Journal of Research of the National Bureau of Standards Vol. 58, No. 6, June 1957 Research Paper 2762

Slotted-Cylinder Antenna With a Dielectric Coating

James R. Wait a nd Walter Mientka

.\na lvs i:; is prese nted for t he fi elds produced by an ar bitrary slot on a circular cylinder \\'hich h as a concentric dielectric coating. Express ions for t he fa r-zone fi elds are developed by evaluatin g t he appropr iate in tegrals usi ng a saddle-poin t m ethod. Xumer-ical res ults are presented for the case' of a narro'" ax ia l slot for a range of values of cylinder diameters a nd electrical constants of t he dielect ric coating. There is some ev idence that t he coati ng pro­v ides a tmp or a duc t for s urface ,,'<1 ,'es, resulting in an increase of over-a ll a mplit ude of t he field in the backward direct ion.

1. Introduction

Slotted-cylinder antennas arc now becoming ex­tensively utilizcd in mi crowave radiating s.\-stems. The great in Lerest in this subj eC'L is evidenced b~- the great number of papers on this subj ect within the last decade [1 to 7].1 In most of !lIe previous theoretical work, the slot is assum ed to be cut on a circular or ellip tical c~TliJ1cl er of perfec t conductivit~· and infinite lengtil . The compu Ultion of the radia­tion patterns is then s traightforward, although it can be very tedio us even if special summat ion techniques are employed. Gsually i t is desirable to program the series formul a 011 an electronic calr ulaLor if ('xtellsive numerical data is required [8, 9].

Several years ago one of the a u LllOrs (J. R. W .) was asked to consider the eiYect of cove ring the slo t wi th a di electric coating such as a fa bri c. A solution was carried out for an infinitel)- long axial slot on a circular cylinder, which itself was covered by a concen tric dielecLric coating of cons tan t t1tickness [10] . It was shown that, if the coating thickness approached zero, the paLLern approached uniformly the pattern expected for the un coated cylinder. It was then concluded that , if the slo L was covered b~- a lossless dielectric coaLing of very small thickness, the pattern would not be modified to any extent. Some related experimental work corroborated this con­clusion [11].

It is the purpose of the presen t paper to pursue this matter further. A solu tion is given for the fields produced by an arbitrary slo t on a circular cylinder which is covered by a concentric dielectric coaLing. At tention is then focused on the special case of an axial slo t where the raLher cumbersome formulas become less forebocling in appearance. Some mIlDel'­jcal results arc presellted for the far field in the equatorial plane of the slot or in the broadside direc­tio n from the c)-linder. It is then possible to give a more quantitaLive viewpoint of the effect of the coating.

It migh t be men liol1cd in passing that the plane wave scattering by a dieleeLric cylinder wiLh a metalli c core has becn considered recently b~' Adey

1 Figures in brackets indicate the Ii teratw'c references at tbe end of this paper.

[12]. Some of the numerical results obtained by him co uld have some application to the reciprocal antenna problem fo r Lhe case when the slot is circumferen tial.

2. Formal Solution

The c~'linder is taken to have a radius a, and the co ncentric dielectric coating has a radius b, as indi­ca ted in figure 1. Cylindri cal coordinates (p,4>, z) are chosen to be coaxial wiLh the cylinder. The electrical constan ts of the coating a re E and M and those of tbe homogeneous (air) space outside are EO

and Mo. The tangent ial elecLri c fields 2 on the cylinder are specified, being fin ite over the area of the slo t and 7, erO elsewhere. Therefore, following the suggestion of Silver and Saunders [7], the tangenLial field is written as a combined azimuthal Fouricr se ries and axial Fourier integral, such tha t

(1)

willt

1 f lO 1"" P (h)= - dz' d¢' E (a 4>' z')eihZ' eim",' m (27r) 2 21 cJ>'l -.J ¢ ), ,

(2)

\d lOre the slot is considered to be bounded by ¢1 < ¢' < 4>2 and Z I < Z ' < Z 2 in terms of the primed coordinates. A similar relation holds fo[, the E z(a,4>, z) component with Pm(h) being replaced by Qm(h) .

In the subsequent equations, the double Fourier representation is considered as an operator to sim­plify the notation. For example, eq (1) is rewritten, operationall~'

(3)

where the r signifies the mul tiplication of P m(h ) by exp[ -ihz-im¢] and then integration with respect to h and summation with respect to m.

In region I , defined by a~p~b , the fi elds can be represented as a superposition of TM (transverse magnetic) and TE (transverse electric) modes [13].

, Tne time factor cxp(iwt) is em ployed throughou t.

421230- 57--1 287

Therefore, the electric and magnetic field components are given by

(4 )

(5)

_:nWjl B J - 'hA OJm] min'/, .,. , p u p

(6)

(7)

'IB oJm+mP A JJ - '/,~ m~ - In m ' up JlWp

(9)

where J m= J m(up), the Bessel function of the first type of order m; where H m=H ;;') (up) , the Hankel function of the second kind of order m; and where U= (P - h2)IA and k = (~Jl) ~2W. The coefficients am, bm, A m, and Bm are independent of p, cp , and z, but are as yet unknown.

