORIAL ILLUSTRATION OF THEOREMS FOR 13= 2 2 189

- THE UNIVERSITY OF MICHIGAN 6633-1-F To evaluate the integral in (A. 27) for general use, the following approximation is made: -jkR r G(x,y:)P ). 1 R o x - -COS R 0 (a- )- Y- sin ( - ) R O + 1 Cos a R 0 sn. jk [xsin (a-3) -ycos (a - ) - sin a+or cos R 2 R 0 0 -jk - x cos (a — +ysin(a-) -]cos- risina+ 1 R 0 (A. 33) Substitution of (A. 33) into (A. 27) leads, after tedious algebra, to the following results: For aL 0, 16 t JmOn, m'0 R m' o a -jkR ( )e ~ m?r kbt sin ( — sin c) L(m'cos a) 2 c kb *( r s sin (a - ). j 1 ka r mr cos (a - (m = odd) (A. 34a) (A. 34b) ka S r os (a - ) (m = even) m7r For a=0 a J 4 t mO, m'O -2 ml7r 0 a (mir ) -jkR bt e 1 190

THE UNIVERSITY OF MICHIGAN 6633-1-F Tka j 0 cos ( —) (m = odd, m' = odd) ( Im A. 35a) kat O sin (-2) (m = even, m' = even) (A. 3! m 2 ka E cos (- ) (m=even, m'=odd) (A. 3 In 2 kat -j E sin ( ) (m= even, m'1 =even), (A. 3 where ka ka *m'r 2 t m'r)2 kt L(m' cos) = (sin 7)2 C(-t c os) - j(cos- 2) S( -cosca) (A. 3 ka ka cos ( 2 cos (Ca-)) C — cos(a-3 =- 2 (A.3 m~t k2a2 m 2 ka 1 r S -- cos(a-P3) = 2 CO (A. 3 k a 1 2 COS (a-) In I7 5b) 5c) 5d) 6a) 6b) 36c) 191

THE UNIVERSITY OF MICHIGAN 6633-1-F kb r ssin (~ sin ) kb ka 0 = 6 -(kb )os( s cosC( -rcos () mn sin P r 2 mv-m r kb + sin ( ~ sin 3)* sin 2 ka r (- cos 1) 2 2 2 k a - 2 cos j3 m 7r kb sin (-r sing E = 2 + (kb )cos mn sin P r kb ( 2r sin ) } (A. 36d)* ka r costS( cos 1) m7r IT kb - sin (- sin P) sin '{ 2 k,' ka ka ka r r r 4( -)cos 3S(( oss ) + ka cos ( cos 3) mrr mr C r 2 2 2 ka r 2 1 -1 2 2 cos 1 m ir (A. 36e)* From eqs. (A. 26) and (A. 34) one determines I(R if VT) is given. The m0 m'0 (T) mode amplitudes of transmitting aperture Vm,0 may be determined by using a m u variational procedure. However, it is usually satisfactory to use incident (T) 7r (R) mode-amplitudes as V 0. For a = 0 and a= 2 the expressions for I are as follows: *) The last terms of (A. 36d) and (A. 36e) should be subject to more discussion (Kwon, 1965) and may be eliminated entirely. -d. % _ 192

THE UNIVERSITY OF MICHIGAN 6633-1-F For a= 2 (H)= I= MnO - 8 -3.w 1 (b b t r 1 1 m R 0 -jkR 0 kb e ~sin (-2) -kb 2 ka 2 C (- sin/3) m7r Z (T) Vt rn'=odd in v(T) Vl in =odd (mn = odd) (in = even) (A. 37a) (A. 37b) For a = 0 I(H) "no 2 73.jw 1 atar 2 btbr 1 in b1 -jkR0 1- e R2 0 I Oxn j ~ MIn in' odd v(T) mlo m'=even ka t sin (- 2 k22 -m=odd) (1 - '2 2 ) (A. 38a) v(T) Vto int ka~ k2a2 (1-2 2 in 7T -jz I m'=even v(T) VitO mlt (in = even) '2 2 in IT (A. 38b) m 193

THE UNIVERSITY OF MICHIGAN 6633-1-F In terms of I(R) the power received W is mOJ r W 2 mo (A. 39) r 2 In 7 MO m=1 m The remaining problem is the determination of Wt. The procedures for calculations are omitted and part of the results will be cited (Kwon 1965, Section IV). 1 (T)2 (P ) t 47r/ 6 mnn LZ mO m mm In m= 1 ^ ^ nm(T) (T) +(1-6) v VV nO Im(P ) (A. 40) mn M=1 n=1 mO nO mnj m=1 n=1 where a ka mm (a4b) ( IT) (t )2 R 1(A. 41a) 4 at kt (2) P =2(4 )( ()2 R() (A. 41b) mm at bt 7 mn and 194

THE UNIVERSITY OF MICHIGAN 6633-1-F bt at R( mm 7f bt dc (- 7r-a) at t kat 2 2 ^p — 2X + a2 (1 2 2 2 2 -m 7r /m 7r2 1 k - m- 2 ) (r -X) cos mX+ (1+ m22 )-sinm k2 2 2 2 m k an k aj (For m=n) (For m = n) (A. 41c) b at b R(2) = da ( t 7-a mn a o o ka j-/ x +a e 12 2+ iX +a C (m+n) sin (m-n) kaO kat Xsin 2 r n( m2 2 1TI -rl (m - n) X (m+ n) X r n cos -- -- sin - 2 1+( llJS(L 2,2 kat kat m+n J (m fn) (A. 41d) I (P ) = imaginary part of P m mn mn If the transmitting and the receiving guides are ordinary waveguides which accommodate only one propagating mode, one can set m = m' = 1 in eqs. (A. 37), (A. 38) and (A. 39), and the summations become unnecessary. The expressions for Wt also become w 195

THE UNIVERSITY OF 6633-1-F W = 1 v(T)2 I (P ) it 4,7rw 10 m[ 10 MICHIGAN 1 (T)12 4 at 4 r Iv ( (-L-) 4irwiA 10 ab -b ' ka 2 (7 ) 71 ( 1) m(R 11 (A. 42) If one further assumes at= ar, bt =br, the following formulas for coupling are obtained fora= - a = 0, and= 2 2 For =: 2 32 c 2 2 7r 1 1 1 2 kb) sin(kb sin ( ka sin 'lb k2 i m(1) 2 2 7 o m 11 (A. 43) for a=0: 2 C= 2 7r 1 1 (51Ro) (kR )2 O b 1 R (R.(1) o I (R m 11 2 02 1 (A. 44) For 3 = 0 32 — t 2 7r 1 22 k R o 1 I 11 m 11 sin2 s)* (ka Cs) 1 2 ( r c' 71 sin (T sin a) -C( sin a) 2 7 (A. 45) These couplings are plotted and compared with experimental results in the Figs. 4-6. m 196

THE UNIVERSITY OF MICHIGAN 6633-1-F A. 3 The Coupling of Two E-Sectoral Horn-Connected Aperture Antennas (Small Flare Angle Approximation) If the flare angle 20 is much smaller than 7r/2 the following approximations are justified for equation (A. 7). a y-0 r. PYl p 20 b (Y = y+ ) p.0 (A. 46) b) The reflection can be ignored, that is, (see eq. 9) for the receiving horn D B nq '- 0 mp C * ' A nq mp c) One can consider only those p's which satisfy: 20 <m p. Henceforth let 20 (A. 47) (A. 48) p 20 In other words those modes whose turning points are in the horn can be ignored, for they only contribute to the non-propagating wave in the receiving waveguide. Imposing the condition of continuity of Et and at z = 0 (horn-guide t t r junction) and z = 0 (aperture) of the receiving system, and (aperture) of the receiving system, and using the approximations described above, one gets, from eqs. (A. 2), (A. 4) and (A. 7), 197

I THE UNIVERSITY 6633-1-F CO OD p1 mfpl vM H1 M (1p) lye M = 1P = 1 n = nq q + y jwA l2 A H (1), I- M l p=O M M OF MICHIGAN 7 e i (x y) =1 nq nq:. 2 (r2 - nirx1 r P ) Cos a r 2 (r) 2 (,y y +(p )) Si m 1 p~ry1 r a r p7ry1 Cosb r n=1 jD 1 nir ()2 () ~~ (-) C H (y) ( y 2+(p r)2 p(r) a 2e0 nq~ m 1) 1 a r sin (A. 49a) 198

THE UNIVERSITY OF MICHIGAN 6633 -1-F m=1 p-O pnm n=o OD(R)t ii q = 1 OD vco 2 2 r 2 1~x ly ) A H (7y y + (p ) sin Cos " im mP p~ mI 1 ar b m=1 p=OPr r ZD O l(m7r)(r 2AMHr11'W2yr p(r) a 20)AM m p (- ~ ar b L j po I OD 0O (1)' 2 r2 m7rx1 q7ry1~ +wz C H + (p )Cos, sin n nq~ 0qn 1 ar bri n=O q=l (A. 49b) 199

THE UNIVERSITY OF MICHIGAN 6633-1-F 00O H )2 mirx1 pry1 x^ 2 H (1)( 2.( )pY xZ L mpH (ymY +2(P 2) sin " cos m Mp m 2 a b m=1 p=0 D OD, H(1) 2c m TX1 p7ry 4- Z ) C )'A H (rY (P)cos2 sin (r) a 20 Mp ( 2' a b m=1 p0O2 r o P r r D 00 (1)' 2 (, m7x q7ry +jWE-[YC H (y y+(P) cos a sinn nq ( n 1 a b n=Oq= q r r (BA (2)A (A)dSA-( B)B B dB A A.). (2 ()dsB (A. 50) As mentioned previously lack of orthogonality of mode-functions with respect to y make it impossible to calculate the coefficients A and C mp nq However, under the present small angle approximation, one may be justified to assume: 200

THE UNIVERSITY 6633-1-F OF MICHIGAN q7ry1 Cos b dy sin b, q0 q=0 b o y 2) f( 2+p12) f( +p172) Jb Jo 2 PrYl cos b. 2 PXrYI sin b (A. 51) and Now, multiplying x H() p ( 2/ (p(r)2) (y +(2) m7rx1 sin a r PiYp cos b b r (2)' 2 (r2;' m7rxl P7ry1 yH +( ) cos ). sin - on to (A. 50) and subsequently integrating O 2 a b p r r over the receiving aperture, on e gets, in view of the approximation (A. 51) (1) 2 ) mp (2 a H(m,p) y a mn r and r (2) C j2 mp mp We a H'(m,p) mrL + (m7r) (H E) L a 3 20 2 o y m 1 p(r) p2 J(1) mp H(mp) M(m, p) H'(m, p) I (A. 52) where - - w 201

I THE UNIVERSITY THEUNIERSTYOF MICHIGAN 6633-1-F (1) J MP - I c Id b r r 2 BA dy H (2) (/Y, 2+ ( (r))2 2( VI m 2 -i(2) A (P)dA dy H (2)1(Y, Y2 ((r))2) p2 2) MP 1 2 b r F2 J b r, U 2 rA~.J y-x p7ry1 m7rx 1 Os sn dx b a 1 r r (A. 53) a piry1, mlrxI An- I dx __co ___ ~1lb I A1 a r j0r (A. 54) (A. 55) P: PA)MVA (A) dSA b r H(m, p) = 2.22 1+ 1 2 b 2 202

I THE UNIVERSITY OF MICHIGAN 6633-1-F b r H'(m, p)= 2 L(pmm ' yy +(p(2)2 2 p r b r M(m,p)=I 2H (1) ( +(p() ), (y y2+(p ))(sin " ) dy y - p r 2 (A. 57) In the above calculations, the approximations given in eqs. (A. 16) to (A. 19) are used. Since A and C are known, the calculations of the amplitudes mp nq (R) (R) V( and Ip from eqs. (A. 49) are a matter of algebra. Assuming the field of mp nap transmitting aperture is given by: - (2) ' 2 (t) 2 Et (pt) = cos 0 H 1 ('Y1 +(2 ) ) sin a (A. 58) One can obtain the following results for coupling (Kwon 1965 Section VII). i) a 0O a) Denoting C ' )as the parts of the coupling which excite TE mode in the receiving waveguide, one has: receiving waveguide, one has: I -- 203

THE UNIVERSITY OF MICHIGAN 6633-1-F (MO) a - ka 2ka 2 (m0) 1 p2 (t) _ ________ c - 16r (-0-) a m2 12 2 mr) 7r bo) t (am (y R ) r r m o M (y b ()) M01(mbr) 1 c Ymbr ) t 2 s i2 2 C (- cos) s sin sin(a - ) L 7 J 2i 2 v (,y b sina) Hcc (7Y b sin(a-/3),0 c 1 t m r C ka 2 C(-r cos(a-0 ) L mr ka( mr cos (S(7t s( cos(a-3))J (m = odd) (m = even) (A. 59) (A. 60) In the above superscript zero on the "b" dimension indicates the dimension is on the horn aperture to distinguish from the waveguide dimension. b) The modal coupling C(0 2n+ 1) which excites only TM 2n1 mode in the receiving guide is: receiving guiide is: 204

THE UNIVERSITY OF 6633-1-F MICHIGAN (0, 2n+1) = 16r C =167r (-) b r MO (kb( ) 2r+ 1) 1 r 2 H (kb, 2r+l1) C r k (-\ (t) a2 at (t) P 2 a r 1 (kar) 1 (kR) o ka 7T 1 r Alt - 2i 2 W c os a) sin IF 7r 12 V c(71 btsinQa) IH (kb sin(a-i), 2n+ 1)I sin( — cos (- )) (A. 61) 205

T1 ) NIVERSTYOF 6633-1F MICHI1GAN ij) a= O a C = ~64ir() b I (t) 1 k a~ n a )2(Y t ("y (r R ka 2 kat 2 (k 7T 0 (yb) 1 ka 2 H(Yr bO) W a=0 c a': sin, 0)silka2 ~~C~y sn (, )P(If) k ~tC S ( b si 3,)si 3cos (3C( MTrcos. + c.yCC(b sin 3,0) P (f) 12 kym (yn i (30ri 3cs( (0 5( ka 2 \~*~fCC(b~i(3~0Ps( I. — y sinP(3,0 sin (3coS C(3S(c rka2 + ~ ~ ~ c yc * pbn sin(3,0) sin (cos (3(losr (in(,b i 0 P()- evenm(A 62 206

THE UNIVERSITY OF MICHIGAN 6633-1-F (0, 2n+ 1) 64 C = 647r a r b(0) r (t) (1 2 2 k at (t) P 2 ( ) a r 1 (ka ) r 1 (kR)4 O ka 2 t) r r bka -12 2 C(-cosa)j v (m bt) L- ja=0C 'Ic 1 'Xcs ka os12 leJ (kb sin,, 2n+l) sin/3. 2qr+l)cos( — cos +r 2 r 2 +4Y C( kb sin A,2r+l)sin q(3)+SY (kb sin3, 2+l)cos( ycos 3) r 2 2 2] kat See ea. A. 36 for definition C(- cos a), etc. 7r (A. 63) Several function forms which have been used are now defined as follows b(0) m r rt-L 2 M01( m (r (0) b b0() m r 2 Yl(T)dt Y (T)dt 1 0 (A. 64) where: (T 2 + (r P 2) *' m l 207

THE UNIVERSITY 6633-1-F OF MICHIGAN (0) M0 1 (kb r J 2n+1). (0) kb r 102 2 n kb (r) k0 r 2 (2n+ 1) iit 2 Jn,(T) (Cos (r) Jo kb Y I(T)(Cos 2n+ )it) dt n kb (r) n kb0 lo (A. 65) where: n' (2n+ 1) At 20 (0) r H (Iy br,0) = H C (kb r, 2 n+1),yb m- r 222 3 fJ[Joc)]2+LY (T)]2dt (A.(E kb i2 I,(T)] + [Y IT] (Co (cS ~)2 dt (A. 6 i 1t 2 k T cos ( —t sin a) dt (A. 6 0 ly1 56) 37) ii ('y b sin a) ci1t 38) 208

THE UNIVERSITY OF 6633-1-F MICHIGAN y1 bt t 2 v (7 =t) 0 H(2)(T) T dt T (A. 69) H (y b sin(a -3), 0) m r b m r 2 (2) k = f H0 (T) cos (- t sin (a - ) ) dt A Tm (A. 70) kb r (cc 2 (2) (2n+ 1)7lt H (kb sin(a-), 2n+ 1) = H(2 (T)cos sin (tsin(a - ) dt 0 r (A. 71) J (Y: b sin 3,0) =ReIH ( b sinI)] m r e m r i (A. 72) (Re: Real part of ) (I: Imaginary part of) YC (my b sin, 0) = I HC( b sin 3) in r m m r (A. 73) b mi r JtC(S b sin i3, 0)= i r k tJ (T) sin ( tsin ) dt nM (A. 74) i 209

THE UNIVERSITY OF MICHIGAN 6633-1-F Yts (,y b sinf3,0) = mr 'cc J (kbr sing3,2n +1) Y c (kb sing3,2n +1) r Ites J (kbr sing3,2n+ 1) r 0 r k ty (T) sin (-~tsin 3) dt 7m = Re [HCc(kb sing3, 2n + 1)] = Im LHIC (kbr sing/, 2n + 1)] (A. 75) (A. 76) (A. 77) kb =2 I ~ (2n+ 1) 7rt tJin, (T) Cos sinJ.L(t sin 3) dt (A. rr.78) (A. 79) k-a r P (in) = Cos 93C (- Cos /3) c mIr 1.2 L( mir' b-in 13 -4 rr cos3C(- Cos 3) - ka sin (-Cos 13)] inir r 2 k2a2 1 - r Co2 2 2 in ir (A. 80) 210

THE UNIVERSITY OF MICHIGAN -- - 6633-1-F ka P (m) = cos 3'S( r cos 3) C inilr - ka ka ka - 4 ) os S(-cos +ka cos (- co -1 insin2r mk. r 2k 4..2..2 K a 2 1 22 1 - 2 2cs,B m 7r (A. 81) ka q(3) = 1- (l+sin2J cosl + 4 ~ts n - cosP ~ k ka cos (-r cos ) 2 (A. 82) bt -7T W' at W = dcr 0 O sin( a ( ( a - CT) r a 2p p t a ka ka ^0 kat \/ 2 2 t sin (-2 +a2) sinX -- x2 + a (A. 83) C ) for the two identical horns are calculated and compared with experiments. 211

THE UNIVERSITY OF MICHIGAN 6633-1-F A. 4 Coupling of H-Sectoral Horn In the present section the following model is considered. i) The incident field in the T 1-region consists only of the TE 0-mode. ii) The flare-angles of horns of the transmitting and receiving structure are large. iii) The distance RH between two antennas is several times greater than the largest of any of the lined dimensions of the apertures. iv) The excitation and coupling of other than TE m-modes are ignored. This is a reasonable assumption in view of the fact that the incident mode is a single TE 10-mode. Due to the assumption iv) and the characteristics of the H-sectoral horn (i. e., b= constant), the only non-vanishing matrix components in eqs. (A. 11) are It if E and H. Then, eq. (A. 11) becomes: (omitting double prime on I and V). qp qp dl y + jut 2 0- = HE. In m (A. 84) dV +- Z IE tV (A. 85) dz mO ) mO, m' (A. 85) From the above 1st order differential equations, the following 2nd order differential equations are obtained: 212

THE UNIVERSITY OF MICHIGAN 6633-1-F d 2 (m - Em0, m'0 Em', mO) VmO dz m'(#m) m( m) m'(#m) dm0 m'O dz m'O m(m) m'm) m;(*m) P(Im'M m) It IT E, E.O mOm'O m'O,pO pO (A. 86) are given by E m' and H are given by: mO, mIO mO, m'O I Im+ml E m 0= (-1) H = - E mO,nm'O mO, m'O 2mm' a' '2 2 a m -m (A. 87) (A. 88) dz By successive approximation the above eq. (A. 86) can be solved up to the desired degree of accuracy (Kwon, 1965, Section IX). After calculating V mo in the T2- and R2-region, one can proceed exactly the same as in the previous two cases. The final results for =a and a = 0 are given: 2 213

THE UNI-VERSITY OF MICHIGAN 6633-1-F For: 7 2 C= 32 ar 1 1 2 kbt s in (-) Ir2 brkR 2 (kat) 2 2 kbr sin Co kra L r.T -dd q= 1 kra r C(-~ sing1) q q q=l1 even q= 2 odd zoo0 nmq ka" ka C(-MI sinl3)C (-q sin/3) mq ka ka r. )S(r S (- sinl3 ( sinl3) M7T q7r e e z m=2 mq I mq +d even e o mq ka ka S(-' sing3) S(-r sin/3) mq odd q=2 ka"I ka o e m7r Qg7r mq mqj (A. 89) w 214

I THE UNIVERSITY OF MICHIGAN 6633-1-F For a=0 2 a C= 2 (b) 7r r 2 R2 o 1 (kR )2 o 1 (ka) 1 Q I[C2 I S1H2] [o - q3 odd q=l m=3 0 0 z00 m q mq mq een +q= q=2 m=2 E E ee m q mq mq odd q=l 1 m=2 E O zeo m q mq mq odd moe m=mq O E m q mq, for 0 see equation (A. 38a) m (A. 90) where 00oo Zmq =g g mq m q - 2 (L(vm) Cm gq 2 2 v -v m 1 - cos * g T(3) (t(0Cm I 1 m 2 2 2 2 m 1 m2 - 1 (A. 91) 215

THE UNIVERSITY OF MICHIGAN 6633-1-F e eF C mg mq = mgq -2L v) 22 in - "1 (3) (0) in 1 - COS ~gq Tm (t ), - q n2 L2 2 2 vm "1v in P 2- lBj (A. 92) e o F,C Igq (3) (0) [C~1m mq VnV1 q in 2 in2 1 +(1-6 ) C G() 2 q + qi in in 2 2 2 2 oe Cm gq 3 Zm = 2M(m 2 +sin ~gqGm~ mq V2 2 1 qG (3) q2 If q -1j) (A. 93) C in 2 ml1 +C G(') 2 m m 2 2 vmlV 2 mi2 1. V =(- -) + —4in in 1 4 a r L G(2) qG(3) CG q 2 '2 2 m'(:Lin) (in -in2) (A.-94) (A. 95) 216

THE UNIVERSITY OF MICHIGAN 6633-1-F C =(-1) nil1 —2 m LM 2-1 z(~ —l 2 p (M#-~1) 2_ (2_2) (p -1) )f (A. 96) (A. 97) 2 mi2 ml 4 1 (=~ m) '2 m (m -in )2 v~'ri (t2) (1) v in d1 L mi - 2 (3) 1ta G (t) (t t~2 (A. 98) (A. 99) J (t V in Ft (A. 100) = 1 W 217

THE UNIVERSITY TH NVEST OF M I C HI GA N qmmmm 6633-1-F f 1 m (t~ r)j) p(# m ~-1) 4=-(t - t )) r r 22 p 2-1 I 2 vI1 2 t(0) r in 2 2 n -p 1 2 2 Vm 1 (A. 101) (A. 102) 1 t r Ii =1 I (l)m'+1 2m' ml( 1 2 2 P2 mni -1 I (A. 103) m'f=odd i I I in ka~ C (ml ) (A. 104) (A.- 105) (A. 106) I Im kat SMI -even(#~-0)in n7 218

I THE UNIVERSITY THEUNIERSTYOF MICHIGAN 6633-1-F sin4~ (0) tr iv r )-o ov( - n- n L(vn) = M(vn) = dt L v jt=tr Cos ~to J M + in ~0 iI(t(0~)) n n r (A. 107) (A. 108) (1) Q=R + 11 4m 2(1) 6..LI2 2 2 2 2 Rmm m=2 (m - 1) ('JA "iA +2(1-6 (-I)~' 2m 1R2 l m=2 m 2-1 A.2 -A I m 1 mn z ~(i ()m+n m= =2 4mn - _ _ _ _ _ _ _ (m2_ ) 2 _)(2 A2) A2_t2) mi1ni1 mn 219

THE UNIVERSITY OF MICHIGAN 6633-1-F A. 5 Conclusions The objective for the present analysis was to find relatively simple and manageable formulas having reasonable accuracy for the coupling between two planar aperture antennas spaced up to a few wave lengths apart. This objective has been accomplished for the case of waveguide terminated aperture antennas. Eqs. (A. 43) and (A. 44) predict experimental results up to a distance of one wavelength or slightly less. This indicates that the assumptions made in Eqs. (A. 16) through (A. 19) are justifiable for power calculations. The significance of the formulas (A. 43) and (A. 44) is that expressions for the far-field (a = 2 ) and near-field (a = 0) coupling are shown to be two special 2 aspects of a unified general treatment, which cannot be accomplished through the use of the directivity concept. The near-field coupling formula for two ordinary waveguide terminated antennas (eq. (A. 44) ) is as far as is known the first to be reported. In the case of the E-Sectoral horn, some accuracy had to be sacrificed to keep the final results within the range of simple computer programming. However, eqs. (A. 59) through (A. 63) predict experimental results accurately in most significant aspects, e. g. power level and points of maxima and minima. For H-Sectoral horns the analysis must be modified in order to adapt to a computer program; this remains to be done. The present analysis can also be applied to higher mode coupling. There is some dependence of the modes of the aperture fields upon the relative location of two antennas. These relations are stated in the form of theorems. 220

