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Optimization of Pyramidal Horn Antennas Directivity Using the Fresnel Integral

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HAL Id: hal-00235445

https://hal.archives-ouvertes.fr/hal-00235445

Preprint submitted on 4 Feb 2008

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Optimization of Pyramidal Horn Antennas Directivity Using the Fresnel Integral

Omar Tekkouk, Ahcene Boualleg, Malek Benslama

To cite this version:

Omar Tekkouk, Ahcene Boualleg, Malek Benslama. Optimization of Pyramidal Horn Antennas Di-

rectivity Using the Fresnel Integral. 2008. �hal-00235445�

(2)

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( ) ( )

( ) ( )

π π

π π

=

= −

"

Å<Ê×ÖÙ½e¿ ÖF¸ïÊe»¾Ø„Ë Äå½Æ“ÉTÊ×Æ’Î’Õ Äå½e»¾Ä T»ï½"ÌÏÊåØ

( ) ( )

! !

" #

$&%('

) * + ,

- .0/

1 2 1 3 3

4θ

θ φ ν σ ν σ

λ

5

+

6

=

79 : 8

( ) ( )

; < ;

=>?@

>A?@ B B

C D

EGF(H

I J K L

M N0O

P Q P R R

Sφ

θ φ ν σ ν σ

λ

T

+

U

=

VX Y W Z\[A]_^

`acbedgfihkjmlcnpo0qnsrtqfujmvwnsrxrzyuh_j|{

}~€ƒ‚„…_†G„ˆ‡‰†‹ŠŒ†&ŒŽ‡€AŒ†‘‡“’†

( )

”

”

”

• –_—

˜ ™ š š

θ φ

θ φ η

› œ

=

Ÿ

+

  ž ¡ Š‡‰~„ˆ‡£¢

( ) ( ) ( ) ( )

¤ ¤

¤

¤

¥ ¦ ¥

¤

§ §¨©ª

« « «

¬_­ ­ ® ¯ ° ±

² ³ ´0µ

θ θ φ ν σ ν σ

ηλ

+

· ¸

=

¹» ¼ º Z\[A½_^

¾

t‰¿ÁÀ†&ŒkÂlj~Ąˆ‡Å‡‰~€ÆÀDŽzȁ†&ÀÉ¿ÁÀʆˌŽ‡€\ŒÁ†Ì‡“’͆ƒ‡Š_ÎDŽA‚…ƒ‡‰~€ÆωŠ‚ÐÎф\‚…c…_†&‚€Òˆ‡‰†‹ŠŒ

¡

‡‰~„ˆ‡Å†(

¡

„ˆ‡ Ó

Ô Ô

Õ Ö

= =

¡ Îр

~Ąˆ×Á€¢

( )

Ø

( ) (

Ø

) ( )

Ø

ÙÛÚ&Ü Ý Þ Ý

Ø

ß

à á_à á_à

â_ã ä å

æ ç èêé ë ë

θ φ σ σ

= ηλ

Z\[ìÛ^

}~€c…_†&‚€Òz‡‰†‹ŠíŒîŠíÏïÀфˆÈ†&Àð¿ÁÀñÂÁ„\†&Œò†ŒÄŠÄ‡ðŒÄ€Ò€Žx„\‚†Ëó̒ò†&Œô‡‰~ĀõωŠí‚‰ÎDŽ\‚ö…÷…_†&‚€Òˆ‡‰†GŠíŒ

¡|ø

¿‡ð†Ì‡ðÀDŽˆ’

ø

€õŒ€„A‚

ø

’úù

}~_†Ç~Ą\ûûÁ€\ŒÁÉ‡“’kû†‹Ò„\óËó̒c΍~Ā\Œ üÛýÌþÛÿ

σ

ù Š ‰‡w…€Ž†ËŒÉ¿ö€ ‡‰~€ÍŠíû_‡‰†&Àð¿ÁÀ ×Á„\ó‘¿ú€

σ =

¡ Îw~_†GÒ\~÷…ŠÁ€Ž~Ąˆ×Á€Í„

ÀDŽzȁ†&ÀÉ¿ÁÀ †ËŒÉ‡‰~€ƒÏ‰Š‚ÐÎф\‚…Ç…_†&‚€Òz‡‰†‹ŠíŒúù †Ì‡‰~ɇ‰~ĀŽ€ï҄ˆ×Á€„z‡†&ŒÀ†ËŒ…

¡

Îǀ€\ϓ†ËŒ€ ‡‰~€ƒŒÄŠí‚ÀDŽ\ó&† €…ÂÁ„\†&Œ¢

( ) ( )

( ) ( ) ( )

( ) ( )

! !