In region II, defined by P? b, the representation is also a superposition of TM and TE modes, but now only the Hankel function is needed because it has the proper asymptotic behavior for large valu es of p. Therefore,

(l0)

(11)

(12)

(13)

(14)

}7 [ 'l d oHm+mkg }7 ] :I p= r -t L m - .,.- -- Cm :I m , u p JloWP

(15)

where H m= H ;;') (uop) , uo= (kg - h2)~2 , and ko = (~O)1o)h w. The coefficients Cm and dm are independent of p, CP. and z.

The E <p and E z components in region I must reduce to the prescribed behavior at p= a, as noted by eq (1). Furthermore, the E q" E ., Hq, and H z com­ponents are continuous at p= b. Together these conditions lead to six linear equations to determine the six unknown coefficients . Symbolically, this set is

a"'lJ(Lm+ b",pbm+A mpA m+ B mpB",+ c",pcm+dmpdm=xmp, (16)

where the coefficients with the double suffix are given conveniently in table 1 for p from 1 to 6. It is now a simple matter to solve for the coefficients in deter­minant form. For example:

0 0 0 Om

0

dm = 0 0 0

0

0 0 0 0

D

Pm 0

0 0 Om 0

0

and Cm= 0 0 0 0 (17)

0

0 0 0

D

where the dots indicate generally finite factors ob­tained from the table. D is the six by six determi­nant of the first six columns in table 1. Expanding these determinants leads to explicit, although lengthy, expressions for the fields.

3 . Far-Zone Fields

The resulting integrals in region II are of the form

if +m 1= -2 _'" Fm(h)H ';;; ) (u op)e-ihZdh, (18)

where the dependence of the geometrical factors of the coated cylinder and the excitation parameters are lumped into Fm(h). For large distances from the cylinder such that kp» 1, the H ankel and

288

~

I

exponential fun ctions arc rapidly varying so the integral can be evaluated by the method of steepest desce nts. The procedure is straio-htforward [6] if it is rem embered that the original contour in the h plane must be inden ted above the branch point at h = k and below the branch point at h=-le . The result is

-ikR

I"' ei1ll~/2 _e - F (k sin (J ) = R lit 0 ,

where R= (p2 +Z2) ~' and (J = tan- I( plz) .

(19)

The far-zon e form of the fields ill region II are then simply obtained by r eplacing the Fourier operator r by its steepest descent form , such that

e -ikR 00 . ' .

rF (h) '" 2i --~ F (k sm (J )e"n,,/2 e- l1nq, (20) If/. = R m=O In 0 ,

where terms which var.,- as 1/R2, l /R3, ete ., fi re neglected .

4. Equatorial-Plane Fields

Even after making the far-zone approximation, the expressions for the fields arc very cumbersome . The situation is simplified somewhat, however , if the obse rver is in the equato rial plane (z= o or (J = 7f/2). Th is ca.se is considered here and, furthermore,

, the slot is considered to b e in the form of a n alTO\'­rectangle wi th its long side parallel to the cylinder axis. The integration , over the slot coordin ates , indicated b.r eq (2) then simplifies to

1 i " P m(h) ~P", (0) ~ -4 . 2 11 (z ' )dz' , 7r a "

where th e ce nLer line of the slo t is at ¢= o V(z') is Lhe transverse voltage along the slo t.

The far-field in the equatorial plane is th en veniently wriLten

where the pattern factor is given by

(21)

and

con-

(22)

P (¢) 2i "£ Em COS m¢ei1U ,, /2

7r(kb )(koa) 111=0 G:) H,;~) (lco b) Tm-H,;~) ' (kob)L1U

(23) with Eo= l , E",=2(m ~o) and

7'111= J:,,(lch)H,;~) ' (lca)-J:,.cka)H,;~) ' (lch), (24)

L m= J ",(Tcb )H,~) ' (lca)-J:,,(lca)H,;~ ) (kb). (25)

The prime oyer Bessel or Hankel fun ction indicates a derivative with respect to its argument. As a par­tial check on this result it can be seen , if b= a or if E= EO and k = ko, that

1 00 A. i m". / 2 P (A.\=_ " Em COS m'f'e (26)

'f') Ie L..J ] ' T (2) , (Tc ' oa m= O L m oa)

which is quite well known [1] . The eq llatorial-plane field is thus proportional to

the integrated voltage moment along the slot , a simple radial factor that r epresents an outgoing spherical wave and a rather complicated azimut h factor. The structure of the solution is elosely re­lated to a two-dim ensional (scalar) problem carried out previously [10] for an infinite axial slot with a uniform transverse voltage, V, througho ut its length. The radiation field for th is problem can be writ ten

E <I>=- i60ELWV (_2_) ~ e-i (kop-" /4)P (¢), (27) 7rkop

which has the form of an outgoing cylindrical wave with the same azimuthal dependence, P (¢ ), as for the finite slot . This two-dimensional counterpart is of further interest because it has a well defined acoustic a nalogy. In this instance, E q, is propor­Lional to the pressure field , in a medium whose wave number is lco, emanating from a cylindrical radiator. The source is a rigid cylinder except for Lhe narrow axial slot where the normal velocity is specified . Surrounding this eylinder is a fi.lm whose acoustic wave number is k and a density, r elative to the outer medium, equal to E/Eo .

