Computation of Input Impedance of Rectangular, Circular and Hexagonal Patch Microstrip Antennas in S and X-Band

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in d m n J .

Phys. 74B (3). 203-208(2000)

u p

m

an international |o u n u l

Computation o f in p u t impedance of rectangular, circular and hexagonal patch m icrostrip antennas in ^S and X-band

Sandhya Mann and P K S Poumsli

M icrow ave L aboratory, D epartm ent o f Physics. Institute of Basic Sciences, L>r H R A m b e d k a r U t m e r s i t y . K l u n d a n . Agra-2 X2 002, U ttar Pradesh. India

Received 15 J u ly / 999, C^cepied 2 November /09O

\b s t r u c t M atching o f input im pedance o f an antenna w ith its fe c i netw ork is one o f the determ ining fa d or to r its best pertorrnan In tins paper, w e h a v e p resen ted im proved fo rm u lae for th e input im portance ut th ree geom etries n am ely re c tan g u lar, e u c u la r and lie sag n.il patch antennas T h e input im pedance for these geom etries has been com puted and plotted as a lu n d io n ol Iced poml location is X (J ( i l l / ) and \ (10 O H /) band o f m icrow ave frequency range. Som e other im portant antenpa p ira m e te rs like quality factor d u e e tm tv . e tlic ie iu x . gain, bandw idth, beumwidth, radiation resistance ctc\ h av e also been com puted and presented A contpariston ol these antenna chaiacteristics has also been made lor two op eratin g trequeneies is S and A'-band respectively

K e y w o rd s M icrostrip antennas, input im pedance, antenna param eters.

I*ACS No. X4 40 I k

1. Introduction

Microstrip antennas arc used in broad range of applications

from

communication systems (radars, navigation and telemetry) to biomedical systems, primarily due to their simplicity, conformability, low manufacturing cost and enormous availability of design and analysis software (1 -7 |.

The radiating element of an antenna and the feed line arc usually photoetched on the dielectric substrate. Normally, printed dipole is used either in its half wave length or full nave length form as a radiating element. However, the patch design has much greater flexibility in that many shapes of patch can be used to realize specific characteristics.

Without adequate attention to feed, the actual antenna cannot function properly, despite the radiating element being designed with care and precision. Since insertion loss of the feed increases drastically with frequency lienee additional care is required for feed design at higher frequencies.

Matching may be achieved by properly selecting the location

of

the feed.

2

. Formulation and computation of the problem Rectangular patch microstrip antenna (RPMA) :

The RPMA is modelled as a cavity with a magnetic wall along the edge and electric walls above and below. The fields ln the antenna are assumed to be those of the cavity. Hcncc the input impedance at any feed point maybe evaluated. The

geometry and coordinate system of RPMA is shown in

Figure 1. Geometry and coordinate system ol RPMA

In RPMA ihe modes of interest arc TM mode, winch have II. = 0 but a non-/.cro value of E-. In this mode the field components are given by |X)

liz = Cnm cos[(///r/ a).v]eos[(m n ! 6 ) v]. ( 1 ) U _ X nm m7t ^

_~ _{i y} _l _-'OS

A 0 Zf) b

nn 1 f m n 1 ( 2 )

tt

nm ntn nn 1 f mn

I f v = :r-TrL---sin — v cos - -r

v k()Z{) a _ a J L (3)

‘O 2000 1ACS

(2)

where n and m arc integers and n =■ m =■ 0 excepted ( is amplitude constant and for Joss free conditions may be written as

ah

^ <'n

j k a

(x ,v )c o s ^

**n* x**

jcos^

k “ ~~ k nr*

w/r

ilxdy>

(4) 1 for n m

2 for

n fh

0 0 , («

k 2 ~ (0)

knnl is resonant wave number of the nnt-th mode defined as

. r .

For RPMA the desired mode is 7T/in so if wc choose k ~ k j„ then this mode is excited with a large amplitude. T hus using cq (5) and taking the dielectric losses (/ c . k 2 - o>~jun (d /£'")), conductor losses, radiation losses along with the effect of energy storage outside the cavity (by modelling the ca\ity walls as walls having a finite complex admittance instead of zero admittance) into account f 8 - 10 |, the expression

for ( ’m becomes / N

2 K Z „ g U)t! /„ cosj

10 _k{()aht; ^<»)

where /<> is tlic total current in the probe of radius r 0 From the defination of quality factor QT% we have