THE UNIVERSITY OF MICHIGAN 6633-1-F APPENDIX B COUPLING DUE TO SCATTERERS B. 1 Coupling Due to An Edge Consider now the effect of a long edge upon the coupling between two antennas. Such an edge could be realized in practice by the intersection of a perfectly conducting ground plane and a flush-mounted radome, canopy or observation windown The length of the edge is assumed finite, but the antennas are close enough to the edge that the dominant return arises from specular scattering; thus the end effects are ignored. A condition upon the length of the edge will be given later. Since the main consideration is the return from the specular regions of the edge, there is essentially one scattering center, and the expression (5. 42) can be used for the coupling. All that remains to do is to compute S 2 The first step is to compute the reflection coefficient for a plane wave travelling along a perfectly conducting semi-infinite face to an infinite wedge and incident obliquely toward the edge. Consider a perfectly conducting wedge, the edge of which lies along the z axis of a cylindrical polar coordinate system (p, 0, z). The faces of the wedge will be given by 0 = 0 and 0 = 0 (where 0 is the exterior wedge angle). The incident radiation will be a plane wave travelling along the surface 0 =0, obliquely toward the edge with the angle between the direction of propagation and z-axis being denoted by a. The incident electric intensity is given by A. a A- jk (-xsina+zcosa) (B.1) E. ye^" ^ (B.I1) For the problem at hand, the electromagnetic field can be represented in terms of a two-dimensional scalar potential V(p, 0) by the relationships 221

THE UNIVERSITY OF MICHIGAN 6633-1-F — jkzcos a E ea Vx (Z V) = - e cos [kVsin - jVVcosa] (B. 2) where V is a solution of the scalar wave equation V+k'2V = 0 (B.3) with k' = k sin a. = - sin a (B. 4) The boundary condition that the tangential component of electric intensity vanishes on the faces of the wedge, gives the condition av -- = O, for 0=0 and =0 (B. 5) an Where n- represents the normal derivative. The incident field can be represented in terms of the potential V. given by V. = E e +j, E =+j/k' (B.6) 1 0 0 a(v.+v ) The problem reduces to finding the scattered scalar V such that = 0 s an on the faces of the wedge, where in particular the scattered field on the back scattered direction (i. e. on the face 0 = 0) is required. Using the result of Oberhettinger (1954), (equation (29) therein) the appropriate Green's function is 222

THE UNIVERSITY OF MICHIGAN 6633-1-F G(p, p') = 2 f 1(k' 1p-p ) 2 si0 2 k'(p+p + 2pp' cosh x) - '2dx' (B. 7) Jo (cosh- - cos -- which gives the total field at a point (p, 0 = 0) on the wedge due to a line source at (p, 0- 0). To obtain the expression for a plane wave incident on the edge, __Pk' multiply expression (B. 7) by jkp ' and take the limit as p' -4 oo. This yields the following expression 2 - sin 2r OD 2A Jkp - __ exp (-jk'p cosh x) dx (B.8) 1o (cosh - cos ) For present purposes the factor A need not be specified. For an incident field of the form given by expression (B. 6), the scattered field is given by the second term of expression (B. 8) multiplied by Eo/(2A), i. e. = sin 7r exp(-jk'pcoshx) (B.9) s dn B) 9 0 sh - - cos \ 0 P P I 223

THE UNIVERSITY OF MICHIGAN 6633-1-F At a sufficiently long distance away from the edge (i. e. k'p~>> 1), the integral expression (B. 9) can be evaluated by the method of stationary phase, giving E 2 j(k'p + r/4) Ve - sin D 2 (B. 1) 0 C (cos 7r Inserting this in the first equation of (B. 2) gives the following expression for the back-scattered field E = y SO exp [-jk'x -jkz cos a] (B. 11) where 2 sin R:= __ e (B. 12) 2 (1 -cosiL ) For the particular case of a right-angled wedge (0 - 3T/2,), the reflection factor R has the value RI= / (B.13) Comparing the direction of propagation of the reflected wave to that of the incident wave, it is seen that the angles formed by the propagation vectors and the edge are the same. Hence, if a transmitter and receiver are a finite distance from the edge, the energy which is reflected from the edge to the receiver arises from a small scattering center on the edge. This scattering center is such that 224

THE UNIVERSITY OF MICHIGAN 6633-1-F the line form the transmitter to the scattering center, and the line from the receiver to the scattering center from the same angle with the edge, (see Fig. B-l). Thus the line TS and SR forms the ray path of the field directed from the transmitter to the receiver by way of the edge. In order to compute the scattering matrix term S, one must take into account the azimuthal curvature of the incident wave front arising from the transmitter. Consider the adjacent ray path going from transmitter to S' to receiver R' in Fig. B-l). Because of the curvature of the incident wave front, the energy incident on the segment of edge between S and S' will be spread out between the lines RS and R'S'. The ratio of the energy density between R and R' to the energy density between S and S' is Rt/(R +R ). Hence the electric intensity R must be t t r multiplied by[Rt/(R +R )1/2 to take into account the curvature of the incident wave front. Thus S is found by multiplying expression (B. 11) by LRt/(Rt+R r)1/2 giving I R __ v =;k R (B. 14) where k' = k sin a, and p = R sin a r r For a right-angled wedge, the above reduces to 2 2 1 v = (: ' -(B. 15) Sw vv 277r (R+R 2 (B.15) t r sin a Hence from expression (5. 42) the coupling due to a right-angled wedge is given by the relation '......-.. - *-. i ',.^.,d n ''.,:. 25:

THE UNIVERSITY 6633-1-F S Edge -4 - a OF MICHIGAN ------ / / / R' T FIG. B-i: SCATTERING CENTER OF THE EDGE 226

THE UNIVERSITY OF MICHIGAN 6633-1-F R 2R C = ()2 C(R V ' ) (B. 16) s t 277r(R +R )kR sin2 a ~ t r r As an example, let the two antennas be the same distance I from the edge, 2 2 -1/2 2 2 and a distance 2d apart. Then sina= ( +d ) /, and R = R = 2 + d 1 t r1 Further, let the antennas be mounted so that their E-planes are directed towards and away from, respectively, the scattering center of the edge: (See Fig. B-2) C(Ro, t' Or ) = C(R 0, 0) (B. 17) where C (R, 0, 0) is the direct coupling between the two antenna a distance R apart, and oriented so that their E-planes lie along the line joining their centers. The coupling due to the edge is given by 1 Ro 2 1 c = 7k ( ) C(R, O, 0) = 2 C( 0, O0) +d1 27r7rk +d1 -- F 22 R 27 [C +d1)] / ( )2 C(R,, 0) (B. 18) 2 The number (27) is (-16. 3 db), so that C can be readily found from graphs available in this report. Note that the coupling is independent of R, which can o be chosen for convenience. As an example of this particular configuration, if the antennas are H-sectoral horns, the results of section 4.5. 1 may be used. For a frequency of 10 GHz and a spacing R of 45. 7 cm, the coupling for 0 = = 0 is given in Fig. C-4c as o r r -39.3 db. Several values of C are tabulated below: S 227

THE Edge UNIVERSITY OF 6633-1-F S MICHIGAN Et FIG. B-2: ORIENTATION OF ANTENNAS WITH RESPECT TO AN EDGE m 228

THE UNIVERSITY OF MICHIGAN 6633-1-F 2d1 C 45.7 cm 22.8 cm -70.9 db 45.7cm 11.4cm -63.7db 22.8 cm 22.8 cm -75.7 db 22.8 cm 11.4 cm -67.9 db I- ~~~...- II Table B-1: Coupling level of two H-sectoral horns due to the presence of a right-angled wedge. B. 2 Coupling Due to the Leading Edge of a Wing B. 2. 1 Part I: Geometric Optics Contribution for Antennas in Near Zone The geometric optics scattering from the leading edge of a wing, taken to extend at right angles to the perfectly conducting common ground plane of two antennas, will now be considered. In the present analysis the termination of the wing will be neglected and taken to be infinite in extent; the contribution that arises from the scattering center at the base of the wing is desired. The leading edge of the wing will be taken to be a portion of a semi-infinite wedge. The associated geometry and orientations of the antennas with respect to the wedge on the ground plane is shown in Fig. B-3. The wing represented by the wedge extends out of the diagram. Since the scattered return from the wing arises from a small region around the base, there is esentally a local scattering center. Eq. (5. 42) may be used, namely tR 2 2 (B. 19) 229

THE UNIVERSITY OF MICHIGAN 6633-1-F T FIG. B-3: GEOMETRY AND ORIENTATIONS OF ANTENNAS WITH RESPECT TO WEDGE ON THE GROUND PLANE 230

THE UNIVERSITY OF MICHIGAN 6633-1-F where C(R, 0, 0t) is the direct coupling between the two antennas a distance R apart with orientation as indicated by Fig. 5-1. The basic problem is to obtain the scattering matrix term S. For a plane wave incident upon a wedge (exterior wedge half-angle 0 ) and polarized parallel to the edge, the scattered field at a sufficient number of wavelengths away from the edge, is given by Crispin, et al (1959): E -jkR -j/4 E e kR (A B 2 (B. 20) where 7r 2 A = cos (at+a + cos 2 (B. 21) Br )] 2 B= pcos - (a -a) -cos (B.22) 2 0 r 2'0 This results holds, provided that 20 > 7r+ a +a and 20 > 20 1 -7r+ ar -t Physically, this means that the position of receiver and transmitter is such that the scattered energy incident upon the receiver arises only from the edge, and not from specular scattering from either face of the wedge. I 231

THE UNIVERSITY OF MICHIGAN 6633-1-F For a thin wedge o ~ 7r, in which case r t r t A ccos ( ) B cos ( ) (B.23) 2 2 However, in order to obtain the scattering matrix term (S ) one must take into account the effect of curvature of the incident wave front (in the vertical plane, i. e. perpendicular to ground plane). Expression (B. 20) must be multiplied by a factor [Rt/(Rt+R )/2. It follows that 2 27T R t 1 1 22 - R )(sin ] (B.24) vv kRr Rr+R t o A 2 -which reduces to RS2 (B. 25) = 87rkR (R +Rt) (A B ) for a thin wedge, 01 r. The coupling is thus given by R2 0) 1 1 12 C 0 1 1 + =(2 C~ (R (B. 26) s R R ) 7rk(R +R ) A B o r' (B26) rt t r In order to illustrate the use of this result, consider two H-sectoral horns, both at the same distance from the wing (i. e. R =Rt). Consider a frequency of r t 10 GHz, and let R = 45. 3 cm. The direct coupling term C(R, 0, 0t) can be obtained from known results. Let both antennas be oriented in the direction of the edge of the wing, i.e. such that 0r = 0~ 0t = 1800. Then suppose the direct coupling C(R 0, 0) has the value 2 x 10-4 and c~~~OupigC I n

THE UNIVERSITY OF MICHIGAN 6633-1-F ( 1 1 1 2 C = ( 0) 1 (11 + )2 s R 167rkRt A B Values of C are computed for various values of the parameters ar, atv and Rt and are given in the table below Rt a a A B C t r s 7.25t" 60~ 0.866.866 57 db 7.25" 90~ 0.707.707 55 db 14.55" 60~ 0.866.866 66 db Table B-2 Coupling Due to the Leading Edge of a Wing (Geometric Optics Cobtribution) B. 2.2 Part II: Contribution from the Traveling Wave Launched from the Wing Tip In the previous section, the termination of the leading edge at the wing tip was neglected. Here consideration will be taken of this termination and the effect on the coupling. The dominant currents producing the field scattered by the edge are edge currents. Besides those that are directly produced by the incident wave, there is a current that is alunched at the wing tip flowing along the edge inwards towards the fuselage (the ground plane of the antennas). This is reflected by the ground plane and a current flows outwards towards the tip. The total current (sum of the currents produced directly by the incident wave, and the traveling wave type currents flowing along the edge) is zero at the wing tip. 233

THE UNIVERSITY OF MICHIGAN 6633-1-F To obtain good estimates of the coupling, the leading edge will be modeled by a thin wire of finite length, for which theoretical results can be obtained. The main emphasis will be placed upon the current traveling from tip in toward the ground plane. The first step will be to represent the far field pattern of the transmitting antenna given by expression (5. 3) in terms of two Hertz vectors. The transmitting antenna coordinate system described in Fig. 5-2 will be used. The electric intensity can be expressed in terms of two potentials Xr and 7r ' by the relation. Et = VV(7Tr) - V - j Vx (r) (B. 27) 2 a In particular, E t = (k +-) 7r (B. 28) For the far field one can set -jkr 7r~- g(0,) (B. 29) r Thus on neglecting higher order terms in r, expression (B. 28) becomes 2-jkr 2 E 2k - sin 0g (0,) (B.30) zt r Equating this to the z component of Eq. (5.3), one obtains g(, 0) = 2s-i[f (O ) (B. 31) 234

THE UNIVERSITY OF MICHIGAN 6633-1-F Consider a vertical wire of length i and radius a, extending out of the ground plane at the point ( P pt.0 z = 0). The component of the incident field parallel to the wire (E ) is represented in terms of the scalar potential zt -jkr r- g(e, 0 t) From henceforth the variable 0 will be dropped and the far field component *f (0 0t) will be written strictly as f (0). The next step will be to determine the current distribution produced on the wire by the incident wave. To simplify the analysis, the geometry of wire, ground plane, and transmitting antenna on the ground plane will be replaced by the equivalent geometry of the wire in free space extending a length I above and below the z = 0 plane, with the radiation field of the transmitting antenna being symmetrical with regards to the plane z = 0. (i. e. f (0) = ft(r -0) ). t t The problem of finding the current distribution on the wire can be treated in a similar manner as the analysis used by Vainshtein [1959] to obtain the current A distribution on a wire of finite length with a plane wave incident. Let J = J z be the current on the wire. Then the component of the electric intensity (at a point outside the wire) can be represented in the following form 1,2 a 2 E +- (k + - ) A (B.32) z jk 2 z where -jkR 1 e A~ = r- JJ ds (B.33) z 4ir R '0 J The integral is over the surface of the wire, and R is the distance from a point of integration to the observation point. Set I 235

F THE UNIVERSITY OF MICHIGAN 6633-1-F 27raJ = J 0 (B. 34) to be the total current per unit length of the wire. It is shown in Vainshtein (1959) that when the observation point approaches the surface of the wire, Eq. (B. 33) can be approximated by 47r A = ln o z a -In sgn(z - ) -jk z -| 0(W) e de (B. 35) for a thin wire. In addition, for a point on the wire o -jkz +jkz -jkrt(z) 4,r A =P e +Q e +S e o z o o o ire can be represented in the form The current on the wire can be represented in the form (B. 36) +-j jkz ' -jkr (z) J (z) = P(z)e + Q(z)e + S(z) e o (B. 37) where P(z), Q(z) and S(z) are slowly varying functions. The first two terms on the right-hand side of Eq. (B. 37) represent the currents flowing along the wire and the third term is the current generated by the incident field. The current must vanish at the end points giving rise to the conditions J (Q) = J (-) = 0 0 0 (B. 38) Equations (B. 35), (B. 36) and (B. 37) form an integral equation for the unknown factors P(z), Q(z), P and Q. Before investigating the solution, the appropriate 0 0 236

I I THE UNIVERSITY OF MICHIGAN 6633-1-F values of S(z) and S must be obtained. o Physically the third terms on the right-hand side of equations (B. 36) and (B. 37) correspond to values that would be obtained for a wire of infinite length. It follows from Eq. (B. 28), (B. 29) and (B. 30) that the potential A. for the incident field has the form -j f(t) -jkrt A eiz krtsine t t Since the tangential components of the total electric field must vanish on the surface of the wire, we have E. =-E and A =-A iz sz iz sz giving -jkrt A = e --- f(0e) (B. 40) sz krt sin t t Inserting this expression into the left-hand side of Eq. (B. 29) for the infinite wire gives the relation o -jkrt ______ 4r (t fe fln(2a sgn(z- - (J ejklz-Ild /2 0kp ta dg 0 0 t (B. 41) where J() = S() exp [-jkrt(gj. I 237

THE UNIVERSITY OF MICHIGAN 6633-1-F The relationship pt rt sin t (B. 42) is used, where pt is the distance of the transmitting antenna from the base of the wire on the ground plane (see Fig. B-4). To obtain the asymptotic expression for S(M), the integral in Eq. (B. 41) is divided into two parts, the ranges of integration being from z to oo, and -co to z respectively. Because of symmetry one can take z > 0. The first integral has the form - In 2 z) S'I -jkS(l + if) exp (-jk( - z) -jk /p+2 ) d S' -jkS(1 + -jkr 2t -ik(l+ cos e t +jk (1+ cost) S(z) e n() e dt, (B. 43) t a which can be reduced to the following form S() I yk (+ cose t)j (B.44) where = 0. 5772 (B. 45) The second integral has the form Z In 2(z -) '-jkS(-l +4)exp-jk(z-)+ Pt+ ]d (B. 46) (B. 46) I 238

THE UNIVERSITY 6633-1-F OF MICHIGAN - wire Ground Plane = n Z - UV T FIG. B-4: GEOMETRY OF WIRE PROTRUDING FROM THE GROUND PLANE 239

THE UNIVERSITY OF MICHIGAN ----- 6633-1-F To obtain an asymptotic evaluation of the integral, let the value of z be in the range | zj <. In addition it will be assumed that for the rangef z <., |S() <<k 1- - (B. 47) S(Z) p + z This assumption will be examined further to determine the requirements that are necessary on the pattern of the transmitting antenna. With assumption (B. 47), expression (B. 46) can be approximated by the relation -jkS(z) In 2(z -l)[-1 + 2 =exp -jk (z-+/p2+) d+2 a L t 2_J (B. 48) Setting a = -E + p 2 2 and C =-z+ + z2 (B 50) and employing the following relation 2 In (2(z-)) = In(-..) - ln (-) + In ( t..-), (B.51) a a a aa one obtains the following expression for relation (B. 48) w 240

THE UNIVERSITY OF MICHIGAN 6633-1-F -jkz -jk o n 2s -2jksds -2 -2jks jkS(z) eZ 2 e In e ds -2 In - je ds a a 0 /2 + ln( (- ) e d (B.52) s ooe a0 edb o t 0 Further reduction is achieved by employing the relation 2jk In 2s -2jks (-_ ) 2jk n e ds = ln ( a -ka Jo (B. 53) and integrating by parts the last two integrals in expression (B. 52). the following expression This yields -jk 2 a 2 k oo 0 -2jka e a da jka r e -j k da u 0 e o (aco +p) J o (B. 54) last integral in the above expression is of the order of (l/krt) and can be neglected In addition the following additional assumption will be made for the range z <(, ko or>>1 0 ' (B. 55) 241

THE UNIVERSITY OF MICHIGAN 6633-1-F If Pt<< i, then the requirement can be written in the following form kpt />>1 (B.56) t This inequality places a restriction upon the minimum distance between the transmitting antenna and the wire. With the above restriction expression (B. 54) can be approximated by the following -j2kr 2 +cr2 -j krt Pt o 0 -jkr S(z)e tin ( 2 ) (B. 57) yka (-cos O where cos t = (B. 59) t 2 2 Pt + Z Combining expressions (B. 41) and (B. 58), and inserting the resulting sum for the right-hand side of equation (B. 41), yields for S(z) 47 ) f(et) = 2S(z) In si] (B. 60) kp t L (Yka 2sin 0 giving j27r -2j - (B. 6) kpt i 0)o It L ka sin0t I I 242 ---

THE UNIVERSITY OF MICHIGAN 6633-1-F From (B. 36) and (B. 40) the following expression is derived for S S (z) 4 0 f(At) (B. 62) The remaining problem is to re-examine the inequality (B. 47). From expression (B. 61), this reduces to ia + ln( lie)L' cote <<t (B. 63) f et ykaosint < < 1+coset The assumption that the antenna pattern is sufficiently slowly varying, or more explicitly, that f(0 t) varies slowly over the distance of a wavelength on a circle of radius pt centered at the transmitter, yields I af 0 <<1 kN (B.64) The maximum value of the modulus of the second term on the left-hand side of ( ) t n c/ ~2 26 (B. 63) for the range 0 < cos0 I/ p+ S. is the order of, or less than /P l Using the restriction given by inequality (B. 56) it follows that this term is small compared to the right-hand side of inequality (B. 63). Under the given restrictions inequality (B. 63) is satisfied. The next step in the process is to obtain solutions for P(z) and Q(z). This involves obtaining solutions of the following equation. -jkz jkz -jkrt A (J, z) = P e + Q e S (B. 65) 243

THE UNIVERSITY OF MICHIGAN 6633-1-F where for convenience the operatorA(J, z) is defined by the relation (J (2z- ) sgn (z - ) d e( -d e oa d o IJ-dt (B. 66) and the appropriate representation of J (z) in terms of Q(z), P(z) etc. is given by relationship (B. 37). From Vainshtein (1959), one can directly obtain the realtions -z -jkz jk(21 - z) 'jkz 2(, A(Pgkz, z) =ng() Pz t- ( P-, Z) = g( z)O - ) P(z) + e In 2z- P'()dS a j-1 (B. 67) -jkz -jkz Q -k(z+21) jkz 2Q -z) A(Qej, z) = g(0) Q(z)ekz -g(z+i) Q(-l) e k(z+2 ) -ejkz a (B. 68) where g(s) = In s + e2jks e da (B. 69) yka To obtain the complete expression for the right-hand side of equation (B. 65) -jkrt the term A(e S, z) has to be considered. From the symmetry relation S(z) = S(-z), this term has the form: I 244

THE UNIVERSITY 6633-1-F OF MICHIGAN -jkrt A(e s, z) = -jkr - e S + d S'() - j-kS() ( 1 + ) oI ^ L /~Pt' -jk krt k 2(- z) e-jkz ( + z)ln - a a J (B. 71) To reduce this expression rt(Q) will be expanded about the point ~ = I, as follows rt =Rt + ( -) - +... Rt - + ~~ where R = P t pt/2 (B. 72) Then since the term S'(I) is small compared to kS, the integral in expression (B. 71) can be approximated by -j (k+w) S(I) e + ek(z+) 3~I {t ek(z-)J I -j(k+w) ( -) In (( z di a 00 -j(k+w) (e +) n 2(+ z) d a s (B. 73) (B. 74) where w = kI /Rt. 245

- THE UNIVERSITY OF MICHIGAN 6633-1-F Expression (B. 71) reduces to the form -jkrt -jkr -jkr F-ik(e - z) -jk(l x z) A g(eo Sz)=e g (-z)+e g+ (+ z+ j (B. 75) where g+(s) = j(k+w) e(k+w)s in 2- ej(k+w)a d (B. 76) s Substitute the sum of expressions (B. 67) (B. 68) and (B. 75) for the right-hand side of equation (B. 65). Then equating terms which have the coefficient ej and terms -jkz which have the coefficient e one obtains, respectively, the relations Z p 9 -2jkl P =g(0)P(z)+ Ina (z- ) P() de -g(z+l) Q(-)e e -jkQ -jkrt -S() e g+(z+Q) (B.77) Q = g(0) Q(z) - In 2_ ( - z) Q'(I) d - g(I - z) P(Q) e2 0 a z -jkQ -jkrt -S(f)e g (f -z) (B.78) -m 246

THE UNIVERSITY OF MICHIGAN 6633-1-F The above two integral equations for P(z) and Q(z) can be further reduced. However, due to symmetry, P(z) = Q(-z), and only one equation need be considered. Considering equation (B. 78), replace P(M) by Q(-I). Then set z =, yielding an expression for Q in terms of Q(M) and Q(-I). Substituting this expression for Q back into the original relation, one obtains the following reduced equation for Q(z): -2j -ln L- ( -z) Q'() d = Q(-)e2 - z) - g( -jkf -jkRt I v + S(l) e -z) - g+(0 (B.79) From Vainshtein (1959) the solution of this equation can be found, and is obtained in terms of two universal functions 0(z) and 0/+(z), by the following relation -jkl -jkRt -2j Q(z) = -+ ( -z) S() e - Q(-) 0 (-z) e (B. 80) and inserting this expression for Q(-() into expression (B. 80) giving -jkI -j kR { - z) (2t) e2jk l Q(z) = -S() e tl -z) - 2) (B. 82) 1 + 1(21) e The appropriate expression for P(z) can be obtained on using the relationship P(z) = Q(-z). The current is now completely determined. 1 247

THE UNIVERSITY OF MICHIGAN 6633-1-F The universal functions q(z) and q+(z) have the property that (O0) = +(0) = 1 (B. 83) Vainshtein (1959 and 1961) gives various approximate expressions for these functions, the accuracy of the approximation depending upon the smallness of the parameter ka. The particular approximate expression that will be used here is given by the following relations below, expressed in terms of a dimensionless variable x where x = kz/q (B. 84) and q = (ka)2 (B. 85) The expression for i(z) is given by ( z) = 2g(0) [l(x) - 0(q) I(x, q] (B. 86) where g(0) = In -_ = In ( ) (B. 87) 'vka -xt e dt 2I; i \(B. 88) I(x, q) e= -2jx e2jq I'(g) dg, (B. 89) X x 248

THE UNIVERSITY OF MICHIGAN 6633-1-F 2-g) and 0(q) = - 2g( (B. 90) 1.. 2 q 2g(0) 1+' q The expression for 0 (z) is similar, given by q+(z) = 2g+(0) I(x) - 0(q+ I(x, q+) (B. 91) where the parameter q+ in the expression f e(q+) and I(x, q ) is given by n + k(k+w) a2 q k(k+)a(B. 92) The factor g+(0) associated with expression (133) and the corresponding expression for (B. 90) is given by g+ (0) = n k(kw (B. 93) It is now possible to compute the scattered field produced by the wire. The scattered field produced by the currents on the wire (0 < z < i), will be computed first, with the effects of the ground plane (and the wire's image) ignored. The scattered field will then be decomposed into two main contributions, that arising from the currents at the base of the wire (near the ground plane) and that arising from the currents in the vicinity of the end of the wire. In the calculations of the coupling with an antenna on the common ground plane, the effects of the ground plane on the scattered field will then be taken into account. 249