Although the prime purpose, stated in Lhe intro­duction, is to evalu aLe th e etrect of the dielectric coating, it docs seem worLhwhile for the sake of completeness to consider the effect wh en /L I/Lo is different from uniLy. Composite materials can be produced whose macroscopic p ermeability differs from unity so th e results may be quiLe signifi.cant in their own right.

K eeping in mind Lhe above points, numerical com­putations of P (¢) ,vere carried for two sizes of cylin­ders (lcoa = 2 and 3) and for a range of values of kob , E/Eo, anel /L I/Lo. Of course , a deLailed study of the interrelation between Lhese parameters would entail a great deal of numerical work. As a compro­mise , onl? a limited number of sets of calculations were carried out for the ampliLude and phase of P (¢) for intervals of ¢ of 10 degrees, using available tables of B essel functions [14]. Case I refers to th e set of calculations where /L I/Lo= l , (c/EO)t = N and case II to the set where E/Eo= l , C/L I/Lo)!=M. Setting A = 1coa and R = kob, results were obtained for case I , taking

A = 2.0 , N = 2.0 , with B = 2. 1, 2.2, 2.3; A = 2.0, N = 1.5 , with R = 2.2; and A = 3.0, N = 2.0, with B = 3.2.

Corresponding values were obtained for case II with M replacing N . For sake of comparison, the two relatively trivial situations, A = B = 2.0 and A = B = 3.0, were also considered.

The numerical results are summarized in tables 1, 2, and 3 with th e appropriate va lu es of A, B , and N or M at th e head of each column of entries. If

289

further values of p c</»~ are required a l smaller inter­\'als of cp, they can be computed directly from the following formula

where Dm is a Fourier coefficient listed in tables 4 and 6 for values of mup to 10. For the uncoated cylin­der, where A=B, it is cOllvenient to redefine the Fourier representa t ion of P (</» by

1 ro

P(</»= (-k ) ~ dm cos m</>, oa ",=0

(29)

where d", is tabulatC'd ill table 7. The amplitude and phase 3 of P (cp) arc shown

plotted in figures 2 to 7. The set in flgures 2, a, and 2, b , indicate, in a graphic \Va~' , the effect of var.\' ing the thicknC'ss of the dielectric coating for a fixed value of a. Figures 3, a, and 3, b , show the influence of the dielectric constant of the coating material and figures 4, a, and4, b , pertain to a larger c~·linder. It appears from these curves t hat the only significant change in the pattern, r esulting from the a ddition of a dielectric coating, is to enhance the ripples in the curves. Physically, it ma y be supposed that the radiation from the slot travels around th e periphery of t he cylinder in both direc­tions. The coating apparently "traps" these pe­ripheral surface waves to some extent and conse­quently enhances the standing wave pat teI'll. The fact that the period of this standing wave pattern docs no t d epend essentially on the dielectric constant would indicate tha t surface wave is guided, with eonsidera ble leakage, along or just above the di­electric-air interface. The only essential effect of increasing the size of t he c.vlinder is to increase the number of ripples and reduce their magnitude somewhat .

The set of curves in figures 5 to 7 correspond to a coating whose permeability relative to free space is 1\([2. The pronounced effect of wave t rapping by a permeable layer is strik·ing. The ripples in the pattern arc mu ch larger than the corresponding ones for the purely dielectric coating. Furthermore, it is apparent that the period of the ripples is modified by the p ermeability ratio M 2, indica ting that th e trapped peripheral surface waves are largely con­fined to within the film . Therefore, their phase velocity is mainl~' determined b.v the wave number k rather than ko•

5. Concluding Remarks

The anal.vtical expressions developed herein arc available for any fu ture calculations of patterns of slotted cylinder antennas with dielectric coveri ngs .

' Actually the quantity plotted is the phase lag, Wllich is simply the negath'c of the pbase.

The formulas co uld probably be programed for machine calculators if further numerical data are required. The complexity of the numerical pro­cedures becomes excessive when the directions are not in the equatorial plane. It would seem to be desirable to search for an approximation technique to supplement this worl;;:. At the moment, the ou tlook is optimistic, using techniques based on simphfied boundary conditions. In any event. the rigorous calculations presented here should provide fl comparative basis for check:ing any new approxi-mate formulations. .

On the basis of the limit.ed calculations presented here it can be concluded t hat a thin dielectric film would have only a small effect. on the radiation pattern for an axially slotted cylinder. On the other hand, a thin permeable film encasing th e c:vlincler would substantially modify the pattern, indicating the presence of trapped peripheral surface waves.

