Sonlu Yer Düzlemi ve Dielektrik Katmanı Üzerine

Sonlu Yer Düzlemi ve Dielektrik Katmanı Üzerine

6 RQOX<HU']OHPLYH'LHOHNWULN.DWPDQÕh]HULQH<HUOHúWLULOPLú
0LNURúHULW$QWHQOHULQ$QDOL]L
Gölge Ögücü, Lale Alatan*, Özlem Aydin Civi*, Tuncay Ege
Gaziantep Üniversitesi
Elektrik-Elektronik Müh. Bölümü
27310 Gaziantep
[email protected], [email protected]
*
ODTU
Elektrik-Elektronik Müh. Bölümü
Ankara
[email protected] ,[email protected]
Özet:
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elektrik dipolünün (H('
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NHQDUGDQ
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VDoÕQÕPÕQÕ
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NHQDU
JHoLULVL
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VRQOX
\DSÕ
LoLQ
\DNODúÕN
ELU
*UHHQ
IRQNVL\RQX NDSDOÕ IRUPGD EXOXQPXúWXU
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GLHOHNWULN WDEDND ]HULQH \HUOHúWLULOHQ PLNURúHULW KDWOD EHVOHQHQ \DPD DQWHQOHULQ DQDOL]L \DSÕOPÕúWÕU $QWHQ
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2. Formülasyon
6RQVX] GLHOHNWULN NDWPDQ YH \HU G]OHPLQLQ ]HULQH \HUOHúWLULOPLú \DWD\ HOHNWULN GLSROQQ VNDODU Ye
vektör
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%X oDOÕúPD 7h%ø7$. WDUD
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DODFDN
úHNLOGH
GH÷LúWLUHUHN
VRQOX
\DSÕ
LoLQ
*UHHQ
IRQNVL\RQODUÕQÕ
EXOPDN
PPNQGU
%X
IRQNVL\RQODUÕQ
EXOXQPDVÕ LoLQ L]OHQHFHN \|QWHP DúD÷ÕGDNL úHNLOGH |]HWOHQHELOLU >@
PEC
2a
z
2b
HED
(x’,y’,d)
d
inc
E sw
y
εr
x
εr
Meq
Yer Düzlemi
y = -a
Yer Düzlemi
y=a
ùHNLO 6RQOX GLHOHNWULN NDWPDQ ]HULQGH GLSRO YH \ D¶GD HúGH÷HU SUREOHP
1) Dielektrik-KDYD DUD\]QGH \
D ]! ùHNLO HOHNWULN DODQÕQ \]H\ GDOJD HOHPDQÕQÕ EXOPDN JHUHNLU
inc
E sw = f ( z )k ρ 0 H 02  k ρ 0 ( x − x ′) 2 + ( a − y ′) 2  aˆ z


(1)
Burada k ρo \]H\ GDOJDVÕQÕQ \D\ÕOÕP VDELWLGLU f(z GDOJDQÕQ z-\|QQGH QDVÕO GH÷LúWL÷LQL J|VWHUHQ IRQNVL\RQ H 02
LVH VÕIÕUÕQFÕ GHUHFH LNLQFL WU +DQNHO IRQNVL\RQXGXU
'H÷LúLN JHOLú DoÕODUÕQGDNL GDOJDODUÕQ HWNLVLQL GDKD L\L DQDOL] HGebilmek
için Hankel fonksiyonunun düzlem
GDOJD DoÕOÕPÕQGDQ ID\IDODQÕODELOLU
− jk y
∞
e y
πH 02  k ρ 0 x 2 + y 2  = ∫ e − jk x x
dk x , k ρ 0 = k x2 + k y2

 −∞
ky
(2)
3) Dielektrik-KDYD DUD\]QH KD\DOL ELU \DUÕ-VRQVX] PNHPPHO HOHNWULN LOHWNHQ 3(& \HUOHúWLULOHUHN \XNDULGD
bulunan gelen \]H\ GDOJD\Õ NXOODQDUDN GLHOHNWULN-KDYD DUD\]QGH \
D ]! HúGH÷HU PDJQHWLN DNÕP
inc
WDQÕPODQDQÕU M eq = E SW × nˆ ).
4) Hayali PEC ve sonlu yer düzlemi 90o¶OLN
o
ELU WDNR] ROXúWXUXU øOHWNHQ WDNR] ]HULQH \HUOHúWLULOPLú
e − jk x x
LOH GH÷LúHQ \DWD\ oL]JLVHO PDQ\HWLN ELU DNÕPÕQ *UHHQ IRQNVL\RQX 'HQNOHP ¶WHNL JLELGLU >@
G xH

