ABSTRACT RADIATION FROM DIELECTRICALLY COATED SPHERICAL ANTENNAS by August Golden, Jr. The edge admittance of a perfectly conducting spherical antenna, which is driven by a voltage source acting across a slot at the Sphere's equatorial plane and which is covered by a layer of homo- geneous dielectric or ferromagnetic material, is defined and derived by solving the related boundary value problem. Subsequently, the radiation field is derived and plotted, along with the edge admittance, for many examples. Then, after the edge admittance formulation has been extended to the case of an N-layered spherical antenna, by means of a transmission line analogy, a method of formulating the antenna's input impedance is presented. Finally, an experimental verification of the theoretical predictions concerning the radiation field of a single-layered structure is presented fo r one example . RADIATION FROM DIELECTRICALLY COA TED SPHERICAL ANTENNAS by August Golden, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1969 To my wife Sha ron Ann Golden ii ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to his major professor, Dr. K. M. Chen, for his encouragement and counseling during the course of this research. He also wishes to thank his committee members, Dr. D. P. Nyquist, Dr. Bong Ho, Dr. D. Yen, and Dr. J. Asmussen for reading the thesis. The author owes a special thanks to his wife, for the typing of the manuscript, and especially for the love and understanding shown by her during the entire course of this study. The author was an NSF Trainee during this educational period. iii Page ACKNOWLEDGEMENTS ................ iii LIST OF FIGURES ................... vi LIST OF TABLES ................... xi CIRCUIT PROPERTIES ................ 1 l. 1 Statement of the Problem ,,,,,,,,,,,, 1 l. 2 Spherical Waves and Impedances ,,,,,,, , 2 l. 3 Definition of Edge Admittance .......... 4 l. 4 Solution of the Boundary Value Problem . . . . 6 l. 5 Edge Admittance for Normal Dielectric and Magnetic Materials. . . . ....... ._ . . . . ll 1. 6 Edge Admittance for Negative Dielectric Constants ......... . . . ......... 23 RADIATION PATTERNS ................ 38 Z. 1 Introduction . . . . . .............. 38 2. 2 External Magnetic Field ------------- 38 2. 3 Far-Field Radiation Intensity . . ........ 4O 2. 4 Radiation Patterns - Positive Dielectric Constants ..................... 42 2. 5 Radiation Patterns - Negative Dielectric Constants ....... . . . ........... 49 TRANSMISSION LINE MODEL ............. 54 3. 1 Introduction . ..... . ............ 54 3. 2 Transmission Line Model ............ 54 3. 3 Real Propagation Constant ............ 58 3. 3. l Assumed: k1a>> n, kob>> n ...... 59 3. 3. 2 Assumed: k1a>> n, kob<< n ...... 68 3. 3. 3 Assumed: k1b<< n ........... 74 TABLE OF CONTENTS iv MULTIPLE LAYERS 77 4. 1 Introduction .................. 77 4. 2 Double Layer . o . . o ............. 77 4. 3 N Layers .................... 88 4. 4 Effect of a Vacuum Gap Between the Metal Sphere and the Dielectric Shell . ....... 91 INPUT IMPEDANCE ................. 97 5.1 Physical Models . . . . . ........... 97 5. 2 Theoretical Results. . . . ........... 101 THE EXPERIMENTAL STUDY OF A DIELECTRICALLY COATED SPHERICAL ANTENNA. . . . . . ................. 108 6.1 Introduction ....... . . . . ........ 108 6.2 Experimental Method. . . . . . . . . ..... 108 6. 3 Experimental Model . . . ........... 110 6. 4 Theoretical and Experimental Results . . . . . 112 6. 5 Possible Improvements. . ........... 116 APPENDIX A - WAVE IMPEDANCES . ....... 121 Part 1 - Recursion Formula for Wave Impedances . 121 Part2 - Relationships between Z;(r) and Zl;(r) . . . 125 Part3 - Curves zzm and zgm . .......... 127 APPENDIX B - CALCULATION OF EDGE ADMITTANCE. . . . . . . .............. 133 Part 1 - Convergence . . . . ............. 133 Part2 - The Errors . ................ 139 Part3 - Proof that Z1n(a) ;£ 0 ........ . . . . 147 REFERENCES. . . . ................. 149 1. 11 LIST OF FIGURES Spherical Antenna; Single Coating . Expanded View of Antenna Gap Region. Theoretical Edge Admittance of an Isolated Spherical Antenna Theoretical Edge Conductance for Various Dielectric Constants as a Function of Frequency . Theoretical Edge Susceptance for Various Dielectric Constants as a Function of Frequency. Theoretical Edge Conductance for Various Dielectric Radii as a Function of Frequency Theoretical Edge Susceptance for Various Dielectric Radii as a Function of Frequency Theoretical Edge Admittance as a Function of Frequency Theoretical Edge Conductance for Various Loss Tangents as a Function of Frequency . Theoretical Edge Susceptance for Various Loss Tangents as a Function of Frequency . Theoretical Edge Admittance of a Sphere Covered by a Lossless Magnetic Material as a Function of Frequency Page 13 15 16 17 18 20 21 22 24 Figure l. 12 1. 14a 1.14b 1. 20 2.1 2.2 Theoretical Edge Admittance of a Sphere Covered by a Lossless Magnetic Material as a Function of Frequency . ..... Theoretical Edge Conductance for Various Shell Thicknesses as a Function of Frequency . Theoretical Edge Susceptance as a Function of Frequency for Various b/a Ratios Theoretical Edge Susceptance as a Function of Frequency for Various b/a Ratios Theoretical Edge Admittance as a Function of Frequency , , , , Theoretical Edge Conductance for Various Dieletric Constants as a Function of Frequency Theoretical Edge Susceptance for Various Dielectric Constants as a Function of Frequency . Theoretical Edge Admittance as a Function of Frequency . Theoretical Edge Admittance for Various Loss Tangents as a Function of Frequency . . Theoretical Edge Admittance for Various Values of Dielectric Constant as a Function of Frequency . . Squares of Associated Legendre Polynomials Radiation Patterns. Radiation Patterns . Radiation Patterns . Radiation Patte rns . Page 25 27 28 29 3O 31 32 34 35 36 43 44 45 47 48 Figure 2.6 Radiation Patterns .................. Radiation Patterns ........... Radiation Patterns .................. Edge Admittance Circuit Model ........... Normal Transmission Line . . . . . ...... Edge Admittance Transmission Line Model. The Numerator and Denominator of a Modal Admittance . ..................... Single Term Approximation for the Edge Conductance . . . . . . ......... Single Term Approximation for the Edge Susceptance ..................... Graphical Solution for the Third and Fifth Mode Resonances ....... . . . . ...... . . Double-Laye red Spherical Antenna ......... Theoretical Edge Conductance of a Double-Layered Sphere as a Function of Frequency . . . . . . . . . Theoretical Edge Susceptance of a Double‘Layered Sphere as a Function of Frequency . . . . ..... N-Layered Spherical Antenna . . .......... Theoretical Edge Conductance as a Function of Frequency as Affected by a Vacuum Gap . . Theoretical Edge Susceptance as a Function of Frequency as Affected by a Vacuum Gap . viii Page 50 52 53 55 55 55 61 66 67 75 78 86 87 89 92 93 Theoretical Edge Conductance as a Function of Frequency as Affected by a Vacuum Gap . . . Theoretical Edge Susceptance as a Function of Frequency as Affected by a Vacuum Gap. Spherical Antenna Models with Source Region Shown Enlarged View of the Gap Region. Theoretical Input Impedance as a Function of Frequency....... ...... Theoretical Input Impedance as a Function of Frequency. . . . . ........ . .. ..... Theoretical Input Impedance ...... Theoretical Input Impedance as a Function of Frequency. . ......... Theoretical Input Impedance as a Function of Frequency . Theoretical Input Impedance as a Function of Frequency........... Cross Section of the Water-Cove red Spherical Antenna... ...... Experimental Test Equipment Arrangement - Dielectric Constant and Loss Tangent of Water as a Function of Frequency ..... Theoretical Edge Admittance of a Water-covered Spherical Antenna . - Radiation Fields . RadiationFields............... ..... ix Pa ge 95 96 98 98 102 103 104 105 106 107 111 113 114 115 117 Figure Pa ge 6. 7 Comparison of Field Patterns of Water-Covered and Isolated Spherical Antennas at a Given Frequency ............ . . . . . . . . . . 119 A-l Normalized Wave Impedance, Real Part . . . . . . 130 A-2 Normalized Wave Impedance, Imaginary Part . . . 131 A-3 Normalized Wave Impedance, Imaginary PrOpagation Constant . . . . . . . . . . . . . . . . 132 B-l Fn and Related Quantities as a Function of n . . . 142 B-2 Fn and Related Quantities as a Function of n , , , 143 Fn B-3 The Quantity as a Function of the n/ka Argument.ka.................... 144 LIST OF TA BLES Table Page 3. 1 Phase Factor Coefficients . . . . . . . . . . . . . . 64 3. 2 Higher Mode Characteristics Near Their Resonances . . . . . . . . . _. . . . . . . . . . . . . 73 xi CHAPTER 1 CIRCUIT PROPERTIES 1. 1 Statement of the Problem The purpose of this chapter is to define and derive the edge admittance of the antenna shown in Figure 1.1. The antenna configuration is that of a perfectly conducting sphere, of radius a , driven by a potential, V0 , across a slot of thickness d , situated at the equatorial plane. The metal sphere is covered by a homogeneous layer of material with a complex permeability and permitivity of F1 and 61, respectively. Fig. 1.1 Spherical Antenna; Single Coating 1.2 Spherical Waves and Impedances Labeling the coating region 1, and free space region 2, allows Maxwell's relationships between the electric and magnetic field vectors to be expressed as Vx Ei : -jLL))-li I-Ii , (1.1) 1 =1, Z V" H1 : jw‘ii E1 where eJWt time dependency is assumed. From the symmetry of the problem, it can be assumed that the a _ 3—0 (1.2) and that the surface current has only an azimuthal component, implying that H=H :E-O (1.3) Then (1. 1) will yield the following relationship for the magnetic field, H") , as shown by Wolffl. 2 __ 1 3 1 3 . 2 arz (er) + r2 a 9 sinG 33(er>i 51110)] + kiwi-1%) : 0 (1.4) 1‘1 ““W‘i The solutions of (1.4) are of the form 1 2 H‘i’in(r’ 9) = ._1_ Prlflcos 9) Ain Hill/2 (kir) + Bin 14:13]]; (kir)] (1.5) Jk-r 1 th where Hg) (x) is the Hankel function of the i degree and order From (1.1), (1. 2) and (l. 3), it can be shown that the azimuthal electric field is given by E = J' B H 1.6 9i we r ar(r (bi) ( a) and therefore (1) (1) ' 1 E91n(r, 9) : we; i1. Pn(COS 9) A1n[kian-1/Z(kir) - an+1/Z “(110] (2) (2) + Bin [kir Hn—1/2(kir) - an+UZ(kir)] (1. 6b) In(1.6)3 z =-_2_z 2 h b d h x p x p + p-l as een use w ere 2 Zp(x) = C1Jp(x) + Csz(x) as given in Jahnke and Emde. Equation (1. 5) can be rewritten in the following manner 1) . (2 H?in(r' 0) - Pln (cos (”[A1n Hn+1/2(ki r) + BmHnd/ZWifl] Jkir Jkir + .- l : P cos 0 H + H l. 4 n( )[ ‘i’in ‘iin ] ( ) The quantity H‘Pin can be pictured as a spherical wave traveling outward toward infinity, and H6”! as a spherical wave converging upon the origin of the coordinate system. If the total magnetic field were H4: , as defined in (1. 7), it 1n would be easy to show from (1. 