We thank R. A. Hurd of the National R esearch Council of Canada for his comments and the verifica­tion of the finn I formulas in reference 10.

6 . References

[1] J. R. vVai t, RadiaLion characteristics of axial slots on a conducti ng cyli nder, Wireless Eng!'. 32, 316 (D ec. 1955).

[2] A. A. Pistolkors, Radiation from a transverse slit on the surface of a cir cular cylinder, J . Tech. Phys., U. S. S. R ., 17,377 (1947) (in Russian).

[3] Gcorge Sinclair, The pattern of slotted-cylinder anten­nas, Proc. l nst. Radio E ngrs. 36, 1487 (1948).

[4] C. H . Papas and Ronold King, Currents on t he surface of an infinite cylinder excited by an axial slot, Quart. App!. Math. 16, 175 (1949).

[5] C. H. Papas, Radia tion from a transverse slo t on an infinite cylinder , J . Math. Phys. 28, 227 (1950).

[6] J . R. Wait, Radiation from a slot on a cylindricall v tipped wedge, Can. J . Phys. 32, 714 (1954). .

[71 S. Silver and W. Ie Saunders, Field produced by a t ransverse slot on a circular cylinder, J. Appl. Phys. 21, 745 (1950) .

[8] L. L. Bailin, Field produced by a slot on a large circular cylinder, T rans. I nst. Radio Engrs. (PGAP) AP- 3, J 28 (July 1955).

[9] J. R. Wait, and J . Kates, Radiation patterns of circum­ferent ial slo ts on moderately la rge co nducting cyl­inders, I nst . E lect. E ngrs. (London), Monograph ]'\0 . 167R, February 1956 (to be republished in P t . C of t he Proceedings).

rIO] J. R. Wait Slotted-cylinder antenna with a dielectri c coating, Addendum t o Radio Physics Laboratory, Project R eport 19- 0- 13 (Otta wa, Nov. 19M).

[11] D. G. Frood and J . R. Wait , An investiga t ion of slot radiators in m etal plates, Proc. Inst. E lec. E ngrs. 103, 103 (Jan. 1956) .

[12] A. W. Adey, Scattering of electromagnetic waves bv co-axial cylinders, Can . J. Phys. 34, 510 (Yray 1956); a lso A. W. Adey (Ph . D . Thesis, Imperial College, Univ. of London, 1954).

[13] J . A. Stratton, E lectromagnetic theory, p. 354 (McGraw­H ill Book Co. , Inc., ~ew York, N. Y., 1941).

[14] Scattering and radiation from circular cylinders and spheres. Math. Tables Proj . an d 1'1. 1. T . Under­water Sound Laboratory, A. }L P . Report, 62- 1- R (1945) .

[15] R. S. E ll io tt, SpllPrical surface antennas, Trans. Inst. Radio Engrs. (PGAP) AP- 4, 422 (July 1956).

290

.. J. 11 ... J.V_

q,

de(f 0

10 20 JO 40

50 60 70 80 90

100 110 120 IJO 140

150 160 170 180

dey 0

10 20 30 40

50 60 70 80 90

100 110 120 130 140

150 160 170 180

'fA RLJi] 1. L ist oj coe.fJicients

"

I ~ • r

0"" )

m" --ll m(u o )

II

i(" . --11 11 rub)

pw '"

/)'''11

m" --.} m(lla ) II

11'1./ '" (I/a )

TI/~ - T,' .[", (11/) )

TARLE 2. Th e pal/ern fli llclion, }> (q, )

Cas<' 1

1.0 2.0 2. 0 2 0 I J.O

2. I :1 . '2 2.:5 2.2 :l. 2 1. n 2. () 2. 11 I. .1 2.0

1 -

.llIlpli lllde IP (<I» 1

J.01542 3. 1905, 3. J9700 3.03u93 3.25498 J. 01288 3. 19604 J. '10803 J. 04000 3.26369 J. 001 24 J.20722 J . 4:5740 J.04 148 3.27721 2.96859 J.20n, J.46905 J. 02249 3.25953 2.8976J J. J6737 :3.4Gu72 2. 96182 3. 16938

2. 77345 3. 05a97 J. :17882 2.8429'1 J.00502 2. 5~731 2.85054 3. 16491 2. G6 1J4 2.84440 2. J995 1 2.58432 2.8J926 2. 43991 2. fi b7" 2.2a553 2. JJ949 2.50987 2. 241 87 2.74122 2. 14J98 1.21210 2.34650 2. 136 14 2.55112

2.09373 2.19945 2. 39723 2. 10868 2. 13809 I. 9924, 2. 16949 2. 47fi21 2.045 17 I. 75636 I. 75627 1. 97774 2. a5044 I. 83122 1. 74087

o ~\;~~6 1 1 I. 57J 8, I. 9J38(j I. 438 11j 1.82952 I. 04402 I. 24227 0. 9607J!) 1. 57059