/ o>~ irr T)r

where Pi - total power dissipated,

H’i ~ total stored energy, obtained as follows,

n; \ ' civ (10)

Wc can define an input resistance as seen from the coaxial line, by the relationship 1/2 |/()|2/<„ = P7 . Hcnccat resonance, R tn is

:

²

a>d h\k„Z()d cos(nx() / a)\~ (Jr

k{()ab

⁽

11

⁾

Thus by choosing xn appropriately, the cavity can be matched

to the input coaxial line. The input impedance

dkiv

given as

2 j ZJi

co s- (

nx(] J a )

__________

w 2

ahcrk()

[rw2 —

atfo

{1 -f / / 2

()r

>- ]

All of the non-resonant modes combine to give the induchu self reactance of the probe, so that the final expression

^{l o r}

Z, is ___

/o>//0/ /

,

\ ah

2

j Z nfi

c o s2

(

7ZVo i a )

- In A

l n y m\~

ⁱ

:

rk{)a b

co-

co2 - a)\0 (1 4- / / 2 Qf )*■

where

^a

and

^h

are the length and width of the rectangular patch respectively. Antenna quality lactor 0] is associated with the system losses, including radiation losses Or> dielectric losses Od and conductor losses 0 C. The expression for qualm lactor may be given as follows :

l (U>

.. > + _ U ± f t l f t/ f t ft where (J

j

1

tan

dP (A ~ o)d h a d jk J Z ^) 2

2 k2 and ft, _2k" ^k{

2 V

Mere, d = J is the skin depth, a - conductivity of mctalization,

. . 2 a xn

kx

--

n! a +

~ — r -— -y

A /r2 ~ ( a a x )2 ' a y

- / Z„ ,

=0.00836-—- + / 0 01668-“ - t:e is wall admittance along x direction

and

k \ =

0,7747 - 0 5977

(1

- h/a) - 0.1638

(1 - May

The length extention (Al) and the effective permittivity u . ) are obtained following Refs. |8,9J.

The input impedance of RPMA has been computed for two cases taking source frequency 3 GHz (.V-band) and K) GHz (A -band) respectively. The variation of impedance v\ ith position of feed point is shown in Figure 2.

S U B S T R A T E R T 0 U R 0 I 0

Figurr 2. Variation of input impedance as a function of feed point location tor RPMA.

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Comput at i on o f input i mpedance o f rectangul ar etc. 205

It is e v i d e n t fr o m t h e f i g u r e t h a t th e i n p u t i n p e n d a n c e m atch es w ith th e i n s e t c o a x i a l f e e d ( 5 0 o h m s ) at t w o fe e d point l o c a t i o n s / e . % x = 0 . 0 8 0 8 c m , y = 1 .0 3 7 5 cm a n d x -

-*1213 c m , y ~ 1 .0 3 7 5 cm f o r 3 G H z a n d x ~ 0 . 2 7 2 3 c m , 0 5 8 1 2 cm a n d x - 0 . 6 0 4 3 c m , y =* 0 . 5 8 1 2 cm fo r 10

01 U

3 .

Antenna

param eters of KPMA

Ihe i m p o r t a n t a n t e n n a p a r a m e t e r s o f R P M A i n c l u d i n g m diation r e s i s t a n c e , e q u i v a l e n t r e s i s t a n c e f o r the c o p p e r loss and d i e l e c t r i c los s R (i, a n t e n n a e f f i c i e n c y ( 7 ) , b a n d w i d t h (B W ), d i r e c t i v i t y ( / 3 M) a n d g a in h a v e b e e n c o m p u t e d in S

and A'-band a n d p r e s e n t e d in T a b l e 1.

( a b l e I. A rile ni u i p a r a m e t e r s o f K P M A ,

s! A n t e n n a D e s i g n f r e q u e n e y ( i n G i l / )

j

No s p a r a m e t e r s

3 ( 5 - B a n d ) 10 ( A' - b a n d )