"

# " ! !

ν σ ν σ

θ φ θ

θ φ θ φ + σ σ

$ %

= =

&( ) ' ZA[+*_^

,

†&À†Ëó‹„\‚ó̒

¡

‡‰~€.-0/„\Œ…213/Ðûó‹„\ŒÄ€ƒÂÁ„\†&ŒÁ Ҋ‚‚€ŽûÁŠíŒÄ…_†&Œ_‡Š 5 4

φ =

„\ŒÄ…

φ =

78 6 „\‚€Ä¢

( ) ( )

( ) ( )

9

9 :: ;

<=?>@

;BA ;

@DCFE

G

A;

H I

J H

I

K L

M M

K

θ ν σ

θ θ ν θ

σ

°

λ

+

N O

=

PR S Q

= =

( ) ( )

( ) ( )

T

T UU V

WX+YZ

V

[\

V

Z^]F_

` \ V

a b

c a

b

d e

f f

d

θ ν σ

θ θ ν θ

σ

°

λ

+

g h

=

ik l j

= =

monqpr

sutvxwzy|{D}0~€  ‚ƒw„†…q~€ˆ‡‰v‹Š

( )

ŒŽ 

‘

σ

~€wz’

( )

“•”– —˜

σ

~€{™všyz›œ^~€ wv?’0žŸ{™y|}¡ 

(7)

( )

( ) ( )

σ σ

σ σ

σ σ

σ σ

σ

=

+ − −

=

mn r

sutv‹Š™vxt~ …|v.œtvx ƒ}†œ w„0…|~€‡‰v‹Šžy|{

σ =

~+w’

σ =

 

( )

( )

"!# ! "!# $

% %

= π =

mnnr

sutv }0~€ wyz›|v'&DŠ^’v€y›qv)(€t~€{™~*( œ^v€{DŠoœ+(‹Š yž2œtv „q~€ w žF‡qw,( œ y|wqŠ

( )

-

.

θ

~€w’

( )

/0

θ

’v21qv+w’•v‹Š Š™v€w‹œ ~+ 43 yw2œ tv.œ650y žF‡qw7( œ ywŠ  

( ) ( )

( )

( ) ( )

( )

8

9

8

9 : 8

9 8

:

8

: : 8

:

; < = >

? < = > ; >

; < @ A

?

<B@ A ; A

σ σ

σ σ σ

σ

=

=

mnDCr

EGFIHKJMLONQP"RTSVUWYXZNV[*J\W^]_[`UaMbdceRf[gS,JM[g[*RTha

i  „j

n

ŠDtzy75 Š œ tzv‹ŠBvxžF‡qw7( œ ywŠuž y{ œ tvxž y y"5†ƒw„†…q~€ˆ‡‰v‹Š œ tzvlkQm^1q~€{™~€}Žv œ^v€{ Šon6p,qMn^rsqMn

C q  

[

t"uwvgxyQzs{,|wugvgx}|,~,uvgx€D~"€D{"uwvgxzg€

]

‚

σ =

~€wz’

[ ]