. (i994 71 O. , 59241i 0.758170 . 706, 1.7 0. 91 8e74

. !J19829 I. 05545 I. 12441 . 936848 . 57548J I. 2118J I. 43974 I. 64774 1 2.)289 I. lag l5 I. J28fil I I. 59211 1.8.13!Ji I. J7900 I. 41 882

------- - -- - - - -- -- -Phase of P(<I»

~---

cleo deo dey dey (l ca 21J.533 213.044 211. J24 212.873 268. 001 211. 872 211. 274 209. 172 2 11. 238 265. 428 206.967 206. 111 203.54J 206. J79 257.900 J99.001 197. 917 194.805 J98.4 10 245. 748 J88. 139 18i. 046 18J.566 187.474 228.942

174.37J 173.583 170.085 173.662 20(j.883 J5i. 447 J 57. 142 153.956 J5(j.842 179. 269 137. 0a2 J36.872 133.942 13(,.562 148. 124 J JJ.415 112. 152 108. 449 11 2.588 117. 146 R8.386 ~4. 411 78.257 8(i.450 87. 094

(i4.621 .~7. 9I 9 ·19. 050 01. 5, 5 53.96 1 4:1.55J a5.268 20.42 1 40. 094 J 1. 569 24.517 J5.904 (j. li44 21. 22 1 -J:3.607

4. \i89 -3.355 - 10.683 I. 840 -(i6.340 -2:5.5!J'i -JO.69J -J4.167 -25.805 -89.882

- 'J.b2J I -83.857 -89. 225 - 76.9Jli - JJ 8.365 - 116.591 - 129.8 18 - 145 . 407 - 122. 107 - 201.6(i l - 13:1. 631i - 14(i. liO, - IG2.9 14 - IJ9. 824 -246.587 - laH.04' -1.\0.9J3 - 11\(;.1)99 -J4'l. a79 - 25a.828

291

eml, d mp

;IJ,WIIJ:,,(1l 0 )

ip.W/I, ; :,, (lIh )

- /I ~ /J", (II() ') )

11/'1 +--,;- II ",(IIII 'J )

T AR L E :3. Th e pol/I'm .rullc/ion, P (q,) Cn:o::p II

~ . 1 2.0 2. 0 2. 0 2. 0 :J.n H 2. I 2.2 2. :3 '2 . '2 :l. 2 .1/ LII 2.0 2. 0 1.,1) 2. 0

q,

Amplilu ti l' P (<I» I

--- -deo

1I J . 24411i 2.3 1J78 2.0:],\(i2 2. 95J9(i 2.21707 10 :],22684 2.39 152 1, 98875 2. !)6226 2.33487 20 J, J70JJ 2,60844 2,01230 2, !)7840 2,6897.1 :)0 :J, 0645(; 2.90218 2, J8(jJI 2. 97777 :3. 1451(i 40 2,90JO(; :5.15747 J , 03508 2. H2()7.') 3, a86(iO

.\0 2. (i9(il(; ;!. :la iOft J. (iOOOO '2 . 7H8 I fi :3 . 15J44 liO 2. 48507 :l.(HO(jS J.75866 2.59546 2.552J8 70 2.33989 2.58J;J4 J. JJ411j 2. J758 1 2. a484, XO 2. Jllj02 2.10452 2. a924J 2.240(j,1 2.938!l0 !JO 2.38889 2.11740,1 1.5J994 1. 249 10 3.32510

100 2. 44442 2 . .1<1 071 2.05J10 2. :]200:) 2.8 1914 110 2.3.\724 2.97488 :l. ooaos 2. :)03a4 I. 7520J 120 2.07477 2.987 13 3.59479 '2 . 0nG52 I. 9J906 IJO I.MJ89 2. ·1170.~ a.22720 I. ,1!J490 2.900n 140 0.9843(;4 I. 41820 2. 03580 1.0 1(j(ili 2.90070

1,)0 .8384,,\ O. 7.\3540 O . . )(iJ5 12 II. 8:J0482 I. 67907 IliO 25J()(i I . liaS5!) I. \;(i202 I. I(j270 O. (j823fJO 170 ti6\iOi I 2.!i1 !i!)7 2. !}(i!09 1 .. \8455 2.241 28 180 82Jl)li 2.84481 J.44 71 2 I. 747 17 2.92864

Phase P (q,] -----

deq dey dey dey dey dey 0 195.74(i 181. 92, 20J.37(j 198.928 244.215

10 194 . 478 179.6J7 J 96.348 197.452 238.73J 20 190.59J 17J.720 176.2(jO J93. 11 0 226.3JI 30 183.829 J 65. 967 152.122 186.000 213. 095 40 17J. SI(j 157.38 1 J34.364 176.340 200.028

50 159.8Sfi 14 7. no 122.656 16J.649 183.65 1 riO 141. 448 la5.613 11 3.362 147.2J8 155.984 70 l J8.679 J 17. 878 J02. !J6\) 126. 198 lIO.576 SO !)J.69 19 SS.8039 85. 1765 100.89J 71. 5498 90 69. 84JO 'J9. 4190 40.03JI 74.6644 47.2J4J