ωε 0π

=
ωε 0π


2 − jk z x
1
e
∑ σ H 2n2 (k t z ′) J 2n (k t z ),
3
n
n
3
z < z′
2 − jk z x
1 2
e
H 2n (k t z ) J 2n (k t z ′),
∑
3
n σn
3
z > z′
3
(3)
3
Burada σ 0 = 2, σ n = 1, n ≠ 0 and k t = k 02 − k x2
toplam manyetik alan ( H ) bulunur.
%X *UHHQ IRQNVL\RQX NXOODQÕODUDN HúGH÷HU DNÕPÕQ ROXúWXUGX÷X
.DUDUOÕ IRUPO VWDWLRQDU\ IRUPXOD >@ NXOODQÕODUDN
ky¶D
ED÷OÕ RODUDN
kenar geçirisi yedge bulunabilir. Yüzey
GDOJDODUÕQÕQ NHQDUGDQ \DQVÕPD NDWVD\ÕODUÕ NHQDU JHoLULVL NXOODQÕODUDN DúD÷ÕGDNL úHNLOGH EXOXQXU:
kρ0
( )
yedge k y =
H , M eq
M eq , M eq
ωµ
, Γ(k y ) =
kρ0
ωµ
− y edge (k y )
(4)
+ y edge (k y )
'LHOHNWULN LoLQGHNL ELU J|]OHP QRNWDVÕQGD \DQVÕ\DQ \]H\ GDOJDODUÕ úX úHNLOGH EXOXQDELOLU
+ jk ( y + y′ )
∞
e y
− jk 2 a
ref
∝ Γ(k y )e y e − jk x x
E sw
dk x
ky
−∞
∫
(5)
7P JHOLú DoÕODUÕ LoLQ EXOXQDQ \DQVÕPD NDWVD\ÕVÕ JHQHOOHúWLULOPLú IRQNVL\on
Function -
*32) \|QWHPL
kalemi (Generalized Pencil of
NXOODQÕODUDN NRPSOHNV VWHOOHU FLQVLQGHQ DoÕOPDVÕ \ROX LOH \DQVÕ\DQ \]H\ GDOJDODUÕ
DúD÷ÕGDNL JLEL EXOXQXU
E zref,sw
= f ( z )k ρ 0
Denklem (6)’daki αi ve βi

∑
i =1, N

β i H 02  k ρ 0




( x − x ′) + (2a − ( y + y ′) − jα i ) 

kaynagin görüntüsü

2
2
(6)
GH÷HUOHUL GLHOHNWULN NDWPDQÕQ NDOÕQOÕ÷Õ GLHOHNWULN VDELWL YH IUHNDQV WDUDIÕQGDQ
EHOLUOHQPHNWHGLU 6RQOX GLHOHNWULN NDWPDQÕQÕQ GL÷HU NHQDUODUÕQGDQ PH\GDQD JHOHFHN \DQVÕPDODU GD \DQVÕPD\D
QHGHQ RODQ VUHNVL]OL÷LQ \DSÕVÕQÕQ GH÷LúPHPHVL QHGHQL LOH D\QÕ αi ve β i GH÷HUOHUL LOH LIDGH HGLOHELOPHNtedir.
Böylelikle dört kenardan (y=a, y=-a, x=b, x=-b \DQVÕ\DQ WRSODP \]H\ GDOJD DúD÷ÕGDNL JLEL LIDGH HGLOHELOLU
{
Ezref , sw = f ( z )k ρ 0 ∑ βi H 02  k ρ ( x − x′)2 + ( 2a − ( y + y′) − jai )2  + H 02  k ρ ( x − x′) 2 + (2a + ( y + y′) − jai )2 



 (7)
i
+ H 02  k ρ (2b − ( x + x′) − jai )2 + ( y − y′) 2 + H 02  k ρ ( 2b + ( x + x′) − jai )2 + ( y − y′)2 




<DPD
DQWHQLQ
DQDOL]LQGH
NDUÕúÕN
SRWDQVL\HO
potansiyelin (Gq) hem de vektör potansiyelin
LQWHJUDO GHQNOHPL NXOODQÕOPDNWDGÕU %X QHGHQOH KHP VNDODU
( G A *UHHQ IRQNVL\RQODUÕQD LKWL\Do YDUGÕU 6RQOX GLHOHNWULN
NDWPDQ YH \HU G]OHPL LoLQ VNDODU YH YHNW|U SRWDQVL\HOOHULQ \HU X]D\ÕQGD VSDWLDO GRPDLQ LIDGH HGLOPHVLQGH
NDSDOÕ IRUPGDNL *UHHQ IRQNVL\RQODUÕ \|QWHPL NXOODQÕOPÕúWÕU >@ %X \|QWHPGH IUHNDQV X]D\ÕQGDNL VSHFWUDO
GRPDLQ *UHHQ IRQNVL\RQODUÕ NRPSOHNV VWHOOHU FLQVLQGHQ DoÕOPDNWD YH \HU X]D\ÕQD G|QúPGH 6RPPHUIHOG
|]HOOL÷L
NXOODQÕOPDNWDGÕU
6NDODU
SRWDQVL\HO
*UHHQ
IRQNVL\RQX
NDSDOÕ
IRUPGD
DúD÷ÕGDNL
úHNLOGH
LIDGH
edilmektedir.