THE UNIVERSITY OF MICHIGAN 6633-1-F With this in mind, the vector potential A associated with the currents on the wire (0 < z < 1) will be first considered. From Eq. (B. 33) the vector potential is given by jkR -()+k ()+() jkr (t)d I 0 - (B. 94) where R=x2 +y2 +(z- )2 (B. 95) The electric intensity of the scattered field is expressed in terms of the potential A = As by the relation 2 -jk E = (VVA) - V 'A (B. 96) Employing cylindrical polar coordinates (p, 0, z) with the z =0 plane being the ground plane, and the wire lying along the z-axis, the components of the electric field intensity are given by 2 a 22 jk E =( + k ) A (B. 97) z z 2 jkE (B. 98) To obtain the values of E and Ep on the plane z =0, the following relations are required 2 _-e 2 _jkr -e e- kp 2 [-j-I]- 1 (B.99) 8pEz R _ R3 kR (kR)2 - 250 I

THE UNIVERSITY OF MICHIGAN 6633-1-F L(k2 k -Rp2k22L+2 2 2 3 2~ ( 2 2 - 3 pkp z z=0 R kRp kpR2 (B. 100) The analysis will be performed for the case kR>> 1, with the following restriction placed upon the length i, 2k 2 p2 2 kp p + >>2~ (B. 101)' which implies that if i >p, then kp / >> 1. Thus with the above consideration, terms of the order of 1/(kR) and 52/(p2kR) can be neglected in expressions (B. 99) and (B. 100). The electric intensity is given by the approximation E 4O J(7)[r: ej2+ R] d] (B. 102) (B. 102) requires the computation of the following three integrals. -jk(R+) 2 [ R ] d, (B. 103) P2o ( -e e-jk(R -) + ] d (B. 104) IO 2 R R L R 251

THE UNIVERSITY OF MICHIGAN 6633-1-F e + S [ + ] d~ (B. 105) R All of these integrals are characterized by the integrand being composed of the product of a rapidly oscillating function times a slowly varying function. Hence, asymptotic techniques such as stationary phase techniques might be used. Integrals (B. 103) and (B. 104), however, have no stationary phase points in the range of integration and hence the dominant return arises from the end points. The third integral does possess a stationary point at z = 0. The contribution arising from the stationary phase point will be evaluated first, since physically it corresponds to the geometric optics contribution. The latter contribution at a distance pr from the wire is given by E - k(R + e S(0)d (B. 106) r Jo where R = 2(B. 107) 2P + 2 (B. 108) rt= t The integral can be evaluated by the stationary phase technique, yielding the expression k 1/2 -r jk(Pt+Pr)-3j7r/4 o Pt Pr I w 252

i THE UNIVERSITY OF MICHIGAN 6633-1-F which, on employing expression (B. 64), becomes - l p -)-1/2 n- -i1 -jk(pt+ Pr) -j7/4 =^,/t-kp p (p +p )0 ] e (B. 110) However this result has to be corrected to take into account the effect of the ground plane. This implies multiplying the right-hand side of expression (B. 110) by 2. Since the amplitude of the field incident at the base of the wire is ft(7r/2)/pt, expression (B. 110) can be normalized to correspond to an incident field of unit intensity. This yields the scattering matrix term S 2 = In^ ),2 (B. 111) w 4 kp (p,+pr *fea It is now possible to make a comparison between the two models of the wing edge. Eq. (B. 25) gives the scattering matrix coefficient when the edge of the wing is modeled by a half-plane, and equation (B. 111) when the leading edge is represented by a wire. The main difference is that the factor 47r2n ( )2 (B. 112) in Eq. (B. 111) corresponds to the factor (_ + 1 )2 (B. 113) In Eq. (B. 25) where A and B are given by relation (B. 23). Typical values of A and B that were given wre A=B=. 866, implying that the factor (B. 113) is the order of 5. For a wire radius such that ka22x101, factor (B. 112) is approximately 8. 253

THE UNIVERSITY OF MICHIGAN 6633-1-F Hence, the modeling of the edge of the wing by a wire, as far as the geometric return from the edge is concerned, yields a good estimate. The main reason for modeling the leading edge of the wing by a wire is to deduce the effect of the ends of the wing upon the coupling. Thus, attention will be returned to considering expressions (B. 103), (B. 104) and the contribution of expression (B. 105) arising from the vicinity of g =I. The evaluation of these integrals involves, mainly, the technique of integration by parts. Performing such a process, expression (B. 103) becomes P(M[sin0 rZ+cose -jkRr -jk P(O) -jkpr 2 e + 2 e z (B.114) -jkR (l+cosO ) jkpr The distance of the receiver (or observation point) from the end of the wing (or wire) is given by R = + 2 (B. 115) r r and the angle subtended at the end of the wire is sin0 = p/R (B. 116) Expression (B. 104) will be split up into two integrals, the ranges of integration being oo < z < I and - oo < z < 0. The second integral will be evaluated by integration by parts. Thus yields the following approximation for expression (B. 104). 254

THE UNIVERSITY OF MICHIGAN ----- 6633-1-F X ( e_ R- )g 0 +R) ' dr (B. 117) 1k- 2 -co jkpr The remaining contribution to consider is that arising from the end point = ~ of expression (B. 105). This is given by S() sine 2+cos0e -jk(Rt+R) r-r-e t r (B. 118) -jkR2 (cos + cos T) where Rt is the distance of the transmitter from the end of the wing, eT is the particular value of et given by Eq. (B. 59) with z= e, namely cos T =/Rt (B. 119) The combination of Eqs. (B. 114), (B. 117) and (B. 118) yields the contribution to the scattered field arising from the currents at the end of the wire, as follows E 47E r R2 e R Z+ pJ d (sine 29+cosO ) -jkrp(l) P jf e'k T r r e r PQ)e + SQ) e R...2. e [(+ cos Or) (cos er + cos e T)J (B. 120) In expression (B. 120) the dominant part arises from the integral. Physically, this represents the contribution from a current launched at the tip of the wire and traveling towards the ground plane. Radiation from such a traveling I 255

THE UNIVERSITY OF MICHIGAN 6633-1-F wave is endfire towards the ground plane. If the receiver is close enough to the wire it may pick up a significant amount of energy from this wave. The dominant contribution to this integral arises from the vicinity of the point ~ =?. With the restriction that kp /1>> 1, and expanding R(Q) as follows R(E)-'R + ( -)) cosO + r r The integral in expression (B. 120) can be approximated by (1+cos ) -jk(R -) O-jk(l-coso )x 2 (sine rZ+cos rp)e r ( e r Q' dx -jk r r (B. 121) As indicated by expression (B. 82) Q(z) is expressed as a linear combination of the universal functions 0+ (Q -z) and (Q - z), i.e. -jk(I +Rt) r 1 Q(z) = -S(l) e L+( - z) - A(I - z), (B. 122) )(-2)jk A = - (B. 123) -2jkl 1+0C(21) e The evaluation of the expression '~ -jk(l-cose )x -jk( +R - e r Q'( - ) d x = - S() e- AI (B. 124) Jo (B. 124) -- - - -- 256

THE UNIVERSITY OF 6633-1-F MICHIGAN requires the computation of the two integrals OCD I1= exp -jk(1-cose0 )zj '(z) dz o 0 12 = exp -jk(l -coser) z ~+ (z) dz 0 (B. 125) (B. 126) From relationships (B. 84) and (B. 86), the first integral I takes the form I = 2g(o) e jq' I( (1 -) +2jqO e+2x o - e-2jq () dt dx, (B. 127) with2q' =(1-cos0 )q. r (B. 128) Integrating by parts the second term in the integral yields the relation Ii= 1 e2jq I'() d -+ - e2jq3 I'(f) e M d] (B. 129) The above can be simply reduced using the following relation given by Vainshtein (1959) -2jqr 1 - e2 I() d = - 1 + 1+ 2 q 1 ln -j/(Typ (B. 130) 257

THE UNIVERSITY OF MICHIGAN 6633-1-F Further simplification will be achieved, in assuming that the wire is 2 2 2 sufficiently thin so that y q = T (ka) <<1. With this approximation, expression (B. 129( reduces to the form F l ln( a)/ln( )] (B. 131) (1+ cos ) L- r 1 'os'fta (l+cosr) -I.j( 2 J.. 2 n- - 2 2 (cos er+cose yka l+cose / a 1 -cos l (B. 132) From Eqns. (B. 121), (B. 122), (B. 131) and (B. 132) the appropriate expression for the integral in Eq. (B. 120) is obtained. Combining all the results, one obtains the following form for the scattered electric field strength. I 0 S() -jk(R t+R A E = — e r (sin0 z+cose p) F (B. 133) 47r kpr r r (l+cosO )(l+cos t) Ln( a +cos 2(yka) /il(a 1 - (B. 134) 258

THE UNIVERSITY OF MICHIGAN 6633-1-F Physically, expression (B. 133) represents a spherical wave originating from the end of the wire, and polarized in a vertical plane (plane containing the wire). The coefficient A arises from the multiple reflections of the traveling waves at the ground plane and end of the wire. Without going into details, it can be shown that for a sufficiently long wire (kI >> 1), A can be approximated by 1+cosO 21n(-j/(Yka)-ln( 2 e-2jk l A-, -'- ----— ]'"- 2 e In [-j 4 / (yka )] Henceforth, the length of the wire will be taken to be sufficiently long so that the term with A as a coefficient, can be neglected. The factor S(e), given by expression (B. 61), contains the amplitude and phase of the field incident at the end of the wire. The scattering matrix coefficient S is obtained by a process of normalization of the modules of vv expression (B. 133), whereby the amplitude of the incident field is divided out. This yields R (l+cos0 )(1+cose0t) - ln(wl 2kp P(cos0 +cos0T ) ln(- 2l * 2kPtpCo9rk +a 1 + cos r -1 |- cn os ln(l -~) (B. 135) Unfortunately, because of the approximations made in the analysis, formula (B. 335) is not symmetric with respect to the angles O and e Expression (B. 135) can be represented in a symmetric form by making the following approximations I 259

I THE UNIVERSITY 6633-1-F OF MICHIGAN in (*ai/ Sr (* yka 1+cos ) 8 In r (B. 136) InIn (ka oka 2 1 -cos0 r ln ( -2j in ka sine r (B. 137) The error will be small for values of 0 and 0 lying between 0 and 45, and ka r T -1 not too large. As an example, taking the value ka = 2 x 10, we obtain in (_ ) = 1.04 yka -i ( In (_ka ~ya 2 1+cos r - ln( -j ) <.08 yka for 0<0 <7r/4. Thus the error is small. The symmetric form of S is given by vv vvl R D(O Or ) kptr T' r (B. 138) (l+cosO )(l+cos T) 2 -1 2-c K(T, r= 2(cos 0T+cose ) Inka,lntasin80T (ka sin)8 r (B. 139) 260

THE UNIVERSITY OF MICHIGAN 6633-1-F For e and e ranging in values between 100 and 45, and ka =.2 (where a is the wire radius), K takes on values between. 2 and. 5. However the results were derived for a wire model of the leading edge of the wing. In the actual case where the scattering is due to currents originating from the tip of the wing, and traveling along the edge towards the ground plane, K would be different. However, taking the value of K associated with the thin wire problem would yield a good estimate. For the values of angles 0 and 0T under consideration, K eafi be taken to be of r T the order of 1/2. Recall that the results were derived under the assumptions that the quantities k, kp, kpt kp2/e, and kpt/I were much gteater than unity. In addition pr / and pt/e were taken to be less than or equal to unity. This restricts the values of 0T and 0 T r With these considerations, the coupling due to the currents launched at the tip of the wing, is given by C =C (Rt) Dt(eT, 0t) D (O 0r) S2 v Pt A i 4K C (pt) Co(pr) Dt(et t) Dr (or ) Pr' ^Pt vi (B. 140) where K is approximately the order of 1/2. The important feature to notice about the coupling is that it depends predominantly upon the distances of the receiver and transmitter from the leading edge of the wing (and not the distances to the tip). If the wing was tilted and not perpendicular to the ground plane, then the distances Pt and p would be perpendicular distances from the transmitter and receiver to the leading edge. Tilting the wing away from the transmitter and receiver would decrease the values of 0r, Pt 0T and p' w 261

THE UNIVERSITY OF MICHIGAN 6633-1-F APPENDIX C RECTANGULAR HORN DATA C. 1 Curves and Tables for Rectangular Horn Coupling Calculation In Section 4. 5. 1 an expression is given for the directivity function of a rectangular horn in terms of a finite integral. This integral was evaluated for values of flare half-angle up to 450, and dimensions up to ten wavelengths, using an IBM - 7090 computer. The results are indicated in Table C-1. Curves are plotted for a number of horns for dimensions up to five wavelengths in Figs. C-la to C-lm, C-2a to C-2m. Also included are the simplified curves obtained from Section 3. 3 for approximate answers. These are given in Table C-2 as well. It should be noted that the flare angle utilized is actually the flare half-angle of the horn, as in Fig. 4-24. Also, the calculations are far-field, and thus applications 2 must satisfy the spacing criterion of Section 3.5. This is discussed further in Sections C. 2 and C. 3. Several points should be noted in using the tables: (1) The angle referred to as "PHI" is related to 0 of the formulas by PHI = 0 + 90. The function SINE-SQUARED PHI becomes cos 0. This discrepancy does not appear in the curves. (2) The functions G and G as utilized are defined as follows: 2 G= U r a()Ig (u) 2 X 2a G(-~) g(v) 2 v 4 X b 262

THE UNIVERSITY OF MICHIGAN 6633-1-F so that the directivity function D(0) is given by D(0) = - G cos2 4 u v or D(0)db= -1.04db + (G )db + (G )db + (cos 0)db u U The three functions G, G, and cos 0 are obtained from table C-1. U v In Figs. C-1 and C-2 to follow, the following points should be noted: (1) The zero level of the chart paper corresponds to + 10 db. Thus readings should be increased by 10 db. (2) Flare half-angles of 00, 200, and 40~ are indicated on the left, while flare half-angles of 100 and 300 are indicated on the right, along with the approximating curve of Section 3. 3. (3) The following scheme is used for identifying the curves 0-o~ 10 -. On the left 20~ - x On the right 30 - x 40 - A Gross - A Q< \ (4) (5) The flare half-angle of Fig. C-1 is 0, while that of Fig. C-2 is 0b. The curves plot the following functions in db: GU vs0 and 2 G cos vs V 263

THE UNIVERSITY OF MICHIGAN 6633-1-F so that D(0) can be obtained from the charts by D(0)db = (-2 Gu)db + (G cos 2)db - 3 db. (6) The extra curve on the right-hand side of the charts is the simplified function of Section 3.3. By comparing this with the other curves, representing various flare angles, its accuracy in a given situation can be estimated. 264

TABLE C-l: HORN COUPLING FUNCTIONS (VALUES IN DB) G-SUB-U (A/L,FLARE ANGLE,PHII c A/LAMBDA - 1.0 Z FLARE A PHI 0 ANGLE O=84 12 18 24 30 36 42 48 54 60 66 72 78 84 0=0 0 -6.69 -6.32 -5.74 -4.98 -4.09 -3.12 -2.13 -1.14 -.22.62 1.35 1.94 2.37 2.63 2.72 5 -6.67 -6.30 -5.72 -4.96 -4.08 -3.12 -2.13 -1.15 -.23.61 1.33 1.92 2.35 2.61 2.70 10 -6.59 -6.23 -5.67 -4.93 -4.06 -3.11 -2.14 -1.17 -.26.57 1.29 1.87 2.30 2.56 2.64 15 -6.47 -6.13 -5.58 -4.86 -4.02 -3.10 -2.15 -1.21 -.31.51 1.21 1.78 2.20 2.46 2.55 20 -6.31 -5.99 -5.47 -4.79 -3.98 -3.09 -2.17 -1.26 -.39.41 1.10 1.66 2.07 2.32 2.41 25 -6.13 -5.83 -5.34 -4.69 -3.93 -3.09 -2.21 -1.33 -.49.28.95 1.49 1.89 2.14 2.22 30 -5.94 -5.66 -5.20 -4.60 -3.89 -3.C9 -2.26 -1.42 -.62.11.76 1.28 1.66 1.90 1.98 C - 35 -5.74 -5.49 -5.07 -4.52 -3.85 -3.12 -2.34 -1.55 -.79 -.09.52 1.02 1.39 1.61 1.69 m 40 -5.57 -5.33 -4.95 -4.45 -3.85 -3.17 -2.44 -1.71 -1.01 -.35.22.69 1.04 1.26 1.33 C C) 45 -5.42 -5.21 -4.87 -4.42 -3.87 -3.26 -2.60 -1.93 -1.28 -.67 -.14.30.63.83.90 e C.-A A/LAMBDA 1.4 0 FLARE PHI ' ANGLE 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 0 -22.31 -20.38 -17.77 -14.89 -12.01 -9.24 -6.66 -4.31 -2.22 -.41 1.11 2.30 3.16 3.68 3.85 5 -21.15 -19.56 -17.26 -14.59 -11.84 4 9.14 -.61 -29 -2.21 -.41 1.09 2.28 3.14 3.66 3.83 10 -18.84 -17.75 -16.01 -13.80 -11.36 -.2 -.46 -21 -19 -.42 1.06 2.23 3.08 3.59 3.76 15 -16.58 -15.81 -14.52 -12.76 -10.68 -8.45 -6.22 -4.10 -2.16 -.45.99 2.14 2.97 3.48 3.65 20 -14.65 -14.06 -13.05 -11.64 -9.90 -7.95 -5.93 -3.96 -2.12 -.49.90 2.02 2.83 3.32 3.48 25 -13.02 -12.54 -11.72 -10.56 -9.09 -7.40 -5.60 -3.80 -2.09 -.54.78 1.85 2.63 3.11 3.27 30 -11.65 -11.24 -10.55 -9.57 -8.32 -6.86 -5.26 -3.63 -2.06 -.62.63 1.64 2.39 2.84 2.99 35 -10.48 -10.12 -9.53 -8.68 -7.61 -6.34 -4.94 -3.48 -2.06 -.73.43 1.38 2.08 2.51 2.66 40 -9.49 -9.17 -8.64 -7.91 -6.98 -5.87 -4.65 -3.36 -2.08 -.88.18 1.06 1.72 2.12 2.25 45 -8.64 -8.36 -7.90 -7.25 -6.44 -5.48 -4.42 -3.29 -2.16 -1.08 -.12.68 1.27 1.64 1.77 A/LAM8DA 1.8 FLARE PHI Z ANGLE 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 0 -18.66 -19.04 -19.96 -22.18 -28.69 -29.91 -17.19 -10.73 -6.14 -2.61.13 2.21 3.67 4.54 4.82 5 -18.41 -18.72 -19.48 -21.17 -24.52 -23.97 -16.38 -10.50 -6.06 -2.59.13 2.19 3.64 4.51 4.80 10 -17.71 -17.87 -18.28 -19.06 -19.98 -18.89 -14.60 -9.87 -5.82 -2.51.13 2.15 3.58 4.43 4.72

15 -16.74 -16.74 -16.83 -16.92 -16.82 -15.61 -12.71 -9.01 -5.46 -2.40.12 2.07 3.47 4.3C 4.58 20 -15.63 -15.5C -15.31 -15.03 -14.47 -13.24 -11.Cl -8.08 -5.03 -2.25.11 1.97 3.31 4.11 4.38 25 -14.50 -14.28 -13.92 -13.40 -12.63 -11.41 -9.56 -7.17 -4.58 -2.09.07 1.82 3.10 3.87 4.13 30 -13.40 -13.11 -12.65 -12.01 -11.14 -9.95 -8.34 -6.34 -4.13 -1.94.02 1.64 2.83 3.56 3.81 35 -12.34 -12.02 -11.51 -10.81 -9.9C -8.75 -7.32 -5.61 -3.72 -1.82 -.06 1.41 2.51 3.19 3.42 - 40 -11.35 -11.02 -10.49 -9.76 -8.86 -7.77 -6.48 -5.00 -3.38 -1.73 -.19 1.12 2.12 2.14 2.95 45 -10.43 -10.10 -9.57 -8.86 -7.99 -6.97 -5.8C -4.51 -3.12 -1.71 -.38.77 1.65 2.21 2.40 A/LAMBDA * 2.2 FLARE PHI C ANGLE 6 12 18 24 3C 36 42 48 54 60 66 72 78 84 90 2 0 -21.15 -20.26 -19.08 -17.97 -17.37 -17.97 -21.8C -33.20 -13.82 -6.48 -1.70 1.64 3.89 5.20 5.64 5 -23.97 -23.13 -18.93 -17.81 -17.13 -17.51 -20.12 -22.54 -13.13 -6.33 -1.67 1.63 3.87 5.17 5.60 10 -20.46 -19.62 -18.48 -17.32 -16.49 -16.36 -17.20 -16.85 -11.56 -5.91 -1.57 1.61 3.80 5.08 5.51 15 -19.67 -18.89 -17.79 -16.59 -15.56 -14.93 -14.63 -13.45 -9.83 -5.32 -1.42 1.58 3.68 4.93 5.34 < 20 -18.68 -17.96 -16.90 -15.68 -14.49 -13.48 -12.55 -11.06 -8.26 -4.65 -1.23 1.53 3.52 4. 1 5.11 25 -17.57 -16.90 -15.89 -14.66 -13.37 -12.12 -10C.86 -9.26 -6.92 -3.98 -1.34 1.46 3.30 4.42 4.80 30 -16.38 -15.77 -14.81 -13.60 -12.26 -10.88 -9.46 -7.84 -5.81 -3.37 -.86 1.35 3.03 4.07 4.42 35 -15.17 -14.60 -13.70 -12.53 -11.19 -9.77 -8.3C -6.71 -4.90 -2.85 -.72 1.20 2.70 3.64 3.96 40 -13.97 -13.44 -12.60 -11.49 -10.19 -8.79 -7.33 -5.80 -4.18 -2.43 -.65.99 2.30 3.14 3.42 C 45 -12.81 -12.32 -11.54 -10.53 -9.27 -7.93 -6.53 -5.10 -3.64 -2.14 -.65.72 1.83 2.55 2.80 - A/LAMBDA = 2.6 FLARE PHI t ANGLE 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 C 3 -33.37 -38.95 -42.43 -28.70 -22.42 -18.59 -16.75 -17.88 -33.14 -13.49 -4.63.57 3.86 5.72 6.33 5 -31.65 -34.27 -34.66 -27.71 -22.C9 -18.35 -16.42 -17T.00 -21.14 -12.51 -4.48.59 3.84 5.69 6.29 10 -28.59 -29.41 -29.08 -25.55 -21.16 -17.68 -15.56 -15.11 -15.38 -10.50 -4.06.64 3.78 5.58 6.17 15 -25.8C -25.97 -25.44 -23.19 -19.84 -16.69 -14.39 -13.10 -11.98 -8.48 -3.48.72 3.67 5.40 5.97 20 -23.41 -23.30 -22.67 -20.97 -18.32 -15.49 -13.1C -11.31 -9.61 -6.76 -2.84.80 3.52 5.15 5.69 25 -21.34 -21.07 -20.38 -18.94 -16.73 -14.19 -11.80 -9.77 -7.83 -5.36 -2.22 88 3.31 4.82 5.33 30 -19.49 -19.13 -18.39 -17.08 -15.17 -12.88 -10.57 -8.45 -6.45 -4.25 -1.67.93 3.06 4.42 4.88 - 35 -17.80 -17.39 -16.62 -15.39 -13.68 -11.60 -9.42 -7.33 -5.36 -3.37 -1.23.92 2.74 3.94 4.35 40 -16.24 -15.80 -15.02 -13.85 -12.29 -10.41 -8.38 -6.39 -4.51 -2.71 -.91.85 2.36 3.37 3.73 45 -14.79 -14.33 -13.55 -12.45 -11.01 -9.31 -7.46 -5.61 -3.87 -2.25 -.73.70 1.90 2.73 3.02 A/LAMBOA z 3. ) FLARE PHI ANGLE 6 12 18 24 30 36 42 -48 54 60 66 72 78 84 90 0 -23.86 -23.74 -24.01 -25.60 -31.73 -33.CC -21.07 -16.66 -16.89 -155.59 -9.24 -1.04 3.60 6.12 6.93 5 -23.73 -23.59 -23.81 -25.22 -29.92 -30.10 -20.66 -16.29 -15.90 -19.27 -8.62 -.96 3.58 6.08 6.88 10 -23.35 -23.16 -23.24 -24.20 -26.72 -25.97 -19.53 -15.31 -13.82 -13.32 -7.21 -.74 3.54 5.96 6.74 15 -22.73 -22.48 -22.38 -22.80 -23.77 -22.59 -17.96 -13.99 -11.70 -9.91 -5.64 -.43 3.46 S.75 6.50 20 -21.92 -21.60 -21.28 -21.21 -21.21 -19.79 -16.2C -12.53 -9.84 -7.58 -4.22 -.07 3.35 5.46 6.17 25 -20.94 -20.54 -20.05 -19.58 -18.97 -17.41 -14.42 -11.08 -8.27 -5.86 -3.05.27 3.20 5.09.74 30 -19.83 -19.37 -18.72 -17.96 -16.97 -15.32 -12.72 -9.71 -6.96 -4.55 -2.12.55 2.99 4.64 5.22 35 -18.63 -18.12 -17.35 -16.39 -15.18 -13.49 -11.16 -8.46 -5.86 -3.56 -1.41.74 2.72 4.11 4.60 40 -17.36 -16.82 -15.98 -14.89 -13.56 -11.87 -9.75 -7.34 -495 -2.81 -.92.83 2.38 3.49 3.89 45 -16.06 -15.51 -14.63 -13.48 -12.10 -10.45 -8.52 -6.37 -4.23 -2.30 -.64 7 1.96 2.80 3.10 { 4 4 ) 4 4 ( 4 f 1 4! A 4 I 4 ) i,