100 49.0554 IS. J85, - 17.5717 .\1. 52J9 24 .8248 JJO J1. 0754 -0.99700·1 - 40.9840 J2.4240 - 16.9857 J20 14 . J408 - 14 . 2664 -52.0521 15.7465 -88. 32(j6 1:30 - 4. 7114() - 26.4508 - .\9.9543 - 1.9Ii822 - 119.898 I'll) -J6.3' 08 -46.099(i - (iO.904J -aO.1l75 - IJ4.839

1.50 -94.454." - 120.53fi - 125.229 - (i8. 10'16 - 150.876 160 - 1:32.229 - 174.099 - 221 .830 - 130. J91 -262. I 13 170 - J4.1. 7!)f) - 186.a3J -2;3 1. 038 - 144.77J -J04.272 180 - 1'J9.3 19 - 188.884 -232. H34 - 148.435 -JOs.·ln

A _____ __ 2.0 B _______ 2. 1 S _

~ --- 2.0 ----

m Real IlJl aginat'~·

0 - 11. 2226 +2. 09515 I - 23.5768 - 2. 2445G 2 - 11. 9489 - 22.4385 :l + 4. 2228 -?885~5 4 + 0 OG9988 +3. (i1[;8-1

5 -, 728316 +0. 0008ti~ 6 +2. I I XIO-' -. 125130 7 +0.018472 -6. 57X to-; 8 -4. 38XlO-' + 0. 002381 9 - 0.000272 0

10 0 -2. 79X 10- 5

---

'" R eal Irn (lgil1ar~'

- - -0 - 10.1144 +5. 170-lli 1 - 23. 4645 + -1 .80230 2 - 20. 9042 - 20.6570 3 + 13.8792 - 4.47222 -I +0. 0718G9 + 3.64883

5 -. 682888 + 0. 00058-1 6 +. 00000177 -, 111720 7 +. 01586Y -.00000034 8 -. 000000034. +- 00 1981 9 -. 0002201 0

10 0 -.000022

TABLE '1. The pattern Junction, P(</»

A _____ 2.0 J.O A __ ___ 2.0

I 3.0

B ___ _ ?Il J.O B _____ 2.0 :1. 0 ---- - --

I <P q,

I --- ------------- -

Amplitude I P (q,) I Phase or P(q,) -- -------------- - - - ---------------

dey 0

10 20 30 40

50 60 70 80 90

100 ItO 120 130 140

150 J60 170 180

Hl'al

- 12.706G -25. 8912 - 15,5227 +7. 9llG + 0.12G520

- .90994:j - \l. JXtO- ; +0 022940

dey derJ deg 2. Y1946 3.0 1228 0 213.310 268. li8 2.91276 3.00388 10 2J I. 744 266.384 2.88927 2.97364 20 207.088 259.2W 2.83980 2. 91104 30 199.443 247. 294 2. 75287 2.81299 40 J88.880 230.567

2.62188 2.69702 50 175.358 209. 146 2.45475 2.59185 60 158.7 19 183.822 2.27965 2.49434 70 138.910 156.249 2. 13482 2. 345i7 80 116.525 127.69 1 2.03659 2.08708 90 93.2lO6 97.6070

1. 95 124 1. 76251 100 70.9499 63.3880 1. 81096 1. 53456 110 50.6797 23.9124 1. 56J92 1.47497 120 31. 6970 - 14.046.5 1. 20474 1. 39522 130 11. 2645 - 43.9il31 0.827194 1. 10650 140 - 17. 8471 -68.8592

. 659909 O. M8455 150 - fX'l. 12G4 - 102.298

.830784 . 488877 160 - 104 . 45J - 176. 50S 1. 058J 1 .815190 J70 - 121. 274 -216.288 1. 14974 . 978586 180 -1 25. iOG - 224. 045

TABLE 5_ R eal and imaginary paTis of D",

Case I

2.0 2.0 2.0 2.2 2.3 2.2 2.0 2.0 1.5

lmaginary Real Imagi nary Real

+1. 96495 - 13. 80G2 + 2.95754 - 8.901G8 - 1. 84874 - 29. 1592 -I. 06811 - 18.707il

Imagi n al'.\-"

+ 1. 68116 - 1. 59443

- 23. H77 - 20. 4830 - 24.2632 - 10.2028 - 17. il962 - 4.83094 +22.2909 -8.60814 + 11. 8857 - 2.78592 + 4.5[,540 +0.224247 +5. 78874 + 0.069761 + 3. 01377

+ 0.001481 - l.1I 81l + 0.002667 - .601487 + 0.000810 -. 155/i9 - 5.06XIO-· -. 18731G - 2.2X IO-6 -. 010257

- 2. 48X lO-; + 0.027259 - 7. 99 X lO- ; + 0.015030 - 6. 4 X 10- ; - 2. 49X10-' + 0.002948 - 5.02X 10- 1 +0. 003456 0 + 0.001922 - 0.000305