68.) that the azimuthal electric field + + + : Zin Hi’in where with Y“ = in/ E1 would be Eein : E 6in (2) Hn-l/Z (kir) _ n (1.8) + _- , (2) k-r Z- - m(r) .1111 Hn+]/2(kir) 1 can be interpreted as the nth outward traveling TM mode wave impedance. As a similiar equation holds for the inward traveling wave, (1.6) can be written as 1 - _ E91n(r,e) = Pn(cos e) [zfilumfinm - Zin(r)H¢in(r)] (1.9) l Zin”) : 1% (1) - kir Hn+1/Z(kir) Now that the solutions of Maxwell's equations, in spherical coordinates and subject to the conditions (1. 2) and (l. 3), have been written in a convenient form, the problem inifially presented shall be continued by first defining what is meant by the edge admittance. l. 3 Definition of Edge Admittance An enlarged cross-sectional view of the metal sphere is shown in Figure l. 2. It was shown by Infeld3 that the edge admittance can be d 1<“\j:_ ' ’7 7 2 q, i i “bx—-1..- I Fig. 1.2 Expanded View of Antenna Gap Region defined as the ratio of the total current transversing the equatorial plane, as given by aZTTH¢ (a, 77/2) , to the driving potential, V0 . Infeld shows that the resultant quantity is finite provided that d a! 0. An equally valid description, again from Infeld, in the absence of the exact form of the electric field across the gap, is to consider the electric field to be of delta function form, acting at 9 277/2 and r = a , and to define the input current in terms of the magnetic field at a small angle, q» , off the equator. This latter definition will be utilized and thus the driving field, input current, and edge admittance are TI" T) (1.10) V 139(51): _a_<_> 6(9 - 1(LZ'_—+):27Ta sin('2_‘- 4,)H4,(a,_'Z_L -4.) (1.11) W I _ _ Y (1 I (2 ‘4’) (1.12) e ge V0 respectively. The problem now is to find the magnetic field at the edge of the gap. This will be done in the customary form of an infinite series expansion for the magnetic field in terms of associated Legendre . 3’ 4’ 5, 6 polynom1als . 1. 4 _Solution of the Boundary Value Problem In region one both outward traveling and inward traveling wave components will exist, thus 00 E61 (1‘, 9) Z 2 PL (COS 9) [21:1 qul-n - an H6111] (1.13) n21 00 l _ + '- H91 (r, 0) — Z Pn (cos 9) [Hiln + Hi’ln] (1.14) n=l In free-space only an outward wave is expected, . oo 1 + + EGZ (r, 0) 2:: Pn(cos 0) Z2n H‘i‘Zn (1.15) n21 oo _ 1 + H4)2 (r, 0) -Z Pn (cos 9) HIZn (1.16) n21 Obviously, an associated Legendre polynomial expansion for the driving electric field is desired, and therefore °° I E0 (a) =2 EnPg‘l(cos 0) (1.17) n=1 where the prime denotes summation over odd n and V0 1 2n+l (1.18) En = '2’ Pn(0) 2n(n+1) One boundary condition which must be satisfied by the fields is that they be continuous across the interface of regions one and two. The other boundary conditions which have already been utilized in deriving (1.10) and (1. 11), are that the tangential component of the electric field be zero at the surface of the conducting antenna, except in the gap, and that the tangential magnetic field be equal to the surface current density on the metal sphere. Application of these boundary conditions to (1.13) - (1.18) yields the following three equations: At r = a + + - - En : Z1n (a)Hb1n (a) - Zln (a)H4’1n (a) (1.19) At r = b + + + + - '- 2211(1)) H¢2n(b) = Zln (b) HI’anD) - 211.1(1)) H¢1n(b) (1. 20) + + - H¢2n(b) 2' H¢1n(b) + H¢1n(b) (1.21) + . . . As Hmn is regarded as an outward travehng spherical wave, magnetic field reflection coefficients can be defined at the outer and inne r radii by H ‘ (b) H + (a) C? (b) 4’1n 9 () h“ In :_-_'F—— 1n a :——..—-—-—- th (b) H¢1n(a) where the one-subscript on the reflection coefficient denotes the source of the incident wave at the interface located at a or b . In addition, if a transmission coefficient is defined at the outer radius by H+ (b) 4’2n T b = In" W 1n equations (1.19) - (1. 21) can be written En = [21:1 (8) - 2111b) /Gln(3)] Hfinb) (1. 22) + + - ') 22.1w) T 1,,(b) = 21.1%) - zln(b)tln 1( )( H ’9 ___ _2 n c030 Pn 0 2n+1) I1 (a ) Zagz1 n(n+1) 1 + K1,, (a, menu.) 2f; (a) - K1,,(a. b)@1n(b) 21;,(a) (1.29) yielding for the edge admittance 277a sin(T{/2 -¢)H¢1(a.lT/2 HP) VO Yedge : 0° ’ l _ 1 Y d ZITCOS+Z Pn(s1nj:)Pn(0)(2n+1) e ge 1 n(n+l) n: 1 + Kln (a, “Gm (b) zit.) - Kln(a.b>@1n(b) 21;,(a) (1. 30) If the gap is very small, then (1. 30) can be approximated as 00 I 1 1 Y :77 Pn(d/2a)Pn(0)(2n+1) edge n (n+1) ' n=1 1 + K1n(a, b)@1n(b) zlt - K1. (a, welsh) 2.; (a) (1. 31) In the absence of any reflection at the interface between regions one and two, the form of (1. 31) is seen to reduce to that given by 11 Stratton and Chu4 and Infe1d3. In fact, the quantity Zn defined by Stratton and Chu for a perfectly conducting sphere is a special case of — + 1 - K1n(3«» 1»)an (b) Zln (an/21.. (a) (1.32) 1 + K1n(a,b)Gln(b) zln(a) = 211(a) where Zln(a) is the total nth mode wave impedance seen, at radius a , by an outward traveling spherical wave propagating through a medium with characteristics P1 and e 1 , and terminated by free space at a radius r = b . Additionally, if a : b , from (1. 27) it is obvious that Kln (a, b) : 1. Then (1. 32) becomes 21:19) ' 2111(a191nm l +0111 (a) Zln (a) z + - + - + + Zln (a) Zln (a) + 2211(3)] - Zln (a)[Zln (a) — Zzn(a)] 211; (a) + 22; (a) + 21; (a) - 22"; (a) + Zzn(a) as required. 1. 5 Edge Admittance for Normal Dielectric and Magnetic Materials The infinite series of (l. 31) is shown to converge, in Appendix B, for d # 0. In this same appendix, the more practical problem of 12 determining the resultant error when only a finite number of terms of (1. 31) are summed is also investigated, as well as the formulation of the terms of (1. 31) actually utilized for the computation of the edge admittance values which follow. In Figure l. 3 the edge admittance, as a function of frequency, for an isolated metalic sphere is presented. There are four susceptance curves due to differing assumptions made regarding how many terms of (1. 31) are important and the effect of the driving gap width. The susceptance curve with the largest values is as found in 6 the liturature and is calculated with d z 0 and only 10 terms of (1. 31). Using onl",r 19 .":rms of (l. 31), but allowing (1 to increase, reduces the susceptance associated with a particular frequency as shown. If all terms of (1. 31) are summed, as approximated in Appendix B, the edge susceptance at any frequency again decreases yielding the final curve shown. For the isolated sphere, the con- ductance is not affected significantly by the gap width, and is well approximated by the first two terms of (1. 31). In Figures 1. 3 thru 1.12, the first 30 terms (all non-zero) of (1.31) are computed before the approximations of Appendix B are introduced. If the metal sphere is covered by a lossless dielectric medium of outer radius b , such that b/a : 1.25 , the edge admittance as a function of frequency for various values of dielectric constant is 13 10 terms d/2a20.0 .____.10 terms d/2a20.025 40 )- --__. 10 terms d/2a=0.05 Appendix B 30 - d/Za20.05 m o "E. C D E .5 95 {520 - II >* «3 o c h (6 :3 E '6 < 10 P 0 L 4 L 1 I l 1 0. 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Radius, a/7\O Fig. 1.3 Theoretical Edge Admittance of an Isolated Spherical Antenna 14 presented in Figures 1. 4 and 1. 5. The most obvious characteristic of these curves is the introduction of resonant points, which are completely absent from the isolated sphere admittance, when the frequency becomes sufficiently high. The same observation was made by Polk for the biconical antenna8. (A resonant point being defined by a conductance peak and associated zero susceptance. ) Apparent also is the shifting of the first resonant point toward lower frequencies, and the increase in number of resonant points in any higher frequency interval, as the value of the dielectric constant increases. The dashed parts of the susceptance curves in Figures 1. 5 thru 1. 12, denote regions of negative susceptance unless other- wise noted. From Figures 1. 4 and 1. 5, it is apparent that the resonant points are extremely frequency sensitive when compared with the lower frequency region of the curves. No attempt has been made to compute the peak values of conductance, or the peak positive and negative susceptance values, for plotting on these figures, but in Chapter 3, a method will be suggested by which their values can be approximated. As the dielectric shell is open to free space, the existence of a finite conductance at all frequencies is required. In Figures 1. 6 and 1. 7 the frequency variation of the edge conductance and edge susceptance is shown for various dielectric G, in mhos Conductance, 15 lo 01- , r. L r 1 10'l‘+L 6r :30 . :20 _ r 6 :10 r L ). 10'2‘* F + r )- 10—3 1 L 1 J 4 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/)\o Fig. 1.4 Theoretical Edge Conductance for Various Dielectric Constants as a Function of Frequency b/a:1.25, d/2a:0.05 16 ',-v —° "’ 10 1 1 | 1 1 | s .1: E 1 ' I .5 1 v u . l 1 7 m '48 I Q; L 8 1 «s 1 . 4-0 G. 0 _h— 0 " \\ __/ a) :3 m -2. )0 2......” 1 10 _ er: 20 l a a * 30 W“- ’ Positive ——-—— ' Negative -" " " - P l l I L 1 1 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/7\o Fig. 1.5 Theoretical Edge Susceptance for Various Dielectric Constants as a Function of Frequency, b/a:1.25, d/2a=0.05 10° :- b/a=1.25 U) 0 .s: E 510-1" 0' I .; 4 U G n) 4..) U _ :3 'U a .. O 0 b/a=1.5 - b/a.=l.75 10 Z--- r- 1 1 1 1 1 m 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/)\o Fig. 1.6 Theoretical Edge Conductance for Various Dielectric Radii as a Function of Frequency, €r230, d/Za:0.05 *——P- 10 .— p—o O l |—‘ Susceptance, B, in mhos 1o-2 Tllll 1 I 1 1 l 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/?\o Fig. 1.7 Theoretical Edge Susceptance for Various Dielectric Radii as a Function of Frequency, 61.230, d/2a=0.05 19 shell thicknesses. Again the properties of Figures 1. 4 and 1. 5 are noted, but in addition, from Figure 1. 6, a periodicity of the curves becomes apparent; i. e. the conductance peaks repeat, at definite frequency intervals, except for some broadening of the peaks. Observation of Figure 1. 8 confirms this repetitiveness. Ignoring the very sharp conductance peaks, the conductance curve resembles a fully rectified sine wave, with conductance peaks which become broader with increasing frequency. Located between the first two broad peaks are four sharp peaks. It will be shown in Chapters 2 and 3 that each of these peaks is associated with a different TM mode, and that each peak will reappear possibly much broader after a frequency interval similiar to that of the broader peaks. The awkward looking susceptance curve after the second peak is now seen to be due to the existence of resonant points associated with higher order TM modes, occuring in conjunction with reoccuring resonance points due to lower order modes. Figures 1. 9 and l. 10 show the admittance of the same structure as is involved in Figure 1. 