ƒs„"ƒQ…}†,‡,†Vˆ‰„‰Šg…‹ƒ"Œ,DŽs„Šg…‹‡,†,„gŠg…4ˆg



σ =

sutv0…|~‡ v‹Š ‘g’}“Q”s•,–

—

σ =

~€w’ ˜g™š"›,œD

ž

σ =

~€{Bv2œtvŽy,1‹œ }†‡ } …‰~‡ v‹Š œ t~ œ }0~ Ÿq} ‚v†œ tv tzy|{^w ’ {Bv*( œD …qœf3

m œ tzv23Ž~€{™vl(y Š™v.œ y œtzvl(y|}Ž}Žyqw¡3 ‡ Š™v’2…‰~ ‡‰v‹Š y|ž ˜g™¢›,›,œD£

¤

σ =

~€w’ ¥¦

σ =

r j

sutv …|~€‡‰v‹Š ˜g™}œD£"œD§

¤

σ =

~€w’ ¨©› šQ™}£,ª,£V«

ž ¤

σ = σ =

~€{Bv œ tv¡y¬1œ } ‡q} …|~€‡‰v‹Ššœt~ œ ~*(€tv …|v œtzv t „tv‹Š œ

’ {™v*( œ …zœ63†žy{ ~­50~ …qv€„‡q’vš~€w’†ty{Dw2œtz~ œ t~ …|v.œ tv2Š™~€}Žvš~‹ŠT1qv*( œ {B~ œ y y|ž ® ® ¯*®"°

±³² ´ µ

= =

j

i y{

· ¸

σ = σ =

¹ œtzv žF‡qw,( œy|wqŠ {Bv’‹‡º(?v œ^y œ tzv Š^ w,(•~€w’ ’yz‡›zv*m Š^ w,( žF‡qw,( œy|wqŠ y|ž

m2*»r

jšsutv …q~€ˆ‡‰v

˜g™}ª,£

ž

σ =

50~‹Š¼(€tzy Š™v€w ›qv (~ ‡ ŠBv†œtvŽžF‡qw,( œyw

( )

½ ¾

¿ À

Á Â

σ

’v …qv+y¬1 Š ~`1~ œ^v~ ‡ ~ œ œtv C mD’wà v …|v€¹ }Ž~2āƒw„ œ tzv

’v€žŸƒwœ ywŽyž œ tv

C

mD’wÃd5†’‹œ tŽ~€}2›z „‡‰yz‡ Š_j

(8)

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vx

f1(Vx,

σ

a)

σ a=0.00 σ a=1.26 σ a=1.37 σ a=1.47 σ a=1.54 3dB

& 0

& 0 3-dB bandedges

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vy

f0(Vy,

σ

b)

σ b=0.00 σ b=0.74 σ b=1.02 σ b=1.37 σ a=1.54 3 dB

0 0

0 3-dB bandedges

i „j

n

~€w žŸ‡qwD( œyw Š ž y|{ ’ž žDv+{Bvw‹œ

σ

m^1q~€{B~+} v œ^v€{ Š

sutv …|~€‡‰v

σ =

5Ž~Š (€tzy Š™v€w ›qv*(~ ‡ ŠBv

(

σ

)

v Ÿtˆ›œBŠ ~ Š™v*(yw’~€{^3•}0~'Ÿ } ‡q} ~ 5Ž~'3 žŸ{By}

(9)

=

!#"%$&')(*$+

σ

, - .'/*$0 0132-$#"%*$4

5

/$!"%/6"%.87*9:; <;=?>A@)B/C

D

σ =

*'/1 E;F?GH/IJ,

σ =

KA"%/3*.'L9M !ONOM1/*$P Q

σ

*'/1 R

σ

'*$0;0OSUT/VW1 XYZ'/1[1( \/$]$0 W6'1'(^"W_"%U/6"%

σ

*$&`2%'/1^2a$0/ bA=:c/B>Ad

e D

=

*'1

GAFIf/g/f

h ,

=

K*'/1*$0W/N3''"%i(/$0*6/4

/ T/V]1 X NOM1 "% ' *')(/9M

θ

*' 1/4"W*$]3'1 2a$0/ o n

= (

p j

λ )

k0l-m

θ

KqNOMZ (/:7

*6/6/$&"W*9sr

∆ = θ

v

( )

t w v

( λ

u x

)

Ky"%z*6/6/$0/"%M/'{*.'(|(}1~2%$ 

€

λ

 ‚&Ki'ƒ$01M*'LZ'/1{'