0

Real -----

-8.52101 - 20.2125 -27. 02\)3 + 24.4545

+ 0.141618

-.8 IGI35 - . 000001l +. 017896

- 3. 51X I0-' - 0.000242

0

- 1.15X1O- s - 0.000388 -7. 4 X 10-' - 0.0002 17

- 0. 42X HI- ' 0 - 4. 15XIO-' 0

--- ---

T A ALE 6. Heal and i maginary parts of D",

Case I[

Im aginaJ'~' Rcal [m nginary Real

+7 J8167 -6. 04711 +7.78317 -8. 06944 + 10.7968 - 15. 8423 + 13.1521 - 18. 4190

- 9.50909 - 22.7941 - 0.004936 -14. 1839 - 14.3722 + 18.3439 - 38.8il82 +12.1601 +4.75266 +0.292332 + 6.37791 + 0.52345

+0.00 1061 -. 975191 + 0.001744 - .520690 -. 128662 - .000065 - . 146022 -. 0000015

-3. 12XlO-; +. 020054 - .00000064 +. 012221 +0 00220il - . 000000037 + .002406 0 - 9. :l X lO-' -. 000266 - . 0000000053 - 0.000170

- 0.0000241 0 - .000027 0

292

0

- 2. 2XlO '

IIIlHg iIl (l r~'

+ 3.6-1 159 + 2. 7645.)

- 16.0117 - 2. 99106 + 2.74666

+ 0.000-18, -. 085703 -. 000000-18 +- 001530 0

-0.00001,

il.O 3.2 2.0

I Keal Imagi nary

-16.5595 - 15.6040 - 28.0280 - 37.0156

- 5. n076 - 49.4498 +44. (j068 - 03.2520 + 14.673,) + 39.1,346

- J:3.0184 I + 0. (i55il36 -0.014469 - il.21999 +. 708769

I -0.000213

+8. -IX 10- 6 +. J37679 - 0.02228a - -I . 09XIO- ·

+ 2. 7X10- ; - 0.003674

I Rc'"l Im agi nary

- l8.5527 - 4.24158 -il6.1706 - 14. -1406 ·- 27 . 5801 - 31. 2589 + 10.0300 - 51. 4962 + 5!J.0508 + 43. 1818

-10.4855 + 1. 23400 -0.015533 - 3. 30210 +.655667 - 0.000158 +. 00OO09ti +. 119019 -. 019108 -. 000045

+- 00000ll2t i -.002903

TABLE 7. Real and imaginary parls oj dm

A=2.0 B = 2.0

111

~1_1 _1 Imaginary --

0 -1. 67622 + 0. 310971 1 -3.50102 -. 400130 2 -1. 44185 -3. 2861 8 3 +1. 82162 - 0.270294 4 +0.006261 +. 454036

.'> -.090604 +.000063 6 0 -.015290 7 +0.002214 0 8 0 + 0.000280 9 -0.000031 0

10 0 -0.000003

z

I

I

=j

.11 = 3.0 I 13= 3.U

Real

- 1.53852 - 2.54238 - 0. 160813 +4.33813 + 0.548382

-.884368 -. 000506 +. 048952 0

-0.00 1603

0

-I Imaginal')

- I. 47332 -3.53029 -4.62823 -2.03 120 +2.81856

+ 0.023621 -. 22303 -.00000

8 8

+. 009409 0

-0.00024.;

{p,rp,zl

z

E ,f-L

{a, cp', z' 1

a

b

FIGUHE 1. Coordinate systems f or the dielectric-coaled cylinder of infinite length.

293

:! 0-

E 0' C

~ (j, "0

~

V> Q)

t 0' Q)

o '= ~ 0-

38

3 6 ~ 34

31 -

30

28

26

2.4

22

10 ,

1 8 ~

1 6 ~ 1.4

12

LO

0.8

0.6

0.4 -

02

000

A ~ 2 0, 8 ~ 23, N ~ 2 0

-­ " A~20, 8~21, N~20

A=2.0,8=2.0 " \

(0) Case I

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Azimuth 4> (Degrees)

200~-------------------------------------------,

50

100

50

A=2.0 , 8=2.3, N=2.0

A=2.0, 8=2.2, N=2.0 -

A = 2.0, 8= 2.1, N= 2.0 -

A =2.0,8=2.0 -

, /

, , , , ,

, / , ,

o -50 /",///'"

0' o

-.J

~ - 100 o r:

- '50

- 100 _/

-150 [ ----~ 20

,... ...... ///

40

//

/'////

//////

(b) Case I

60 80 100 120 140 160 180

AZim uth 4> (Degrees )

F IGURE 2. Pattern of slotted-cylinder antenna with a dielectric coating showing effect of thickness.

~ 0-

r: 0. c ~

(j, "0 Qj u:

V> Q)

t 0' Q)

o '=

0' o -.J

Q) V> o

L 0-

3.8

36

34

3.2

3.0

1.8

2.S

24

~2

2.0

18

I.S

14

12

to

OB

O.S

04

0.1

0.00

"" . "". ,," """"', ..... .