8, but a lossy dielectric is considered. The results are as to be expected, i. e. conductance peaks are reduced, valleys are increased. These curves do show the presence of conductance peaks that are apparent in the lossless case only from modnmmhu .ménafin .mmnuw roosqmuh mo ~530ch .m we oocmufigm... owpm Hmofiouooafir w; .mrm o<\.m .mapmm voufidguoz 3‘6 find mmd 0N6 mud 3.10 wad Mad mod I _ I q aw ML J a J I . . _ 1 4 .0 .0.0 . . .0 .0 .0 0 0 a a“ a c. .7... z z ... ... C4 L2 C.. "D. 9 Nu. I—M\Lr/m % . _ _ ., _ . _ _ _ . . . . _ _ . . _ . . . . _ _ _ . . . _ . . . _ . m _ . . . . . . .. . U . _ _ . . _ _ _ _ N I ennui TIT ‘ :3 2111911 rnm u 21 100 :' 810"1 .C E .5 L5 03 U I: 13 U :3 "U C: O o 10'2 f I L l I l 0.07 0.11 0.15 0.19 0.23 0.27 Normalized Radius, a/Ao Fig. 1.9 Theoretical Edge Conductance for Various Loss Tangents As a Function of Frequency, b/a=1.5, €1,225, d/2a=0.05 22 0.0 tan 6 = 0.05 0.10 ———-——-- 0.23 0.27 0.19 Normalized Radius, a/>\o Fig. 1.10 Theoretical Edge Susceptance for Various Loss Tangents 0.15 As a Function of Frequency, b/a=l.5, 6r 0.11 0.07 mean 5 .m .olwseumoomsm 10" 0.05 25, d/Za: 23 the susceptance curves. This latter is of course only due to the courseness of the number of points computed, which is at intervals of a/)\0 = .002 for the case cited. The effect of losses upon the different conductance peaks; their relative broadening, suggests that the Q of the first peak is lower than that of the second peak, and-so on. If the dielectric material is replaced by a lossless magnetic material, the effect upon the edge admittance for two different cases is as shown in Figures 1.11 and 1.12. These curves are similiar in shape to those for a dielectric shell, the major difference being a much lower Q associated with the higher order resonant points. 1. 6 Edge Admittance for Negative Dielectric Constants If the medium surrounding the metal sphere is characterized by a negative dielectric constant, the edge admittance as a function of frequency curves which result will have little if any resemblance to those of the previous section. The major difference between the curves of the two cases will be the lack of any resonant points for sufficiently high frequencies, or shell thicknesses, in the negative dielectric case. In the following figures, Figures 1.13 thru 1. 20, many examples will sometimes be plotted on the same set of axis. The reason for this complicating procedure is to show, in as little space possible, how the frequency variation of the edge admittance of a spherical 24 100 f 10'2 I I I m I O I I: I E I I .5 ' I I m" I "—1 P' ‘ C5 1 I .1: . ~. I a I I m - 310 3 r I i S r ‘, 1 ct: 1" ' I E 1- I I re , I <1 " I I . 1 10'4 1 1 L L L 0.07 0.11 0.15 0.19 0.23 0.27 Normalized Radius, a//\0 Fig. 1.11 Theoretical Edge Admittance of a Sphere Covered by a Lossless Magnetic Material as a Function of Frequency b/a=1.25, [Jr-=10. d/2a=0.05 25 10-11. )- L 10"2 )— b L (D I- O .2 E b .5 5‘5 P -3 610 :— U C: I. (d I- 33 E . '0 < 1' )- 0.03 0.07 0.11 0.15 0.19 0.23 Normalized Radius, a/}\o Fig. 1.12 Theoretical Edge Admittance of a Sphere Covered by a Lossless Magnetic Material as a Function of Frequency b/a=1.25, ,ur220, d/Za=0.05 26 antenna changes as the surrounding dielectric region changes from one which is electrically thin, |k1(b-a)<<1, to one which is relatively thick, 1k1(b-a)¢'~l. In the following curves, at least 15 non-zero terms of (1. 31) have been utilized. In Figures 1.13, 1.14a and 1.14b, the dielectric constant is held fixed at --1, while the ratio b/a varies such that IS b/aS 1. 5. An increase in conductance is effected by increasing the shell thickness for frequencies such that a/ ROS 0.04. From Figure 1.14a, the addition of the shell can be considered to correspond to the addition of a negative susceptance to the existing susceptance of the metal sphere producing a downward displacement of the susceptance curve. Resonances, which are very dependent upon the shell thickness, also become apparent due to the finite shell size. From these curves, these resonances appear to move toward lower frequencies as the shell electrical thickness increases. As the ratio, b/a , continues to increase, a relatively flat conductance and susceptance begins to predominate over the frequency interval constructed, and the curves imply that the resonances dissappear eventually as they move toward lower frequencies. Figure 1.15 shows the final case of Figures 1.13, 1.14a and 1.14b. It is presented here to serve as a reference, as in Figures 1.16 and 1.17 the affect of increasing the magnitude of the dielectric constant will be presented in a somewhat unconventional form. Figure l. 17 10 Conductance, G, in mhos .—a o I w 10'4 Li I—di—ll—‘b—Il—ll—‘D—‘D—‘l—‘H N O l I l l 0.02 0.04 0.06 0.08 0.10 0.12 Fig. Normalized Radius, a/A0 1.13 Theoretical Edge Conductance for Various Shell Thicknesses as a Function of Frequency, erz—l.0, d/2a20.05 28 1 0 - 1 (+3 \(-I TfTIWI \ \ 1- U) o - .1: e I .5 " _ 1.00 “310 3T 1.05 - (+1 _____ 8 * b/a : 1.10 ' g " I 1.15 _ .‘3 1.20 __________. e - d) U {-1 U) .- 5 \ U) \ I ” 1 10-4 1 1 1 l l 0.02 0.04 0.06 0.08 0.10 0.12 Normalized Radius, a/ A Fig. 1.14a Theoretical Edge Susceptance as a Function of Frequency for Various b/a Ratios, €r=—1.0, d/2a=0.05 29 (-) I 1+) I \ . \ ' (+1 ‘ (-7 10"1 f .. I I P .0 . I \ I— \_ (+1 10'3 CD 0 '1: I- E \ ,I .5 °, ' 2 1.25 A” m - I / 1.30-———-—-——-— I ' b a = I 35—-'-—-——- (D I . 1. —-----—-- g ‘,I 1.58 S (o) I; e '. ID _3 | 1:10 :- =3 .- m D " I _ I ' I r I 10-4 1 l L I l 0.02 0.04 0.06 0.08 0.10 0.12 Normalized Radius, a/>\ Fig. 1.14b Theoretical Edge Susceptance as a F‘iinction of Frequency for Various b/a Ratios, Er: -1,0, d/2az0.05 30 10‘1 2' )- L I ' \ : \ \ ' .— I- ' \ \ ‘ s _, g ........ -2 10 _ )- (n O .C: E - .5 s“ 6.5.10"3 L >. C a; h- o .. I: d I- 3 <1: .. L. 0-4 I J J I L 0.02 0.06 0.10 0.14 0.18 0.22 Normalized Radius, a/Ao Fig. 1.15 Theoretical Edge Admittance as a Function of Frequency b/a=1.5, Er: -1.0, d/2a=0.05 Conductance, G, in mhos 31 J I I //’ I ' : /' ’ /' I: 10‘3 - . " 3125 ——----— ’ I er : “-2.0 "r“ '— ' ~2.5 -----—- " ~3.0 -—- -—-— . -4.0 P . '. '0 I '/,I.‘ 10"4 I 1 l n 1 n 0.02 0.06 0.10 0.14 0.18 0.22 ' Normalized Radius, a//\0 Fig. 1.16 Theoretical Edge Conductance for Various Dielectric Constants as aFunction of Frequency, b/a=1.5, d/2a=0.05 Normalized Radius, a/7\O .l. .I. ozfimom moss.“ 5 .m .eecmumoomdm 0>Mudmoz 1.5, d/2a:0.05 Fig. 1.17 Theoretical Edge Susceptance for Various Dielectric Constants as a Function of Frequency, b/a 10‘11'. 33 tries to show how the edge susceptance would appear on a linear graph, where zero crossings are possible, as a function of frequency. Due to the large variations in susceptance magnitude, a semi-log plot is required, resulting in the appearence of some discontinuities near the zero crossings. This adverse effect is felt to be outweighed by the increased clarity given by Figure 1.17 to the visualization of the variation of susceptance as a function of frequency. From Figure 1.16, the increase of the dielectric constant to -1. 5 is seen to produce two additional resonances which are of a different form from those observed in Section 1. 5. An increase in the dielectric constant tends to shift downward the left hand portion of the susceptance curve, introducing a few resonances, and eventually smoothing out high frequency variations. Corresponding to the susceptance variations with changing dielectric constant, the peaks of the conductance curves shift in position and become broader. Again, in order to provide a means to understand the unconventional Figure 1.17, Figure 1.18 is presented. The effect of any dielectric losses are shown in Figure 1.19. Finally, Figure 1. 20 shows the broadband nature of the edge admittance for sufficiently high values of dielectric constant. The antenna conductance decreases as [61.) increases, while the reverse is true of the susceptance. This corresponds to the adding of an 34 10'l _ P p I ______________ I I. I " l -3 ........ - x I” -’ I I I I 10'3r- ' L I I- :f") : I' G l _ I 8 (+1 I .1: I g I :3 I .Fq ‘ ,. I CO I. I .._, I 6 I II I >4 I .. I 810'3f’ I 8 . ' t: _ I .H | E ' I Q: I- r 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/A0 Fig. 1.18 Theoretical Edge Admittance as a Function of Frequency b/a=l.5, 6 = 2.0, d/2a=0.05 r - in mhos G+jB, Admittance , Y 35 10 IfitIT r B “fig/4 ‘x‘ I “\"\_ I. I, ‘ G “‘~. . av” \\ ~-_- \ o' \\‘\. \\‘ \ I I I I I I '1 I I I I - .I 101:“ I) I p ' I - I I I- \I ' 0.0 " tané = -0.05 ------ -0.10 ' 10-41 I 1 I 1 W 1 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/Ao Fig. 1.19 Theoretical Edge Admittance for Various Loss Tangents As a Function of Frequency, b/a=1.5, er: -2.0, d/2a20.05 36 = G + jB, in mhos Admittance , Y L 4L 1 l l l J 10 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/A0 Fig. 1.20 Theoretical Edge Admittance for Various Values of Dielectric Constant as a Function of Frequency, b/a=l.5, d/2a20.05 37 increasingly large negative susceptance to the isolated sphere's normal value, while the conductance decreases as less energy is able to propagate through the dielectric to eventually become radiated into free space. CHAPTER 2 RA DIA TION PA TT ERNS 2.1 Introduction In the preceding chapter, the edge admittance of a spherical antenna covered by a layer of homogeneous material was investi- gated. It is now of interest to know how the radiation pattern of this same antenna will vary as a function of frequency. From the work of Polk8, it is expected that the pattern will be strongly dependent upon frequency near the conductance peaks of the previous chapter. 2.2 External Magnetic Field From Chapter 1, the magnetic field external to the antenna can be expressed as oo H¢2(r,6) =2 P111(cose)H¢2+n(r) (2.1) n=1 where (2) H¢+ (r) = Cann+W(k°r) <2. 2) 2n ’kor From (1. 20) and (1.25) at r : b + _ + _ + . thw) _ Tln(b) H¢1n(b) _ [1. +@1n(b)] H¢ln(b) (2. 5) 38 39 where 2 H¢+ (r) = BlnHri+i/2(kr) 1 n \lkr (2.4) Additionally, (I. 28) is En Hana) = + - (2. 5) Zln(a) - K1n(a, MGln (b) 2111(3) with En as defined in Chapter 1. Combining (2. 4) and (2. 5) yeilds B1 1' EnI/ka “ (2) + _ (2. 6) Hn+u2 [2mm - Klnwflnm 2mm] Then (2. 2.), (2.3), (2. 4) and (2. 6) yeild [1 +an (bj En hflkbhfi/Tr C = 2 h£2)(k0b)h§12)(ka) [21:19) — K1n(a,b)C-?1n(b)zl;1(a)] n (2.7) 2 where h: )(x) 2 JE/Trx Hi12+)]/2 (x) is a spherical Hankel function. Finally, inserting (2. 7) into (2.1) and (Z. 2) yields, with the value of En included, an external magnetic field of 4O 2 H4) (r,9) =13 00’P,1(0)P,{(cos 6)(2n+l) ( ) 2 2a n- 2 hn (kor) hi1 )(kb) n(n+l) (Z) (2) hn (kob) hn (ka) 1 +Gln(b) + - (3- 8) 2mm - Klnm. b) Gmw) 2mm) 2. 3 Far-Field Radiation Intensity For x-s» (I) and n odd n-l 7T 1 _— _. 22)) 2 2 '5" ’5 4 hn (3052:1182 (UH-1) e e (3.9) Inserting (2. 9) into (2. 8) yields the magnetic field in the far-zone as 'k 300’ 1 1 H (r,9l:::j V0 “ 0r €342 Pn(0)Pn(cose)(2n+1) . (1)2 Tra kor _ n(n+l) n—1 n-1 (~1)T Cn(a,b) (2.10) where (2) h kb 1+ b Cn(a.b) = n ( ) 01’“ ) (2) (2) hn (kob)hn (ka) 211(3) - Kln(a, b)Gln(b)Zl;1(a) 41 In the far-field E = 7" H , and therefore, the far—field 9 0 ¢ radiation intensity is 2 dP 1 - 4-. A 2 rue 2 2 2 mziRe\o : O. 232 in Figure l. 4, the third order associated Legendre polynomial dominates 45 Fig. 2.3 Radiation Patterns 46 the distribution of the radiated power. The pattern then changes, as shown in Figure 2. 