1/*(/$& …„;N†)"W*'"%ˆ‡V#6/9‰*'/†*'1Š‹V#6/9‰*'/†/6"%[T/V]1 XƒNOM1 "%}Œ



Ž;s‘/’/“ ”•As/•

–}s•—”— ’—As˜‘

™š › ›

œ œ

λ λ

θ

λ λ

θ

∆ = =

∆ = =

%ž/Ÿ/ 

/z'/1M"W1Z'(/9M ¡‚%"¢£$0Z6/9M1~r ¤/¤As¥/¦/§…*'1_¨)TA©/©)§ª2«"%£'*$¬6"%­

σ

*$0®‚&1

'#"W1/K"%"¯-0K °;±³²/²/´µ

σ =

*'/1 ·¸

σ =

Xy‚&¢/2¹"%/ºT/V]1 X[6/9M"W»/2 À ¼

(

Á ¾ ½

σ

¿

)

"Â/$ˆ'*$ °;±?Ã/µ

Ä

σ =

KA"%/Z'/1¡1(1/*2a'/;!"WÅ3'¢"%

3‰1/19M†2A"%i6/9M"W¯;Æi" Ç;È?É/Ê

Ë

σ =

K"%†/36/"W1*'11(i- ÌA±?Í}Î)Ï/Ì

Ð Ä

=

K*'/13-Ñ]/NO'3'

5

.(‚

ž

/ªT/V]1 XÒ*'/1z1(;32%/$"%/Ó6Z$0*"WZ$… °;±?´µ´Í

σ =

*'1 Ô)Õ ÖA×sØ/Ù/Ø}Ú

Û Ü

σ = σ =

$]$& ]6/'/1')(®"W

W6"¯$0"%‰¢/2ˆÝ…Þ

ž

*$0 ßAàá;âßã

ä å

=

 ]/NO''­"%/i2a$…%"¯(/$0*6/

 

Z'/1 ÌA±´/´Í/Í

Ð Ä

=

H-plane gain

0 o

180 o

90 o 90 o

θ

θ 30 o

150 o

60 o

120 o 30 o

150 o 60 o

120 o

-10 -20 -30

dB

(10)

E-plane gain

0 o

180 o

90 o 90 o

θ

θ 30 o

150 o

60 o

120 o 30 o

150 o 60 o

120 o

-10 -20 -30

dB

5

M(‚:T‡V *'/1ŠÂV#6/9M'/i(M'A+2%$

λ =

K

λ =

K}*'1

σ =

K

σ =

5

(‚‚T»]/Nºˆ"%‡Vi*'1 ŠÂV#6/9M*'3(*'}P/2Ñ6r)$0Z3M1/*9‚//$]'¢NO:"%ª]‰1/

λ =

*'1

λ =

*'/1L2%/$

"%†6"%­7*9:;+2A"%

σ

V#6*$0*"W*$… /†T/V]1 X`*')(/9MˆN3‰1 "%} N*$0†/36/"W132a$0/

#ž/Ÿ/ 

"W3/Œ

∆θ

! ‹ÝZ¥A©/¨)§…*'1

∆θ

"# ‹Ý©:¦/¥/§

/¢(/$0*6/3]N *9-& LT/V]1 X (*'»Z$0*9ML;†:"P' "WZ$…04"0i"%/¢(*.'»$#7 ­"‹"%/LT/VW1 X *'(/9M;0K NOM*

*$& $%

&

θ

*'/1

∆ θ

( $'%

)+*,.-0/21#3547698;:</=1

>

'|"%-¬6*6*$0KU%"a1 r^/2¢"%/ 6"%­ 2L6r)$0Z3M1/*93$W'^*';"W*'' »1$&"%:7:"ar -ª6/$0;0*';"W1*'1

1/7*9M/61r­‚]')(3"%/

5

$0;]'/Z9

>

' "W*(/$&*9/"%}1A

/L/6"% 7Z9: ­/2¹"%/0?AV#6*$&*"W*$ ZŒA2%'1Å"%/"Â/.3A@;3"%/«$W'ª1$&"%:7:"ar¬*$0º*9MA0"WÓ"%

/3/'9sr{‚&1{7*9:;4CBƒ/$07*$0K­/'/ /;¢"W1/"WZ$]3'®"%/z/6"%Y7Z9: "%/"º*M47 "%®)(// %"