1 a (ill c m ) 3 1 1 1 1 0 . 8 7 6 6

2 bt i n c m ) 3 . 8 7 4 9 1 . 1 6 5

\ £t 2 . 2 2 2 5 2 . 0 9 7 2

1 _{Q i} 2 5 1 7 8 3 6 6 1 2 5

s /„ ( 1 7 1 0 1 2 9 - / 1 6 9 8 1 1 1 5 9 . 1 8 2 9 - / 6 0 1 8 1

i ( m o ^ 2 ( * * 0 ^

c o s — - c o s — -

l

^*

J l

^« ^J

2 ( }

6 li,„ (i n o h m s ) i7i 0 1 2 9 c o s y a

J

1 5 9 . 1 8 2 9 C 0 S [ Q J

7 k, (h i o h m s ) 0 2 3 8 0 0 . 0 2 8 1 5

8 R j ( i n o h m s ) 0 7 1 0 5 3 0 . 1 7 3 9

>) kr (i n o h m s ) 599 39 5 9 9 4

to _{k i} (in o h m s ) 3 0 0 6 4 8 2 9 9 9

11 // (i n %) ^{99 68} ^{9 9 9 3}

12 D* ( i n d 13) 7 78 7 7 8

t J Gam 7 7 5 7. 8

1 1 ^{H I V} ( m %) 1 62 6 17

C i r c u l a r p a t c h m i c r o s t r i p a n t e n n a ( C P M A ) :

1 h e g e o m e t r y a n d c o o r d i n a t e s y s t e m o f C P M A is s h o w n in I igure 3 (a).

.CftOUNO

Hr u t c 3 . ( a ) G e o m e t r y a n d c o o r d i n a t e s y s t e m o f C P M A . a n d ( b ) E q u i v a l e n t r e s o n a n t p a r a l l e l L - C - R c i r c u i t .

T h e f a r - z o n e f i e ld s o f th is g e o m e t r y o b t a i n e d u s i n g c a v it y m o d e l [8] a re as f o l l o w s :

Ej

=

j"

e~Jk«rcos 0sin

ruf>

[/«+! (oasin 8) + (*oosin O)],

_{( 1 5 )}

E 0 = j n ^ ~ ^ e - J kor cos n<j>

[ ^ i t A b o s i n ^ - ^ ^ A o u s i n ^ ] , ( 1 6 ) w h e r e c o m p o n e n t s o f to t a l e l e c t r i c field v e c t o r fo r EM w a v e ,

- ( n ♦ 1 )-th a n d (/» 1 )-th o r d e r B e s s e l 's f u n c t i o n s o f first k in d r e s p e c t i v e l y .

V ~ h E $ J n (Aa ) is th e e d g e v o l t a g e at 0 - 0.

A t r e s o n a n c e , th e i n p u t i m p e d a n c e o f a m ic r o s tr ip a n t e n n a is real. If the d is k is fed at an a r b i t r a r y p o i n t (P o,0,0), th e r e s i s t a n c e at r e s o n a n c e is

( 1 7 )

R 2 P ,

w h e r e P j - P r -t P t + P j .

T h e r a d i a t e d p o w e r P r is w r i t t e n as f o l l o w s [8]

P

- (hE0J„(ka)ak0y .

r ~ 1920 1

H e re ,

h

= J t ^ + i ( Ao " s in 6») + (A0 a s i n 6>)]2

+ c o s 2 <?[j„+|(/t0osin£?) + J „ _ ,( ioflsin 0 )]2 sin O d d . T h e p o w e r d i s s i p a t e d d u e to c o n d u c t o r lo s s / ’, is g i v e n as

( 1 8 )

1/2 En rr>2

(

com

Y \ k a ) { k a ) 2 - V } ]

_{( 1 9 )}

T h e d i e l e c t r i c lo ss is d e t e r m i n e d by i n t e g r a t i n g th e e le c tric field in s id e th e c a v it y o v e r th e v o l u m e V a n d m a y be g iv e n as f o l l o w s :

P“ = h ^ - J i W { ( k a ) 2 - » 2 ].

(2 0) F r o m eq. ( 1 7 ) , R m a y be c a l c u l a t e d fo r a n y p o s i t i o n o f the s o u r c e . T h e a p p r o x i m a t e v a l u e o f i n p u t i m p e d a n c e can be d e t e r m i n e d b y u s i n g a s i m p l e p a r a l l e l L C R r e s o n a n t c i r c u i t s h o w n in F i g u r e 3 (b ) . A t r e s o n a n c e the c i r c u i t m a y c o n s i s t o f r e s i s t a n c e a n d i n d u c t a n c e in p a r a l l e l w ith e a c h o t h e r /.e.,

^ (ol l}R + jcoLR2 _ B Z “ R l + a t l ? “ Ri jX] ’

^ ~ Ri ~ c o l ■

L =

2 n f t Q r -

(21)

(22)

( 2 3 )

(4)

A l t e r n a t i v e l y , in t h e s i m i l a r c o n d i t i o n , th e c i r c u it m a y c o n s i s t o f r e s i s t a n c e a n d c a p a c i t a n c e in p a r a l l e l w ith each o t h e r a n d t h e r e f o r e , th e c a p a c i t a n c e (C ) is o b t a i n e d

C =, . . . .