N TE + N TM


e − jkrn
+ (− j 2π ) ∑ k ρp ( i ) H 02 (k ρp (i ) ρ ) Res i 
∑ bn
rn
i
n



yüzey da lg a
1
Gq =
4πε
Burada ρ = ( x − x ′) 2 + ( y − y ′) 2
ve rn = ρ 2 − a n2 ’dir. bn ve an
VÕUDVÕ\OD
IUHNDQV
IRQNVL\RQXQ NRPSOHNV VWHOOHU FLQVLQGHQ DoÕOÕPÕQGDNL NDWVD\Õ YH VWHOL LIDGH HWPHNWHdir,
GDOJD VD\ÕVÕ
(8)
X]D\ÕQGDNL
k ρp
*UHHQ
\]H\ GDOJDQÕQ
Res ise rezidüsüdür.
6RQOX \HU G]OHPL YH GLHOHNWULN NDWPDQ ]HULQH \HUOHúWLULOPLú \DPD DQWHQLQ DQDOL]LQGH 'HQNOHP ¶GHNL LIDGH\H
'HQNOHP
¶GHNL
\DQVÕPD
WHULPOHUL
HNOHQPHNWHGLU
0R0
QXQGD \XNDUÕGD EXOXQDQ *UHHQ
IRUPODV\R
IRQNVL\RQODUÕ NXOODQÕODUDN PLNURúHULW KDWOD EHVOHQHQ YH VRQOX GLHOHNWULN NDWPDQ ]HULQH \HUOHúWLULOPLú ELU \DPD
DQWHQ
]HULQGHNL
DNÕP
GD÷ÕOÕPÕ
EXOXQPXúWXU
(OGH
HGLOHQ
DNÕP
GD÷ÕOÕPÕQGDQ
KHVDSODQPDVÕ LoLQ >@¶GD |QHULOHQ \|QWHP NXOODQÕOPÕúWÕU 0LNURúHULW KDWWÕQ
\|QWHPGH DQWHQL EHVOH\HQ PLNURúHULW KDW ]HULQGHNL DNÕP \ JHOHQ YH
–\
y
DQWHQLQ
JLULú
HPSHGDQVÕQÕQ
\|QQGH X]DQGÕ÷Õ GúQOUVH EX
\DQVÕ\DQ \|QOHULQGH
LOHUOH\HQ LNL
dalga cinsinden ifade edilmektedir.
I = C + e −γ 1 y + C − e γ 2 y
Burada γ1 ve γ2
VÕUDVÕ\OD \ YH
–\
\|QQGH KDUHNHW HGHQ GDOJDODUÕQ GDOJD VD\ÕVÕ ZDYH QXPEHU
(9)
C + ve C − de
VÕUDVÕ\OD EX \|QOHUGH KDUHNHW HGHQ GDOJDODUÕQ NDWVD\ÕODUÕGÕU 'HQNOHP ¶GDNL GDOJD VD\ÕVÕQGDQ KHU LNL \|QH
KDUHNHW HGHQ GDOJDQÕQ \D\ÕOÕP VDELWL EXOXQDELOLU %XOXQDQ EX \D\ÕOÕP VDELWLQGHQ HIHNWLI GLHOHNWULN VDELWL εeff ) ve
karakteristik empedans (Zc) bulunur. Zc ve εeff GH÷HUOHUL NXOODQÕODUDN YH C + ve C − NDWVD\ÕODUÕQGDQ VDoÕOÕP
SDUDPHWUHVL YH JLULú HPSHGDQVÕ EXOXQDELOLU
3. Sonuçlar
*HOLúWLULOHQ
\D]ÕOÕP
NXOODQÕODUDN
GH÷LúLN
E\NONWHNL
GLHOHNWULN
NDWPDQ
YH
\HU
G]OHPL
]HULQH
\HUOHúWLULOHQ ùHNLO D¶GDNL PLNURúHULW \DPD DQWHQLQ ]HULQGHNL DNÕP GD÷ÕOÕPÕ YH JLULú HPSHGDQVÕ *+]¶WH
IDUNOÕ D YH E GH÷HUOHUL LoLQ DQDOL] HGLOPLúWLU
'LHOHNWUL÷LQ NDOÕQOÕ÷Õ FP YH GLHOHNWULN VDELWL ¶GLU 7DEOR
lmek için
¶GH DQWHQLQ VDoÕOÕP SDUDPHWUHVL YH QRUPDOL]H HGLOPLú JLULú HPSHGDQVÕ YHULOPLúWLU .DUúÕODúWÕUPD \DSDEL
GLHOHNWUL÷LQ
YH
\HU
G]OHPLQLQ
VRQVX]
ROGX÷X
GXUXP
GD
WDEORGD
VXQXOPXúWXU
7DEORGDQ GD J|UOG÷ JLEL
GLHOHNWULN NDWPDQÕ E\GNoH DQWHQLQ JLULú HPSHGDQVÕ GLHOHNWUL÷LQ VRQVX] ROGX÷X GXUXPD \DNODúPDNWDGÕU