A/LAHBCA a 3.4 FLARE ANGLE 6 0 -34.32 5 -33.41 10 -31.39 15 -29.15 20 -27.02 25 -25.04 30 -23.17 35 -21.40 40 -19.71 45 -18.07 A/LAMBDA a 3.8 12 -30.92 -30.45 -29.22 -27.61 -25.88 -24.12 -22.38 -20.68 -19.02 -17.41 18 -27.34 -27.09 -26.38 -25.32 -24.02 -22.56 -21.02 -19.44 -17.86 -16.30 24 -24.53 -24.35 -23.83 -23.01 -21.93 -20.66 -19.26 -17.78 -16.28 -14.80 30 -23.21 -23.01 -22.42 -21.48 -20.29 -18.92 -17.45 -15.94 -14.45 -13.02 36 -24.85 -24.37 -23.11 -21.40 -19.53 -17.64 -15.82 -14.11 -12.54 -II.1C FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 -29.33 -29.10 -28.45 -27.48 -26.29 -24.96 -23.55 -22.08 -20.57 -19.04 12 -30.76 -30.41 -29.44 -28.10 -26.58 -24.98 -23.35 -21.72 -20.09 -18.48 18 -35.36 -34.22 -31.86 -29.37 -27.07 -24.94 -22.95 -21.07 -19.27 -17.54 24 -46.50 -38.89 -33.28 -29.55 -26.65 -24.1 7 -21.95 -19.89 -17.98 -16.18 30 -28.87 -28.43 -27.23 -25.57 -23.70 -21.75 -19.80 -17.89 -16.05 -14.33 36 -23.62 -23.38 -22.70 -21.63 -20.25 -18.66 -16.98 -15.28 -13.62 -12.C5 PHI 42 48 -41.87 -22.18 -31.88 -21.51 -25.86 -19.82 -21.9C -17.66 -18.82 -15.42 -16.26 -13.30 -14.07 -11.38 -12.17 -9.69 -10.53 -8.22 -9.11 -6.98 PHI 42 48 -23.26 -37.44 -22.85 -30.07 -21.70 -24.26 -20.08 -20.22 -18.22 -17.04 -16.3C -14.40 -14.44 -12.16 -12.68 -10.26 -11.08 -8.64 -9.64 -7.29 PHI 42 48 -23.87 -22.99 -23.56 -22.43 -22.68 -20.94 -21.31 -18.91 -19.60 -16.71 -17.70 -14.55 -15.74 -12.52 -13.83 -10.69 -12.03 -9.06 -10.40 -7.66 PHI 42 48 -36.37 -22.85 -33.60 -22.47 -29.44 -21.40 -25.84 -19.77 -22.70 -17.79 54 -16.04 -15.55 -14.29 -12.68 -11.02 -9.43 -7.98 -6.70 -5.60 -4.69 60 -17.62 -15.68 -12.52 -9.87 -7 80 -6.16 -4 86 -3.83 -3.04 -2.45 66 -17.87 -14.22 -9.98 -7.03 -4.92 -3.38 -2.24 -1.41 -.86 -.55 72 -3.32 -3.09 -2.49 -1.70 -.91 -.21.34.73.92.93 78 3.11 3.11 3.11 3.11 3.08 3.02 2.90 2.71 2.43 2.05 84 6.42 6.37 6.23 6.00 5.68 5.27 4.77 4.19 3.54 2.81 90 7.46 7.40 7.23 6.95 6.56 6.06 5.45 4.74 3.95 3.09 54 -20.20 -19.52 -17.76 -15.52 -13.21 -11.05 -9.14 -7.49 -6.11 -4.99 60 -15.07 -14.24 -12.39 -10.36 -8.50 -6.87 -5.48 -4.32 -3.38 -2.66 66 -27.61 -16.35 -10.68 -7.37 -5.12 -3.50 -2.31 -1.45 -.8 7 -.53 72 -6.53 -5.88 -4.42 -2.85 -1.50 -.44.33.83 1.08 1.09 78 2.39 2.42 2.51 2.64 2.76 2.84 2.84 2.74 2.52 2.16 84 6.62 6.57 6.42 6.17 5.82 5.38 4.84 4.23 3. 54 2.80 90 7.93 7.87 1.67 7.34 6.88 6.30 5.60 4.80 3.94 3.03 cS CA) I I H C C) ci 0 P —l 0 I —4 0 A/LAM8DA = 4.2 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 -29.80 -29.63 -29.14 -28 36 -27.35 -26 16 -24.84 -23.42 -21.92 -20.36 12 -28.60 -28.46 -28.03 -27.34 -26.42 -25.31 -24.03 -22.64 -21.15 -19.59 18 -27.60 -27.45 -27.02 -26.33 -25.39 -24.25 -22.94 -21.50 -19.97 -18.39 24 -28.08 -27.86 -27.21 -26.21 -24.94 -23.48 -21.90 -20.24 -18.55 -16.86 30 -33.51 -32.51 -30.28 -27.8C -25.4C -23.11 -20.94 -18.87 -16.92 -15.09 36 -34.75 -33.05 -29.89 -26.81 -24.01 -21.44 -19.C5 -16.85 -14.81 -12.96 54 -39.17 -28.68 -22.36 -18.07 -14 72 -11.99 -9.73 -7.86 -6.33 -5. 11 60 -16.49 -15.81 -14.11 -12.00 -9.88 -7.93 -6.23 -4.80 -3.64 -2.77 66 -16.51 -13.92 -10.26 -7.45 -5.34 -3.74 -2.52 -1.61 -.96 -.57 72 -11.35 -9.29 -6.11 -3.61 -1.78 -.48.41.96 1.21 1.20 78 1.42 1.51 1.77 2.11 2.45 2.72 2.86 2.84 2.65 2.29 84 6.76 6.70 6.54 6.27 5.90 5.43 4.87 4.24 3.54 2.80 90 8.36 8.28 6.05 7.67 7.14 6.47 5.68 4.80 3.87 2.94 A/LAH80A, 4.6 FLARE ANGLE 0 5 10 15 20 6 -42.25 -40.22 -36.86 -33.91 -31.41 12 -85.01 -44.10 -37.94 -34.17 18 -37.14 -36.25 -34.23 -31.95 24 -29.99 -29.75 -29.07 -28.02 30 -27.17 -26.98 -26.41 -25.50 -24.28 36 -29.82 -29.27 -27.84 -25.92 -23.79 54 -24.74 -23.46 -20.65 -17.57 -14.71 60 -21.95 -20.57 -17.54 -14.28 -11.32 66 -14.26 -12.93 -10.38 -7.94 -5.89 72 -20.49 -12.38 -7.05 -3.92 -1.85 78.18.39.93 1.60 2.22 84 6.81 6.76 6.59 6.32 5.93 90 8.75 8.66 8.39 7.95 7.34 -31.32 -29.74 -26.69

25 30 35 40 45 -29.21 -27.21 -25.32 -23.50 -21.72 -28.93 -26.80 -24.80 -22.90 -21.05 -27.63 -25.60 -23.64 -21.72 -19.83 -25.15 -23.46 -21.69 - 19.86 -18.03 -22.81 -21.16 -19.41 -17.6C -15.81 -21.60 -19.42 -17.31 -15.31 -13.45 -19.86 -17.28 -14.93 -12.82 -10.96 -15.66 -13.54 -11.55 -9.74 -8. 15 -12.17 -9.98 -8.11 -6.55 -5.28 -8.81 -6.73 -5.05 -3.73 -2.77 -4.22 -2.87 -1.82 -1.06 -.58 -.45.48 1.03 1.28 1.25 2.68 2.94 2.98 2.79 2.38 5.45 4.89 4.24 3.54 2.80 6.58 5.70 4.75 3.78 2.86 H A/LAMBDA = 5.0 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 -30.73 -30.62 -30.24 -29.63 -28.81 -27.79 -26.62 -25.30 -23.86 -22.32 12 -30.93 -30.79 -30.34 -29.62 -28.67 -27.52 -26.22 -24.78 -23.24 -21.61 18 -32.78 -32.50 -3 1.69 -30.50 -29.08 -27.50 -25.83 -24.10 -22.31 -20.51 24 -42.71 -40.00 -36.04 -32.76 -30.01 -27.56 -25.28 -23.11 -21.01 -18.99 30 -34. 14 -33.50 -31.91 -29.88 -27.70 -25.48 -23.26 -21.07 -18.93 -16.86 36 -27. 16 -26.93 -26.25 -25.16 -23.71 -21.99 -20.07 -18.08 -16.C7 -14.14 PHI 42 48 -29.62 -30.34 -28.94 -29.11 -27.19 -26.39 -24.91 -23.31 -22.45 -20.27 -19.97 -17.39 -17.56 -14.74 -15.29 -12.37 -13.19 -10.28 -11.29 -8.49 54 -21.56 -21.02 -19. 54 -17.43 -15.07 -12.73 -10. 55 -8.61 -6.93 -5.54 60 -153.23 -26.93 -20.04 -15.46 -11.95 -9.15 -6.91 -5.14 -3.77 -2.76 66 -14.77 -13.75 -11.48 -9.01 -6.74 -4.80 -3.22 -1.98 -1.3 8 -.52 72 -26.53 -12.93 -7.22 -4.00 -1.90 -.47.47 1.04 1.29 1.27 78 -1.39 -.96.04 1.15 2.08 2.73 3.07 3. 12 2.90 2.43 84 6.81 6.75 6.59 6.32 5.94 5.47 4.90 4.26 3.55 2.81 90 9.11 9.00 8.70 8.19 7.50 6.65 5.68 4.67 3.69 2.82 A/LAMBDA = 5.4 00 00 Iz C) 0 0 —^ 0 0 FLARE ANGLE 5 10 15 20 25 30 35 40 45 6 -39.76 -39.00 -37.20 -35.12 -33.07 -31.10 -29.22 -27.38 -25.56 -23.74 12 -35.40 -35.09 -34.21 -32.94 -31.46 -29.86 -28.18 -26.46 -24.68 -22.87 18 -31.75 -31.59 -31. 10 -33.30 -29.25 -27.98 -26.53 -24.93 -23.21 -21.41 24 -30.42 -30.26 -29.75 -28.93 -27.82 -26.48 -24.93 -23.23 -21.41 -19.53 30 -34.86 -34.21 -32.59 -30.54 -28.35 -26.13 -23.91 -21.70 -19.53 -17.42 36 -36.03 -34.91 -32.46 -29.74 -27.05 -24.44 -21.90 -19.44 -17.10 -14.91 PHI 42 48 -26.60 -33.90 -26.32 -31.69 -25.48 -27.87 -24.14 -24.23 -22.39 -20.92 -20.35 -17.89 -18.16 -15.15 -15.94 -12.69 -13.80 -10.53 -11.81 -8.68 54 -23.81 -23.23 -21.60 -19.26 -16.60 -13.92 -11.42 -9.20 -7.29 -5.73 60 -24.16 -22.37 -18.78 -15.17 -11.99 -9.31 -7.09 -5.29 -3.88 -2.83 66 -17.57 -16.35 -13.54 -10.43 -7.62 -5.27 -3.42 -2.3 1 -1.02 -.43 72 78 84 90 -16.60 -3.35 6.74 9.44 -11.80 -2.51 6.69 9.32 -7.14 -.79 6.54 8.97 -4.11.83 6.29 8.40 -2.04 2.05 5.93 7.62 -.59 2.83 5.47 6.67.40 3.22 4.92 5.63 1.02 3.24 4.28 4.58 1.30 2.96 3.57 3.62 1.30 2.43 2.82 2.81 A/LAMBOA - 5.8 FLARE ANGLE 6 0 -35.20 5 -34.98 10 -34.32 15 -33.32 20 -32.09 25 -30.72 30 -29.24 35 -27.68 40 -26.04 45 -24.34 12 -38.09 -37.60 -36.30 -34.63 -32.84 -31.02 -29.20 -27.39 -25.55 -23.69 18 -57.78 -46.13 -40.17 -36.41 -33.51 -31.04 -28.79 -26.65 -24.57 -22.51 24 -35.74 -35.32 -34.17 -32.58 -30.78 -28.85 -26.85 -24.80 -22.71 -20.60 30 -30.26 -30.08 -29.52 -28.61 -27.37 -25.84 -24.08 -22. 15 -20.11 -18.03 36 -33. 95 -33. 30 -31.63 -29.47 -27. 13 -24. 71 -22.27 -19. 86 -17. 52 -15.30 PHI 42 48 -32.56 -26.32 -31.84 -25.89 -30.00 -24.64 -27.61 -22.76 -24.99 -20.45 -22.29 -17.95 -19.60 -15.45 -17.00 -13.07 -14.57 -10.89 -12.35 -8.98 54 -34.41 -31.02 -26.18 -21.94 -18.20 -14.89 -12.00 -9.53 -7.47 -5.80 60 -20.96 -20.14 -17.99 -15.20 -12.36 -9.76 -7*48 -5.58 -4.05 -2.90 66 -24.16 -20.92 -15.88 -11.58 -8.16 -5.50 -3.47 -1.98 -.96 -.37 72 -13.72 -11.01 -7.26 -4.42 -2.32 -.79.30 1.00 1.34 1.35 78 -5.86 -4.20 -1.48.66 2.10 2.96 3.34 3.32 2.98 2.40 84 6.61 6.57 6.45 6.24 5.92 5.49 4.95 4.31 3.59 2.82 90 9.75 9.61 9.22 8.57 7.70 6.66 5.56 4.50 3.58 2.83 0 z

A/LAPBUA = FLARL ANGLE 0 5 10 15 20 25 30 35 40 45 6.2 6 -34.78 -34.62 -34. 14 -33. 3tf -32.4C -31.22 -28. 43 -26.at -25. 19 12 -i3,43 - 33. Y) - 321.9i -32.24 - 31. 36 - 3C.2 1 -29.*0() -2 1.5.. - 25 o 9 11 -24o2*L is -33.23 -3113.07 - 32. 59 -31.82 -30.79 -2I.54 -23a.1 - 26.51 -24. 71 -22.92 24 -38.20 -37I. 6 -35.A?? -33.98 -31.89 -29.79 -27.bl -2 5.54 -23.303 - 2 1.2 2 30 -39.1C -38. IC -35.86 -33.37 -30.89 -2 8.45 -26.04 -23.64 -21.25 -18.92 30 -36.CC -35.53 -34.29 -32.57 -30.61 -28.49 -26.28 -23.95 -21.66 -19.33 -29.94 -29.71 -29. C3 -27.5,g3 -26.45 -24. 64 -22.517 -20. 35 -18.C7 -15.E3 -37.C7 -36. 26 -34. 32 -31.53 -29.41 -26. 82 -24.2C -21.5 7 -18.58 -16.45 F,9J 42 -38.03 -3t.08 -32.60 -29.21 - -26.C5 -23.05 -20. 18 -17.46 --14.93 -12.63 PHI 42 -29.66 -29.34 -28.42 -26.95 -2 5.02 -22.76 -2C.3C -17.77 -15.29 --12.97 48 -27. 57 -27. 15 -25* 94 -24.06 -21.70 -19.08 -16.39 -13.80 -11.42 -9.32 54 -31.47 -29.49 -25.74 -21.92 -113. 33 -15.08 - 12. 19 -9.69 -7.57 -5.88 60 -21.86 -21.13 -1;.13 -i6.36 -13.36 -10.51 -7.98 -5.87 -4.17 -2.ql 66 -41 U 2 -23.9 1 -16.79 -11.94 -8. 3 -5.5 7 -3.49 -1.91 -.9 4.35 72 -12.99 -11.05 -7.78 -4.95 -2.71 -1.cc -.4 1.03 1.42 14 42 78 -9.22 -5.8~7 - 1.96.6C 2.18S 3.42 3.36 2.917 2.317 84 6..42 6.40 6.33 6,6: 6.17 5.92 5. 53 5.00 4.34 3.59 2. 80 90 10.03 9,88 8.72 7.75 6.63 5.47 4.43 3. 58 2. 89 A/LAMBUA a 6.6 0 7) 0 FLARE ANGLE 0 5 10 15 20 2 5 30 35 40 45 6 -48. 37 -45.82 -42.G4 -36.34 -324* C -30. Cd -28. 17 -26.27 A. 4.~C 12 - 51.2'4 -46.9> -42. 22 - 38. 77 - 36.03 - 33.6 7 -31, 527 -27.4g - 7 7o 4 - I5. 4 7 1i -37.39 -37.07 -36.15 -34.84 -33.29 -31.60 -29. 81 -27.93 -25.97 -23.93 24 -32.92 - 32.76 - 32.2 8 -31.419 -3C.42 -29.08 -2 7.52 -25. 15 -23.81 -21.*76 48 -39.12 -35,99 -31.46 -27.47 -23.84 -2C0.44 -17.27 -14.36 -11.77 -9.52 54 -25. 77 -i'S. 18 -23.52 -21.08 -18.23 -15.30 -12.52 -10.00 -7.83 -6.04 60 -26.88 -25.43 -22.11 -1E.25 -14.51 -L1.17 -8.33 -6.C1 -4.19 -2.87 66 -23.8 1 -20.95 -16.14 -8.4 2 -5.b9 -3 59 -2.0 3.9 6 -.3 4 72 -13.64 -11.9 5 -8.72 - 5.64 -3.10 -1.15.24 1.11 1.I0 1.4e 78 -14.15 -7. 2C -2.21.60 2.25 3. 14 3.47 3.37 2.55 2.34 84 6. 18 6. 18 6. 18 6. 11 ). 93 5. 58 5. 05 4. 36 3. 58 2.78O 90 10. 3C 10. 14 9. 64 8. 84 7. 78 6. 58 5. 39 3. 60 2. 95 H z 'C,, ci~, V) M "~ W0. I 0 z Og C) A/LAMBCA z 7.C FLARE ANGLE 0 5 10 6 -35.20 -35.C -34.7L 12 -35.89 -35.72 — i.22 is -40.83 -40.20 -38.66 24 -432.43 -42.13 -39.50 30 -32.89 -32. 70 -32.12 PHI 36 42 48 -37.72'- -3r.92 -33.45 -36.E7 -31.53 -32.39 -34.86 -30.42 -29.91 54 -26.49 -25.97 -24.44 60 -156.49 -31.61 -24.45 66 -20.42 -19.00 -15.69 72 -15.60 -13.73 -9.97 78 -24.00 -7.e9 - 2.30 84 5.87 5.92 6.01 90 10.56 10.37 9.82

15 -34. C -34.43 -'6.74 -36.81 -31.19 -32.43 -28.71 -26. 89 -2 2.09 -19.31 -12.01 -6.3 3.62 6.C6 8.c 20 -3.26 - 3*3, 4 -4,74 -34.30 -29.92 -29.E -26. 5 5 -23.72 -19.22 -15.03 -b.72 -3.40 2.29 5.96 7. 25 -32.' -32.l -32.75 -31.927 -28.34 -21.28 -24.06 -2C.54 -16.1 - 1144 -5.91 -1.22 3.11 5.64 6., 30 -31.0z -3C..78 -30.7i -29.62 -26.50 -24.64 -21.3 -17.45 -13.18 -8.46 -3.7 8.29 3.4e 5.09 5. 35 -29.69 - 2'.23 -23.75 -27.34 -24.43 -21.98 -18.63 -14.56 -10.46 -6.06 -2.13 120 3.36 4. 3 7 4. 40 -8.81 -21.8;' -?6. 7I -25.04 -22.2C -19. 36 -15.95 -11.95 -8p. 11 -4.2 -.92 1. 57 2.94 3.55 3. 45 -16.s5 -25.7V) -24.6b -22.72 -19.89 -16.E3 -13.45 -9.67 -6.17 -2.85 -.3 0 1.53 2.33 2. 75 3.( A/LAPI4UA x 7,4 FLARL Fhi ANGLE 6 12 18 24 30 38 47. -R 54 72 7R 0784. 90 0 -43.% -3r. 28 -35.13 -37.12 -44.41 -32.27 -8337 -29.04 -33.49 -~8.42 -20.14 -19.32 26.11 5. S oa 5 -4?. C -38.04 -305.01 - 36.82 -42.6C -32.04 -43.13 -28.66 -3.85 -,6.10 -19.04 -16.35 -7.95 5.61 10.' 10 -40..t -31.3I -'4...8 -35.96 -39. 37 -31.36 -36.6 -27.53 -28.42 -23.00 -16.19 -11.23 -?.32 5.84 9.5 15 -39.04 -36.33 -33.88 -34.70 -36.3C -30.25 -32.38 -25.73 -24.61 -18.91 -12.6 7 -6.8'.61 6.02 9.C 20 -37.0V -35,T- -32.93 -33.15 -33.52 -28. 74 -28.81 -i 3. 38 -2C.79 -15.01 -9. -3.56 2.29 6.00 7.1 25 -35.16 -33.61 -31.75 -31.41 -30.89 -26.86 -25.51 -20.66 -17.1I -II.'"4 -6.3i -1.20 3.18 5.11 6.4 30 -33.2fi -33.04 -3C.35 -29.51 -28.33 -24.68 -22.33 -17.73 -13.75 -8.58 - 3.9.37 3.49 5.13 5., 35 -31.45 -30.36 -2.175 -27.49 -25.78 -27.28 -19.28 -14.94 -10.78 -6. 16 -2.1 e 127 3. 37 4.35 4.3 40 - &A. -2 3.57 -2.I8 -25.33 -23.21 -19.77 -16.40 -12.29 -8.25 -4.26 -.')4 1.61 2.34 3.52 3.1 45 -21.?1 -26.6I -25.04 -23.0') -20.65 -17.25 -13.74 -9.92 -6.21 -2.81' -.24 1.49 2.?3 2.74 3. I\3 -5 A/LAMRtA 7.8 rLARL PHI ANGLL 6 12 18 24 30 36 42 48 54 60 GG 72 78 84 90 0 -39. 3 -44.12 -47.68 -35.77 -37.02 -37.97 -313.3e -31.88 -38.95 -2'4.94 -22.19 -27.03 -17.IC 5.08 11.0 5 -39.13 -43. -45.84 -3.39 -36.66 -37, 36 -32.95 -31.35 -35.01 -24.13 -20.94 -19.20 -7.68 5.26 1C 8 10 -38.42 -4. -42.o4 -35.05 -35.66 -35.71 -31.74 -29.66 -29.8 3 -21.8 -17.12 -12.07 -2.36 5.67 10.1 15 -37.40; -39.45 -39.7 -34.17 -34.20 -33.66 -29.93 -Z 1.64 -25.34 -18.12 -13.14 -1.11 7II 6.01 9.0 20 -36.1.3 -37.35 -37.12 -33.00 -32.42 -31.30 -27.61 -24.90 -21.22 -15.22 -9.94 -3.60 2.25 C.07 7.7 25 -34.72 -3..3 -34. 1-) -31.56 -30.43 -28.77 -2 5.09 -l.84 -17.41 -11.86 -6.6 -1.17 3.17 5.77 6.3 30 -33 23 -3'.34 -32.53 -29.89 -28.25 -26.12 -22.30 -18.67 -13.95 -8.87 -4.). 42 5C 5.15 5.2 35 -31 oC -31.4 -30.42 -28.0O -25.92 -23.36 -19.44 -15.58 -10.91 -.36 -2.17 1.31 3.39 4.33 4.4 40 -29. 9 -?9.'44 -28.24 -2 5.92 -23.47 -20.58 -16.62 -12.70 -8.33 -4. 36 -.8 P 1.62 2.95 3.48 3.7 45 — 2i.:? -21.4 4 -26.05 -23.70 -20.97 -17.84 -13.97 -10.16 -6.25 -2.90 -.18 1.49 2.33 2.74 3.0 A/LAPBCA = 8.2 FLARE PHI ANGLE 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 0 -38.25; -36.9'4 -38.30 -66.77 -35.25 -41.39 -32.51 -57.98 -29.49 -~5.61 -27 b3 -36.06 -1 3.72 4.59 11.2 5 -3o.14 -36. 2 -38.08 -49.91 -35.05 -40.24 -32.21 -41.60 -28.81 -24.81 -24.90 -19.9 -7.42 4.88 10.9C 10 -37.68 -36.41 -37.44 -43.74 -34.45 -37.74 -31.33 -35.05 -26.91 -22.75 -19.81 -12.26 -'.4e 5. 50 10.24 15 -36.34 -35.80 -36.45 -39.*8 -33.47 -34.98 -2c.9C -30.50 -14,.15 -19.61 -14o77 -7.17.42 6.01 9.1: 20 -35.o9 -34o33 -35.19 -36.86 -32.15 -32.22 -27.96 -26.55 -20.88 -16.01 -10.40 -3.62 2.1e 6.14 7.6 25 -34.E2 -33.86 -33.73 -34.24 -30.51 -29.46 -25.58 -22.84 -17.45 -12.45 -6.85 -1.17 3.16 5.83 6.2 30 -33.50 -32.59 -32.11 -31.78 -28.58 -26.66 -22.89 -19.29 -14.13 -9.24 -4.11.42 3.53 515 5.21 35 -32.03 -31.14 -30.33 -29.37 -26.40 -23.81 -2C.C1 -15.96 -11.11 -6.55 -2.13 1.32 3.42 4.28 4.4d 40 -30.43 -23.51 -28.41 -26.95 -24.01 -20.96 -17.12 -12.93 -8.49 -4.42 -.82 1.63 2.96 3.45 3.8( 45 -28.71 -27.73 -'6.35 -24.48 -21.5C -18.16 -14.35 -10.29 -6.35 -2.87 -.15 1.48 2.31 2.75 3.0' t8 78 18 55 18 i'1 06 - )1 5d?6 5 ) - W 0. I I 2 3 16 0 -I CI 1 1 9 6 1 5