2 n j r R (24)

T h u s , k n o w i n g th e v a l u e s o f R, L a n d C , th e i n p u t i m p e n d a n c e c a n be f o u n d fr om th e r e l a ti o n

Zm ~ Rm + JX m

= +

j & C

+

j

^{(2 5 )}

( a ) In p u t i m p e d a n c e at r e s o n a n c e :

A t r e s o n a n c e th e i n p u t i m p e d a n c e I m - R ,„ 4 R . ( 2 6 ) ( b ) I n p u t i m p e d a n c e at o f f r e s o n a n c e :

T h e q u a l i t y f a c t o r Qt n e c e s s a r y f o r c a l c u l a t i n g th e i n p u t i m p e d a n c e at f r e q u e n c i e s o f f t h e r e s o n a n c e is d e f i n e d by eq ( 9 ) w h e r e th e t o t a l s t o r e d e n e r g y W/ is g iv e n by [8] *

Wr

=

j

0

j

]

J Z

(

k a i ( k a '>2 - n 2 l (27) F u r t h e r , f o l l o w i n g e q. ( 9 ) a n d ( 2 7 ) , e x p r e s s i o n s fo r Q (i

a n d Q r c a n be o b t a i n e d as

Q . = h ( n f j u c r y 1 2 . ( 2 8 )

~ t a n 8 ' ( 2 9 )

2 4 0( ( k a ) 2 - n 2 }

( 3 0 ) T h u s k n o w ing th e to tal q u a l i t y f a c t o r Q j, the in p u t im p e d a n c c at a ny f r e q u e n c y c a n b e f o u n d from eq. (2 5 ) .

T h e a v o v e f o r m u l a f o r i n p u t i m p e d a n c e can be i m p r o v e d by c o n s i d e r i n g (i) th e e f f e c t o f w a ll a d m i t t a n c e d u e to w h i c h th e e i g e n v a l u e b e c o m e s c o m p l e x a n d h a s a real v a lu e s l i g h t l y less th a n 1 .8 4 1 1 8 a n d (ii) th e e f f e c t o f se r ie s i n d u c t i v e r e a c t a n c e d u e to p r o b e . T h e r e f o r e , c o n s i d e r i n g t h e s e e f f e c t s a n d u s i n g m o d a l e x p a n s i o n m o d e l in th e v i c i n i t y o f d o m i n a n t m o d e r e s o n a n t f r e q u e n c y [8 ,9 ] , the m o d i f i e d f o r m u l a f o r th e i n p u t i m p e d a n c e o f C P M A m a y be w r i t t e n as f o l l o w s :

w h e r e

X / ~ s e r i e s i n d u c t i v e r e a c t a n c e a s s o c i a t e d w ith the p r o b e ,

P o f e e d p o i n t from th e c e n t r e o f th e c i r c u l a r d is k ,

s i

_£p <sy ^

C

~ 2 . 7 7 5

h

is th e p a t c h c a p a c i t a n c e ,

k u a = 1 . 8 41 1 8 - ^ 5 = R e(A n a ) + y im ( A ||< z ) ,

^ 5 c a n b e o b t a i n e d fro m i t e r a t iv e a l g o r i t h m d e f i n e d as [81.

1 =

1 . 8 4 1 0 9 6 9 ( 1 - a a ) + 4 . 0 2 6 0 9 5 2 / f p (1 - a a ) - 1. 8 4 1 18 3 . 3 2 6 3 8 3 9 ( 1 - a a ) — 1

w i t h (<^0 — 0 ) ,

a = y + j B w ).

H e r e , 7o is th e free s p a c e c h a r a c t e r i s t i c i m p e d a n c e and a

a n d BH are w a ll c o n d u c t a n c e a n d s u s c e p t a n c e r e s p e c t i v e ^ a nd g iv e n as

G w = 0 . 0 1 2 5 4

B w = 0 . 0 0 8 3 4

m

I o mhos*

n a e r

n>) l " )

T h e in p u t i m p e d a n c e o f C P M A h a s b e e n c o m p u t e d at two f r e q u e n c i e s i . e . , 3 G H z a n d 10 G H z . T h e v a r i a t i o n o f input i m p e d a n c e is p l o t t e d in F i g u r e 4,

700 T

Figure 4. V u r i a l i o n o f i n p u t i m p e d a n c e as a f u n c t i o n o f f e e d p o i n t l o c a t i o n f or C P M A

F ro m fig u re it is o b s e r v e d t h a t th e i n p u t i m p e d a n c e of the C P M A m a t c h e s w ith th e in s et c o a x i a l f e e d / e., 50 ohm s at P o 0 . 3 8 4 7 cm for 3 G H z a n d P 0 - 0 . 0 9 9 9 cm fo r 10 G H z .

