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8CIENCE~DIRECT e Applied Mathematics Letters 17 (2004) 721-726

Applied Mathematics Letters

www.elsevier.com/locate/aml

R a d i a t i o n from a D i e l e c t r i c - C o a t e d C y l i n d e r w i t h T w o S lots

M. A. MUSHREF P.O. Box 9772

Jeddah 21423, Saudi Arabia mmushref ©yahoo. co. uk

(Received April 2003; accepted May 2003)

Communica ted by C. Bardos

Abstract--The transverse electric (TE) radiation patterns are investigated for a circular cylinder

with two slots coated by a dielectric material. In this paper, field characteristics are determined when another axial slot is added to the cylinder. Radiations are found by applying the boundary conditions to the cylindrical wave functions of the fields. Numerical results are obtained by truncating the infinite

series to a finite number of terms and comparing them to the one slot case. @ 2004 Elsevier Ltd. All rights reserved.

Keywords--Antennas, Radiation patterns, Slotted cylinders, Boundary value analysis.

I N T R O D U C T I O N

Radiations from cylindrical structures have been the subject of various studies [1-6]. However, all these studies did not consider the radiation effects when an additional slot is added at ¢ = ~r.

The field investigated in this paper is found for two axial slots in a circular cylinder coated with a pure dielectric material, as illustrated in Figure 1. Th e cylinder is assumed to be a perfect electric conductor with radius a and with infinite extent along the z-axis. On the surface of the cylinder two slots are axially cut with an angular opening of ¢0 located at ¢ = 0 and ~b = ~r, respectively, with respect to the x-axis. The cylinder is entirely coated by Region I which is a dielectric layer with radius b. This region is assumed to be linear, homogeneous, and isotropic and characterized by permittivity ~1 and permeability #1. Region II is assumed to be free space with G2 and #2.

F O R M U L A T I O N O F A L G O R I T H M S

Due to the even symmetry of the proposed structure, the magnetic field in the dielectric coating can be expressed in a Fourier-Bessel cosine series given by (see [2])

Hlz : ~ [A~Jn(lqr) + BnYn(klr)] cos(n@. (I) ~0

0893-9659/04/$ - see front matter @ 2004 Elsevier Ltd. All rights reserved. Typeset by ,A3,45-~X doi:10.1016/j.aml.2003.05.011

722 M.A. MUSHREF

yY

~ x ¢o t ~o

Figure 1. Geometry of the problem.

Also, in free space the magnetic field is (see [2])

Hy = ~ GH~(~)(k~) eos(~¢), (2) n = 0

where kt and k2 are the dielectric coating and free space wave numbers, respectively, given by k = 27r/;~, and )~ is the wavelength. Y~(x) and E~(x) are Bessel 's functions of the first and the

second type, respectively, with order n and argument x. H~ (2) (x) is the outgoing Hankel function of the second type with order n and argument x.

According to Figure 1, the tangential magnetic and electric fields are continuous at r = b, for all ¢ and the tangential electric field is zero at r = a, for all ¢ except at the slots where it is assumed to be E0. Tha t is,

H I = H H, ~=b, 0 < ¢ < 2 ~ , (3)

E ~ = E 5 ~=b, 0 < ¢ < 2 ~ , (4) 1 l

where E 4 is derived from Maxwell 's equations as E¢ =- ( i /ws)(°o@) [3]. By applying the boundary conditions in equations (3)-(5) to the field equations in (1) and (2)

[2,5], the C , coefficients are found to be

2E0~c1¢o 1 + cos(n~) c~ = iS~b~2k~ ~J ' (k l a ) + Z~Y'(kla)' (a)

where 5n equals 2 for n = 0 and 1 for all other values, and the prime notat ion designates differentiation with respect to the argument. Also, an and/3n are found as

an = H(~ 2) (k2b)Y~(kib) - k2Q H(2)' (k2b)Yn(klb), ~i£2

~ -= k~lH(2) '(k~b)J,~(klb) - H (~) (k2b)J'~(klb). ~lZ2

D i e l e c t r i c - C o a t e d C y l i n d e r 723

3 .... I

\ \

z5 \

/ / 4 \

~ / ~ /

15- / / ....... b , / .