3. 2, to some combination of the terms of (2. 11) such that P31 (cos 9) continues to remain quite important. Near the next resonant point, the fifth order associated Legendre polynomial becomes most important, as in Figure 2. 3. 3. This pattern is seen to change very rapidly with frequency, as does the associated admittance of Figures 1. 4 and 1. 5. The pattern reduces again to one that depends strongly upon the first two terms of (2. 10) as the frequency increases. The example of Chapter 1 that was carried to the highest value of a/ )\0 was for b/a =1. 5 and 6 r = 25 , as shown in Figure 1.8. Figures 2. 4 and 2. 5 show the radiation pattern variation over much of the same frequency interval. The basic observations made from Figures 2. 2 and 2.3 still hold, with the first resonance at a/ ho : 0.135 due to the P%(cos 6) term, and the second one at a/ X0 = 0.181 due to the P15(cos 6) term. The next two resonances of Figure 1. 8 can be concluded to be due to the P%(cos 6) and P;(cos 6) terms, respectively. The Q of these two resonances is concluded to be so high that they are not detected when a/ )\o is incremented by 0. 001, as was performed to plot the given antenna patterns. An interesting observation is that in Figure 2. 4, the patterns tend toward a dipole pattern after every resonance. But in Figure 2. 5, the 4.8 a/)( =O.23O - - - - o 0 260—— 0.290 ....... Fig. 2.5 Radiation Patterns a/AO=O.323 — - — - o.325————— 90° 00 J a/)\o = 0.331 — — - — 0.340____....._ ‘/\ . \ li/ / /\ x" // 90 49 patterns tend to become the major lobe of a third order associated Legendre polynomial away from resonances. 2. 5 Radiation Patterns - Negative Dielectric Constants The following radiation patterns are plotted for antennas for which edge admittance characteristics have already been obtained in Chapter 1. The frequencies, at which the patterns are displayed, are chosen to correspond to frequency sensitive edge admittance regions. The lack of many resonances for the negative dielectric case, necessarily means that few field patterns will need to be plotted. For the case b/a = 1.5 and er = -1, the radiation pattern is that of a small electric dipole over the entire interval 0. 02 S a/ hog O. 25 , as in Figure 2.6.1. For a large negative dielectric constant, 6 r = -10 , Figure 1. 20 shows a broadband edge admittance. There is a small ripple in the admittance curves corresponding to similiar stronger variations present on the admittance curves corresponding to smaller dielectric constant values. The associated radiation pattern is that of a dipole at the lower frequencies, as shown in Figure 2. 6. 2, but changes to reflect a strong third order associated Legendre polynomial component at higher frequencies. If the relative permitivity is changed to 6 r :-l. 5 , Figures 1. 16 and 1. 17 show two sharp resonances near a/ >‘o : 0.144 and 50 a/)(0=0.02 - - - - a/) = 0.050 — — - - 0.10————— ‘\ ° 0.090—-——— I, 0.25 ....... / \ 0.190 ....... , l/ \ f \ ,/ / ‘\ . /\\ WT? ' a a a - 0 90 ' 2 * 90 Fig. 2.6.2 'Eel b/a 21.5, Er: -10.0 \ . o .I’ \ /" a/ AO=O.Z6 -—-———-— Fig. 2.6 Radiation Patterns 51 a/ >‘o : 0. 246. Figure 2. 7.1 shows that the radiation pattern away from these frequency points is mainly dipolar. Figure 2.7.2 and 2. 7. 3 show that at a/ AC 2 0.144 and a/ )(O : 0. 248, the P31(cos a) and then P51 (cos 9) terms become the dominant field components, respectively. With E r : ~2, the antenna pattern is again that of a dipole at lower frequencies, but gradually changes to reflect a strong third order associated Legendre polynomial component with increasing frequency, as shown in Figures 2.8.1 and 2.8.2. The fifth order polynomial becomes dominant at a somewhat higher frequency, as in Figure 2. 8. 3, and it is again expected that the roughly dipole pattern will predominate at higher frequencies. 52 0.064 - — - - 0.300——————— a/)\0=0.020 - - - - . a/AO: 0.134 - - — - O.l40-————— 0.160 ....... b/a :1.5 er 2 -l.5 Fig. 2. 7. 3 Ed Fig. 2.7 Radiation Patterns 53 a/>(O=O.050 - - - - a/A 20,170 _ _ _ - 0.082 - - - - 0.278 _— al)‘ 20.318 0 0.322 oooooooo b/a : 1.5 Fig. 2.8 Radiation Patterns CHAPTER 3 TRANSMISSION LINE MODEL 3 . 1 Introduction Two conclusions, which can be reached after observation of the curves of Chapters 1 and 2, are that for the lower frequency region and except in some narrow frequency intervals, the radiated power is essentially distributed in the pattern of a small electric dipole and the edge conductance is determined to a large degree by the first term of (1. 31). In the narrow frequency intervals, other terms of (l. 31) predominate in the edge admittance, and corresponding terms of (2. 11) determine the radiation field. In this chapter, a transmission line model will be created to represent the edge admittance and to thereby give some "physical feeling" for the characteristics of the curves of Chapter 1. 3. 2 Transmission Line Model Following the example of Chull, the edge admittance of the spherical antennas of Chapters 1 and 2 can be written as the sum of an infinite number of modal admittances, modified by a coupling network, as shown in Figure 3. 1 and stated by (3.1) 00 / :"’ 3.1 .2 t , n=1 54 55 , network _I *———1 coupling ’ Yedge . Yl Fig. 3.1 Edge Admittance Circuit Model vvw Z1n Zo‘tl/Yo AAA k1 v Fig. 3.2 Normal Transmission Line - + :11 $221M Y F— coupling L , + edge:— network Y13 ‘1 223(1)) 3 ‘ + _Fln 2'2n(b) Fig. 3.3 Edge Admittance Transmission Line Model 56 Rewriting (l. 31) as 0° ’ 1 1 pH (0) PD (d/2a) (2n+1) n(n+l) Zln(a) Y =")T edge (3. 2) n=1 it is obvious that 1 1 On = Pn(0)Pn(d/2a)(2n+l) Y z 1 : Yln(a) The analogy can be carried one step further, as was hinted in Chapter 1, by digres sing from the point and reviewing some basic uniform transmission line equations. Following normal analysis , the current reflection and current transmission coefficients for the line shown in Figure 3. 2 are: 2 .2 GI: 22...}: (3.3) O+ZL ~ 22 II21+GI:-Z-(—)-—.:_QZ—— (3.4) L The resulting input admittance will be -'2k1 Y = Y H GIeJ (3 5) 1n 0 -j2kl ' 1" 918 57 where k = 2 7T/)( and A is the line wavelength. Rewriting Zln(a) here, - + 1 - (9mm K1n(a, b) zlntawzlne) Zln(a) = 21:10)) 0 l + 1n(b)K1n(a, b) (3.6) it is clear from comparing (3. 5) and (3. 6) that Z1;(a) can be th considered as the n modal characteristic wave impedance at radius a for the given medium. The quantity Gln(b) is comparable h to the nt modal current reflection coefficient both from comparison of (3. 5) to (3. 6) and (3. 3) to (3.7) below. + + 91 (b) = 21,,(b) ' 22‘1”” n - + Zln(b) + ZZn(b) (3. 7) A similar equation would justify comparison of Tln(b) to the current transmission coefficient, ’TI. Therefore, Figure 3. 1 as presented by Chu, can be extended. The extension is to consider the lumped admittance which characterizes each mode as the input admittance of a nonuniform transmission line a o o c o o o v + With an initial characteristic impedance at radius r = a of Z1n(a) , + and with a terminating load at r = b of Zzn(b). The line is non- uniform as Kln(a” b), the phase factor, is not linear with distance. 58 The analogy, although imperfect as the numerator of (3. 6) contains the quantity Z1;(a)/Z;n(a) # l , is useful as will be shown in the following sections. This final physical model, which can be used to characterize the antenna, is shown in Figure 3. 3. If the variation of each Zln(a) with frequency can be determined, the frequency dependence of the edge admittance can be visualized as well. The examples of Chapters 1 and 2 can be separated into two basic cases. They are when k1, the propagation constant of medium one, is real, and when k1 is purely imaginary. The case with a real propagation constant is treated below. 3. 3 Real Propagation Constant For a real propagation constant, Appendix A shows that (3.8) From Appendix B, part 3, and the definition of K1n(a, b), respectively lGIn(b)i< 1 lK1n(a'b)| =1 (3-9) , + + . + - From (3. 8), With Zln(a) = R1n(a) + len(a) , (3. 6) can be rewritten as 1 - K1n(a.b)G1n(b) + + j X (a) (3.10) l + Kln (a,b)Gln(b) 1n 21,1121) = Riga) 59 Thus each modal input impedance is the sum of a pure reactance, jX1:1(a), and the input impedance of a nonuniform transmission line. The line characteristic impedance is real, R1:(a) , while the phase factor, K1n(a, b), is still nonuniform. The form of Z1n(a) will be investigated under a variety of con- ditions . 3.3.1 Assumed: k1a>>n, kob>>n Under these conditions, it is obvious that the wavelength in the medium is quite small compared to the radius of curvature of the medium-free space interface as n is greater than or equal to one. With this in mind it is not hard to accept the following approximation for Glnu’) which is arrived at from observation of the z;(x) curves in Appendix A. + + Z (b) - Z (b) T1 _ 1 Glue) = 1f 2+“ 7x 4-10 2 Oman) (3.11) 2111(1)) + 2211(5) 111 + no Equation (3. 11) is the result that plane wave analysis would have yielded for a reflection coefficient. + + Equation (3. 10) is approximately, as Xln(a)5:0 , and R1n(a) 2111 l - Kln(a, b)(,)lln(b) 1 + Kln(a,b)Q'ln(b) (3.12) Zln(a) 2 n1 60 (Z) If kla 4- co , the approximation made in (2. 9) for hn (x) could be applied to K1n(a, b) , but as n << kla << 00 , a better approximation for this quantity will be utilized. Before deriving such an approximation, it is easy to see what effect a term such as (3.12) will have upon the edge admittance. Assuming that for some N1 the initial assumptions hold, (3. 2) can be separated into two parts N Y _ fl 1’ cn 1+K1n(a,b)G'1n(b) edge—“2 ’11 1-K (a b) ' (b) n=1 ln ' Gln 00 I + H z CnYln (3.13) n=Nl+2 From (3. 9). (3.13) can be written with _ -j2¢n _ -j29 G'lnw) - 'Gni e K1n(a,b) ' e n and +n=6n+¢n,as N1 00 ’ on 1 + (Q l e'JZ‘l’n I _. '— n ~ Y d - | , + I Z G Y (3.14) e ge n1 1 _ (G ‘ e'JZ‘I’n n ln n=1 n n=N1+2 _ + ~ ' In the present case, (In — 0 or - I /2 as from (3.11), Glnw) is greater than or less than zero as the medium is ferromagnetic or dielectric , respectively. 61 The variation of any one of the first Nl terms can be discerned from a comprehension of Figure 3. 4 or equation (3. 15). Yl (a):—I— l .- K’Dnl z —j2 lGnl sin (2+n) n 1'11 1 + lenl 2 -2 lGnl cos (2%!) (3.15) Im lfiep‘jzi’" —\ /’ \ Re lee-qu’n Fig. 3.4 The Numerator and Denominator of a Modal Admittance In Figure 3. 4, as the driving frequency varies, so does the angle and the ratio of the complex quantities 1 + '0 ‘ e-j 24,11 h. n -j2+ . . . . . . and 1 - Gui e n, This quotient has been diVided into its real and imaginary parts in (3. 15), and from Figure 3. 4 or this last equation it is apparent that if IGnl 2:: 1, a very large value of conductance would occur whenever ‘l’n = mTF , m = O, l, 2,. . . . Additionally, when «fin = mTl' + TF/z , m = 0,1,2, . . . . , the associated susceptance will be zero. 62 From (3. 