A/LA?'eLA 6:. 6 FLARt ANGLL 6 0,3 4 20 - 3.17)1 2 5 - 37. 64. 3 0 -J9.l 3 5-33 4 0 -31.6-1 4 5 -429.64 p,.F I 12 -470 211 - 44. 3,.) -4 1.6A -39. Li - 34. Th-3j2..8 - -'31,. 07 -2 * 7 t 18 - 38. 51 - 383.3 3 - 37. 76 -6 M. 60 - 35.70 - 314 32" -22-. 7 4 - 3 0.99 - 29.0 7 - 261.99 24 - 3 7.27 - 37.08 - 36.4 9 -35 *"55 - 34.321 - 32.032 -3 1.1 C -29.1 6 - 27.0 3 - 24 7 73 30 - 51. 17 -47.C9 -42.34 - 3da.?? - 3 5.70 - 32. G4 -30. 28 - 27. 61 -24.89 -2 2.153 36 -314. 21 -34.05 -33. 35 - 32. 2 6 - 30. 72 -?8. /9 - 26. 51 -23.51; -21.25 - 18.419 42 -42.2 3 -4C.45 - 31.11 - 3 0. S c -2 7.23 - 24.C 3 - 20C. 7 -1 7. 63 -14.6ts -31!81 - 33.1 5 - 31. 3 7 -2. 836 - i5.9 1 -2 2. 6 h -19. 35 -16. it - 13. 09 - 10.42 54 a-29.7 8 -28.28 -26b. 8 -'-4.4 3 -2 1. 3') -10 * (11 -14.62 -1 1.4o) -8.7 1 - 6. * 44 60 - 30. 42 - 28. 94 -Z 5.4 9 -21.29 - 16.9 9 - 1 2 * 99 -9. c -4.4 1 - 2 tl 3 66 -6 5.9 1 -28.6A -20.9 3 -15.! 9 -10.'66 -6.9 1 - 4.12'L -2.11I -.$t0 -.1 4 72 -.3. 7 2 - 18.54 - 12. 07 - 7.21I - 3.70 - 1.2 3.40 1. 32 1.64 1.4 9 78 - 12.0C6 - 7.3 3 - 2.70.24 2. 1 3.1 7 3. 58 3.4 5 2. 55 2. 28 84 4.03 4.47 5. 36 6.04 6.2 3 5. 88 5. 14 4. 24 3.44 2.78 no 11. 45 11.1la 10. 38 9. 13 7.6 3 6.2 3 5. 21 4. 51 3. 83 3. 02 A/LAMMLA = 9.0 FLARE ANG LL 0-'? 5 -3 a 1 0 -38. 2 0 -36. 2 5 - 35. 30 - 34. 3 5 - 32. 4 0 -11I. 4 5 -29, A/LAVI'PA =9.4 FLARE A NG LE 0 -45. 5 -45. 10 -43. 15 -41. 20 - 40. 25 3f 30 -36. 35 -34. 40 -32. 45 -30. A/LAMbsUA =9.'! F LA RE ANGLL 0 -4?4 5 _1'.2 10 -41. 15 -40. 20 -39,1 6 6)4 61 '57b 12 - 3. 92 - 39.173 -36 7. - 34.217 - 3?2. 6'5 - 30-,. F -? 21. -49. 25 - 41. 83 -?6.69 -314.4 4 -3 2.2 4 - M.0~ - 2 7.97 5 24 -39 * 26 - 3,9.n1 - 38 * 26 -3 7.11 -3-15. 67 - 33.99 - 3 2.1 2 - 3 0.0 -217.31I -If j.4 1 -3b. '35 -3. 58 - 3 3. 6 - 32. U7 - 2 9.93 -21.69 Y -25.06C -12 2.40 36 -38. -lE -38. 29 - 3 6 *lp 6 - 35. C'; - 32. P; - 30. 43 -27. 16 -24'4. 2 -2 1.51 -19.02 -.42 314 -31.621 -31 *17 -35.64 -30.08 -33.10 - 28.2S". -30.30 -25.87 -27.31 -~?.96 -24.1b -19.76 -2iC.98 -16.51 - /.83 -13.41 -14.85 -10.64 54 - 33. 50 - 32.48 -? 9". 9 3 -26,*5 1 -2 2.8 1 -18. 9~ -1).11f -I1. 77 -8. 814 -6.4 7 60 - 1', 2.6 b - 35.01.17.7 4 -.'2.2 7 - 17.4 6 -1I3..,71 -9. * 6 -t6.66 -4.40 -2. (I1 66 -29). O -2 5.* P2 -20-.3j1 -4.1 7 - 2.1 3 -. h -. 1? 72 - 20C.1 1 - 1 7.2 0 - 11.96 - 7. 37 - 3. 6 -1 I 32Z. 3,, 1.35. 1.6-8 1. a," 78 -11. 36f -17. 45 - 3.03. 03 2. 05 3.?21 3. 64 3. 48 27. 93 2.-25 84 3.40 4.04 5. 24 6.0C9 6. 31 5. 91 5. 11 4.19 3.44 2. 82 90 11.64 11. 35 IC.49 9. 14 7. 57 6. 18 5.2 3 4. 56 3. 83 2. 99 H z P-4 C)H" F-h.. 6. to '. & L,94.11C 12 - '.C %* 45:-4'1.?? -) 41 7 1 -1i8 *:",4 - 3 7.7/ - 36.4 i - 34. 9.? - 13. 2') - 3 1.53 -291. 62 18 -38.38 -30P. 2 5 - 37 * 80 - 317.00,, - 36C.11 - 34. )C - 33. 46 - 31.8 2 -?9.9 7 - 27.93 24 -48..8 -4 7.0 6 -4 3.8 5 - 4 C.." - 3e. 2 1 - 35.75 - 3 3.3 7 -30.98 - 28. 53 - 2 5.9 9 30 - 37.4 7 -3 7.26 - 36. 60 - 35.56) ( - 34. 16 -32.4 4 - 30.4?2 -28.1 3 -2 5.6 1 -22.91 36 -45. 21 -4 3.52 -4 0.3 5 - 37. 21 -34.2'g - 3 1.35 -28e.4 3 -2 5. 42 -22. 36 - 19. 32 Phi1 42. 48 -3.3.93 -.,~6.33 -33.65 -35.55 -32.80 -323.53 -31.41 -30.79 -25.49 -27.63 -21.10 -24.17 -24.31 -2C.581 -21.27 -17.05 -18.16 -13.74 -15.14 -10.80 r54 - 51. 86 - 39. 68 -3 3.00 -28.05 -2f."3.58 - 19.3 5 -1 5.41 -11.90 - 8. 90 -6.40 SO - 31.60 - e-54.92 *11 1 9 - C'-1.*81 -17. 39 -13.2 8 -5;.70 -6. 74 -4.44 -2.81 - 24.7 6 -2 3.217 -1 9.5 5 - 15.0 5 - 10.7 8 -7. 1 ' -4.2 9 -2.1 8 -.7'9 -.09 72 -1I8. V6 - 16.-12 - 12.-1 7 - 7.70 - 4. i C -1I. 442.3p 1.40 I. 72 1.50c 78 - 11 * 34 -17. 93 - 3.46 -. 17 2.-C 2 3.-2 7 3. 70C 3.45S 2. 30 2.2I33 84 2. 69 3. 59 5. 14 6. 16 6. 39 5. 93 5. 06 4. 16 3.46 2. 85 n0 11. 83 11. 51 10c.5 7 9. 14 7. 51 6.1 5 5.2 7 4. 60 3 * 82 2. 96 0-4 0 z 6. ()..621 12 - 50. 74 -49. 11 -46. 12 - 4 3.3 1 -40, 834 is8 - 44.20 -43. 74 -42.49 -40.8H3 - 39.00o 24 - 38.16 -- 31. 52 - 36.72 - 3 5.62.30 -64.94 - 50. 76 -44.60 -40. 65 - 37. 49 38 -36. 17 - 35. 94 - 35.2 3 - 3 4.01 -32. 48 PHI 42 48 -38.47 -47.55 -37.88 -42.37 -36.?8 -36.90 -34.06 -32.57 -3-1.44 -28.63 54 - 33. 11 -32. 26 -.30.00 -26. 86 -23. 19 60 -27.97 -27.17 -24.819 -21.49 -17. 51 66 -23.9 1 -22.7 8 -19.6 7 -15.46 -11.1 9 72 - 19.06 - 17.1 7 - 12.79 -8.1I8.-4.-34 78 - 1 1.591 -89. 65 - 3.94 -. 35?. 03 84 1. 89 3. 14 5.ce b.2 3 6.45 90 -12.01 11. 66 10.65 9.12 7. 45

25 30 35 40 45. -37.82 -36.27 -34.62 -32.89 -31.06 -38.61 -36.5.' -34.49 -32.45 -3C.36 -37.1C -35.15 -33.12 -30.99 -28.75 -34.2i -32.61 -30.71 -28.57 -26.21 -34.67 -31.95 -29.20 -26.37 -23.48 -30.5C -28.14 -25.47 -22.58 -19.58 -28.51 -25.34 -22.01 -18.67 -15.47 -24.78 -20.97 -17.29 -13.89 -10.89 -19.32 -15.52 -12.0 -9.00 -6.54 -13.53 -9.92 -6.8E -4.50 -2.81 -7.43 -4.42 -2.19 -.75 -.05 -1.49.42 1.46 1.74 1.48 3.34 3.74 3.4e 2.86 2.23 5.93 5.02 4.14 3.49 2.87 6.13 5.32 4.62 3.80 2.94 H A/LANOA - 10.2 FLARE ANGLE 6 0 -4C.99 5 -40.86 10 -40.42 15 -39. 70 20 -38.75 25 -37.61 30 -36.30 35 -34.84 40 -33.23 45 -31.48 A/LAF6LA = 10.6 12 -39.80 -39.70 -39.31 -38. 7 -37.60 -36.72 -3.44 -33.9T -32.31 -30.47 18 -43.88 -43.48 -42.38 -40.88 -39.17 -37.35 - 5.45 -33.47 -31.37 -29.14 24 -43.93 -43.43 -42.10 -4C.34 -38.38 -36.32 -34.11 -31.90 -29.50 -26.94 30 -38.91 -38.67 -37.94 -36.80 -35.3C -33.50 -31.42 -29.06 -26.47 -23.68 36 -39.54 -39.14 -37.S9 -36. 3C -34.24 -31.66 -29.2C -26.3C -23.22 -2C.C6 PH' 42 48 -47.07 -34.54 -44.15 -34.06 -39.86 -32.66 -36.11 -30.50 -32.67 -c7.14 -29.30 -24.51 -25.87 -21.C0 -22.38 -17.44 -18.93 -14.06 -15.65 -11.02 54 -30.90 -30.40 -28.91 -26.50 -23.30 -19.64 -15.86 -12.30 -9.17 -6.61 60 -28.50 -27.75 -25.58 -22.22 -18.17 -14.02 -10.22 -7.02 -4.52 -2.77 66 -25.42 -24.19 -20.84 -16.31 -11.71 -7.69 -4.49 -2.16 -.69 -.02 72 -20.6C -18.53 -13.74 -8.69 -4.54 -1.50.48 1.51 1.75 1.45 78 -13.05 -9.63 -4.43 -.47 2.07 3.41 3.77 3.45 2.82 2.24 84 1.00 2.70 5.04 6.31 6.51 5.91 4.97 4.14 3.53 2.88 90 12.19 11.81 10.72 9.10 7.39 6.13 5.36 4.63 3.77 2.94 cr 0-' C3H CO3 I C I FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 -58.20 -53. 7( -4 P.. e -43.44 -42.72 -40.4( -38. 2' -36.29 -34.32 -32.32 12 -48.20 -47.44 -45.62 -43.51 -41.4 -39.39 -37.42 -35.4 -33.44 -31.35 18 -40.08 -39.96 -39.51 -38.78 -37.79 -36.56 -35.11 -33.44 -31.55 -29.45 24 -44.36 -42.85 -42.50 -4C.71 -38.7 5 -36.69 -34.54 -32.27 -29.86 -27.29 30 -39.63 -39.4C -36.67 -37.52 -36.01 -34.19 -32.08 -29.69 -27.04 -24.19 36 -49. -46.79 -42.69 -39.16 -36.02 -33.02 -2 9.S -26.85 -23.62 -20.36 PD 1 42 48 -36.16 -34.04 -35.84 -33.66 -34.87 -32.52 -33.33 -30.65 -31.25 -28.11 -28.7C -24.99 -25.74 -21.49 -22.49 -17.86 -19.13 -14.36 -15.85 -11.20 54 -33.91 -33.17 -31.13 -28.13 -24.47 -20.43 -16.37 -12.59 -9.30 -6.64 60 -33.19 -31.69 -28.17 -23.77 -19.05 -14.49 -10.43 -7.08 -4.50 -2.73 66 -30.01 -27.66 -22.73 -17.25 -12.13 -7.83 -4.50 -2.12 -.65 -.01 72 -23.85 -20.80 -14.81 -9.10 -4.64 -1.47.54 1.54 1.74 1.42 78 -14.87 -10.83 -4.86 -.54 2.13 3.47 3.78 3.42 2.81 2.26 84.OC 2.27 5.04 6.40 6.55 5.88 4.93 4.16 3.56 2.87 90 12.35 11.95 10.78 9.07 7.33 6.15 5.40 4.63 3.74 2.94 -j 0 C) 0 z A/LAMBDA s 11.0 FLARE ANGLE 6 0 -41.21 5 -41.12 10 -40.74 15 -4C.11 20 - 39.27 25 -38.23 30 -37.C1 35 -35. 6. 40 -34.Cc 45 -32.34 12 -43.52 -43.30 -42.58 -41.51 -40.20 -38.73 -37.13 -3.41 -33.53 -31.6( 18 -58.67 -53.40 -48.3C -44.73 -41.88 -39.41 -37.12 -34.88 -32.61 -30.24 24 -39.68 -39.54 -39.06 -38.26 -37.17 -35.79 -34.14 -32.22 -30.03 -27.59 30 -60. 32 -51.76 -45.96 -42.08 -38.95 -36.13 -33.39 -30.6C -27.69 -24.66 36 -37.51 -37.67 -36.92 -35.72 -34.C8 -32.C4 -29.61 -26.83 -23.78 -20.58 4JHI 48 -37.36 -41.59 -37.01 -40.13 -35.97 -37.04 -34.32 -33.58 -32.15 -29.96 -29.49 -26.16 -26.43 -22.22 -23.06 -18.30 -19.57 -14.61 -16.15 -11.32 54 -51.24 -41.62 -35.11 -30.13 -25.51 -20.99 -16.66 -12.73 -9.35 -6.64 60 -157.1C -37.69 -30.36 -24.69 -19.47 -14.68 -10.51 -7.09 -4.49 -2.71 66 -47.2 -31.92 -24.04 -17.72 -12.31 -7.88 -4.49 -2.10 -.64 -.0 1 72 -30.72 -23.52 -15.60 -9.33 -4.67 -1.44.57 1.55 1.72 1.41 78 -17.61 -12.12 -5.19 -.56: 2.18 3.51 3.78 3.39 2.eC 2.28 84 -1.13 1.88 5.05 6.47 6.57 5.85 4.91 4.18 3.58 2.85 90 12.51 12.08 10.83 9.03 7.29 6.17 5.44 4.61 3.72 2.96

G-SUB-V (I/LFLARE ANGLE,PHII 8/L ABOA *.4 FLARE ANGLE 6 12 18 24 30 36 42 48 54 60 66 0 2.02 1.94 1.82 1.66 1.46 1.24.99 *74 49 925.04 - 2.3 -. -0 -. S5 2.00 1.93 1.81 1.65 1.45 1.22.98 473.48.24.03 -.14 -.2 -*5 -.3 10 1.97 1.89 1.77 1.61 1.42 1.19.95.70.45.21.03 -1.1 -.37 -.36 -*3 io. 3 -.17 -. -. 39 -.42 1 15 1.90 1.63 1.71 1.51 1. 1.1.13.89.64.39.16 -.05 -.22 -.35 -.44 -.46 20 1.4 1.27 1.05:.8 56.32.09 -.1- -.5 -.46 25 1.69 1.62 1.51 1.35 1.16.94.70.46.22 -.01 -.22 -.39 -.52 -.6C -.62 56.32.08 -.14 -.34 -.51 -.64 -.PH1 -.4 35 1.35 1.28 1.17 1.01.83.62.39.15 -.08 -.30 -.50 -.66 -.79 -.86 -.69 40 1.11 1.04.93.78.60.39.17 -.06 -.29 -.50- -.07 45.82.75.64.50.32.12 -.IC -.32 -.54 -.75 -.94 -.0w -.21 -1.28 -1.09 6/LAMBDA..8 FLARE ANLt: 6 12 18 24 30 36 42PH 48 54 60 9 0 4.95 4.65 4.16 3.48 2.63 1.62.48 -.76 -2.08 66 -7.36 75 5 4-93 4.64 4.14 3.47 -3.41 -4.68 -5.83 -6.76 -7.36 -7.57 4.59 10 0 3.43 2.62 1.61.48 -.77 -2.07 -3.34 -4.67 -5.7. 15 4.81 4.51 4.03 3.36 2.52 1.53.42 -.79 -2.06 -3.33 -4.5' -5.61 -6.46 -7.01 -7.20 20 4.69 4.40 3.92 3.26 2.44 1.46.37 -.82 -2.05 -3.28 -4.44 -5.45 -6.25 -6.76 -6.93 0 25 4.54 4.25 3.78 3.13 2.32 1.37.3C -.85 -2.04 -3.22 -4.31 -5.27 -6.CO -6.47 -6.63 30 4.35 4.07 3.60 2.96 2.17 1.24.21 -.90 -2.04 -3.15 -4.18 -5.06 -5.73 -6.15 -6.30 35 4.11 3.83 3.38 2.76 1.99 1.09 -.97 -2.05 -3.09 -4.34 -4.84 -5.45 -5.83 -5.96 40381 356.89 -.06 -1.06 -2.07 -3.04 -3.91 -4.63 -5.18 -5.51 -5.62 FLARI PiH ANGLE 6 12 18 24 30 36 42 48 54 60 66 78 4 9 11.76 - 35 5.45 4.84 3.83 2.43.69 -1.30 -3.33 -5.04 -6.06 -6.40 -6.35 -6.19 -6.05 -5.97 5 40 5.06 4.48 3.51 2e2O.59 -1.18 -2.89 -4.26 -5.09 -5.42 -5.48 -5*44 -5.39 -5.36 45-362 -4.219 -4.62 -4.74 -4.79 -4.80 -4.61 -4.81 6/LtAM#OA 1.6 FLARE ANGLE 6 12 16 24 30 36 42 48 54 60 66 72 78 84 90 0 7.66 6.42 4.23.84 -4.56 -15o96 -15.62 -8.43 -6.07 -5.29 -5.24 -5.55 -5.95 -6.28 -6.40

10 5 15 20 25 30 35 40 45 7.63 7.54 7.40 7.19 6.91 6.56 6.13 5.60 4.97 6.39 6.32 6.19 6.01 5.76 5.46 5.08 4.63 4.08 4.22 4.17 4.09 3.97 3.82 3.62 3.38 3.08 2.72.85 -4.43 -14.17 -14.C9 -8.16 -5.97 -5.22 -5.20 -5.51 -5.92 -6.25 -6.37.88 -4.06 -11.16 -11.35 -7.47 -5.66 -5.04 -5.07 -5.41 -5.83 -6.16 -6.29.94 -3.52 -8.59 -8.89 -6.53 -5.20 -4.76 -4.87 -5.25 -5.69 -6.03 -6.15 l.00 -2.9C -6.54 -6.89 -5.52 -4.64 -4.41 -4.62 -5.04 -5.50 -5.84 -5.96 1.07 -2.26 -4.92 -5.29 -4.54 -4.04 -4.01 -4.31 -4.79 -5.26 -5.61 -5.74 1.13 -1.64 -3.61 -3.98 -3.64 -3.44 -3.58 -3.98 -4.50 -5.0C -5.35 -5.48 1.17 -1.39 -2.55 -2.92 -2.84 -2.87 -3.16 -3.64 -4.20 -4.71 -5.06 -5.19 1.17 -.62 -1.71 -2.C6 -2.16 -2.36 -2.76 -3.30 -3.89 -4.40 -4.76 -4.88 1.13 -.25 -1.05 -1.38 -1.59 -1.91 -2.40 -2.98 -3.59 -4.11 -4.45 -4.58 -1 B/LAMBDA = 2.0 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 8.40 8.36.24 8.05 7.77 7.40 6.94 6.37 5.68 4.86 12 6.4' 6.37 6.28 6.14 5.93 5.65 5.30 4.88 4.37 3.76 18 2.66 2.66 2.68 2.70 2.72 2.73 2.72 2.67 2.57 2.40 24 30 36 -4.26 -137.42 -7.93 -4.03 -16.09 -7.44 -3.40 -10.13 -6.25 -2.55 -6.71 -4.84 -1.64 -4.36 -3.48 -.78 -2.62 -2.27 -.02 -1.29 -1.25.61 -.26 -.41 I.C1.51.26 1.44 1.07.75 PHI 42 48 54 60 66 72 78 84 90 -4.62 -4.36 -5.70 -8.23 -11.88 -16.87 -24.01 -36.14 -141.67 -4.47 -4.28 -5.64 -8.16 -11.75 -16.53 -22.53 -27.42 -28.09 -4.06 -4.06 -5.48 -7.97 -11.40 -15.62 -19.79 -21.82 -22.03 -3.47 -3.73 -5.21 -7.65 -10.85 -14.40 -17.22 -18.31 -18.44 -2.79 -3.31 -4.86 -7.23 -10.15 -13.06 -15.05 -15.74 -15.84 -2.09 -2.84 -4.46 -6.74 -9.36 -11.73 -13.19 -13.70 -13.78 -1.43 -2.36 -4.01 -6.18 -8.51 -10.45 -11.57 -11.98 -12.06 -.84 -1.90 -3.55 -5.59 -7.64 -9.25 -10.15 -1C.50 -10.57 -.33 -1.48 -3.10 -4.99 -6.78 -8.12 -8.88 -9.19 -9.26.06 -1.11 -2.67 -4.39 -5.96 -7.09 -7.74 -8.02 -8.10 -d c I-4 Td C) I — 0 B/LAMBCA = 2.4 FLARE ANLLE 5 10 15 20 25 30 35 40 45 6 8.90 8.85 8.71 8.46 8.10 7.63 7.04 6.31 5.44 4.41 12 5.92 5.89 5.81 5.67 5.48 5.23 4.90 4.51 4.03 3.47 18 24 30 36 -.31 -22.43 -6.32 -3.45 -.19 -13.16 -5.75 -3.29.16 -7.62 -4.42 -2.85.64 -4.32 -2.9C -2.22 1.19 -2.06 -1.50 -1.51 1.72 -.40 -.30 -.81 2.18.83.67 -.16 2.54 1.73 1.44.41 2.78 2.36 1.99.86 2.86 2.73 2.35 1.18 PHI 42 48 54 60 66 72 78 84 90 -4.72 -9.17 -20.66 -18.72 -11.84 -9.49 -8.57 -8.24 -8.16 -4.63 -9.04 -19.43 -17.97 -11.70 -9.42 -8.52 -8.20 -8.12 -4.36 -8.66 -16.96 -16.24 -11.28 -9.21 -8.37 -8.08 -8.01 -3.94 -8.07 -14.48 -14.26 -10.65 -8.88 -8.14 -7.88 -7.82 -3.43 -7.32 -12.28 -12.35 -9.85 -8.43 -7.82 -7.61 -7.57 -2.87 -6.47 -10.35 -10.61 -8.95 -7.88 -7.42 -7.28 -7.25 -2.29 -5.58 -8.64 -9.03 -8.00 -7.26 -6.96 -6.88 -6.87 -1.73 -4.67 -7.13 -7.60 -7.04 -6.60 -6.44 -6.43 -6.45 -1.21 -3.80 -5.79 -6.32 -6.10 -5.91 -5.89 -5.95 -5.99 -.76 -3.CO -4.62 -5.17 -5.20 -5.22 -5.33 -5.45 -5.50 B/LAMBDA = 2.8 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 9.23 9.17 8.99 8.68 8.24 7.65 6.92 6.04 4.98 3.78 12 4.95 4.95 4.93 4.89 4.82 4.71 4.55 4.31 3.98 3.54 PH 18 24 30 36 42 48 54 60 66 72 78 84 90 -5.92 -8.06 -2.81 -4.72 -13.15 -17.81 -9.16 -7.39 -7.75 -9.15 -10.92 -12.41 -13.01 -4.98 -6.67 -2.57 -4.59 -12.73 -1684 -9.04 -7.33 -7.71 -9.10 -10.86 -12.34 -12.92 -3.03 -4.13 -1.93 -4.21 -11.63 -14.72 -8.68 -7.15 -7.58 -8.97 -10.69 -12.12 -12.68 -1.13 -1.85 -1.09 -3.65 -10.15 -12.41 -8.11 -6.87 -7 -. -10.41 -116 -.1229.50 -.05 -.21 -2.97 -8.54 -10.26 -7.37 -6.47 -7.06 -8.44 -10.04 -11.29 -11.77 1.75 1.32.61 -2.25 -6.94 -8.32 -6.51 -5.98 -. 6.06 -9.57 -10.72 -II.15 2.68 2.34 1.31 -1.52 -5.43 -6.59 -5.58 -5.41 -6.23 -7.60 -9.02 -10.0T -10.44 3.33 3.06 1.86 -.85 -4.06 -5.06 -4.63 -4.79 -5.72 -7.07 -8.41 -9.35 -9.68 3.71 3.51 2.25 -.24 -2.84 -3.72 -3.71 -4.14 -5.17 -6.50 -7.74 -8.59 -8.88 3.82 3.73 2.46.26 -1.79 -2.57 -2.84 -3.49 -4.59 -5.88 -7.04 -7.80 -8.06 0 z