1 \ / '" ' ,

05

0

,...~ / I " 1 \ I

\ I \ i \ I \ I \ / \ I

\ I \ / \ I

\ /,- ....... I / \ . - ! ...... \ /.. /

",.. \.,./ '.., . .

I I I 45 90 135

¢o

( a ) R a d i a t i o n p a t t e r n s for 8 1 / Z 2 = 2.56 , /~1 /#2 = 1.

- - R e f e r e n c e [2] fo r a = 0.358)~2 a n d b = 0 .4217A2 fo r o n e s lo t .

. . . . . C a l c u l a t e d fo r a = 0 . 3 5 8 A 2 a n d b = 0 .4217A2 fo r t w o s l o t s .

- - - C a l c u l a t e d fo r a = 0.5A2 a n d b = 0.8A2 fo r t w o s l o t s .

Ip(O)l 4 I t i

,

/ /

3.5 / \

/ \ / \ / 3 / \ / / / \

/ \ / \ / 25 / / / / / \

2J. .... I I I I I ~ ' \ ' " ' , . I I I .... I I \, . . " ' / / "

\ ""../ / /.,'"" "'",./ / \.."" .'

0.5 "

0 80

¢o

(b ) R a d i a t i o n p a t t e r n s for el~S2 = 1, # 1 / # 2 = 4.

- - R e f e r e n c e [3] fo r a --- 0 .31SA2 a n d b = 0 .350A2 fo r o n e s lo t .

. . . . . C a l c u l a t e d fo r a = 0 .318A2 a n d b = 0 .350A2 fo r t w o s lo t s .

- - - C a l c u l a t e d fo r a = 0.5A2 a n d b ---- 0 .8A2 fo r t w o s logs .

F i g u r e 2.

From equation (6), we can notice the new factor 1 + cos(n~r) which indicates the effects of the additional assumed slot located at ¢ = ~r to the radiated fields. This means tha t all radiation pat terns for two slots are the same pat terns for one slot multiplied by tha t factor.

Applying the asymptot ic expression of the Hankel function [3,4] to equation (2), the radiated

724 M . A . MusHREF

Gain, dB 8 - - I I I

6 p -

4 // , ? ~

- ~ ~ _ / ~ ~ - ' , ,~

0

- - 2 I I I 2 2 . 0 5 2.1 2 1 5 2 .2

b

(a) An tenna gain versus coating thickness for Z l /~2 = 4, /~1//~2 = 1, and a = 2A2. - - Reference [6] at s]ot loca t ion for one slot.

- - - Calculated at slot locat ion ¢ = 0 or 9 = ~ for two slots.

G a , ms 50

4O

30

20

10

0 i i 2 2 .05 2.1

b

t \ J~ \ ~ J \

I

2 1 5 2.2

(b) Aper ture conductance versus coat ing thickness for Sl/Z2 = 4, /zl/F2 = 1, and a = 2~2.

- - Reference [6] for one slot. - - - Calculated for two slots.

Figure 3.

f ie ld c a n b e e v a l u a t e d a t a f a r p o i n t a s

H I I = p(¢)e-~(k2r-Tr/n) i~2r ,

w h e r e P ( ¢ ) is t h e f a r f ie ld p a t t e r n g i v e n b y

o o

p ( ¢ ) = n = O

(r)

(8)

Dielectric-Coated Cylinder 725

In addit ion, s imilar to the derivations in reference [6], the an tenna gain and the aper ture

conductance per uni t length A2 for the two slots are found as

G ( ¢ ) - ~_p(@[2, (9)

E IC,,I 2 n = O

oo

E IC, ? n = O

A2 -- 2a2¢0287cEg ' (10)

where r]2 is the intr insic impedance in free space approximate ly equal to 1207c [6].