11), as (b) '1. to 1' “(Jr/Er 6 (f = = - 3.1 ln 01+ no 1+VHr/Er ( ) a very large relative permitivity or permeability would be required for the magnitude of the reflection coefficient to be close to one. The desired better approximation for the phase constant can be obtained by writing it as it was first derived in Chapter 1. (1) (2) (2 (11 Hn+)l/Z(kla) Hn+1l2(klb) Kln(a, b) = From tables for p fixed, x large and positive, and q = 4p H11) p (x) = M ej0p(x) P where + ELL. + fi'1111-25)+ (q-1)(q2-1149+ 1073) 2(4") 6(4):)3r 5(4x)5 ep(x)zx-(§+7})Tr + ...... (3.17) Thus Kln(a’ b) becomes , with p = n + 1/2 ej0p(k1a)) e-j9p(k1b) _- ka) ' Kln(avb) : : e-jz [6P(klb) - 9p(kla)] : e'j26n(a’ b) 63 where 9( b) 6( ~ (b q-1 k1(b-a) (q-1)(q-25) Pk1 - pkla)~kl “ab 8 klbkla - 384 z 2 kl(b-a) (klb) + klbkla + (kla) ( o 3.18) klaklb (klaklb)2 With 2 2 a a 7- (klb) + klbkla + (kla) 1+ «5 + (5) 2 ' 2 (klaklb) (kla) (3. 18) can be written q-l a/b (q-1)(q-25) a/b 0 ,b = b- 1 - .. n‘a ) kl( a) 8 (kla)z 384 (kla)2 2 a a 1 _— o + b+ 2(3) + ..... (3.19) (1.1a) Table 3. l tabulates the coefficients of the three terms of (3.19) for a few values of n . 64 TABLE 3.1 Phas e Factor Coefficients 3;; (q-1)(q-25) n _ P q 8 384 1 1. 5 9 1 -0. 333 3 3. 5 49 6 3. 0 5 5. 5 121 15 30. 0 7 7. 5 225 28 116. 667 As an example, let b/a = l. 5 . first four values of n Then 6n(a,b) will be, for the 91(a,b) = k1(b-a) [l- all—:36; + 211:: +. . . .] (3.20a) 03(a,b) = k1(b-a) [1 - “(1:)2 - if): +. . . .} (3.20b) 05(a,b) = kl(b-a) [ 1 - “(11:)2 - (:3: + . . . . ] (3. 206) e,(a.b) =k1(b-a)[ 1 - 1:372 .. (11:): +. . . .J (3.204) 65 These equations will be more useful in a later case, but under the existing conditions, kla >> n , the leading term of (3. 20a-d) should be sufficient considering the previous approximations. Thus, using this roughest approximation for 0n(a, b) and the th definition of Wu, the admittance of the n term can be written from (3.15) as Y1n(a)::—1- l - 10111 2 _j2 iGnl sin(2k1(b-a) + 2¢n) (3. 21) 11? 1+ lGnl 2 -2 (en) mailmamg) The restrictions, kla >>n and kob >>n, are not in general satisfied for any of the examples of Chapters 1 and 2. These con- ditions are satisfied for the case n = 1, Q r = 30 , and b/a = l. 5 of Figures 1. 6 and l. 7 if a/ )( 0>> 0.029. For this case, ()1 = - lT/Z , ‘01! = 0. 6913 , and the first term contribution to the edge admittance will be TFC1Y11(a)::0.0685 1‘ '01) 2 '12 K31) Si“(2k1(b'a) ‘71—) (3.22) 1+ 101' 2 -2 191' cos(2k1(b-a) - 1T) Figures 1. 6 and l. 7 are reproduced, in part, as Figures 3.5 and 3. 6. Upon these figures is plotted the contribution to the total admittance of (3. 22). It is clear that, neglecting the sharp conductance peaks, the contribution of (3. 22) to the total edge admittance is large enough that a close comparison is made in Figure 3. 5. The degree of 66 6223036500 ompm of new cofimeflxoamaq. EpoH 3,6055% m.m .ofim o/\m .mapmm pomflwEqu 0N6 mmd HNd N170 mic 00.0 mod ~ _ T 4 q . 1 n l N' Huse; 14 l H- OH OH OH soqw u; ‘9 ‘eoueionpuog 67 oocmumoomsm ompm 65 Mom coflwccflxohnaxw EHMH oamcflm o 6.m .WE A\m .mdflpmm UoNEmEpoZ 0N0 mN.0 Hm.0 >70 m~.0 00.0 m0.0 q u d _ . _ q a . . 1 a. . - __ , 1 _ I 1“ 0H .. , I 1. . N .. i _ _ . , . n x l _ _ x _ a q . / . v. _ _ _ / . _ . 1 _ , _ m. _ . x _ _; . _ x . ._ 1+ _ _ / - __ _ z. I . . . .. 7 _u_ . . . _/ .,_. // ~ _ — — I. ‘_ 7.x . z . E _ . _ X . .. . . . . 013/ ‘ 1x \a f <\ . . .. xx \s /I . n . ax — ' l\ \ _ . c . 1 . 1 a 1 . . \ _ _ . a . a d m. _ a i . L .7 _ m , .. I a. _ . . .. ’ 1 _ __ , \ _. _ _ r \ . ,. _ L C _ . a _ C 1 ., _ M _ . .n . E 1 ' _ . . _ oIv— .1 . L _ . _. ._ 1 : a u _ 1m 00H .1 . 50mm 111 ‘g ‘aoueidaasns 68 dissimilarity of the curves of Figure 3. 6 point out the large susceptance contribution of the highest order terms. 3. 3. 2 Assumed kla >> 11 kob << n This condition is similar to that of Polk8, and better represents the conditions in some regions of the curves of Chapter 1. It is noted that a very large dielectric constant will be required to satisfy both of the assumed conditions simultaneously, but that this restriction will be relaxed substantially when results are constructed for comparison to the curves of Chapter 1. Under the above conditions, it is relatively easy to see that the steps leading to (3. 15) still hold, provided that a new approximation is made for Gln(b) . From the wave impedance curves of + Appendix A, with 22n(b) = R2203) + jX§n(b) and for Rob <>n/kob >>1. . . . . . + + If this condition is not strictly adhered to R2n(b) << T\1<< IX2n(b)‘ and + - + + (G ((3)) ,0 n1‘R2n‘JX211 ~1_ 2R2n(b)nl (3 25) 1n C 711 R+ ' + ~ 2 xJr b2 . + 2n+JX2n T11 + 2n() where use has been made of 1/2 , [i'x] a: /1-2x 2::l-x for x<<1 +x 6 ' ‘ ' f z + b The value of n can be computed if an approx1mation or 2n( ) is made for the case kob <\o = 0.118, and for n = 5 , a/ A0 = 0.153. These values compare favorably with those values determined from Figure 3. 5, namely for n = 3 , a/>\0=0.125, andfor 11:5, a/>\0=0.164. 3. 3. 3 Assumed: klb <= 3‘17: ‘lln ¢ln 2 equations (4. 10) and (4. ll) become + 1 + - Zzn(r2)- 2n(r2n)/Gz(r :1n(r——————Z[r2) 1n(r2) - Zln(r2)Gln(r2)] (4.19) and Tlnhz) (1 + l/Gzn(r2)J ‘ 1 +an(r2) (4' 20) where (4.17) has also been utilized. 83 Solving for the reflection and transmission coefficients at r = r ields 2 Y 1 + (r ) Tln(r2) = GI" Z (4.21) 1+ K2n(r2' 1.3)GZn(r3) and - + Z1+(r ) - 2+ (1- ) 1 ' K211”? r3192niriZZn(r2)/Z2n(fl2) n 2 2n 2 Gm‘rz) = 1 + K2n(r2: r3)Gz,,(r3) - + _ + 1 - K2n(r2’ r3)GZn(r3)zzn(r2)/ZZn(r2) Zln(r2) + ZZn(r2) l + K2n(r2, r3)Gzn(r3) (4.22) It is now clear that the next step is to define the reflection coefficient at r = r1 such that + H (r) G (r1)= —-:¢—1-9——L = 1 (4.23) 1“ H-(r) (r)K (r r) ‘11,, 1 Gln 2 1n 1’ 2 and then from (4. 9) H + (r ) = E“ h“ 1 zfn(rl) " Zlhhl) [Gln(rl) 84 and finally + En H (r ) = 1 + - 1’111 2111(5) — Kln(r1, r2)Gln(r2) zlnul) (4.24) Inserting (4. 24) and (4. 8) into (4.6) gives a magnetic field at the metal surface of CD I 1 1 H (r10) = Vo P11(COS91Pn(0)(2n+1) 2r1 n=1 n(n+l) 1 + Kln(r1. 51911102) Zin(r1) ' Kln(r1’ r2)Gln(r2)Zln(rl) and an edge admittance of co Yedge : '2 I n l I 1 1 Pn(d/2a)Pn(0)(2n+1) (4. 25) n(n+l) Zln(r1) where 21:01) - Kln(rl’ r2)Gln(r2) Zlhhn) l + Kln(r1, 12)an(rz) At this point, if (4. 25) and (4. 26) are compared to their equivalents in Chapter 1, it will be noted that the form of the equations is identical. 85 The actual difference between the two sets of equations is in the definition of the magnetic field reflection coefficient, Glnhz), at r = r2. In Chapter 1, the total nth mode wave impedance looking into Region 2 was just Zz+n(r2). Due to the existence of another interface at r = r3 , the nth mode wave impedance looking into Region 2 is now given by (4. 22), and depends upon the electrical characteristics of Regions 2 and 3. Drawing upon the developments of Chapter 3, for each TM mode, the two layers can be looked upon as two waveguide or transmission line sections connected in series. The final terminating impedance is just the outward traveling wave impedance of free space, Z3+n(r3) . This load, after transformation by the two transmission line sections, becomes the total nth mode wave impedance seen at the antenna surface, Zln(rl) . As an example of an antenna covered by two layers of dielectric, consider the following set of parameters as they relate to Figure 4.1. Let rZ/r1 = 1.2, r3/r1 = 1.5, and GI = 40, 62 = 20. The edge conductance and susceptance are then as plotted in Figures 4.2 and 4. 3, respectively. Upon the same figures, the admittance for a single layered structure from Chapter 1 is plotted for comparison. It is apparent that the two sets of curves are similar in form. A point to notice is that the first conductance peak has become wider, Conductance, G, in mhos 10 -1 86 l 1 '1 'I 'I 1| H H II I) double layer II Single layer ___________ ll b/a : 1.5 H er : 25 H H H II II I \ '1 \‘l I l \ \ a \ ‘___,_/’ \\ \ \ \ l I 1 J 0.05 0.13 0.17 0.21 0.25 Normalized Radius , a / A0 Fig. 4.2 Theoretical Edge Conductance of a Double-Layered Sphere as a Function of Frequency 87 rr 0 ywss .4312 .11. e :: lbsa mos/6 o.mb ds _h l 2 . 0 0 1 l .oosmaoomdm 0.17 0.21 0.25 Normalized Radius , a/?\0 13 Fig. 4.3 Theoretical Edge Susceptance of a Double—Layered Sphere 0.09 0. .05 0 as a Function of Frequency 88 even though the total antenna size has not been increased. This suggests that the appropriate choice of parameters, or possibly a structure containing more dielectric layers with properly chosen radii and dielectric constants, could cause the dielectric coatings to act as an impedance transformer between the metal sphere and free space in a manner similar to that of an n-section quarter wave trans mis sion line trans fo rme r. 4. 3 N Layers With the aid of this transmission line analogy, it is relatively easy to derive the form of the edge admittance of a metal sphere of radius r1, covered by N layers of a dielectric or ferromagnetic material as shown in Figure 4. 4. It is Yedge ___ 7T: Pn (d/2a) Pn (0) (2n+l) (4. 27) n=1 n(n+l) Zln(r1) where the nth mode wave impedance at r : r1 211,01) - K1,,(rl. 12)Q,n(1-2) 211101) 1 + Kln(r1, r2)Gln(r2) Zln(rl) : is defined in terms of the phase factor of medium 1 89 Fig. 4.4 N-Layered Spherical Antenna 90 (1) 2 hn (k1r1)h1(1) 11‘2) h;2)(k1r1) hill)(k1r2) (k Kln( r1, r2) = and the magnetic field reflection coefficient at r = r2 + Z (r ) - Z (r ) Gln(r2) = m 2 2n 2 23111(1’2) + Zznhzl th mode This reflection coefficient is, in turn, defined by the n wave impedance of Region 2, and so on. In gene ral + ( ) _ Zjn(rj+l) ‘ Zj+1,n(rj+1) jn rj+l Zjn(rj+l) l Zj+l.n(rj+1) K- (r-, r. )2 J . 1n 3 1+1 2 1 J h; )(kjrj)hf1)(kjrj+1) 111(11)(kjrj) hfl21(kjr.+1) 1,2,uuoo,N and + _ Zjnuj) - antrj. rj,,)C>J-n(rj,,)zjn(rj) 1 + anh'j' rj+l)Gjn(rj+l) + : Z‘N+1,n(rN+1) J 2 NH 91 4. 4 Effect of a Vacuum Gap Between the Metal Sphere and the Dielectric Shell If one of the antennas of Chapter 1 were to be built using a solid dielectric material, it would be of interest to know the effects of an inaccurate fit between the dielectric shell and the metal hemisphere. To accomplish this, in Figure 4.1, imagine that Region 1 is a vacuum and that it characterizes the imperfect fit. Then Figures 4. 5 and 4. 6 show the effect of this small vacuum region upon the edge conductance and susceptance, respectively. As has been shown in the previous chapters, the general shape of the admittance curve is determined by the first term of the series in (l. 31). The more frequency sensitive sections of the admittance curve are then due to higher order terms of (l. 31). It is apparent from Figures 4. 5 and 4. 6 that this first term contribution is only slightly modified by the vacuum gap. As would be expected, the highly frequency dependent regions are strongly affected by the thickness of the vacuum region. From Figures 4. 5 and 4. 6, the gap's effect upon these higher order modes is to cause the antenna to appear to be covered by a layer of material with a much lower dielectric constant. This is exactly the effect of an air gap between the dielectric and conducting surfaces of a parallel plate condens e r . Conductance, G, in mhos 92 H '1 100 l :I 1. I - l - , 1 . l _ - . 1 '- I )- 'I I ' .. 