B/LAMBDA - 3.2 FLARE ANGLE 6 0 9.41 5 9.34 10 9.12 15 8.75 20 8.22 25 7.53 30 6.67 35 5.65 40 4.46 45 3.17 8/LAMBCA = 3.6 12 18 3.44 -27.88 3.51 -9.92 3.69 -4.09 3.94 -.82 4.20 1.33 4.41 2.80 4.54 3.80 4.53 4.41 4.35 4.67 3.98 4.61 PHI 24 30 36 42 48 54 60 66 72 78 84 90 -2.97 -3.39 -13.10 -12.83 -7.05 -7.47 -11.35 -20.63 -25.87 -16.79 -14.24 -13.59 -2.46 -3.20 -12.50 -12.34 -6.95 -7.41 -11.26 -20.02 -24.19 -16.56 -14.11 -13.48 -1.27 -2.69 -11.00 -11.07 -6.64 -7.22 -10.97 -18.53 -21.21 -15.92 -13.76 -13.18.10 -1.97 -9.10 -9.40 -6.15 -6.90 -10.52 -16.73 -18.50 -15.00 -13.21 -12.71 1.34 -1.17 -7.16 -7.62 -5.50 -6.46 -9.90 -14.94 -16.21 -13.91 -12.51 -12.10 2.35 -.37 -5.32 -5.88 -4.72 -5.91 -9.16 -13.24 -14.24 -12 -2.75 -11.70 -11.38 3.11.36 -3.66 -4.28 -3.86 -5.27 -8.32 -11.64 -12.49 -11.57 -10.82 -10.58 3.62.99 -2.20 -2.84 -2.98 -4.57 -7.40 -10.14 -10.92 -10.40 -9.90 -9.73 3.88 1.49 -.97 -1.60 -2.13 -3.83 -6.44 -8.73 -9.47 -9.24 -8.95 -8.85 3.87 1.84.03 -.56 -1.35 -3.10 -5.47 -7.41 -8.13 -8.13 -8.00 -7.95 M CJ FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 9.45 9.37 9.12 8.70 8.10 7.32 6.36 5.25 4.02 2.78 12 1.19 1.48 2.21 3.10 3.92 4.56 4.96 5.10 4.94 4.48 18 -8.50 -5.65 -1.90.80 2.67 3.95 4.77 5.18 5.23 4.91 PHI 24 30 36 42 48 54 60 66 72 78 84 90 -1.73 -8.08 -13.83 -6.36 -8.30 -18.96 -17.07 -10.77 -9.30 -9.31 -9.71 -9.92 -1.40 -7.74 -12.96 -6.23 -8.20 -18.35 -16.71 -10.69 -9.25 -9.28 -9.68 -9.88 -.58 -6.80 -10.95 -5.86 -7.92 -16.86 -15.74 -10.47 -9.13 -9.17 -9.58 -9.78.44 -5.48 -8.61 -5.26 -7.47 -15.02 -14.42 -10.10 -8.91 -8.99 -9.41 -9.61 1.43 -3.99 -6.35 -4.47 -6.84 -13.14 -12.93 -9.59 -8.61 -8.75 -9.17 -9.37 2.27 -2.51 -4.31 -3.55 -6.08 -11.31 -11.41 -8.96 -8.22 -8.43 -8.87 -9.07 2.91 -1.14 -2.53 -2.58 -5.22 -9.57 -9.89 -8.23 -7.74 -8.04 -8.49 -8.69 3.34.06 -1.01 -1.61 -4.29 -7.92 -8.42 -7.41 -7.19 -7.58 -8.05 -8.25 3.54 1.05.23 -.72 -3.35 -6.38 -7.01 -6.53 -6.57 -7.05 -7.54 -7.74 3.50 1.79 1.19.05 -2.44 -4.97 -5.68 -5.62 -5.90 -6.46 -6.98 -7.18 < arCa C" — ] C, I z -1 cn Il B/LAMBDA - 4.0 FLARE ANSLE 0 5 10 15 20 25 30 35 40 45 6 9.38 9.29 9.01 8.55 7.91 7.08 6.08 4.96 3.78 2.68 12 -2.24 -1.18.88 2.77 4.19 5.14 5.66 5.76 5.47 4.76 18 -3.15 -2.00.18 2.15 3.64 4.66 5.25 5.46 5.28 4.72 PHI 24 30 36 42 48 54 60 66 72 78 84 90 -2.84 -134.30 -6.31 -7.86 -28.66 -11.51 -8.75 -10.22 -14.28 -21.08 -33.14 -138,67 -2.54 -18.93 -6.12 -7.74 -24.12 -11.37 -8.69 -10.16 -14.18 -20.65 -28.96 -31.10 -1.77 -12.49 -5.54 -7.38 -19.26 -10.95 -8.52 -10.01 -13.88 -19.55 -24.36 -25.02 -.77 -8.41 -4.64 -6.79 -15.72 -10.29 -8.24 -9.75 -13.40 -18.12 -21.07 -21.40.28 -5.32 -3.50 -5.99 -12.91 -9.42 -7.84 -9.38 -12.77 -16.60 -18.54 -18.76 1.24 -2.86 -2.27 -5.04 -10.50 -8.38 -7.33 -8.91 -12.01 -15.11 -16.47 -16.63 2.05 -.88 -1.06 -4.00 -8.36 -7.22 -6.71 -8.34 -11.15 -13.66 -14.68 -14.82 2.68.68.05 -2.92 -6.44 -6.00 -5.98 -7.67 -10.21 -12.27 -13.08 -13.21 3.10 1.87.99 -1.88 -4.71 -4.77 -5.18 -6.92 -9.21 -10.93 -11.61 -11.73 3.27 2.69 1.71 -.93 -3.19 -3.58 -4.34 -6.10 -8.17 -9.64 -10.24 -10.36 0 0 z.-4 O B/LAMBDA s 4.4 FLARE ANGLE 0 5 10 6 12 9.19 -8.27 9.09 -3.99 8.81.37 18 -1.03 -.31 1.22 PHI 24 30 36 42 48 54 60 66 72 78 84 90 -6.78 -8.95 -6.06 -22.03 -10.28 -8.67 -14.13 -33.78 -15.14 -11.93 -10.98 -10.79 -6.30 -8.41 -5.91 -20.14 -10.13 -8.60 -13.99 -28.40 -15.01 -11.87 -10.94 -10.75 -5.03 -6.95 -5.47 -16.82 -9.70 -8.40 -13.60 -23.35 -14.62 -11.70 -10.82 -10.64

15 d.35 3.13 2.75 -3.34 -5.02 -4.76 -13.74 -9.00 -8.06 -12.97 -19.90 -14.02 -11.42 -10.61 -10.45 20 7.69 4.87 3.97 -1.58 -2.98 -3.83 -11.03 -8.06 -7.59 -12.15 -17.25 -13.25 -11.04 -1C.33 -10.19 25 6.87 5.90 4.79.06 -1.10 -2.76 -8.60 -6.92 -6.97 -11.17 -15.04 -12.35 -10.55 -9.96 -9.85 30 5.90 6.35 5.23 1.45.52 -1.65 -6.41 -5.66 -6.22 -10.07 -13.11 -11.36 -9.98 -9.51 -9.43 H 35 4.83 6.30 5.29 2.55 1.82 -.59 -4.44 -4.34 -5.37 -8.89 -11.34 -10.29 -9.31 -8.S9 -8.94 40 3.76 5.74 4.97 3.31 2.78.35 -2.70 -3.05 -4.44 -7.64 -9.70 -9.18 -8.57 -8.39 -8.38 45 2.78 4.67 4.28 3.71 3.37 1.13 -1.22 -1.83 -3.47 -6.37 -8.14 -8.04 -7.76 -7.72 -7.75 T 8/LAMBOA = 4.8 FLAR E PHI Z ANGLE 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90, 0 8.88 -41.02 -.55 -19.44 -5.15 -11.60 -11.43 -8.35 -17.80 -15.98 -10.53 -10.48 -12.29 -14.39 -15.35 5 8.79 -4.77.01 -14.64 -4.89 -11.26 -11.17 -8.26 -17.43 -15.76 -10.47 -10.44 -12.23 -14.32 -15.26 10 8.54.94 1.25 -9.33 -4.15 -10.32 -10.43 -8.01 -16.44 -15.14 -10.31 -10.32 -12.09 -14.10 -15.00 tT1 15 8.11 3.94 2.58 -5.36 -3.01 -8.92 -9.28 -7.58 -15.07 -14.21 -10.34 -10.11 -11.84 -13.75 -14.58 20 7.51 5.68 3.67 -2.28 -1.64 -7.23 -7.83 -6.97 -13.51 -13.07 -9.65 -9.83 -11.50 -13.28 -14.03 25 6.76 6.61 4.44.12 -.25 -5.42 -6.2C -6.18 -11.86 -11.78 -9.16 -9.45 -11.07 -12.70 -13.36 ( 30 5.86 6.88 4.85 1.95 1.02 -3.62 -4.52 -5.25 -10.18 -10.40 -8.55 -8.98 -10.54 -12.02 -12.60 35 4.87 6.57 4.92 3.26 2.07 -1.94 -2.89 -4.21 -8.50 -8.97 -7.82 -8.42 -9.92 -11.26 -11.77 C 40 3.85 5.68 4.64 4.08 2.83 -.45 -1.39 -3.12 -6.84 -7.52 -7.00 -7.77 -9.21 -10.42 -10.87 Co-H 45 2.89 4.21 4.01 4.42 3.27.77 -.10 -2.05 -5.25 -6.09 -6.09 -7.04 -8.43 -9.52 -9.91 w0< 8/LAMBDA = 5.2 0 FLARE PHI ANGLE 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 0 8.45 -9.40 -1.38 -12.30 -5.50 -21.53 -7.61 -16.17 -13.60 -9.83 -13.32 -26.01 -22.32 -16.83 -15.69 5 8.39 -2.83 -.87 -10.55 -5.28 -19.01 -7.49 -15.84 -13.43 -9.77 -13.23 -24.92 -21.85 -16.69 -15.59 r 10 8.20 2.07.33 -7.13 -4.64 -15.06 -7.15 -14.91 -12.95 -9.61 -13.00 -22.66 -20.65 -16.31 -15.30 15 7.88 4.83 1.68 -3.79 -3.63 -11.57 -6.56 -13.59 -12.18 -9.33 -12.61 -20.33 -19.13 -15.72 -14.83 20 7.41 6.40 2.89 -.98 -2.37 -8.53 -5.74 -12.01 -11.17 -8.93 -12.38 -18.21 -17.54 -14.97 -14.23 0 25 6.77 7.12 3.83 1.26 -1.03 -5.84 -4.71 -10.30 -9.97 -8.41 -11.42 -16.31 -15.99 -14.10 -13.50 30 5.96 7.16 4.45 2.94.25 -3.48 -3.54 -8.51 -8.62 -7.75 -10.62 -14.57 -14.49 -13.14 -12.69 35 5.00 6.55 4.73 4.08 1.37 -1.44 -2.30 -6.72 -7.17 -6.96 -9.71 -12.94 -13.05 -12.13 -11.80 40 3.94 5.30 4.65 4.70 2.25.25 -1.10 -4.97 -5.68 -6.06 -8.69 -11.38 -11.65 -11.08 -10.85 45 2.88 3.46 4.17 4.78 2.84 1.56 -.02 -3.32 -4.21 -5.07 -7.59 -9.88 -10.28 -9.99 -9.86 Z SINE-SQUARED PHI VS. PHI 6 12 18 24 30 36 42 48 54 60 -19.61 -13.64 -10.20 -7.81 -6.02 -4.61 -3.49 -2.58 -1.84 -1.25 66 72 78 84 90 -.79 -.44 -.19 -.05 -.00

ts/LAMMO8A a FLARE ANG LL 0 5 1 0 1 5 20 2 5 30 35 40 4 5 5.6 6 7.8e4 7. 71 * 91 6. 1 3 *,; 2. 70 -.62 *, 175 6.89 7 * I4C 6. 26 4.72 2. 70 * is -3.71 - 3.09 -14.56. 26 1.99 3. 41 4.42 4.98 5.*04 4.5S6 24 -5.91 - 5.34 -3.80 -1.74.3 5 2.14 3.51 4.41 4.7 9 4.*64 30 -10.00 -9.60 -8.48 -6. 82 -4.85 -2. 83 -.9. 74 2.06 2.98 36 -8. 56 -7.87 -5.26 - 3. 58 -.2E 2. 14 42Ph -10.84 -10.68 -1I0c.1I9 -9. 39 -8. 30 -6. 97 -5.47 -3.*90 -2. 36 -.94 -1 -1 - 1 -1I -1 48 54.5.C9 -9. 60.4.79 -9.53.3. 96 -9. 33 2. 12 -$3*99 1. 2 0 -8.49 -9.5S1 -7.84 -7. 71 -7.02 -5. 9 -6.0'. -4.12 -4.9 7 -2. 41 -3.81 60 -1 7.00 -16. 79 -16.20 -1b.32 -14.21 -12.96 -1LI1.60. -10.16 -E.66 -7.13 66 -19.5 9 -19.26 -18.3 1 -17.1 7 -15.7 9 -14.3 3 - 12.8 5 -11.3 5 -9.b4 -8.3 3 72 - 12. 34 - 12. 28 -12. 11 -11.8H3 - 11.44 - 10. 94 -10.3 3 -9.6 1 -8. 78 -7. 86 78 ~I1.23 -1 I 19.I -11 I. 09 -1C..9C -10.64 -IC0. 3C - 9. 88 -9.3 7 - 8. 76 -8.C7 84 - 11. 56 -11. 53 -11.42 -11.25 -11.01 -1C.69 -1C.29 -9.81 -9.25 -8. 61 90 -11.*84 -11.80 -11.70 -11.52 -11.27 -10. 95 -10. 55 -10.07 -9.51 -8. 86 H M~AM'8A a6.0 FLARE ANGLE 0 5 1 0 1 5 20 2 5 30 4 4 3 5 40 45 B/LAPOCA a FLARE ANGLE 0 5 1 0 1 5 20 2 5 30 35 40 45 6 7. 2L 1. 29 7.49 7.13 I1.56 7. 15 6. ~3 5. 13 3 * 6,9 2.4 1 12 -1.14. 91 3.98 6.03 7.14 7.43 7.00 5.86 4.913 2.25 is - 7. 11 -4.131 -1.00 1.61 3.58 4.89 5.55 5.52 4.79 24 30 24 -4.04 -132.58 -7.12 -3.67 -20.65 -6.96 -2.59 -14.01 -6.5C -1.03 -9.68 -5.10.66 -6.16 -4.61 2.19 -3.1') -3.29 3.38 -.71 -1.66 4.16 1.27 -.46 4.4? 22.74.79 4.27 3.67 1.19 PHI1 42 -34.87 -25.*25 -19.32 -15.37 -12.21 -9.43 -6.89 -4.5 7 -2.49 -.7C 48 54 -9.-11 -16.69 -8.84 -15.98 -8. 38 -14.-92 -7. 72 -13.62 -6.84 -12. 16 -5. 78 -10. 57 -4. 55 -8.91 -3.24 -7. 19 -1.-94 -5.48 60 -15.21 -15. C6 -!4.tb2 - 13.-9 3 -13.C2 -11.94 -10.71 -9.936 -7.92 -6.43 66 -1C.93 -10.8 8 -10.74 -10.5 1 -10.1 7 -9.7 2 -9.1 6 -8.4 7 -7.6 6 -6.7 3 72 -13.23 -13. 18 - 13.00 - 12. 72 - 12.32 -11 I.81 - 11.19 -10.46 -9.62 -8.67 78 -19. 46 -19.2 5 -1 3. fLe -1 7.P4 -16. 82 -1 5.69 -14.951 - 13. 30 - I2. c -10.179 84 -31.39 - 29.03 -25.45 -22.49 -2C. IC -18.08 - 16. 31 - 14.70C -13. 19 -1 1.75 90 - 135. 19 - 32. 86 -26. 78 -2 3.1 5 -20. 5r -18. 36 -16. 52 - 14. 88 -13. 36 -11 I.91I C:1 z IM4 411 0,l Cl) CDz 6.4 6 6.34 6.061 7.1,9 7.*68i 7.8C 1 * 42 6.46 5.cc 3 e*33 2.1If 12.37 1.94 4.43 6.21 7.14 7.,*t28 6.69) 5.44 3.76 2.26 is -24.88 - 12.2 7 -S.49 -1.04 2. 10 4.25 5.55 6.06 5. 77 4.65 24 -4.63 -4.29 -1.81 -.14 1.45 2.7 7 3.71 4.21 4.21 -10.58 -8.58 -6.46 -4.05 -1.68.43 2. 15 3.38 4.07 24 -10. 71 -10.47 -9.177 -8.*64 -7. 14 -5. 38 -3. 5C -1.66. cc 1. 37 PHI 42 48 -lC.7C -12.58 -1C.53 -12.45 -lC.CC -12.C3 -9.112 -11.36 -7.92 -10.42 -6.43 -9.2'5 -4.75 -7.87 -3.00 -6.33 -1.32 -4.71.17 -3.09 54 -15.66 -15.45 -14.85 -13.93 -12. 75 -11. 38 -q).85 -8.20 -6.49 -4. 77 60 -10.81 -10. 75 -10.59 -10.31 -9.90 -9.36 -8.68 -7.84 -6.o86 - 5.75 66 - 17.88e -17.7C - 17.19 -16.4 1 -15.4 3 -14.3 1 -13.0 9 -11.7 8 -10.3 9 -8.9 4 72 -22.94 - 22.44 -21.20 - 19.62 -17. 97 -16. 35 -14. 78 - 13.24 - 11. 73 - 10. l22 78 -14.-5 3 - 14. 46 -14.2 5 -11.92 - 13.46: - 12. 38 -1.'.2C -1 1.4 1 - 1 C 5 4 -9q.5S7 84 -12. 74 - 12.170 -12.51I - 12. 36 - 12.07 - 11.69 - 11.22 - I C.6 7 - IC.02 -9.29 90 -12. 42 -12. 38 -12.2 7 -12.08 - I11.8a1 -11.4 7 -1 1.05 -10. 54 -Co. 95"' -9. 27 8/LAMOCA a 6.8 FLARE ANG LE 0 6 13 is 5.30 I.Cl -12.14 -7?.90 -6.66 36- 42 PI 46 54 60 66 72 78 84 91) -47.96 -8.86 -25.21 -1C.40 -20.32 -15.64 -11.86 -13.07 -15.57 -16.86

5 10 15 20 25 30 35 40 45 5.87 6.98 7.85 8.11 7.67 6.48 4.7L 2.90 7.13 2.35 4.52 6.13 6.94 7.01 6.39 5.19 3.71 2. 5 - -H.86 -3.94.06 2.99 4.96 6.06 6.30 5.67 4.16 -7.40 -5.99 -3.95 -1.67.49 2.32 3.68 4.47 4.60 -6.41 -5.63 -4. 35 -2.68 -.86.86 2.31 3.34 3.87 -24.C5 -17.63 1 3.42 -9.97 -6.91 -4.16 -1.73.31 1. 1C 1. 9C. -8.75 -8.41 -7.81 -6.95 -5.83 -4.49 -3.CC -1.51 -.12 -23.38 -20.15 -17. 14 -14.45 -1 1.96 -9.59 -7.30 -5.11 -3.08 -10.34 -10.13 -9. 78 -9.27 -8.59 -7.72 -6.67 -5.45 -4.13 -19.96 -19.00 -17.67 -16.16 -14.57 -12.93 -11.25 -9.52 -7.77 -15.52 -15.18 -14.64 -13.92 -13.04 -12.02 -10.87 -9.62 -8.2 6 -11.81 -11.69 -11.48 -11.19 -10.80 -10.31 -9.71 -9.00 -8.16 -13.C3 -12.90 -12.68 -12.38 -11.98 -11.49 -10.90 -10.21 -9.42 -15.50 -15.29 -14.96 -14.50 - 13.94 -13.27 -12.52 -11.67 -10.74 -16.77 -16.50 -16.07 -15.51 -14.83 -14.05 -13.19 -12.25 -11.25 H B/LAMHCA = 7.2 FLARE ANGLE 0 5 10 15 20 5 30 35 40 45 -00 00 6 4.C4 5. 1(: 6.S1 8. 1'8.45 7.8e6 6. 37 4.24 2.55 2.42 12 1.15 2.27 4.26 5.30 6.63 6.76 6.25. 20 3.87 2.70 18 -6.05 -4.80 -1.88 1.22 3.71 5.39 6.23 6.21 5.3C 3.57 -24 -17.72 -14.84 -10.28 -6.14 -2.56.39 2.66 4.22 5.02 5.01 -6.91 -6.68 -5.99 -4.82 -3.28 -1.53.20 1.72 2.87 3.56 -%6 -11.41 -11.11 -10.22 -8.81 -6.97 -4. 5 -2.67 -.61 1.15 2.5C Phi 42f 48 -14.33 -10.91 -14.06 -10.80 -13.30 -1C.48 -12.11 -9.93 -1C.57 -9. 13 -8.76 -8.C9 -6. 77 -6. 80 -4.69 -5. 31 -2.67 -3. 71 -.82 -2.10 54 -16.29 -16.10 -15.55 -14.71 -13.61 -12.31 -10.83 -9.21 -7.47 -5.67 60 -14.67 -14.56 -14.22 -13.67 -12.93 - 12.CO -1C.9C -9.64 -8.23 -6.72 66 -11.97 -11.92 -11.78 -11.53 -1 1.19 -10.73 -10.14 -9.4 3 -8.57 -7.56 72 -18.79 -18.62 -18.15 -17.42 -16.51 -15.47 -14.34 -13.13 -11.85 -10.5C 78 -29.78 -28.15 -25.24 -22.56 -20.28 -1 8.3C -16.53 -14.89 -13.33 -11.82 84 -18.76 -18.61 -18.19 -17.54 -16.73 -15.80 -14.79 -13.72 -12.60 -11.44 90 -17.11 -17.01 -16.72 -16.26 -15.65 -14.93 -14.11 -13.21 -12.24 -11.21 B/LAMBUA = 7.6 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 B/LAMHCA = FLARE ANGLE 0 5 LQ 15 20 25 30 35 4Q 45 PHI 6 2.47 4.37 6.9SF 8.1' 7 8.17 7.9 6. 1C 3.7C 2.47 2.95 12.63 1.70 3.67 5.31 6.32 6.63 6.33 5.39 4.00 2.63 -.51.61 2.80 4.78 6.15 6.77 6.59 5.59 3.92 2.31 18 -3.55 -2.76 -.66 1.85 3.98 5.43 6.08 5.90 4.90 3.25 24 - 15.89 -13.53 -9.33 -5.25 -1.68 1.24 3.42 4.82 5.39 5.05 30 -11.32 -10.9C -9.7C -7.85 -5.58 -3.13 -.79 1.25?.81 3.79 36 -8.1 -8.CC -7.5C -6. e 5 -5.43 -3.eG -2.14 -.41 1.13 2.33 42 -20.77 -19.73 -17.43 -14.82 -12.20 -9.59 -7.03 -4.55 -2.26 -.26 48 -11.12 -11.03 -10.73 -10.23 -9.51 -8.55 -7.35 -5.93 -4.35 -2.73 54 -17.82 -17.55 -16.78 -15.64 -14.25 -12.67 -10.96 -9. 13 -7.22 -5.28 80 -11.77 -11.71 -11.54 -11.25 -10.83 -10.27 -9.55 -8.66 -7.60 -6.38 U6 -26.92 -25.83 -23.56 -21.1 9 -19.02 -17.04 -15.19 -13.42 -11.67 -9.92 72 - 15.94 -15.85 -1.58 -15.14 -14.55 -13.82 -12.97 -12.01 -10.93 -9.74 768 -12.72 -12.68 -12.57 -12.38 -12. 1C -11. 74 -11. 29 -10.75 -10. 1C -9.34 84 -12.83 -12.79 -12.69 -12.51 -12.27 -11.94 -11.54 -11.05 -10.47 -9.80 90 -13.16 -13.13 -13.02 -12.84 -12.59 -12.26 -11.85 -11.35 -10.77 -10.09 z.2 -04 < m Pd 0) CA) I-4 Q0 z 8.0 6I.5C 3.77 7.20 8.86 9.C5 1.94 5.68 3.20 3.53 3.53 18 -2.79 -2.14 -.39 1.81 3.77 5.13 5.76 5.63 4.75 3.33 24 -8.04 -7.43 -5.68 -3.14 -.40 2.04 3.90 5.03 5.34 4.73 30 -131. 29 -21.88 -15.24 -10.7C -6.87 -3.51 —.61 1.72 3.4C 4.34 36 -10.2 2 -10.C3 -9.44 -8.45 -7.C6 -5.32 -3.37. —3^ -1.39.42 1.5C PHI 42 48 -10.42 -25.69 -10.28 -24.01 -9.86 -20.95 -9. 13 -18.01 -8.08 -15.35 -6.70 -12.85 -5.03 -10.43 -3.20 -8.04 -1 36 -569.31 -3.46 '4 -11. 14 -11.07 -10. 86 -10.51 -9,98 -9.28 -8.37 -7.24 -5 9Z -4.44 Of -24.71 -23.91 -22.08 -19.98 -17.92 -15.95 -14.04 -12.15 -10.24 -8.31 66 -13.84 -13.77 -13.57 -13.24 -12.76 -12.15 -1 1.3 9 -10.49 -9.43 --8.2 2 72 -13.03 -12.98 -12.85 -12.63 -12.32 -11.91 -11.39 -10.76 -10.00 -9.11 76 -18.40 -18.28 -17.92 -17.37 -16.65 -15.80 -14.85 -13.81 -12.70 -11.52 -30.15 -28.67 -25.92 -23.33 -21.11 -19.17 -17.44 -15.84 -14.33 -12.87 90 -135.66 -34.12 -28.03 -24.40 -21.74 -19.60 -17.76 -16.10 -14.56 -13.08