ùHNLO E¶GH DQWHQL EHVOH\HQ PLNURúHULW KDW ]HULQGHNL DNÕP GD÷ÕOÕPÕ IDUNOÕ GLHOHNWULN E\NONOHUL LoLQ
oL]LOPLúWLU *|UOG÷ JLEL GLHOHNWULN NDWPDQÕ E\GNoH VRQXoODU PRQRWRQLN RODUDN GH÷LO VDOÕQDUDN VRQVX]
GH÷HUH \DNODúPDNWDGÕU YH EX J|]OHPGH >@¶GHNL VRQXFODUOD |UWúPHNWHGLU
b
1.25 cm
a
a
0.179 cm
1.6 cm
1.875 cm
(a)
(b)
b
ùHNLO D 0LNURúHULW DQWHQ E PLNURúHULW KDW ]HULQGHNL DNÕP GD÷ÕOÕPÕ
7DEOR (IHNWLI 'LHOHNWULN 6DELWL YH .DUDNWHULVWLN (PSHGDQV *LULú (PSHGDQVÕ
a=b=0.1λ
a=b=0.25λ
a=b=0.4λ
a=b=0.5λ
a=b=λ
a=b=2λ
a=b= ∞
Gerçek(Zin)
0.871949
0.905642
0.793759
0.923155
0.98733
0.847602
0.885981
Sanal(Zin)
-0.394871
-0.527866
-0.380465
-0.321198
-0.418489
-0.488117
-0.446258
| s11|
0.216981
0.271181
0.236013
0.169383
0.206151
0.267586
0.237657
∠s11 (0)
83,944284
-15,537515
-106.48573
-93.97288
-22,54696
-92.54034
-91.02005
Kaynaklar
[1] A. K. Bhattacharyya, “Effects of Ground Plane Truncation on the Impedance of a Patch Antenna”, IEE
Proceedings-H, 138(6), s.560-564, 1991.
[2] A. K. Bhattacharya, “Effects of Ground Plane and Dielectric Truncations on the Efficiency of a Printed
Structure”, IEEE Trans. Antennas Propagat.,39(3), s.303-308, 1991.
[3] Lale Alatan, Özlem Aydin Civi, Gölge Ögücü, “An Approximate Green’s Function for a Finite Grounded
Dielectric Slab”, 2001 IEEE International Antennas and Propagation Symposium and URSI Radio Science
Meeting, Boston-USA, Proc. URSI, s.43, 8-13 Temmuz 2001.
>@ / $ODWDQ g $\GLQ &LYL * g÷F ³$QDO\VLV RI 3ULQWHG 6WUXFWXUHV RQ 7UXQFDWHG 'LHOHFWULF 6ODE DQG )LQLWH
Ground Plane”, 2002 IEEE International Antennas and Propagation Symposium and URSI Radio Science
Meeting
[5] L.W. Pearson, “The Electromagnetic Edge Wave Due to a Point Source of Current Radiating in the Presence
of a Conducting Wedge”, IEEE Trans. Antennas Propagat., 34(9), s.1125-1131, 1986.
[6] G. Dural ,M. I. Aksun, “Closed form Green’s Functions for General Sources and Stratified Media”, IEEE
Trans. On Microwave Theory and Techniques, MTT-43(7), s.1545-1552, 1995.
[7] T.K. Sarkar, Z.A. Maricevic and M. Kahvizi, “An Accurate De-embedding Procedure for Characterizing
Discontinuities”, Int. Journal of Milimeter-Wave Computer-Aided Engineering, 2(3), s.135-143, 1992.