1 I a 1 ‘ )— ’_ ' 1 “ l’ ‘2} ' 1 / \ l 1 / \ 1 l / \ I l I f V ‘ ’ I —l )— ’ \ 10 1 / a , \ )— I \\ _ ’ \ I \ .. / \\ I \ - / I / - I I I - I .' / 104... .' ’ 1 . / d/2r120.05 )— ' / _ .' I :: - .' / Er 3O ..' // r3/rl: 1.25 b .o/ l.(’ ....... a" rZ/rl :l.01 ------ ~ . / 1.02 -I .'l - .'I .'I .‘I l" -/ 10- {I 1 l l 1 I J 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/ >10 Fig. 4.5 Theoretical Edge Conductance as a Function of Frequency as Affected by a Vacuum Gap 93 1 {-I (+1 (D o .C E .9. 111 ' -2_. E §10 L r g " r3/1‘1 Q _ a) . g (_ r /r1 :3 2 m — 10-3 I _I L J I 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/7\O Fig. 4.6 Theoretical Edge Susceptance as a Function of Frequency as Affected by a Vacuum Gap 94 If a plasma were used to achieve a negative dielectric constant coating, the effect of the plasma sheath would be of interest. Modeling this sheath by a vacuum, and assuming that this negative dielectric constant is independent of frequency, the sheath effect upon the admittance is as shown in Figures 4. 7 and 4. 8. As these admittance properties are determined mainly by the first term of (l. 31), the effect of the vacuum gap appears to be very small. 10‘l Conductance, G, in mhos 10'2 10' I o I : I l a I I 95 d/Zrl = 0 05 Er - -l.0 r3/rl =1 1.00 rZ/rl — l 01 -------- .02 oooooooooo ‘ ‘ .0 ‘ O m 0 ~ 0 5 ‘ ‘ I -- .. ‘ .I ~ .0 ~ .0 ‘ ‘0 ~ ‘ ~ l h' I. '.' l. 31‘? l l l L l 0.02 0.06 0.10 0.14 0.18 0.22 Normalized Radius, a/7\o Fig.4.7 Theoretical Edge Conductance as a Function of Frequency As Affected by a Vacuum Gap 96 ., .\'. ....... :-.;__:; . . \ ...... _','_;;.—--' ‘3 . I: ‘ .2.4—3;:-:;-.-;:-,.,_.‘,:_,;______ .1: :1'1 , “ E 1.: ; .5 '1) ' .. :1 m i ' d/Zrl = 0 05 2’1: I 6 r — -1.0 F g: ; r3/rl — 1 5 a”. . 1.00s 0:; _ rz/rl - 1.01 ....... 1.02 ......... 1. 10'3 1 J L 1 , 0.02 0.06 0.10 0.14 0.18 022 Normalized Radius, a/>\o Fig. 4.8 Theoretical Edge Susceptance as a Function of Frequency As Affected by a Vacuum Cap CHAPTER 5 INPUT IMPEDANCE 5. 1 Physical Models Necessarily, any physically constructed spherical antenna will not be driven by a delta function electric field placed at its outer surface as described in the previous chapters. Rather, the antenna will be composed of either two solid hemispheres with the source placed at their geometric center as in Figure 5.1. l, or one solid hemisphere driven against a ground plane as in Figure 5.1. 2. In the first case, if the source region dimensions are much smaller than a free space wavelength, assuming that the gap is free space, the input impedance can be defined at a nearby imaginary cylindrical surface as the potential difference between the two hemispheres divided by the current transversing this surface on either hemisphere. In the later case, the input impedance is just half the value found for the first case. This input impedance can be found by considering the gap between the hemispheres to be a radial transmission line terminated in the edge admittance as previously found. Accordingly, consider Figure 5. 2, where it is desired to find the input impedance at the radius r = e , as given by Z(e) = V(e)/I(e) , in terms of a load impedance at r = a , given by 2(a) = l/Yedge. 97 98 v Fig. 5.1.1 Fig. 5.1.2 Fig. 5.1 Spherical Antenna Models with Source Region Shown n—Q—u I —" Fig. 5.2 Enlarged View of the Gap Region 99 Following the development of Ramo, Winnery and Van Duzer6, between the circular disks, jweoE (5.1) (7.11 VXE = -jw,uoH In cylindrical coordinates, assuming EzEQ fi=H$ 02) d z o a : 0 B7. 75$ it follows that 815: 1 a - : - 'w H _ (rH) = we E (5- 3) _5? J H0 r .517 J o and therefore 2 H - 1 8E E 1 [ OE + r B E] ... jwrlo '5'; ‘ - kozr Sr 8 r (5. 4') . . 7- 2 Accordingly, With ko =00 ,Uo 50 2 a E 1 8F. 2 + ... k E : 5. 5 5 r2 r a r + 9 ( ) 100 The solutions of (5.5) are, from Jahnke and Emdez, E = AJO(kOr) + B Y0(k0r) (5. 6) _ J. H - ’1 [AJl(kor) + BY1(kOr)] where A and B are constants to be derived. The impedance at r = a is defined as Z(a) = V(a)/1(a) where V(a) = -E(a)d 1(a) = 27TaH(a) Therefore, with C = B/A V(a) d 13(3) . d J0(koa) + CYO(kOa) Z(a): 1(a) : ‘ 2’)Ta H(a) : Jrlo 27Ta J1(k0a)+CYl(koa) (5. 7) Solving for C yields 12mm ) . C : - LanTl'a 0 Jl(koa) - JJo(kOa’) (Z(a)/710 Ld/ZTI'a (5.8) 1 Yl(k0a) - j Y0(k0a) It is apparent that (5. 7) holds at any radius, and therefore, using (5. 8), the input impedance is 101 d Jo(koe) + C Y0(koe) ZTTe J1(koe) + c Yl (koe) (5.9) 5. 2 Theoretical Results In the following curves of input impedance as a function of frequency, e/a = 0.135 , therefore, if a/ )\ o < 0. 3 , it follows that e/ )‘O < 0. 045 and the approximations requiring the input region to be a small fraction of a wavelength in any dimension should be valid. As the edge admittance is required to formulate the input impedance, the accuracy of the following curves is directly related to that of the corresponsing edge admittance from which they are formulated. Figure 5. 3 shows the input impedance of an isolated antenna, while Figures 5. 4 and 5. 5 show the same antenna after it is covered by a dielectric and then permeable shell respectively. Figure 5. 6 is the result if the dielectric shell contains losses. From Figures 5. 4 and 5. 5, it appears that the effect of the shell upon the reactance curve of Figure 5. 3 is to shift it up or down, and then add resonances to it. Figures 5. 7 and 5.8 are for shells having negative dielectric constants of -l and -10 respectively. Comparison of these curves to those of Chapter 1 shows the effect of the radial transmission line model as the antenna radius increases. It is also obvious that the input resistance is much larger in this case, compared to the normal dielectric or permeable case of the above. 102 0N0 1b NN.0 4)- 0N0 q» 13:630on 00 cofloc5h m we oocmpoQEH 5&5 HwoCoHOoLH m.m mm~.003m\o .H 0A0 W m0.0ns.wm.:U 1! ' 67m 1 ON+ 1. 0m+ 'Z ‘aauepodLuI induI 801111011va :1 103 1") .AoCodwouh mo soflogfm m mam oochoQEH “3&5 HmoflonoofiH .....m ..fih mmdo u .20 ms 1 m3 mm“ o m0.0 n mm\p 1 0m: So o~.o 3.0 one 0:. 0N+ 0m+ z ‘aouepaduil indul sumo u: ‘x .f+ H 104- oocmpomcfi 0:9: Hmofiouooxh m.m mam m m m .1 ON: 1 0H1 vwho mm. 0Nw0 00.0 1 4i)“ 4 {I O a. _. ”m .1 ”a m _ l 02. ..1 0N+ .... 1 o... sumo at ‘X f+ H : 'Z ‘aouepoduil induI 105 saucodvouh mo Gowuucdh m we oocmfioaflb page: HmofiuouooLH wim .mfm mm~.0 n w\0 m4 .1. p.33 m0.0 .1. aces mm 0 w m0.0 n .mN.\p cl CH. of. ombo 030 07% «67)0 ward 0W0 00.10 00.. 0 ,‘/‘\ l 0N+ suiqo 111 ‘X I). H : Z ‘aouepodwl 106 10‘Z 1: T \ ‘\ 1 101 r- m : g _ .5 _ .5 _ >6 ./ 1, m ,’ II P N a; U 5 610° 1‘ a I. E r ’- o.02 0.06 0.10 0.14 0.18 0.22 Normalized Radius, a/)\O Fig. 5.7 Theoretical Input Impedance as a Function of Frequency b/a =1.5, e/a = 0.135, d/Za = 0.05, er: -l.0 107 103 r- C )— ‘\ H \ r— \G—I L- \ (+1 \ \ \ \ \ 102 '" X \ '- \ '- \ '- \ '- \ \ ' \ \ m P \ a R x .8 )- \ o \ .S ‘ >6 + m .1 101 N t )1 0” _ U h' 1: Id ,_ 'o (D Q _ ..E. 100 l l 1 l m 0.05 0.09 0.13 0.17 0.21 0.25 Normalized Radius, a/>\ Fig. 5.8 Theoretical Input Impedance as a Function of Frequency b/a = 1.5, e/a = 0.135, d/Za = 0.05, 6,. = -10.0 CHAPTER 6 THE EXPERIMENTAL STUDY OF A DIELECTRICALLY COA TED SPHERICAL ANT ENNA 6. 1 Introduction To prove the validity of the theoretical results of the previous chapters, it is necessary to provide experimental verification of the theory for at least one example. This does not mean that the theory will then be correct in all instances, but it would lend support to theoretical predictions for similar examples. Accordingly, as there are two basic classes of antennas discussed in the previous chapters, corresponding to whether the coating material is described by a positive or negative dielectric constant, two basic experiments need to be considered. The case of a positive dielectric constant is treated below. Two approaches to experimental verification exist. One method is to measure the antenna input impedance variation with frequency, the other involves the measurement of the radiation fields. The method of verification utilized, and the reasons for its use, are discussed in the next section. 6. 2 Experimental Method A major consideration involved in the choice of measurement technique is the available equipment. In the present case, available 108 109 is a rectangular anechoic chamber measuring 6' x 8' x 6'. One 6' x 8' wall is covered by an aluminum ground plane, and the other walls are covered with B. F. Goodrich VHP-8 microwave absorber. Attached to the ground plane is a 75 ohm coaxial slotted line, the center conductor of which is to be utilized to excite the antenna. If the input impedance of the antenna were to be measured, from Chapter 5 it is apparent that this would require the measurement of extremely high standing wave ratios on the existing coaxial slotted line. From Chapter 1, the phenomenon of interest occurs at frequencies such that the dielectric shell is at least a quarter wave- length in thickness, where the wavelength noted is that in the material. To prevent the physical antenna size from being excessive, and to fully utilize the absorbing properties of the microwave absorber, both a high dielectric constant in the shell material, and a driving frequency in the range of ZGHz are desirable. Measurement of the input impedance has been attempted for a copper hemisphere covered by both a layer of water and powdered strontium titanate near ZGHZ . It was found that the standing wave ratios are too large to be accurately measured due to the inherent error associated with locating a point on the line and the amount of minimum shift observed. For a lossless material, this minimum shift should be very large near a resonance, but as shown in Chapter 5, any losses will drastically reduce any such shift. 110 Because of the forgoing discussion, it was decided to measure radiation patterns as a function of frequency. To measure a far field pattern, the microwave absorber properties and the anechoic chamber dimensions require a driving frequency of approximately 2C‘1Hz , which in turn allows the use of a reasonably sized antenna model. The major problem now is to find a material which is suitable for the dielectric shell. Preferably, one that has a small loss tangent in the frequency range of interest in order that a measureable amount of power can be transmitted through the shell, and a large dielectric constant which will allow overall size minimization, is desired. Much time and effort was spent trying to utilize strontium titanate for the shell material as it has very desirable characteristics, 6r: 230 and tan 6 < 0. 00289’10’12’13. Due to problems involved in forming the ceramic shell, no progress has been made in this direction, and the experimental model constructed utilized water as shown below. 6. 3 Experimental Model The physical construction of the antenna utilized is shown in Figure 6.1, as well as its connection to the 75 ohm coaxial source and ground plane. The teflon shell is ignored in the theoretical analysis because of its thinness and small dielectric constant and loss tangent. Petroleum jelly was used in the joints to seal in the , 14 water because of its low loss properties. 