A=2.0,8=2.0 -\,

,

'--

(0) Case 1

1

10 20 30 40 . 50 60 70 80 90 ~O 110 120 130 140 150 ISO 170 180

Azimuth 4> (Degrees )

200 ,-------------,----------------------------,

150

I 100 ,-

50

-50 I

- 100 I

- 150

- 200

'" I -300 0 20 40 60 80

A=2.0, 8=2.2, N=2 .0 ~

A= 2.0,8=2 .2, N= 1 5

A=2 .3,8=2.0

(b) Case I

100 120 140 160

AZimuth 4> (Deg rees)

~

J 180

FIGURE 3. Pattern of slotted-cylindel' antenna with a dielectric coaling showing e:O'ecl of Tef raclive index.

294

38

3£

J4

32

30

28

26

::&. 2.4

CL 22

~ 2.0

~ 18

(ji 16 "0

~ 1.4

12

1.0

08

06

04

02

00 0 20 40

,

, ,

,

-\

A 0 30, 8 0 3.0 ~ ,

60 80 100 120 140

Az imuth ¢ (Degrees)

, , ,

"~-'

(a) Cose I

160 180

3oo,---,----,---,,---,----.--~------------_,

250

200

150

V> <J) 100 ~ 0' <J)

o

'0 0'

50

..5 -50

<J) </> o £: -100 Q

-200

-250

A=3.0, 8=3.2, N=20

A=30,8=3.0

(b) Cose I

-]00'-_ ______________________ ----"

o 10 40 60 80 100 120 140 160 180

AZimuth ¢ (Degrees)

lerG UlUJ 4, Pattern of slotted-cylinder antenna with a dielectric coating for a larger cylinder.

38 ,-~--~~--~----------~--------~------

3.6

3.4

3.2

3.0

2.8

2.6

~ 2.4 ... A

.!h.. 22

:& 2.0

~ 1.8 (j,

16 "0 a; LL 14

V>

12

10

0.8

0.6

0.4

0.2

0.00

300

250

200

150

<J) 100 ~ 0' Q)

o ~

'0 0'

50

..5 -50

Q) V> o if -100

-150

-100

-250

-300

20

20

40 60 80 100 120

Azimuth ¢ (Degrees)

A=20, 8=2.3, M=2.0

A=20,8=2.2 , M=20

///

140

" /

I

I

/ I

(a) Cose 1I

I I

I

160

/ /

/ /

/

/,,'>//

//~ A=2.0, 8=2.1, M=2 0

/ A=2 0,8=2.0

/ -///

40

/////

60

/ /

80 100 120

AZimuth ¢ (Degrees)

(b) Case II

140 160

180

180

FIGCRE 5, Pattern of slotted-cylinder antenna with a permeable coating showing effect of coating thickness,

295

16 L

~ 14 '-

~ 12 .<:: 10. "& c ~ 18 en 16 "0 a;

14 ~

11

0.8

0.6

0. 4

0..1

DO. 0. 10. 40.

",

A ~20, B ~2 2 , M ~ 15 ""'\\

'~", e ~ " \\,~" "~

60. 80. 100 120. 140.

Azimuth cp (Degrees )

', , -,'

(a) Case II

160. 180.

10.0..--------------------------------------------,

150.

10.0.

lG 50. '" 0. '" o

.<;

-B-

~ -50.

'0 CJ'

..'3 - 10.0.

~ o if -150.

-20.0.

-250.

---

--/ / '/

A ~ 2.0, B ~ 2.2, M ~ 15

",,/'

~ ~/,/"'" /

1"/' ,

,

(b) Case IT

-30.0. '----------'------------------' 0. 10. 40. 60. 80. 10.0. 120. 140. 160. 180.

Azimuth cp ( Degrees)

F lGl-RE 6. P attern of slotted-cylinder antenna with a permeable coating showing efJect of refrocl ive~i nde.r.

B OULDER, COLO" October 25, 1956.

296

38l 36

3.4

:: ~ 2J ~ 2.4

CL 2.1

:& 2.0 c ~ 18

(jj 16 "0

~ 1.4

11

10.

DB

0..6

0..4 -

0..2

Go. 0. 10. 40. 60. 80.

>'\\

,

\\, ' ,,- ,'

(a) Cose IT

---- -- -----~ 100 120. 140. 160. 180.

Azimut h cp ( Degrees)

350. .---------,---------

'" '"

300.

250

20.0.

150.

~ 100

'" o

~

'0 CJ' o

-.J

~ o

.<:: 0..

50. ~

-50

-10.0.

-150.

-10.0.

-150.

- 30.0. . 0. 10. 40. 60. 80.

A~30, B ~32 , M =2.o

A = 3.0, B ~ 30

10.0. 110. 140.

Azimuth cp (Degre es)

(b) Case l[

160. 180

FIGU RE 7. P attern of slotted-cylinder antenna with a penneable coating for a larger cylinder,

.,

)

l'