4. Antenna parameters of CPMA

T h e i m p o r t a n t a n t e n n a p a r a m e t e r s s u c h as r a d i u s ( a ) , effective r a d i u s ( a , . ) , e f f i c i e n c y (*7), b a n d w i d t h ( B W ) , d ir e c t i v i t y ⁽l) ) ,

g a in (O’ ) e t c . h a v e b e e n c o m p u t e d f o r C P M A in S and \- b a n d a n d are p r e s e n t e d in T a b l e 2.

T a b le 2. A n ten n a p aram eters o f CPM A

SI A ntenna D esign frequency (in G H z)

Nos param eters

3 (5 -B an d ) 10 (A'-band)

1

a t>n cm ) l 8255 0 5123

2

Or (in cm ) 1 9241 0 .5813

3 R esonant freq f, (in G I I/)

2.99 9 907

4 C H (in m hos) 0 0072 0.0067

5. B„ (in m hos)

0

⁰¹¹¹ ^{0 .0 1 0 4}

6

Q i 42.6 7 7 6 14 68 5 9

7 R,„ (in ohm s) 1455 128 y f ( 0 .9 8 I A ) 1907.577 j \ (3.282/1i )

8

^. ftr (in ohm s) 455 79 4 9 9 9 .0 4

9 n (in % ) 93 97 98.43

10

^BW^{(in % )}

0

⁹⁵⁶⁶ ^{2 .7 7 9 9}

11

D (in d ti)

6

⁹⁹

6

⁸

12

G (in dB )

6

⁷ ^6.7 ^___

(5)

Comput at i on o f input impedance o f rectangul ar etc 207

H e x a g o n a l p a t c h m i c r o s t r i p a n t e n n a ( H P M A )

I he g e o m e t r y a n d c o o r d i n a t e s y s t e m o f H P M A is s h o w n in

Figure 5. ?

i

I i g u r c 5. G e o m e t r y u n d c o o r d i n a t e s y s t e m o f l l l ’ M A

B ecause a c i r c u l a r p a tc h is a li m i t i n g c a s e o f p o l y g o n w ith a large n u m b e r o f s i d e s , t h e r e s o n a n t f r e q u e n c y fo r the d o m i n a n t as w e ll as fo r th e h i g h e r o r d e r m o d e s m a y be c a lc u lated u s i n g th e s i m i l a r e x p r e s s i o n as fo r C P M A by re p la c in g th e r a d i u s a o f c i r c u l a r p a tc h by e q u i v a l e n t r a d i u s T h e e q u i v a l e n t r a d i u s </cq is d e t e r m i n e d by c o m p a r i n g areas o f a h e x a g o n a n d a c i r c u l a r d is k o f r a d i u s a cq, i e

a cq - 0 9 0 9 4 S,

w here S is o n e s i d e o f h e x a g o n as s h o w n in F ig u r e 5 U s in g the a b o v e c o n c e p t a n d r e p l a c i n g th e r a d i u s a o f ( ’PMA by a cl] in e q u a t i o n s 31 to 33 , an e x p r e s s i o n for in p u t im p e d a n c e o f H P M A m a y be o b t a i n e d . W e h a v e c o m p u t e d the v a l u e s o f in p u t i m p e d a n c e fo r v a r i o u s v a l u e s o f feed point l o c a t io n fo r I1 P M A at 3 G H z a n d 10 G H z a n d the icsults are p l o t t e d in F i g u r e 6.

J>, (m om) Dtetano* from tfw contr* ——»

H g u r c 6 , V a r i a t i o n o f in p u t im p e d a n c e as a f u n c t io n o f fe e d p o i n t lo c a tio n

h>« 11 I'M A

It is o b s e r v e d fr om th e f i g u r e th a t the input i m p e d a n c e o f H P M A m a t c h e s w ith th e i m p e d a n c e o f inset c o a x i a l feed (50 o h m s ) at p 0 - 0 . 4 3 6 5 cm a n d 0 . 1 3 6 2 cm for 3 a n d 10 G H z r e s p e c t i v e l y .