111 r— ground plane / / .x / ’1 //— styrofoam spacer I T ///—— /,-- teflon support / coaxial line teflon shell ” “ 0.279 0 635 / 2.223 copper hemisphere / dimensions 1:;\\\\ 1) water shell _.__ '--~~ — in centimeters Fig. 6.1 Cross Section of the Water Covered Spherical Antenna N 112 In Figure 6. 2 is shown the arrangement of equipment for the excitation of the antenna and the measurement of the resulting field patterns. The small dipole utilized to measure the far zone field can only rotate through an angle of 90°. Thus, the assumption of symmetric fields within the chamber requires a driving frequency that is sufficiently high to insure the minimization of reflections from the absorber covered walls. 6. 4 Theoretical and Experimental Results From Chapter 1, it is discerned that, to a very rough approxi- mation, the broad conductance peaks occur when n=l,3,5,"" (6.1) a/7\ = n O 4(2-1)./€r With E r = 76 , a broad conductance peak should then occur near a/ A0 = 0. 0854 and a/ A0 = 0.256. The loss tangent and dielectric constant of water are as shown in Figure 6. 3. These values, as opposed to the above assumed constant value, are utilized in the following calculations. The theoretical edge admittance of the water-cove red spherical antenna is shown in Figure 6. 4, and it is obvious that a broad conductance peak does occur near a/ X0 = 0. 256. From (6.1), this peak is taken to be the second broad conductance peak, and the SWR Meter Filter 113 —-£:L «Z—g Diode Det. X-Y Recorder Sliding Shorts Frfequency Meter .11-] R. F. Osc. _, 1.7 — 4 GHz ““ Amp. Mod. l__.___. l 11111, Sq. Wave Anechoic Chambe r 6 'x8'x6' Fig. 6.2 Experimental Test Equipment Arrangement Rec. 1AA/jiiyifx1531 Absorber I \/\/\/K /\7 ——._-- W.--—.——( Ant. ,2 / C/\\., ' \“xo/\\/\/\. L/ 114 4‘1 Log10[tan 6 x10 OOmN© >ocoscouh mo coauocsh m mm .5an mo usowcmH mood was 33950 3.36230 mo .wah NE Ca $38.90lo 0H01H1 1 . 1 J . . 00~1_1 moa nod . . 4 J1 J u d A a d A q )— < < . N I: L 0 CS usomsmH mood I. L m l J osmomdou uwhuuoaoflfl 00 on ow iueisuog otiioolatq 115 0) O .c /\/\/ E __ 5 - /,,- m ,’ a, I + ) ,’ O I II I\ ’ H '1 ’ >0 ‘, ‘I ‘13 10--1 ' '1 B ‘ 3: ° I , ' :9, I ) _ ,' .ti 1 ’1 E I 2 I : b/a = 1.337 I. 1 I d/2a : 0.11 I I I ' I l l I I l I ' l 1 l 1 I; " I 10"2 '15 l l l l n 0.15 0.19 0. 23 0.27 0.31 Normalized Radius, a/)\O Fig. 6.4 Theoretical Edge Admittance of a Water—Covered Sphc r11 11 Ante-m1 1 116 initial ripples are then explained as being higher order resonances that have occured after the first broad peak. From Chapter 2, it is expected that in the frequency interval presented in Figure 6. 4, the third order associated Legendre polynomial will dominate the field pattern throughout that part of the interval associated with the second conductance peak. Figures 6.5 and 6. 6 show both the predicted and measured field patterns at definite frequencies within the frequency interval of Figure 6. 4. It is apparent that a third order polynomial term is excited due to the presence of the water shell, as from Figure 6.7, the measured and predicted fields in the absence of such a shell are basically dipolar. Apparent also in the figures is that this resonance pattern occurs well after it is predicted, and that it experimentally disappears before expected. These discrepancies can be explained, drawing upon Chapter 2 and other work not presented, if both a larger loss tangent and a larger dielectric constant were assumed. 6. 5 Possible Improvements It may be possible to measure the input impedance by standing wave ratio methods, if a much larger antenna structure and correspondingly lower frequency, were used. This would require, using the given anechoic chamber, the assumption that the input 117 ;‘ a/x = 0.176 O 2 lEel O o 90 Fig. 6.5.2 00 Theoretical ----------- Expe rimental :‘I Fig. 6.5 Radiation Fields "I f ,X ‘ ' ./‘ [Ila I x ’ I 4f; I 90° 118 a/>\o = 0.268 \Eiz o 90 a/XO = 0.278 Fig. 6.6.2 90 Theoretical ---------- Experimental Fig. 6.6 Radiation Fields "a 119 a/)\0 = 0.278 2 H A >_,- ./I «p 90° Theoretical --o--—.o.--...o.- No Water Experimental ——.————o_—o— Theoretical .............. Water-Covered Experimental Fig. 6.7 Comparison of Field Patterns of Water-Covered and Isolated Spherical Antennas at a Given Frequency 120 impedance can still be measured in the chamber even though the microwave absorber does not work as well as it does at higher frequencies. The longer slotted—line wavelengths will allow, if the line is lossless, the measurement of the large standing wave ratios. Better field patterns could be obtained if a low loss material were utilized in the dielectric shell. The use of ceramics appears to be complicated by their inherent hardness and the tolerances required in the shell construction as shown in Chapter 4. The use of a plastic material, such as Stycast HiKlSlé, would appear to yield more fruitful results. Finally, the field pattern could be measured while using a negative dielectric constant material, a plasma, in the shell. In this case, some of the results could be compared to the 17 work of other authors. A PPEN DIX A WAVE IMPEDANCES Part 1 - Recursion Formula for Wave Impedances The following development is, in general, similar to that of Infeld3. It is different in that it is an extension into the domain of complex permitivities and permeabilities, and in that it yields a recursion formula for the converging wave impedance, 28h) , . + as well as for the outward travelling wave, Zn(r). In Chapter 1, (l. 8) and (1. 9). the wave impedances were defined as FHu) (k ) 2 + _ 1‘ Z;(r>=im rim --‘-‘- ...-4,2,3 (1) kr Hn+U2(kr) J 1 —- i = H for H (A-l) With the spherical hankel functions defined by hilikz) = /;: HELL/ab) (A-l) can be rewritten as + hfi:_)1(kr) + ' _t- -.I_1_ = "k 211”" m m .. “g1“ ., “[12] H M 121 122 From the Handbook of Mathematical Functions , (. . . iii-113m +£24wa = hfl’dz) 1 =1.2 which allows (A-2) to be written, with z = kr , as (i) - d (1) 1L + hn (z) + ”32% 1‘6””) :1; Zhn (z) Z'nh.) : '1“ (i) : :371 (i) (A_4) Zhn (Z) zhn (Z) + where, unless noted otherwise hereafter, for Zn(r) , i : 2 , and so on. The function will): zhghz) is called a Riccati-Bessel function7, and is a solution of d2 (i) mm (1) an +1———2—— wn :0 121,2 (A-S) dz Z Rewriting (A—3), with n replaced by n-1, and with will)(z) : zhill)(z) will yield (1) n (i) d (i) . wn (z) — ”z" wn_1(z) - a—z'wn_l(z) 1 — 1,2 (A-6) Using (A-5) and (A-6), d (i) “a; “’n-I‘Z’ . nz . - n + z .21. WW WE) dz n n-l . : . i : 1:2 (A-7) (1) _d_ Wm Wn 2 dz n—l nz - z _ (1) Wn-l 123 According to (A-7), the wave impedance of any order can be found if the wave impedance of the previous order is known, as from (A-4) d (i) + w (Z) _ 2 H Zn(r) _ J“ dz n 1 -[1] for i A knowledge of ZO(r) will therefore allow the calculation of all wave impedances. Accordingly, d (1) d h(i) d (i) EWO‘W.EZ( ‘2’]_i+3?h°‘z’ Win”) zhmm z hmm where from tables , (2) h(1)(z) : sin(z) i j COS(Z) 0 z — z e+JZ 1+ yielding for a final result 2%; 2?; W0 (Z) = ;-.,- who) 124 This initial ratio will yield first order impedances of, with z = kr , 1 1 z sz + ._ _ 2;“) T1 Z4-zz+l These values are identical with those of Infeld in the case of Zl(r) and conform to results derived in Part 2 of this appendix. By defining f(i)(z) = i w(i)(z) / wéli)(z) , (A-4) and (A-7) can be n dz n written, respectively, as i. . Zn(r) : ijq f;1)(z) (A-9) and (1) 2 2 (i) nzf _1 - n + z f r: n . - 1 .. n 2 (i) 1 — [Z] for L} (A-10) nz - z fn—l Under the conditions z = x+jy 2 2 2 2 n >>x ;n >>y ; n>>1 (A-lO) is approximately (1) nzfgjl 2 2 (i) nz - z fn_1(z) (z)-n fn (Z): (A-ll) 125 Infeld found, for the case y = 0 , that (2) in m7..- 2 If a similiar result is assumed for complex 2 , {mm 2: — E i = 1, 2 n z the assumption is shown valid upon its insertion into (A—ll), as it yields f(i)(z)~ -n(n-l) - n2 z _ g 1 = 1,2 n ~ nz + z(n-1) z Part 2 - Relationships between 2:1(r) and Z-n(r) The outward and inward traveling wave impedances are, from (A-Z) with z = kr, i 1‘ Zn(r) = T] gn(Z) With an asterisk denoting the complex conjugate z;(r) *= Tfi g§ * (A-lZ) (A-l3) 126 From (A-2) 3': TH“) (2) :g - €32) -.i “-1” .. .... (2) * za‘ I-In+]/2(z) _ _J 7 Using the identity 2 =:: 1 g, H;)(z) = Hfflz) (A-14) becomes [— (1) * .— ::: H __ a, g;(z) : 'j n-1/2(Z ) - 9.3,: ___ gn(z«.~) High/dz > 2 Thus d; + ... 9.: + at :z: _ J, znm = q gnu) =n gnaw») and if z = kr and n are both real * Z;(r) = Z;(r) Using a second identity (1)(zejT\') : _ e-jP“ H(2)(z) H P P (A-14) (A-lS) (A-l6) (A-l?) 127 for p: nil/2 , yields m , n (2) Hn+l/2(Z) = J(-1) Hn+1/2( _z) (2) (1) n "j ('1) Hn-l/Z(-z) Hn_yz(Z) Insertion of this last result into (A-l) yields FHB) ( ) + _ _ "Z Zn(r)=jTl (1211/2 - (f2) -zn(-r) (A-18) Hn+1/2('z) — If the medium is characterized by a purely imaginary propagation constant Zak) ‘ g'(z) - +(-z) - g+(z*) - 22(4) (A 19) n n n ' Part 3 - Curves of 23m and zgu) i :I' In this section, Zn(r)/T\ or gn(kr) , will be plotted as functions of kr for various values of n, where t (i) z (r) 1' . hn_l(kr) n . _ [I T) n - J hghkr) - 15 1+ 128 For kr real, ZI';(r) is just the complex conjugate of 2:1(r) from (A-16). Therefore, with z,’;(r) = Rim + jxgm , Figures A1 and A2 show R;(r)/T\ and x;(r)/Yl , respectively, as functions of positive kr. i If kr is purely imaginary, Zn(r) is purely imaginary. To show this, the spherical Hankel functions are - n hg)(Z) = ‘51:}:— 2 (n+l/Z.k) (-J'2Z)-k (A-ZI) j Z k=0 where (n+1/2,k) = fl and 115.2%) = in”; Jz :3; (n+1/2,k) (1221* k;0 For imaginary z , the terms in the summation are real for both (A-21) and (A-22), and thus (1) 8J3 ' (2) _n+1e-jz h (z):c-—nn—- hn(z)=D J —— I). n j z n z where Cn and Dn are real. 129 Therefore (1) (2) hn-l(z) , Cn-l hn—1(Z) j Dn-l ——————— _J —_ .__.-—__ Z — 1 C 2 D hfflz) n hf, ’m>> ponfldEqu Hide .mflh .Hx m: n: «l NH OH w o v N o Ma : U 11/(1)+‘d 131 NH imam >pmcwwmcfi .oocfloomcfi o>m>> powflmguoz N14. .mfim Rx OH m o v N o u x- + h/(I) 1) ucmumcoU cofimw mQOgnH \Cmcwm mEH .oocmmVoQEH erred? pvmfimezaoz muxi. .mfm 4 4 .v w- l. or .4 VI H N. m Lr NI .5: . om W b v N N- v- on w- 2 B i 111 \O 4 APPENDIX B CALCULATION OF EDGE ADMITTANCE Part 1 - Convergence For a perfectly conducting spherical antenna, covered with a layer of lossless material of relative permitivity E r and relative permeability (Jr , driven by a voltage source across a gap of thickness d/2a at the equator, the derived form for the edge admittanc e is 0° P1(d/2a)Pl(0) (Zn-+1) 71.2 n n Y = 13-1 edge n=1 H(n+1)Zln(a) ( ) nodd where - + 1 - K ,b b 2 z z (a):Z+(a) 1n(a 10m” 1n(a)/ ln(a) (B_2) 1“ 1n 1 + K1n(a, memw) It has been assumed in (B—l) that d/2a is small enough for replacement of cos (d/2a) by 1. (d/Za S 0.1) As shown in Appendix A, for n greater than some Nl such that 2 2 N1 >>Ikal and N1 >> 1, the wave impedances at r = 3 become 1- _ . n n 2N1 k zfg/g‘rko 133 134 Insertion of (B-3) into (B-2) allows (B—l) to be written as Z n(n+1) Zln(a) =1 :5 + - (B-4) J n2(n+l) 0° / 1 1 ka’fi‘ 2 Pn(x) 1311(0) (2n+1) n=N1+2 where x = d/2a and the sum is over odd n for the reasons given in Chapter 1. A more exact approximation for Z1n(a) , as given in Section 3. 