1$&"%:7:"ar32%/$+ˆN7*(‰1/†*'/13//$]'"%"¯/7ˆ"%/&*† W6"¯$0"%M‚

D+EFGEHEICJEK

L

ÝNM_ÆO P†9:7*$ :KRQ! SX TÂ9‰*$]$]M}"0 sK{Æ?Æ U­-WV‚sKXWPZY//*2%ZasK\[GBzMZ$0N7 //$]'}[Z'/1 2%1}<];K_^<`a`

`cbed;fhgjiGklnmopdgrqeftsvuwmx;dtszydtiGq{dtsc|}K

>

YXC~ ¦¢©/¨

ž

¥ ©)¦/¥

Ÿ

K ÝZ¥/¥

Ÿ

Lž

M€T†‚ $&*'/"WKƒ[GQ¯$0/2a9M/6"%M/'}\2%$\2%1 $W'81/ W(/'] Kw„CiGkf;d;dN…q†posZk7‡ˆgj‰d‹ŠŒsq†mŽ„Cmf;q‡q{f+qeftik‘’mx“d

” Ÿ/Ÿ Ÿ ž ž

(11)

TM€YASP†$]2%'/M1/ `cbed“fgjik5lnmopdg†q{f0uvmx“dhszmp… Š p˜g†dNppms /"%(*$ O'/7*$…0:"%rKSÝZ¥/¥/¤A

LŸ

MCSO ‡';"W*$P*'/1

>  Bƒ/$](*'K=[ ~¢*$0V

5

‰*9M1 *.' Tˆ/$]$0"%M/'¢2%/$‹S$&*'}W3-…WM/'«4"aNZ'«‡$]'ºÆ†' "W*')'/ <];K

^ `9`9`i mps ^7;–~º‚ ÝZK/6/6‚¨/©)V  ÝZKBƒ*$0ZªÝZ¥/¤/¤A

L

¨M

5

SrY Bƒ/$0*.$0K0dtsGqop mp… iGd o!kiGksZmpmbsq†s k7‡ o!dtpdNi mbeqtdt… Šq†mb{b ylnl dgAiGq{f … mb"!cdA‡b{d;fg†ki

Š p˜g†dtppms# Q OL;O3…0Z$0"%M/'/K$3'):7‚)/2aY/"%/*$W'‚T‹*92%/$]'‰K WÑA… Ɔ'(/9M &KÆi(‚aÝZ¥/¥/¤

L

M

5

SrY Bƒ/$0*.$0¢*'1ÓÆ Q¯$0"WK% *'*$0*9.A@;1 T!9M;…]‰*9 ÆiM*99sr‹Yr)3"%$]Mº1 *9

*2a9M"W$‹Æ†';"W*'' .^ `9`a`

i mps;Š p˜g†dNppms.„CiGk&Cmom˜g'-K7/9a

Ÿ

¥/K~¢‚

Ÿ

K6/6A¨

Ÿ

¤/V ¨/¨

Ÿ

K/Ɔ6/$4

ž

¦/¦ Ý

L

¤M

T‹*M/9asK Q Z9M WasK

7*'/Z99asK [ZÆ }1/ª4"W*')(V#';"W*(/$0*9P)( "%‰' "WZ'*( 2%/$3"%/»*'*9:r]-

*'11/ ](/'2¯/$]$#("W1$W'} ] KtŠ dtiGks&Cmf;d yf;q{dtpf;d mp…+d;ft‰pkbek;o) K

ž

K/6/6‚ Ý ž ÝZVZÝ

ž

©)KSÝZ¥/¥}©

L

©Mv«

5

1‰K BƒZ'/"a"% Y‚sK « Q *9MA]:K Y}Z99M*$]:K9[*Ɔ$#"%.2aM*M*9+'/4$0*9+'/"aN$<V '»//$]'\1/ W(/'»6/$09M*¢<];K

q{beb{dtppqlŽŠ p˜g†dNppm-, „Cik&Cmom˜g†q{kp

”

kp•‡dtidtpf;dt– ‚mx;ks –!y‘ q7g.NdtiGbrmp…–

Ÿ

6*( ‹/'«"%Æ Q ž

¦/¦/¦/V TaOKAƆ6/$].9

ž

¦/¦/¦A

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