R E S U L T S A N D D I S C U S S I O N S

In order to ascer ta in the accuracy of the expressions derived above, a number of numerical

computa t ions are performed. The results obta ined are only calculated for the first 30 terms of

the series produced in the solution due to the convergence of the summat ion. The field pa t t e rn

described by equat ion (8) is compared with similar numerical d a t a in references [2,3] for a slot

size of ¢0 = Tr/100. In addit ion, all pa t t e rns are normal ized to the d a t a in references [2,3] to

facil i tate the s tudy of the generated new pa t te rns and unders tand the differences tha t may occur

when two slots exist. F igure 2a shows the rad ia ted pa t t e rn compared with da t a in reference [2] for z l / e2 = 2.56,

ffl/P2 = i, a = 0.358A2, and b = 0.4217A2. The pattern is also calculated for a = 0.5A2 and

b = 0.8A2. As shown, the field is enhanced and a symmetry is maintained around ¢ = :r/2 due to

the two slots. In Figure 2b, the field pattern is also compared with similar values in reference [3],

Case II for Zl/C2 = i, #i/#2 = 4, a = 0.318A2, and b = 0.350),2. In reference [3], A = ak2

and B = bk2, and in all figures, A2 = A0, #2 = #0, and ~2 = Co, which lead to ~1/z2 = zr and

#i/#2 = #r. As expected, the radiation is also enhanced with a symmetry around ¢ = Ir/2.

Furthermore, the antenna gain and the aperture conductance per unit length A2 are compared

to the one slot case in reference [6]. The antenna gain in equation (9) is estimated versus the

coating thickness as shown in Figure 3a at the slot locations ¢ = 0 or q~ = 7r for el/a2 -- 4,

#i/#2 = i, and a = 2A2. It appears that the gain is changed but it is not significantly enhanced

for the two slots case. Moreover, the aperture conductance in equation (i0) is shown versus the

coating thickness for ¢I/E2 = 4, #i/#2 = i, and a = 2A2 in Figure 3b which indicates that the

aperture conductance is smaller for the two slots case.

C O N C L U S I O N

A solution was derived for the problem of two axial slots in a circular cylinder coated with a

dielectric material . The T E case was considered based on the b o u n d a r y value method, and the

r ad ia t ed fields were represented in terms of an infinite series of cyl indrical waves. The solution

explained the effects of the proposed addi t ional slot located at ¢ = 7r to the far field pa t te rns

in addi t ion to the influences tha t can arise to the an tenna gain and the aper ture conductance.

I t was found tha t the addi t ional slot can increase the r ad ia ted fields wi th a symmet ry around ¢ = 7r/2. Also, the an tenna gain was not enhanced. However, the aper tu re conductance was

found to be smaller.

R E F E R E N C E S 1. S. Silver and W. Saunders, The external field produced by a slot in an infinite circular cylinder, Journal of

Applied Physics 21, 153-158, (February 1950). 2. R.A. Hurd, Radiation patterns of a dielectric-coated axially slotted cylinder, Canadian Journal of Physics

34, 638-642, (1956). 3. J. Wait and W. Mienteka, Slotted-cylinder antenna with a dielectric coating, Journal of Research of the

National Bureau of Standards 58 (6), 287-296, (June 1957).

726 M . A . MUSHm~F

4. L. Shafai, Radiation from an axially slot antenna coated with a homogeneous material, Canadian Journal of Physics 50, 3072-3077, (1972).

5. M. Hinders and A. Yaghjian, Dual-series solution to scattering from a semicircular channel in a ground plane, IEEE Microwaves and Guided Waves Letters 1 (9), 239-242, (September 1991).

6. J. Richmond, Axial slot antenna on a dielectric-coated elliptic cylinder, IEEE Transactions on Antennas and Propagation 37 (10), 1235-1241, (October 1989).