3. 3, will yield this same result. Replacing 11 by 2m+1 allows (B-4) to be rewritten as M1 1 edge — m=O (2m+l) (2m+2)Z1’ 2m+1(a) ~ on 1 l +. kall _ Z P2m+1(X)P2m+1(0)(4m+3) (B 5) J ...—.... _ Y) (2m+l)2 (2m+2) sz1+1 = Y1(O' M1) + Y2(M1+1, 00) From the form of (B-5), Infeld's result Zka Mld Y2(Ml+1,oo) 2: -j"fi‘" Ci (-—a— ) (B-6) 135 asM x 1-.» 00 and where Ci(x) :Lo cos(v)/v dv can be concluded. Thus, provided that Zln(a’) # O which is shown in Part 3, it has been shown that (B-4) converges for d/2a greater than zero. To the practical problem of choosing an N , which is not too 1 large, and knowing roughly what the resultant error will be, the remainder of Part 1 and Part 2 is devoted. This is achieved by following Infeld's development, noting where approximations are made. From Abramowitz and Ste gun7, for all n Prim) = nPn_1(0) (13-7) 2 From Jahnke and Emde m (2m-l)!! (B—8) (2m)!! with m 21,2, 3, """ n!! = n(n-2)(n-4) ' ' ' ' Combining (B-7) and (B-8) yields 2 +1)!! Pimuw) =(-1)m LIE—- m =1.2."" (B-9) 136 Again from Jahnke and Emde for n>>m and e> 1 x+l/2 - x! = ./2T|_ x e X (B-13) and it is obvious that if m > M3>> 1 , 2 M212.._=4+ 4 +2.12 :::4 (13-14) (2m+l) 2m+1 (Zm+l) With m >M4 such that (3-13) is valid, (B-13)and (B-14) allow the approximation (4m+3)2(2m+1)! ! __‘_1___ (13-15) (2m+1) 2(2m+2)! ! $le 138 Then for M greater than the largest of the Mi’ i = l, 2, 3, 4 , (B-12) is 00 OS[4m+3 x] Y (M, 00) :j —— “1‘3sz 2 (B-16) m=M With x<<1, as already assumed, 4m+3 2 cos ( x ) 2: cos (2mx) — —§£— sin (2mx) and the summation of (B-l6) can be transformed into an integration, yielding (I) (I) 2 C03 (2mx) _ I M dv — - Ci (ZMx) m ' v m=M ZMX (B-l7) 00 oo 2 imam") = I 331—“ (V) dv= 1'— -Si(2Mx) m 2Mx V 2 m=M Finally the contribution to the antenna susceptance due to terms after some large M is Y (M, 00) =j 359- [ é’i (Si(2Mx) - I211) - Ci (2Mx)] (B-18) ’1 where as Ma» 00 the leading term is, as given by Infeld, 139 Y2(M,oo) = -j 5‘53 c1 (ZMx) *1 with x = d/Za. Part 2 - The Errors No attempt is made here to calculate the error associated with a general set of Mi (i = l, 2, 3, 4), but rather the method actually utilized for calculating the edge admittance is presented and for the particular set of Mi used, the resultant error is approximated. With no little algebra (B-l) can be written as oo . W P1(d/2a)Pl(0)(2n+l) 1 Y = — —— n n _ edge J (B 19) n n(n+l) F _ .9— n=1 n ka n odd with F A1n+RnA2n n = A3n+RnA4n (1) (2) (Z) (1) Aln = Hn_1/2 (kb)Hn_1/Z (ka) — Hn_1/Z(kb)Hn_]/2 (ka) (1) (2) (2) (1) Zn - Hn-m(kalHn+1/z(kb) ‘ Hn-1f2(ka)Hn+l/Z(kb) 11> l (1) (2) (1) (2) 3n — Hn_1/z(kb)Hn+1/2(ka) - Hn+l/2(ka)Hn-l/Z A (kb) 140 (1) (2) (Z) (1) A4n = Hn+m(ka)Hn+]/Z(kb) — Hn+v2(ka)Hn+Uz(kb) E (2) R = n(l- r) + E 52 Hn_vz(kob) n kb r k H(a) (k b) n+l/2 0 With n replaced by 2m+l , the actual quantities computed are Y = Y1(O,29) + Y2(30,49) + Y3(50,oo) (B-ZO) edge where Tl— P2m+l(d/23)P2m+l(0)(4m+3) Y (0,29) = -' _— 1 J T\ (2m+1)(2m+2) m=0 1 21 ' imfir (B' ) F2m+l kg, 49 1 r“ P (d/Za)P (O)(4m+3) Y2(30’49):j Ilka Z Jm+l 2 mm (B_22) “1:30 (2m+l) (2m+2) . Zka. 3 d . d W . d = ——- — —_ _ -— - ._ 13-23 Y3(50,oo) J n 2 (2a) [81(50 a 2] C1 (50a) ( ) The above division of the summation (B-ZO) is found to be advantageous, as well as necessary. The calculation of the required Bessel functions on the digital computer is time consuming, and it 141 is advantageous to reduce the number of terms in Y1 as much as possible. Conversely, it will be shown in Figure B-3 that to maintan a constant accuracy in the computed admittance, as ka increases, the number of terms in Y1 must increase. The number of terms actually utilized in the preceding equations (B—Zl), (B-ZZ), and (B—23), should limit the total error of (B-ZO) to less than ten percent for an antenna with a/ )\o< O. 25 and Wr< 18 according to the reasoning which follows. The second term of (B-ZO), Y2 , which does not contain large order Bessel functions, is composed of only as many terms as is deemed necessary before the approxi- mations utilized to form Y3 can be considered to be valid. The sum of the first 30 terms is an exact partial sum and therefore Y1(0, 29) has a negligible error, due only to computing inaccuracies. The sum of the‘second 20 terms involves the approxi- mation Fn - n/ka 7:: - n/ka in each term. The quality of this approximation can be understood by observation of Figures B-l, B-2, and B-3. Figures B-1 and B-2 show how Fn varies, for a given set of parameters, as the integer 11 increases. Fn is a point function and the solid lines are merely aids to the observation of its values. These figures show that the imaginary part of Fn , which contributes to the edge conductance, is negligible with respect to the real part of Fn after the first one or two terms unless for some integer n the “It‘ll-Ill! 142 101 " n/ka 10 , 0.15 ' a/kO 10‘1 I FTTT I r -2 1 10 0 5 10 15 20 25 n Fig. B-l Fn and Related Quantities as a Function of n 143 1 _. 10 n/ka , \ ,’ I l” \_'Re(Fn -n/ka)‘ I \ / / 100 '- v, a/7\o = 0.15 Gr =15 b/a : 2 mew,» 10'1? : IIm(Fn)l ‘\ P l \ b l l I 10-2 i 1 1 L 1 1 0 5 10 15 20 25 n Fig. B-Z Fn and Related Quantities as a Function of n 144 mmCCoucm Sm H3 230$ 8:: cm mm H a £33 .mx .usoaswumw 05 mo aoflocdrm m mm mcmfipmu cm :3” ON A: Na w v o a a d u I < d 1 N I 33526 BC. Tm .OE 8.0 3.0 smug E d— wod m Nio 145 real part of I?n is equal to n/ka. This observation is a graphical proof that Zn #- 0 for any real frequency as Zn = jTl [Fn - n/ka] As n is increased, Fn decreases as required in (B—3) as shown in Figures B-1 and B-2. In the present computation, Fn is computed up to n = 59 and the fractional error in making the approximation Fn - n/ka z-n/ka is shown in Figure B-3 for n = 59 as a function of ka . Obviously the error for n = 61 is less than the error for n = 59, and the total fractional error in the computation of Y2 (30, 49) will be equal to that of the 6lst term. In the last term, Y3 , when it is related back to (B-ll), Mz , M3 and M4 have all been set equal to 50 and the accuracy of the approximations to l P1 (x) P (0) (4m+3) J _l_<_a_f_l_oo _2m+l 2m+l (B-Z4) (2mm2 (2m+2) are in question. 1 The equation (B-24) becomes, approximating sz+l(x) by (B-lO) co 2 4m+3 . Jfika 2 km”) (2m+l)!! cos ( Z X) (B—ZS) J l" (Zm+l) 2(2m+2)! ! ,/ 4m+3 m=M l where P40 (0. l) , approximated by (B-lO) is accurate to l. 5 “/c. 146 Equation (B-ZS) becomes after applying (B-13) and (B-14), accurate to roughly 6% for m = 50. The error is transforming the summation to an integration is directly proportional to the magnitude of x, and for x = O. 1 it can be shown that 60 12 z 323 2mx_ ~ I cos (u) du m 10 u m=50 to within 8%. Taking the above errors into account, as well as the fact that x = O. 05 is actually used, the error expected in Y3 should not exceed 10%. As identical analysis, performed in the case of a purely imaginary propagation constant, would produce curves similar to Figures B-l, B-2, and B-3. In these new curves, the real part of Fn provides any conductance, and goes to zero most rapidly. The convergence requirements remain unchanged, d/Za # 0 , and the error associated with (B-ZO) would be on the same order. 147 The result of the above work is that the error associated with (B—ZO) is approximately the error associated with the approximation Fn - n/ka £1: - n/ka drawn on Figure B-3. This is because Y2 is of the order of Y1 in magnitude only when ka becomes increasingly large, while Y is always a small fraction of Y for the chosen 3 2 gap thickne s s . Part 3 - Proof that Zln(a) # O The edge admittance, as shown in (B-l), will become infinite if, for any n, Zln(a‘) = O . To show that this can never happen for .. + :fi: a lossless coating medium, note that in this case Zln(a) = Zln(a) . The zero condition requires that 1 = K1,,(a. b) anm gym/2132111 (3-27) From its definition in Chapter 1, and the identity used to derive 'j29n(a-9 b) (A-15), for the lossless case, K1n(a,b) : e , while from _ + _. above, Zln(a)/Zln(a) = e 12¢(a) . Therefore, for equality in (B-Z?) to hold, 101nm] = 1 (B-28) can be From its definition in Chapter 1, and Appendix A, ‘Gln(b) written 148 (a+jb) - (c+jd) (a-c) +j (b-d) (a+c) +j (d-b) 1611.“)! = (a-jb) + (C+jd) 2 2 : (a-c) + (b-d) (B-29) (a+c)2 + (d-b)2 where a,b, c and d are all real; a = R1;(b) Therefore, either a = 0; c = O; or a = c = 0 for ’Gln(b)l : 1 This requires that for some r + + Re [Zn(r)] = Rn(r) : 0 where from (A-l) + R (r) = n Tl Ji+vz(kr) + Yfi+vz(kr) But the quantity? Jn.vz(x1Yn_l/z(x) - Yn+UZ (x) Jn_ 1,2 (x) = 77332 and thus + R ( )= ,— 2 2 showing that Zln(a) # 0 for any non—zero frequency. (l) (2) (3) (4) (5) (6) (7) (8) (9) REFERENCES E. A. Wolff, "Antenna Analysis," John Wiley and Sons, Inc. , 1965 Jahnke F. , and E. Emde, ”Tables of Functions," 4th ed. Dover Publications, Inc. , New York, N. Y. , 1945 L. Infeld, "The Influence of the Width of the Gap upon the Theory of Antennas, " Quarterly of Applied Mathematics, Vol. 5, No. 2, pp. 113-132, July 1947 J. A. Stratton and L. J. Chu, "Steady-State Solutions of Electromagnetic Field Problems; II. Forced Oscillations of a Conducting Sphere, " J. Appl. Phys. , Vol. 12., pp. 236-240, March 1941 J. Galejs, ”Dielectric Loading of Electric Dipole Antennas," J. Res. NBS, Vol. 66D, No. 5, pp. 557-562, Sept. 1962 S. Ramo, J. R. Whinnery and T. Van Duzer, ” Fields and Waves in Communication Electronics, " John Wiley and Sons, Inc. , 1965 M. Abramowitz and I. A. Stegun, " Handbook of Mathematical Functions,” Dover Publications, Inc.., New York, N. Y. 1965 C. Polk, " Resonance and Supergain Effects in Small Ferromagnetically or Dielectrically Loaded Biconical Antennas, " IRE Trans. on Antennas and Propagation, AP-7, pp. 5414-5423, Dec. 1959 E. N. Bunting, G. R. Shelton and A. S. Creamer, ”Properties of Barium-Strontium Titanate Dielectrics, " J. Am. Ceram. Soc., Vol. 30, pp. 114-125, 1947 149 (10) (11) (12) (13) (14) (15) (16) (17) " Insulation Directory/ Encyclopedia Issue, " Lake Publishing Corp. , Libertyville, 111., 1961 L. J. Chu, "Physical Limitations of Omni-Directional Antennas," J. Appl. Phys., Vol. 19, pp. 1163—1175, December 1948 S. Marzullo and E. N. Bunting, " Note on Dielectric Properties of Magnesium-Strontium Titanates, " J. Am. Ceram. Soc., Vol. 40, No. 9, pp. 285-286, 1957 T. Y. Tien and C. J. Moratis, “Dielectric Properties of Strontium Titanate Solid Solutions Containing Niobia, " J. Am. Ceram. Soc., Vol. 50, p. 392, 1967 A. R. Von Hippel, " Dielectric Materials and Applications, " The Technology Press of M. I. T. , and John Wiley and Sons, Inc., New York, 1954 D. D. King, "Measurements at Centimeter Wavelengths," D. Van Nostrand Co. , Inc. , New York, 1952 D. Lamensdorf, " An Experimental Investigation of Dielectric Coated Antennas, " Harvard Univ. , June 1966 C. C. Lin, "Radiation of Spherical and Cylindrical Antennas in Incompressible and Compressible Plasmas, " A Thesis, Mich. State Univ., E. Lansing, Mich., 1969 150 TATE ”111111111111 3 1 2 93 lllfllflflljllllllllllll“