H/LAtlUA = 1.4 FLA R ANGLE 0 5 10 15 20 25 30 35 40 45 6 -2.1'l 1. 3 j 9.24 9.25 5. 15.1) '.93 3.3c, 3. ""s 12 -2.44 -1.0 1. I7 t,.-40 6f.22 7.n0 6. )9) 5.65 3 5 1 1. 6 18 -3. 36 -2.75 -1.* 0 1.15 3.11 4. t 5r.594 4.13 3.$2 24 -5.65 -5.26 -4.0 -2.1 2.18 3.82 4.03 5.02 4.4(0 5.( 2 30 -11. 76 -11.24 -9.77 -7.36 -4.91 -2.15.38 2.45 3. 8 4.55 36 -22.73 -20.72 -17.19 -13.78 -10.58 -7.5C -4.57 -1.68.42 2.2C -1 -1 - I - I P I 42 48 54 0.31 -13.81 -15.78 C.19 -13.64 -15.63 9.82 -13.17 -15.19 9.19 -12.38 -14.49 8.25 -11.30 -13.54 7.01 -9.93 -12.33 5.47 -8.29 -11.CO 3.73 -6.44 -9.43 1.92 -4.46 -7.68 -.22 -2.49 -5.82 60 -14.30 -14.21 -13.93 -13.48 -12.85 -12.04 -11.05 -9.87 -8 50 -6.98 66 -13.14 -13.63J -13.50 -13.2 1 -12.19 -12.25 -11.56 -10.74 -9.15 -8.[,0 72 -41.85 -32.e -27.21 -23.62 -20. 94 -18.1 4 -16.80 -15.02 -13.32 -11.65 78 -17.CC -16.9C -16.63 -16. 2C -15.62 -14.12 -14.10 -13.18 -12.17 -11.07 84 -14.06 -14.02 -13.89 -13.67 -13.37 -12.98 -12.49 -11.92 -11.25 -1C.48 90 -13.60 -13.56 -13.44 -13.26 -12.99 -12.65 -12.22 -11.71 -11.10 -10.40 H Z B/LAMtUA = d.8 FLARt ANGLc 0 5 10 15 20 25 30 35 40 45 PHI 6 12 18 24 30 36 42 -5.82 3.31 7.93.5 ) '. 3! 7. 56 4.57 3.93 4.22 -5.5 1 - *.30.dZj 4.32 6.54 7.4 ' 1.13 5.4:) 3. 2 1.83 -5.42 -4.68 -2.68 -.04 2.49 4.46 5. 6 5. 3 5.?0 3.49 -5. 392 -j.47 -4.37 -2,6! -.47 1. 4.3 3.33 4.44 4.86 4.43 -7.78 -7.53 -6.74 -5.38 -3.51 -1.35.76 2. 5' 3. T7 4.3C -14.5C -14.C5 -12.81 -10.96 -8.66 -6. C -3.45 -.95 1.1 2.19 -19.15 -18.55 -17.02 -14.99 -12.73 -10.32 -7.81 -5.26 -2.81 -.60 48 54 60 66 72 78 -1C.85 -20.25 -12.76 -30.79 -13.68 -13.6e -i0.77 -19.85 -12.70 -28.70 -13.63 -13.64 -10.50 -18.79 -12.51 -25.31 -13.48 -13.52 -10.C5 -17.34 -12.20 -22.39 -13.24 -13.32 -9.37 -15.66 -11.76 -19.93 -12.90 -13.04 -8.46 -13.86 -11.16 -17.79 -12.44 -12.67 -7.27 -11.96 -10.40 -15.d4 -11.8f -12.2C -5.83 -9.96 -9.46 -13.99 -11.19 -11.64 -4.19 -7.88 -8.32 -12.16 -10.37 -10.97 -2.46 -5.75 6 -6.9 -10.33 -9.41 -10.19 84 - 16. 38 -16.31 -16.11 -15.79 - 15.36 -14.82 -14.18 -13.43 -12.60 -11.67 90 -17.98 -17.88 -17.61 -17.18 -16.61 -15.92 -15.14 -14.26 -13.31 -12.28 c( Ciz I - r{,-j CD B/L AIBUA = ).2 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 a,. 3 5 3. 53 9.09 9.42 7.21 4.C 3.32 4.4 '; 4. 9 i2 -11.21 -5.77.4 4.61 7.0, 7. 4 7.2') 1.. 1 ), 1:3 2.4.?. *. 18 -9.85 -P.4C -5.C -1. 16 2.2C 4.66 6.06 6.30 5.28 3. C ) 24 -8.54 -.06 -6.64 -4.43 -1.19.80 2.97 4.46 5.11 4.75 '30, -7. 9 7 -7. 14 -7.02 -5.77 -4.02 -1.94.17 2.C2 3.39 4.C8 PHI it w -42r -9.31 -9.14 -8. l -1. 68 -6.32 -4.55 -2.52 -.45 1.39 2.77 -16.61 -16.22 -1 5.15 -13.57 -11.62 -9.39 -6.97 -4.46 -2.04.10 48 -17.17 -16.91 -16.18 -15.06 -13.63 -11.94 -10.04 -7.94 -5.72 -3.50 54 60 -11.86 -11.78 -11.57 -11.20 -10.66 -9.93 -8.98 -7.79 -6.36 -4.75 -32.12 -29.17 -25.12 -21.91 -19.28 -16.99 -14.88 -i2.84 -10.81 -d.74 66 72 -13.17 -16.12 -13.11 -16.05 -12.96 -15.3 -12.7C -15.47 -12.32 -14.99 -11.83 -14.36 -11.20 -13.61 -10.42 -12.74 -9.47 -11.74 -8.35 -1C.61 78 -63.05 -34. 34 -2P.25 -24.6C -21.92 -19.74 - 1 7.e -16.14 -14.51 -12.94 84 -2C.39 -2C.22 -19.75 -19.04 -18.15 -17.16 -16.09 -14.96 -13.79 -12.58 90 -18.17 -18.07 -17.78 -17.33 -16. 72 -16.00 -15.18 -14.28 -13.30 -12.25 P-4 C) 0 z) B/LAIbLA = 9.6 FLARE ANGLt 6 0 -33.37 5 3.'1; 10 38.17 12 -3P.0n -6. ' 7.76 18 -23.13 -13.61 -6.56 24 -16.' 3 -14.tC -1C.78 30 -12.34 -11.9C -10. 65 36 -1C.C3 -9.86 -9.33 PHI 42 48 -10.51 -18.92 -1C.38 -18.53 -1C.01 -17.47 54 -15.39 -15.26 -14.89 60 -14.07 -13.99 -13.76 66 -16.55 -16.45 -16.18 72 -21.32 -21.OH -20.42 78 -14.C8 -14.C3 -13.91 84 -13.79 -13.75 -13.65 90 -14.18 -14.14 -14.03

15 20 25 30 35 40 45 IU.14 9.3U 6.77 3.61 3.d9 4.87 3.,G 5.13 -1.35 7.46 2.55 J.09 5.17 7.08 6.51 4.55 6.51 2.22 5.06 2.87 2.47 -6.70 -2.80.58 3.21 4.90 5.53 4.92 -8.69 -6.2C -3.42 -.7C 1.66 3.41 4.36 -8.42 -9.36 -7.C8 -8.39 - 5.33 - 7.07 -3.2e -5.41 -1.15 -3.49.81 -1.5C 2.36.35 -15.97 -14.19 -12.20 -10.04 -7.73 -5.34 -3.00 -14.29 -13.46 -12.40 -11.12 -9.60 -7.86 -5.95 -13.37 -12.82 -12.09 -11.18 -10.07 -8.75 -7.22 -15.13 -14.35 -13.43 -12.36 -11.14 -9.7 6 -19.47 -18.33 -17.08 -15.77 -14.41 -12.99 -11.5C -13.7C -13.4C -13.01 -12.53 -11.95 -11.25 -1C.43 -13.47 -13.23 -12.90 -12.50 -12.01 -11.42 -1C.73 -13.85 -13.59 -13.26 -12.84 -12.34 -11.75 -11.05 H b/LAMBCA = 10.0 c z 60 66 72 78 84 90 e4 FLARE ANGLE 0 5 10 15 20 25 30 35 40 45 6 -11.: 4.53 10.3 J 'J. 2'" 6.2 d 3. 3i 4.5C 5.C3 3.39 12 -12.46 -5.23 I 54 2.42 3.46 18 24 30 -14.79 -19.66 -130.34 -17.E4 -11.06 -16.17 -22.85 -17.17 -5.42 -11.07 -16.17 -15.45 -.51 -6.42 -11.53 -13.15 3.24 -2.25 -7.5C -1C.52 5.69 1.23 -3.8C -7.67 6.79 3.82 -.54 -4.74 6.45 5.37 2.C9 -1.91 4.59 5.73 3.93.57 2.00 4.73 4.79 2.49 PHi 42 -1Z. 10 -11.96 -11.53 -10.79 -9.72 -8.3C -6.52 -4.47 -2.32 -.28 48 -11.54 -11.45 -11.17 -10.69 -9.98 -9.00 -7.73 -6.15 -4.33 -2.41 54 -23.15 -22.51 -20.94 -19.01 -16.99 -14.95 -17.86 -10.71 -8.47 -6.17 -13.82 -13.75 -13.55 -13.21 -12.73 -12.09 -11.27 -10.25 -9.02 -7.58 -20.82 -20.57 -19.90 -18.92 -17.73 -16.42 -15.01 -13.53 -11.95 -10.2 6 -13.49 -13.44 -13.32 -13.12 -12.82 -12.43 -11.94 -11.33 -10.58 -9.69 -17.69 -17.6C -17.35 -16.95 -16.41 -15.74 -14.96 -14.08 -13.1C -12.01 -29.20 -28.19 -26.06 -23.83 -21.79 -19.95 -18.28 -16.72 -15.22 -13.76 -133.62 -35.10 -29.01 -25.38 -22.72 -20.58 -18.73 -17.06 -15.51 -14.02 or) cn 0) " W0 Ig B/LA^RCA = 10.4 FLARE ANGLE 0.F5 10 15 20 25 30 35 40 45 6 -5.a4 5.14 9 49 10.43 9.C4 5.79 3.78 5.01 4.98 2.78 12 -6.he 3.06 2.40 6.17 8.03 8.09 6.38 3.60 2.91 3.03 1t -8.01 -6.71 -3.38.54 3.84 5.99 6.80 6.16 4.11 2.01 24 -9.87 -9.19 -7.25 -4.33 -1.03 1.98 4.25 5.52 5.58 4.31 30 -12.6S -12.17 -10.71 -8.45 -5.62 -2.55.34 2.71 4..29 4.86 PHI 36 42 48 -18.65 -28.41 -13.98 -17.e1 -25.31 -13.85 -15.76 -21.03 -13.47 -13.16 -17.48 -12.83 -10.29 -14.37 -11.93 -7.25 -11.42 -10.74 -4.17 -8.5C -9.24 -1.27 -5.59 -7.45 1.21 -2.80 -5.42 3.C4 -.31 -3.27 54 -12.57 -12.49 -12.27 -11.89 -11.33 -10.57 -9.58 -8.33 -6.80 -5.05 60 -46.42 -32.55 -26.54 -22.81 -19.99 -17.60 -15.43 -13.33 -11.23 -9.08 66 -13.3C -13.25 -13.11 -12. 7 -12.53 -12.0 7 -11.48 -10. 4 -9.H3 -8.73 72 -23.13 -22.81 -21.98 -20.82 -19.49 -18.09 -16.67 -15.22 -13.74 -12.19 78 -19.64 -19.50 -19.11 -18.51 -17.74 -16.85 -15.85 -14.77 -13.61 -12.37 84 -15.14 -15.09 -14.96 -14.73 -14.42 -14.02 -13.52 -12.93 -12.24 -11.44 g0 -14.53 -14.48 -14.37 -14.18 -13.91 -13.57 -13.14 -12.62 -12.01 -11.30 1-4 C) 0 z

- wrl TABLE C-2: GROSS APPROXIMATION G-SUB-U(A/L, PHI) A/L 1.0 1.4 1.8 2.2 2.6 3.0 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6 -6.69 -12.73 -16.62 -19.53 -21.88 -23.85 -25.55 -27.04 -28.38 -29.59 -30. 70 -31.72 -32.66 12 -6.32 -12.40 -16.31 -19.23 -21.58 -23.55 -25.25 -26.75 -28.09 -29.30 -30.41 -31.43 -32.37 18 -5.74 -11.84 -15.78 -18.71 -21.07 -23.05 -24.75 -26.25 -27.59 -28.81 -29.92 -30.93 -31.88 24 -4.98 -11.02 -15.01 -17.97 -20.34 -22.33 -24.04 -25.54 -26.88 -28.10 -29.21 -30.23 -31.17 3C -4.09 -9.91 -13.98 -16.98 -19.37 -21.36 -23.08 -24.59 -25.94 -27. 16 -28.27 -29.29 -30.24 36 -3.12 -8.46 -12.65 -15.70 -18.12 -20.13 -21.86 -23.38 -24.73 -25.95 -27.07 -28.09 -29.04 42 -2.13 -6.59 -10.96 -14.C9 -16.55 -18.59 -20.33 -21.86 -23.22 -24.45 -25.57 -26.6C -27.55 PHI 48 54 -1.14 -.22 -4.31 -2.22 -8.80 -6.00 -12.05 -9.46 -14.58 -12.09 -16.66 -14.23 -18.43 -16.04 -19.97 -17.62 -21.35 -19.01 -22.58 -20.26 -23.71 -21.40 -24.74 -22.44 -25.7C -23.40 60 66.62 1.35 -.41 1.11 -2.61.13 -6.05 -1.70 -8.88 -4.48 -11.13 -6.97 -13.01 -8.99 -14.63 -10.71 -16.06 -12.20 -17.33 -13.53 -18.49 -14.72 -19.55 -15.80 -20.52 -16.83 72 1.94 2.30 2.21 1.64.57 -1.04 -3.21 -5.16 -6.81 -8.25 -9.53 -10.68 -11.72 78 2.37 3.16 3.67 3.89 3.86 3.60 3.11 2.39 1.42.18 -1.32 -2.69 -3.91 84 2.63 3.68 4.54 5.2C 5.72 6.12 6.42 6.62 6.76 6.81 6.81 6.74 6.61 90 2.72 3.85 4.82 5.64 6.33 6.93 7.46 7.93 8.36 8.75 9.11 9 44l z Pd C, m "c 0') C3H $-3 I Ili 9.75 r -4 C) 0 z

- w6 A/L 6.2 6.6 7.0 7.4 3D —.i — ___ 0. a c 7.8 3 8.2 8.6 9.0 9.4 9.8 10.2 10.6 11.0 6 12 -33.54 -33.25 -34.36 -34.07 -35.14 -34.85 -35.87 -35.58 -36.56 -36.27 -37.21 -36.92 -37.84 -37.55 -38.43 -38.14 -39.C0 -38.71 -39.55 -39.26 -40.07 -39.78 -40.57 -40.28 -41.06 -4C.77 18 -32.76 -33.58 -34.36 -35.09 -35.78 -36.43 -37.06 -37.65 -38.22 -38.77 - 3'9.29 -39.80 -40.28 24 -32.05 -32.88 -33.65 -34.38 -35.08 -35.73 -36.36 -36.95 -37.52 -38.0'7 -38.59 -395.C9 -39.58 30 36 -31.12 -29.93 -31.94 -3C. 75 -32.72 -31.53 -33.45 -32.26 -34.14 -32.95 -34.80 -33.61 -35.42 -34.24 -36.02 -34.83 -36.59 -35.40 -37.14 -35.95 -37.66 -36.47 -38.16 -36.98 -38.65 -37.46 _42 -28.43 -29.26 -30.04 -30.77 -31.47 -32.13 -32.75 -33.35 -33.92 -34.47 -34.99 -35.5C -35.98 PHI 48 -26.59 -27.42 -28.20 -28.94 -29.63 -3C.29 -3C.92 -31.52 -32.09 -32.64 -33.16 -33.67 -34.15 54 -24.3C -25.13 -25.92 -26.66 -27.36 -28.02 -28.65 -29.25 -29.82 -30.37 -30.90 -31.40 -31.89 60 -21.43 -22.27 -23.06 -23.80 -24.51 -25.17 -25.81 -26.41 -26.98 -27.54 -2R.06 -28.57 -29.06 66 -17.72 -18.58 -19.38 -20.14 -20.85 -21.52 -22.16 -22.77 -23.35 -23.90 -24.44 -24.95 -25.44 78 -12.68 -13.57 -14.40 -15.18 -15.91 -16.60 -17.25 -17.87 -18.46 -19.03 -19.57 -20.09 -20.59 78 -5.00 -5.99 -6.91 -7.75 -8. 54 -9.28 -9.98 -10.63 -11.26 -11.85 -12.41 -12.95 -13.47 84 6.42 6.18 5.87 5.50 5.08 4.59 4.03 3.40 2.69 1.92 1.19.51 -.13 90 10.03 10.30 10.56 10.80 11.03 11.24 11.45 11.64 11.83 12.01 12.19 12.35 12.51 cO ( - fa) cWH I Pz I 0 P7l C) 0 z

G-SUB-V(B/L, PHI) B/L PHI 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90.4 2.02 1.94 1.82 1.66 1.46 1.24.99.74.49.25.04 -.13 -.27 -.35 -.38.8 4.95 4.65 4.16 3.48 2.63 1.62.54 -.37 -1.11 -1.70 -2.17 -2.52 -2.76 -2.90 -2. 15 1.2 6.59 5.90 4.75 3.09 1.31 -.10 -1.22 -2.14 -2.87 -3.46 -3.93 -4.28 -4.52 -4.67 -4.1 1.6 7.66 6.42 4.23 1.85.06 -1.35 -2.47 -3.38 -4.12 -4.71 -5.18 -5.53 -5.77 -5.91 -5.96 2.0 8.40 6.40 3.27.88 -.91 -2.32 -3.44 -4.35 -5.09 -5.68 -6.15 -6.50 -6.74 -6.88 -6.93 2.4 8.90 5.92 2.48.09 -1.70 -3.11 -4.23 -5.14 -5.88 -6.47 -6.94 -7.29 -7.53 -7.68 -7.72 2.8 9.23 5.25 1.81 -.58 -2.37 -3.78 -4.9C -5.81 -6.55 -7.14 -7.61 -7.96 -8.20 -8.34 -8.39 3.2 9.41 4.67 1.23 -1.16 -2.95 -4.36 -5.48 -6.39 -7.13 -7.72 -8.19 -8.54 -8.78 -8.92 -8.97 3.6 9.45 4.16.71 -1.67 -3.46 -4.87 -5.99 -6.91 -7.64 -8.23 -8.70 -9.05 -9.29 -9.44 -9.48 4.0 9.38 3.70.26 -2.13 -3.92 -5.33 -6.45 -7.36 -8.10 -8.69 -9.16 -9.51 -9.75 -9.89 -9.94 4.4 9.19 3.28 -.16 -2.54 -4.34 -5.74 -6.87 -7.78 -8.51 -9.11 -9.57 -9.92 -10.16 -10.31 -10.35 4.8 8.88 2.91 -.53 -2.92 -4.71 -6.12 -7.24 -8.15 -8.89 -9.48 -9.95 -10.30 -10.54 -10.68 -10.73 5.2 8.53 2.56 -.88 -3.27 -5.06 -6.47 -7.59 -8.50 -9.24 -9.83 -10.29 -10.64 -10.89 -11.03 -11.08 00 CO t/L Phi 6 12 18 3 30 42 4C 54 60 66 72 78 84 90 5.b 8.21 2.24 -1.2C -3.59 -5.38 -6.79 -7.91 -8.82 -9.56 -10.15 -10.62 -10.97 -11.21 -11.35 -11.4( 6.0 7.91 1.94 -1.50 -3.89 -5.68 -7.C9 -8.21 -9.12 -9.n6 -1n.45 -10.92 -11.27 -11.51 -11.65 -11.7( 6.4 7.63 1.66 -1.78 -4.17 -5.96 -7.37 -e.49 -9.40 -IC.14 -10.73 -11.20 -11.55 -11.79 -11.93 -11.9E 6.8 7.37 1.39 -2.05 -4.43 -6.23 - 7.63 -8.76 -9.67 -1C.40 -11.CC -11.46 -11.81 -12.05 -12.20 -12.24 7.2 7.12 1.15 -2.29 -4.68 -6.47 -7.8e -9.CC -9.92 -10.^5 -11.24 -11.71 -12.06 -12.30 -12.45 -12.49 7.6 6.88.91 -2.53 -4.92 -6.71 -8.11 -9.24 -10.15 -10.89 -11.48 -11.94 -12.29 -12.54 -12.68 -12.71 8.0 6.66.69 -2.75 -5.14 -6.93 -8.34 -9.46 -1C.37 -11.11 -11.70 -12.17 -12.51 -12.76 -12.9C -12.95 8.4 6.45.49 -2.96 -5.35 -7.14 -8.55 -9.67 -IC.58 -11.32 -11.91 -12.38 -12.73 -12.97 -13.11 -13.1E 8.8 6.25.28 -3.17 -5.55 -7.35 -8.15 -9.88 -1C.79 -11.,2 -12.12 -12.58 -12.93 -13.17 -13.32 -13.3E 9.2 6.05.08 -3.36 -5.75 -7.54 -e.94 -1C.07 -IC.98 -11.72 -12.31 -12.77 -13.12 -13.37 -13.51 -13.56 9.6 5.87 -.1' -3.54 -5.9 -7.72 -9.13 -10.25 -11.16 - 11.90 -12.49 -12.96 -13.31 -13.55 -13.69 -13.74 10.0 5.59 -.28 -3.72 -6.11 -7.9C -9.3C -1C.43 -11.34 -12.08 -12.67 -13.13 -13.48 -13.73 -13.87 -13.92 10.4 5.52 -.45 -3.89 -6.26 -8.07 -9.48 -10.6C -11.51 -12.25 -12.84 -13.30 -13.65 -13.90 -14.04 -14.09 H z C4;d CI, Th 0). CA0 IT P" D 4 0 i Q 5 i p

THE UNIVERSITY OF MICHIGAN 6633-1-F +11 C C, I I I I I.. i. I I... I i I i - I I I - -,I -r NO. DATE a/X = 1.0 L. I - I -- - I I I i. i 1- I - M - - ~ 7.... -2C LIl -40 -- ~~ --- —- —. —. — --- - -. —.. —..- -- -- -.- __ 9 pU L I' 1 - _I _ _ _ _ 0 1 i I.r.1900 284k

THE UNIVERSITY OF MICHIGAN 6633-1-F + 10. T I I -. / I.- - I NO. DATE a/X = 1. 4. 0 I j I '-lo i I I I I I I I I I I II a'. ot, z.0 -",g - I - - - ---- 2.... A- - - - - - -. --- arFIG. 0-1(b): (-r G) in db vs.0 2u -40i __ ___. ____ _ _ 0 1 717111.... oI I I I I. a so - - i I I. I I I 1 -, I -— L 4..I. + 1 i i i — ~ — + i 285

THE UNIVERSITY OF MICHIGAN 6633-1-F +t0 -........ DATE...___.;T- --- a/X= 1.8 -1o — -- rI -;- - --— { —0- - " - ^vs<i;^"'S -i-4 -........... _i I ~ ~_ I - -!,.:' | | - -i '. - -. I. - ' --- — 1..... —. --- —. ------ ------ I - _....! -;....., - i — -- i-. ' _ -. 1 -: -,, FIG. C-l(c): (fG) indbvs. - - - - -- 40- 1 -"1 ' ': 900 I 1 - -.11 '! I. i i -.... - -t- - ' I i _, 1. i- I I -I-~ -I. I I I: I I I, I I -' -I - - i I I i - i 1 - i a 0; i 1! 1 i 1 1:; 1 i 1 1 i i I i! 1; I .- - 286

THE UNIVERSITY OF MICHIGAN 6633-1-F +10 _ _ _ _ _ ___ _ _ _ _ _ _ NO. DATE _ — - -a/X =2. 2 LAJ -3C0___ FIG. C-l(cD: (G in dbvs. ~ __ __ -401 90 0 'II 01 90 -I — -- -t- i i - -1 287

THE UNIVERSITY OF MICHIGAN 6633-1-F +10 - _ _ _ _ _ _ _ NO. DATE - a/X 2.6 00 -20 CL -30U i I. —I I i i 0 90 ( idb vs. 0 FIG. C-l(e): (-! G ) 2 u in -40 a 2 i ---- i IL- i i 0 -1. 90 0 288

THE UNIVERSITY OF MICHIGAN 6633-1-F ad% I -. +1. -- -- - - ~-: I. --- - -. --- -. - -. - -~U — I- l. LX-' _ __~ -c I- c-. -.- - I ~ —t- ~- -t-~ — t-i- ~ -11'.-,- -..,-+~ - -~ ~ --- NO. DATE a/X = 3.0 -I -?- -t-+ i~~t1 ~-; -4 ---1~-. — — / --- r 1 —1~_~. ~ 1.-. — -10 --20 > i ~-r-( i- — i 4 --- -i~-I - -~ I~- - -—; --- — I-~ ---- -- -- wI --— 4-, t ----' --- —--- I ---— ~~ -. ---- — ~ -_____ - - ___ -303 FIG. C-l(f): (! G )in db vs. 2 u 4.I j __ __ 9Q9P 0 - I I _ _ i i ___. __ —t i I.,, I L._' i. i I..I. I - I- - j. 900 i I- - I I - +- e - ~ I I I 289