A n t e n n a p a r a m e t e r s o f H P M A

I h e i m p o r t a n t a n t e n n a p a r a m e t e r s o f H P M A h a v e been e s t i m a t e d on b o th the f r e q u e n c i e s / e , 3 G H z a n d 10 ( i l l / a n d are s u m m a r i s e d in T a b l e 3.

T a b l e 2. A n t e n n a p a r a m e t e r s o f H P M A

SI A n l c n n a D e s i g n ( r e q u e u e s ( i n t i l l / ) N o s p a r a m e t e r s

3 ( . 9 - B a n d ) 10 ( \ - b a n d )

5 (i n c m ) 2 1 1 1 7 0 6 3 3 5

<7rt| (i n cm) 1 9 2 0 4 0 5 761

R e s o n a n t f r e q f, (in G H z )

2 9 9 8 9 9 9 0 7

4 G H ( i n m h o s ) 0 0 0 7 6 0 0 0 7 6

5 (i n m h o s ) 0 0 1 1 7 0 0 1 17

6 ( ?/ 41 4 2 9 9 13 0 2 0 3

7 R m (i n o h m s ) 1 2 7 6 4 3 J \( 0 9 2 5 6 A O 1 3 3 7 16 . / J( 2 8 9 6 A i )

8 R r(i n o h m s ) 4 4 1 64 4 4 1 6 4

9 H ( i n % ) 9 4 14 9 9 8 6 0 S

10 R I V (i n % ) 0 9 8 5 3 1 15

1 1 D ( i n d B ) 7 2 9 7 2 9

12 C, ( m d B ) 7 0 2 8 7 2 2 9

5. Conclusion

I m p r o v e d f o r m u l a e fo r th e in p u t i m p e d a n c e o f r e c t a n g u l a r , c i r c u l a r a n d h e x a g o n a l a n t e n n a s h a v e b e e n o b t a i n e d I h e se e x p r e s s i o n s are v e r y m u c h r e q u i r e d in m a t c h i n g th e a n t e n n a to th e c o a x i a l line. K x p r e s s i o n fo r th e in p u t i m p e d a n c e o f r e c t a n g u l a r p a tc h m i c r o s t r i p a n t e n n a h a s b e e n d e v e l o p e d by u s i n g c a v i t y m o d e l , w h ile for c i r c u l a r a n d h e x a g o n a l p a tc h a n t e n n a , c a v i t y m o d e l a l o n g w ith m o d e l e x p a n s i o n m o d e l a n d e q u i v a l e n t r e s o n a n t p a r a l l e l I.C R c i r c u i t h a v e b e e n u se d T h e v a r i a ti o n o f in p u t i m p e d a n c e at r e s o n a n c e w ith lo c a t io n o f feed p o i n t fo r all th e th r e e g e o m e t r i e s in the S a n d A - b a n d are s h o w n in F i g u r e s 2, 4 a n d 6 r e s p e c t i v e l y I m p o r t a n t a n t e n n a p a r a m e t e r s o f th e s e g e o m e t r i e s h a v e a ls o b e e n c o m p u t e d a n d p r e s e n t e d in T a b l e s 1 3. F o r this a n a l y s i s , all th e th r e e a n t e n n a g e o m e t r i e s are d e s i g n e d on d o u b l e s i d e d c o p p e r c l a d e d R T d u r o i d s u b s t r a t e ( £ r 2 . 3 3 ) o f h e i g h t , h - 0 159 cm a n d loss t a n g e n t , tan - 0 . 0 0 0 6 6 . O u r re s u lts a re c a p a b l e to p r o v i d e th e e x a c t p o s i t i o n o f fe ed p o i n t s for th e s e a n t e n n a g e o m e t r i e s o v e r to S a n d A'-band for d i f f e r e n t in s e t c o a x i a l fe ed line O v e r a l l , th e r e s u l t s o f the p r e s e n t in v e s tig a tio n m a y be u s e f u l for a p ro s p e c t i v e a n t e n n a d e s i g n e r

Acknowledgment

T h e a u th o r s e x p r e s s th e i r p r o f o u n d g r a t i t u d e to P rof. M a n z o o r A h m e d , V i c e - C h a n c e l l o r , D r . B R A m b e d k a r U n i v e r s i t y , A g r a , f o r p r o v i d i n g n e c e s s a r y f a c i l i t i e s a n d c o n s t a n t e n c o u r a g e m e n t .

(6)

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