THE UNIVERSITY OF 6633-1-F MICHIGAN 290

THE UNIVERSITY OF MICHIGAN 6633- 1-F 291

THE UNIVERSITY OF MICHIGAN 6633-1-F I; 292

THE UNIVERSITY OF MICHIGAN 6633-1-F NO. DATE a/X = 4.6 0 293

THE UNIVERSITY OF MICHIGAN 6633-1-F a/X= 5.0 0 I 294

THE UNIVERSITY OF MICHIGAN 6633-1-F NO. DATE a/X = 5. 4 0 K a. i I I 295

THE UNIVERSITY OF MICHIGAN 6633-1-F NO. DATE a/X = 5. 8 0 296

THE UNIVERSITY OF 6633-1-F MICHIGAN 297

THE UNIVERSITY OF MICHIGAN 6633-1-F NO. DATE b/X=.8 0 I 0' -20 — J LUJ -30 -~~~~~i — - -- ---------- ------ -- - - - - _ - i C-2(b): (G cos 0)in db vs.0 V ~ ~ ~ ~ ~ ~ ~ ____ - ----------- -40 l SO ~ ~ - I i ~ ~ I ~ ~ - ~ F go,, 10 I A oh a'o I I...: I 0. I I _ i I.. i I -. I I! I i -K I - --------- - t - I. i i ~i i - I 298

-I 0 + r') H z rci z

THE UNIVERSITY OF 6633-1-F MICHIGAN 300

THE UNIVERSITY OF MICHIGAN 6633.-1-F I +1o - NO. DATE I...- -b/X =2.0 I-20 1-30 C-2(e): (G cos 0) in db vs.0!__ 40. I -vw I I I I I I 900?1 1-i- -. 900l 301

THE UNIVERSITY OF MICHIGAN 6633-1-F I 302

THE UNIVERSITY OF MICHIGAN 6633-1-F +10 --- —---------------- NO. DATE -. —.... -................ -. -. -.................... -. --- b/X = 2.8 0.........._.................................._....... 0 I -- ---- — ( -c- 0-) n — - vs..40-1 1 ----111V-' --- —--— ' —1 --- —------------------------- -20............ 4 — 0..C-2(: ( cos... in.. vs.t......................... C-2()4Ocs0)idv.~..... - 90.. 0... 7... —7 — 906:! -! —I -- - --- I I,! I i I I 303

THE UNIVERSITY OF MICHIGAN 6633-1-F +10 _ _ _ _ _ _ _ _ _ _N O. - - 4- -DATE - - - ------- -17 7 ~ 77 ~ '- ~ -b/X = 3.2 -20 -30T C-02(h): (G COB 0) in dbv 1 -40 oi 4 -~ I,!! I I I -I t --- - I I I I I I i.. I I -I p i i I I i l a i 1; d! 1 ;,! 1 i... - -T. o I; i i i I i i i I I i-~ I 50)4

THE UNIVERSITY OF MICHIGAN 6633-1-F I I 305

THE UNIVERSITY OF MICHIGAN 6633-1-F +10 NO. DATE - - --- ---.. ---. - —.. - -. -- b/X 4. 0 0 -2C LU. -30 -.. --- —...- - - - -- 2 - - - - C-2(j): (G cos 0) in dbvs.0 -4C1 90 v 0 1 - i -- -- —.- -T — I - I goo 506

THE UNIVERSITY OF MICHIGAN 6633-1-F I 307

THE UNIVERSITY OF MICHIGAN 6633-1-F +10 NO. DATE ___ b/X= 4.8 0 LiJ -22 C-() ( o 0 nbJs 9.0'.. I! I I I!. I 0 I j,.;:, I I '- - -7 ---t ---- -,: I. I..... t, 1 4 i — n I I i i I I i. - 1 I ii~1 i, i! i I I. I I i i I I I I I --- --: i 1 1 1 i 'm, i f i: I -F-F- — TI III Iif I I:.! 1 i;! i I: i,. -;;! i, I I. I I t I I i r! I 3o8

THE UNIVERSITY OF 6633-1-F MICHIGAN 309

-- - THE UNIVERSITY OF MICHIGAN 6633-1-F C. 2 Experimental Data: E-plane Coupling With reference to Fig. 4-24 the E-plane coupling for two identical E-Sectoral horns at a center-to-center spacing R of 45. 7 cm is given in Figs. C-3a through C-3e, for five frequencies: 8 GHz, 9 GHz, 10 GHz, 11 GHz, and 12 GHz. Wedges were used to obtain several flare angles. These wedges were silverpainted and held in place by adhesion. The coupling patterns for 0b=10 and 150 are dropped in the figures for readability. The flare angles and aperture dimensions are listed in Table C-3. E-plane coupling for H-sectoral horn is indicated in Figs. C-4a through C-4e for the same five frequencies as above. Wedges were used here to obtain several different flare angles. (See Table C-4), The coupling patterns for 0 =10 and 200 are dropped by 10 db for readability. a TABLE C-3 b a b 10~ 2.3 cm 3.9 cm 15~ 2.3 cm 5.9 cm 23~ 2.3cm 8.2cm 30~ 2.3 cm 8.2 cm 40~ 2.3 cm 8.2 cm TABLE C-4 a a b 100 3.4 cm 1.0 cm 20~ 5.1cm 1.0cm 35~ 8.2 cm 1.0 cm 50~ 8.2 cm 1.0 cm -- 310 -- --

THE UNIVERSITY OF MICHIGAN 6633-1-F Figure C-5 demonstrates the variation of coupling with spacing. The spacings of 11.43 cm, 22. 86 cm, and 45.72 cm are in the ratio of 1:2:4 and thus are successively a distance octave apart. In the far-field, with no scatterers present, the corresponding couplings should be spaced at 6 db apart. These curves demonstrate the filling-in of the nulls as the spacing is reduced within the Fresnel zone. In all the figures that follow, Ot and Or are as indicated in Fig. 5-1 where the base of the horn should be interpreted as "slot". 311

THE UNIVERSITY OF MICHIGAN 6633-1-F ol 'N ol M4 -IT Tl Tj 4A 1910+f+~H44196r IihI7 2T —T 9 I I 1 l2 1 f* l I I I I 11 i -H i I 14 - _ 'O I 0 1 I I 5 I 1 W I II I.1 I I I I I I_ I I I l ] I 1 - I l II I 1 I 1 I a 1i ANGLE il LI FIG. C-3a: E-PLANE COUPLING FOR E-SECTORAL HORNS f=8.03 GHz, D=45.72 cm, t=0~, -180~- 0- 180~ ( — ) 0=10~, 0=-40db; ( A) 0b=15~, 0=-20db; (-) b=23~ 0=-50db; (-G-) 6 300, 0=-50b; (-+-) 3=40~0 0=-50d; () =4, 0=-50db. b b 312

THE UNIVERSITY OF MICHIGAN 6633-1-F FIG.C-3: E-LAN COPLIN FO E-ECTOAL ORN b b 313..LA

THE UNIVERSITY OF MICHIGAN 6633-1-F o - --- — - - - - - 4 I -, 42. 2+ - 2 - I 6 I __ 1T T I- -I i; I --— II — - -- JL i V XI Ail k 71 1w X TT WI rw. AL T-77 Jj -FFT 1-T-T 7 t41 -A _L ANGiLE FIG. C-3c: E-PLANE COUPLING FOR E-SECTORAL HORNS f= 10. 03 GHz, D=45. 72 cm, 0t00o, -180O. —* 0. — 1800 (*) e ioo,00-40db; (- 3 —) 0 150, -40db; (- 9 =230 0-50db; (-a e — 3Q0 000-5Odb; (+)e 4Q0, 0=-50db 314

THE UNIVERSITY OF MICHIGAN 6633-1-F IL4ZYN0 i 44 1 72 iw I ANGLE FIG.- C -3d: E-PLANE COUPLING FOR E-SECTORAL HORNS f= 11. 03 GHz, D=45. 72 cm, ot= 00., -1800 —4 0r-4+ 1800 00 0 (-*-)e = io 0=-4Odb; (-a —) e=15 00=-40db; ( e23I 0=-50db; b 0 b b (-0-) 0 30 0 0=-50db; (-+-) 0 40, 0=-5Odb; b '. b 315

THE UNIVERSITY OF MICHIGAN 6633-1-F -4-4 10 I 100 VI Oa, NN v i WL iA I an I ANGLE I I FIG. C-3e: E-PLANE COUPLING FOR E-SECTORAL HORNS f=12.03 GHz., D=45.72 cm, =Ot, r1O- - 8 ( — 10 0 0=-40 db; A- -) 6 =1500 0-4Odb; (- 6 230 0= 50dI1; b b 'b (e)0 =30,0 0-50db; ( e- 0 400, 0=-50db; b b 316

THE UNIVERSITY OF 6633-1-F MICHIGAN I7 272 j-4KK I 0 r274i I I uvo *yr I ~ t m'hly f t7ZTll I -- ttr 1 ' 1 I T' I 1~oU........... I. I............ I -,L — l - - a I..,. j.......... I I I.. 60 I'll - 1. ANGLE Ii FIG. C-4a: E-ILANE COUPLING FOR H-SECTORAL HORNS f=8.03 GHz, D=45.72 cm, t=00~, -1800 — 0r - 1800 ( —X) e =oo10 0=-20db; ( - ) 0 =20~, 0=-20db; a a ( —)e =35~0 0=-30db; ( —) 0 =50~, 0=-30db; a a 317

THE UNIVERSITY OF MICHIGAN 6633-1-F i —K 9j 6-4-t 41I jj~5 #7I Io~ t720I Ttt-tj-16 I t72tttHill 14.t-H I w.......... I. I.......................I...... I.. I..... w ANGLE I t tt. FIG. C-4b: E-PLANE COUPLING FOR H-SECTORAL HORNS f=9.03 GHz, D=45.72 cm, Ot00, -i0- r~+10 (**)0 =100, 0=-20db (- 0 =200, 0=-2Odb; a.0a. (-.) e =350, 0=-30db; (-.) 6 =50, 0=-30db; a a. 318

THE UNIVERSITY OF MICHIGAN 6633-1-F 1~t -T 44 7?~~ 36 11I FIG.- C -4c: '~ I ANGLE E-PLANE COUPLING FOR H-SECTORAL HORNS f= 10.O 3GHz, D=45. 72 cm, t~ -1800-*~ 0 - 1800 (-*-) 0i0, 0=-20db; ( —)0 =200, 0= 120db; (- =~350, 0=-3Odb; ( ---e =500, 0= -30db. 319

THE UNIVERSITY OF MICHIGAN 6633-1-F t —4 'PN 1 T I r ivi I I i I -loll I Til 1 1171. 1 1 1 1 1 1 11 I Iii I I I I- - 19O'-1 14N I i I 9s I F2?-6 II s6 qsI I I r-Hj-l2tthfl195Hjj1t44 - I~o~ I I I II I I II 1I I-M L L. I I'i I 1 1 II I II II I I I I I I I I I I I I I Y 11L4 4 1 I I I L 1 1 1 L I r FIG. C-4d: ANGLE E-PLANE COUPLING FOR H-SECTORAL HORNS f= 11.03 GHz, D=45. 72 cm, Ot0o, r1800* 0-' (-K-) )=10 n, 0=-20db; ( ) =200, 0=-20db; a a.9 0 =)35 0, 0=-30db; ( — =50 50 0=-3Odb. a a. 1800 320

THE UNIVERSITY OF MICHIGAN 6633-1-F - - - - - - e - - - - - - -- - - I-,- _ _ _ _ I I [T- IW I~~~~~~ Ae i S ShE i2EHd~ I yi - - - - - — I - - --- - - - - - - - - - _ _ I i 1, 1 1 IA I I / V I ' sI I - -^ -- WII I a l\ IT I i1 T I I T I I IT - 1 I i"-H Uli ll A,i| t I:T I Il VA I_ lZIZZt~4-ttiZZ"4^ 'ci "^ 'I -*ff~~ — S-C-C'I-CW7~F-C --- — -- l —fi --- 4 1u. ANGLE I I I~ FIG. C-4e: E-PLANE COUPLING FOR H-SECTORAL HORNS f=12.03 GHz, D=45.72 cm, t-0~, -180~ — 0 -- 180~ ( ---) a =10o, 0=-20db; ( -) 0 =20~, 0=-20db; a a, 0=-30db. ( —) 0 =35~, 0=-30db; ( e[ ) 0 =50~, 0=-30db. a ' a

THE UNIVERSITY OF MICHIGAN 6633-1-F 00 —t —t 1ll 1 -IeT r 1 I - \AI l rm -40. l~f-i~ —k~ CCC~~~ ILi~1 11-? Ir rr I 31r ''75 t +1B~tt I ANGLE FIG. C-5: VARIATION OF E-PLANE COUPLING WITH SPACING FOR E-SECTORAL HORNS =230 180 b 23, f=10.03 GHz, 0t=0, -180~- 0r- 180~ (- -)D=11.43 cm, 0=-40 db; (- -) D=22. 86 cm, 0=-40 db; (-)D=45.72 cm, 0=-50 db. 322

THE UNIVERSITY OF MICHIGAN 6633-1-F C. 3 Experimental Data: H-plane coupling Figs. C-6a through C-6e show the H-plane coupling for two identical E-sectoral horns. The coupling is only near-field, in the sense that the receiving antenna is at a null of the transmitting antenna's far-field pattern. Minimum coupling occurs for the T-formation, i.e. when t = + 90 and 0 = 1800r 00or t r vice versa. This applies to any rectangular aperture using linear polarization. The center-to-center spacing is 11.43 cm. The designations of table C-3 apply here as well. For the H-sectoral horns, Table C-4 applies; the coupling patterns are shown in Figs. C-7a through C-7e. These antennas have a fan-shaped beam, and consequently have high energy concentrations in certain regions of the ground plane. Thus the sensitivity to scattering objects is considerable. Since the direct coupling is so low, the interference can dominate. Fig. 8 shows the variation of coupling with spacing, where, as in section C. 2, the spacings are in the ration 1:2:4. Ideally, these curves would be about 12 db apart. The sensitivity of the H-plane coupling to interference is demonstrated quite vividly. The anechoic chamber had been tested and improved (see Section 7.1) far beyond what would be encountered in the field. Thus it would be expected that except for the very close spacings used here, the H-plane coupling will be largely determined by scattering. This is particularly true of H-sectoral horns, or any antenna that radiates a significant amount of its energy along the surface, or in the direction of a reflecting object. 323

THE UNIVERSITY OF MICHIGAN 6633-1 -F --- -- -— 2 - ---- -- -----: [ - - - -i- -- - - _- - - -. - - - - - - - I II?,t X\: ili i 1:ijli i f- [il I:\\ FIG. C-6a: H-PLANE COUPLING FOR E-SECTORAL HORNS f=8.03.GHz, D=11.43 cm, 0.!90~, -180O.. 0 l 1800 ( —) \ 1 "1i, 0=-3db; (- / -) 1 l5 0=-3I db; 1 1 1 11% d 0=-4db,\ =3, 0=-4Odb; (-+-) e4o =-4Odb -— 7 f=8.03 GHzi, 4 cm, D=1.4 r p - F i ~ 18 ( I ).=10I 0=-30db; (.....15. 0=.-30db; (.)..=23. 0=-40db, I, b,_LP:t l~ i... 324

THE UNIVERSITY OF MICHIGAN 6633-1-F I I'A i M VI i I v 1 /17 I I V11,_1 i A I / v T I- I X7 I 'II Tv I X H I Li I J\.1N OC I v N, I I 1A k v UA I I k I I W I I I-IL J IT I L 11 11/1 71 - *T I -— I L CL W T1 1 I T I 1 t I A 72 42! - I I I. XNGLE FIG.- C -6b: H-PLANE COUPLING FOR E-SECTORAL HORNS f= 9.03 GH z, D= 11. 43 cm, Ot 90180 \T L1b 0.~ 0= ''; 1, 0=-30db; 0 0 e 300=-40db;(-) 0 =300, 0=-4Odb; 0=b 325

THE UNIVERSITY OF MICHIGAN 6633-1-F I 0 N\W IF A I 'K I V11117I y k4\1 -1 I NAR /F"'N ll' N U N f I 'M i %(II -2 IIFI Allflill II I k I I I IT IJr L Ito / I I — I I i I I I " - -.4 — -4- -4-t 1 t I I Hill Tt_ I 7-11 -42- -72!-+ AN A De FIG. C -6c: H-PLANE COUPLING FOR E-SECTORAL HORNS f= 10. 03 GHz, D= 11. 43 cm, Ot r~-8~- -p10 e 2300=-40db; ( 0G~ e30 0 0=-40db; ( —, 0 0=-4Odb. 326

THE UNIVERSITY OF 6633-1 -F MICHIGAN yL I v I I U i 7 IN 7d 1 I L"L I N I I V-1 I X 1.7 Al I IX I IT FT I 11 44f-L —L-J I i I ililIl'Il t-ml I I I i7-25 1 i 11 I I I I I I I i 14. t, ho....-...., _,, 6.. b..... -.............. I a I I I I I I I I I I I I I I i I at) - -lb... - - - -..L4 AN~LE I FIG. C-6d: H-PLANE COUPLING FOR E-SECTORAL HORNS f= 1.03 GHz, D= 11. 43 cm, 0~Otr~80~ ~10 (- b= e 100 0=30d7D -s- e =150 0=-3Odb; -)e =230, 0-4Odb(e) 0 =300 0=-4Odb; (-+-)& 400, 0-40db. b 327

THE UNIVERSITY OF MICHIGAN 6633-1-F 10T rT T L ' l a d V ti"N N i6 -Ad_______ 2V -4 --- 4. 4 7A-4-I f% ~I I FIG. C -6e: H-PLANE COUPLING FOR E-SECTORAL HORNS f=12.03 GHz, D= 11. 43 cm, t r - 10 A 0 100, 0=-3Odb;(&- 0 150, 0=-3Odb; b Yb -)0 =23, 0=-4Odb;( —) 0 300, 0=-4Odb; b b 0b =400, 0=-40db. 328

THE UNIVERSITY OF MICHIGAN 6633-1-F 61......... N 20 -I'm I 1h, i I I ki 11 I I X I /V I I I I I v I I I /17 I 11A I W, I Nl I I I A I I 11 I v ji > I AwAA 84 I I I I 1. I I I 'O! PWI f-72* I I I -f61 1 144 -t I 1! 1 I -I H W,11 'I 'I H, [- I L I I I-6.... W L B. I I 1-1 I I I B- I I I I I 2 I I I I I I I I I I -1 I I I I M FIG. C -7a: ANGLE H-PLANE COUPLING FOR H-SECTORAL HORNS f=8.03 GHz, D=11. 43 cm, 0=-4Odb, 0 =90r.10...~~10 ( — e 100; ( e =200; ( —)O =350 (i-) =500. a a a a 329

THE UNIVERSITY OF MICHIGAN 6633-1-F I 1 I I I _ I I I I V. I i i.20 2'o VIYIX 2 -y 10 1w 2. -+ - 72' t-36% '''- ' '$`t+ ---72S-+- ~tSt ANGLEr o0 1 FIG. C-7b: H-PLANE COUPLING FOR H-SECTORAL HORNS f=9.03 GHz, D=11.43 cm, 0=40db, 0t=90~ -180~- 0 180 (-X —) =10, (- O) 6 =20~; ( —) 0 =350; (-+-) 0 =50~ a a a a 330

THE UNIVERSITY OF MICHIGAN 6633-1-F Hill A X I I IN 1 I IMI 1 7 LLI air \1 lilt I I I y IN I I N I uj "-t* -7 I TI T I I 1-4 kr f ANIPLE I I III FIG. C-7c: H-PLANE COUPLING FOR H-SECTORAL HORNS 0 00 ( — =100; (-e- 9 =200; (-~) e =350; (i- 9 =500 a a a a 331

THE UNIVERSITY OF MICHIGAN 6633-1-F 16 —7 18 10 LI 4-1 Al 20 — HII -40 30 IV i 100 ++ "law l 0' FIG. C-7d: H-PLANE COUPLING FOR H-SECTORAL HORNS f=11.03 GHz, D=11.43 cm, 0=-40db, t=90~, -180 ~ — 0- 180~ (-)0 =10~; ( — ) 6 =20~; ( ) 0 =35~; ( -) =50~. a a a a 332

THE UNIVERSITY OF MICHIGAN 6633-1 -F -7-7 A 10 7 1, I F Lj 71-1 I L F T)'-i I L I 7 7, I ---Al IV rw I _TP ]ILI Mal I I I i I I I I I IWI I I I t I II R I 5I-H t -H tul 1 1-72 11 1,4 1 1 1 i i -11 I I i i H I t-I 0* -!I I I I I I I I-....-... -.. -..... -.-....... -. -... -.... -1 I I m 4 ANGLE FIG. C-7e: H-PLANE COUPLING FOR H-SECTORAL HORNS f=12.O3 GHz, D=11. 43 cm, O=-40Odb, Ot=90, -1r-~~10 (-4) =100; (-e-) =200; (~)9 0350; (-+-) =500 a a a a 333

THE UNIVERSITY OF MICHIGAN 6633-1-F 10 -I2 -I I _ 2' II I 11 T 00 I I I III 44N 11111', t i ft Hl"h ti2" I I I 1 36" 1 1 1 1 1 1. tZ-wIh 1 [93 m e I1 1144 ioll I L I i I -I -6 I I I I I I I I m I I I I I I - I -i -I_ I Iu.. AN, I LE FIG. C-8: VARIATION OF H-PLANE COUPLING WITH SPACING FOR E-SECTORAL HORNS f=8. 03 GHz, 9 =300 0= -40db, t=900 -180 r- 0 — )1800 b 0 0 Top, D= 11. 43 cm; Middle, D=22. 86 cm; Bottom, D=45. 72 cm. 334

: UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA R&D (Security claeailication of title, body of abstract and indexing annotation must be entered when the overall report is claasilied) 1. ORIGINATING ACTIVITY (Corporate author) 2. REPORT SECURITY C LASSIFICATION The University of Michigan Radiation Laboratory UNCLASSIFIED Department of Electrical Engineering 2 b GROUP Ann Arbor, Michigan 48108. 3. REPORT TITLE DERIVATION OF AEROSPACE ANTENNA COUPLING-FACTOR INTERFERENCE PREDICTION TECHNIQUES 4. DESCRIPTIVE NOTES (Type of report and inclusive detee) Final Report, June 1964 through September 1965 5. AUTHOR(S) (Last name, first name, inltisl) ' Lyon, John A. M., Kalafus, Rudolph M., Kwon, Yong-Kuk. Digenis, Constantine J. Ibrahim, Medhat A. H. Chen, Chao-Chun. 6. REPORT ATS. 7. TOTAL NO. OF PAGES 7b. NO. OF REFS April 1966 334 21 Se. CONTRACT OR GRANT NO. *4. ORIGINATOR'S REPORT NUMSBRS) AF 33(615)-1761 6633-1-F b. PROJECT NO. 4357, Task 435705 c. Ib. OTHER RPPORT N 0) (A4ny othor nmbera hat may he casJtded this reportO d. AFAL-TR-66-57 10. AV A IL ABILITY/.IMITA ION NOTIC.ES Qualified requesters may obtain copies of this report from DDC 11. SUPPI.EMENTARY MOTES I.' SPONSORING M1l61TARY ACTIVITY Air Force Avionics Laboratory, AVWC Research and Technology Division, AFSC...____Wright-Patterson Air Force Base, Ohio 45433 13. ABSTRACT This report emphasizes procedures to determine the power interference coupling from one antenna to another. The two antennas need not be similar. Simplified methods including graphs showing the variation of coupling vs angular orientation, frequency or distance are presented for a number of antenna types. Nomographs have been designed to make possible the rapid calculation of the coupling between two antennas. The last chapter of the report (VI) shows by example the manner of calculation using the nomographs. In the main body of the report, details are given concerning the methods of obtaining the coupling between two antennas by analysis and also by measurements in the laboratory. Some of the more mathematical aspects of the analysis have been introduced as appendices to the report. For a reader interested primarily in the system interference problem, Chapters I and VII will be helpful. The simplified formulas for coupling in the far region will prove interesting and the numerous experimental data graphs will be helpful for the various types of antennas represented. The presence of a third antenna or acattering object may occasionally influence substantially the coupling between two antennas, especially in the case of very weak coupling. Some discussion of the influence of scattering objects is given. DI FR 43UCASFE I DD I JAN 64 1473 UNCLASSIFIED Security Classification

UNCLASSIFIED Securitv Classification.... -........ I 14. LINK A LINK B LINK C -KEY WORDS. ROLE WT ROLE WT ROLE WT INTERFERENCE COUPLING COUPLING NOMOGRAPHS ELECTROMAGNETIC COUPLING ELECTROMAGNETIC INTERFERENCE ARCHIMEDIAN SPIRAL SPIRAL COUPLING RECTANGULAR SLOT COUPLING INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. 3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 6. REPORT DATE: Enter the date of the report as day, month, year; or month, year. If more than one date appears on the report, use date of publication, 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information. 7b. NUMBER OF REFERENCES: Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by thq originating activity. This number must be unique to this report. 9b. OTHER REPORT NUMBER(S): If the report has been assigned any other repcrt mumbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those imposed by security classification, using standard statements such as: (1) "Qualified requesters may obtain copies of this report from DDC." (2) "Foreign announcement and dissemination of this report by DDC is not authorized." (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through,i (4) "U. S. military agencies may obtain copies of this report directly from 1?DC. Other qualified users shall request through (5) "All distribution of this report is controlled. Qualified DDC users shall request through If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 11 SUPPLEMENTARY NOTES: Use for additional explanatory notes. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S). (C), or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. I I -I UNCLASSIFIED Security Classification I