5(=21$162:(:=0$&1,$&=(02&<:,(/.,(-
&=6727/,:2&,
:352:$'=(1,(
5H]RQDQVRZH Z]PDFQLDF]H PRF\ ZF] VWDQRZL QDMF]FLHM
VWRSQLH NRFRZH XU]G]H JHQHUDF\MQ\FK OXE QDGDZF]\FK D LFK
]DGDQLHP MHVW GRVWDUF]HQLH GR RGELRUQLND HQHUJLL RGSRZLHGQLR GX*HM
PRF\ X*\WHF]QHM V\JQDáX SU]\ PDá\FK ]QLHNV]WDáFHQLDFK L GX*HM
VSUDZQRFLRHQHUJHW\F]QHMXNáDGX0RJRQHSUDFRZDüZNODVLH$$%
% D WDN*H Z NODVLH & SU]\ NFLH SU]HSá\ZX 2
Θ
VSHáQLDMF\P
QLHUyZQRü 0 2
180
≤
<
Θ
$
0LPRVLOQLHQLHOLQLRZ\FKZDUXQNyZSUDF\
HOHPHQWXDNW\ZQHJRSU]HELHJQDSLFLDZ\MFLRZHJRPR*HE\ü]EOL*RQ\
GR VLQXVRLGDOQHJR G]LNL ILOWUDF\MQ\P ZáDFLZRFLRP VHOHNW\ZQHJR
REZRGX UH]RQDQVRZHJR :H Z]PDFQLDF]DFK PRF\ ZF] SUDFXMF\FK
ZNODVLH&DZLFEGF\FKXNáDGDPLVLOQLHQLHOLQLRZ\PLVWRVRZDQLH
DQDOL]\ PDáRV\JQDáRZHM MHVW Z ]DVDG]LH QLHGRSXV]F]DOQH 1DMF]FLHM
NRU]\VWD VL ] PHWRG NRPSXWHURZ\FK OXE JUDILF]QRDQDOLW\F]Q\FK
VWRVXMF OLQHDU\]DFM RGFLQNRZ QLHOLQLRZ\FK FKDUDNWHU\VW\N :H
Z]PDFQLDF]DFK PRF\ ZF] GX*\FK L UHGQLFK PRF\ SRZ\*HM N:
]UHJXá\VVWRVRZDQHODPS\HOHNWURQRZHWHWURG\VWUXPLHQLRZHSU]\
PRFDFK SRQL*HM G]LHVLWNyZ N: RUD] WULRG\ SU]\ ZLNV]\FK PRFDFK
%LSRODUQH L XQLSRODUQH WUDQ]\VWRU\ PRF\ Z F] ]QDMGXM ]DVWRVRZDQLD
ZH Z]PDFQLDF]DFK PQLHMV]\FK PRF\ Z ]DNUHVLH G]LHVLWNyZ : GR
N:
1DV]HUR]ZD*DQLDRJUDQLF]\P\GRNUyWNLHJRVFKDUDNWHU\]RZDQLD
Z]PDFQLDF]\ PRF\ ZF] SUDFXMF\FK Z NODVLH & UHDOL]RZDQ\FK QD
HSLSODQDUQ\FK WUDQ]\VWRUDFK PRF\ ZF] RUD] GR RSLVX ]DVDG G]LDáDQLD
Z]PDFQLDF]\ZF]SUDFXMF\FKZNODVLH'L($QDOL]DLSURMHNWRZDQLH
W\FK Z]PDFQLDF]\ V RSDUWH ] NRQLHF]QRFL QD ]QDF]Q\FK
XSURV]F]HQLDFK L SU]\EOL*HQLDFK SR]ZDODMF\FK UDF]HM QD MDNRFLRZH
Z\MDQLHQLH]MDZLVNIL]\F]Q\FKDZPQLHMV]\PVWRSQLXQDLFKLORFLRZH
XMFLH 3U]\F]\Q WHJR MHVW ZSá\Z HIHNWyZ F]VWRWOLZRFLRZ\FK
SDUDPHWU\F]Q\FK ]DOH*QRü SRMHPQRFL ]áF]RZ\FK L G\IX]\MQ\FK RG
QDSLü L SUGyZ ZSá\Z\ WHPSHUDWXURZH SU]\ GX*\FK PRFDFK
WUDFRQ\FK Z WUDQ]\VWRU]H RJUDQLF]HQLD PRF\ SUGyZ L QDSLü PDá\
SR]LRP LPSHGDQFML ZHMFLRZHM Z SU]\SDGNX WUDQ]\VWRUD ELSRODUQHJR
RUD] VLOQLH QLHOLQLRZH ]DOH*QRFL SRPLG]\ SUGDPL L QDSLFLDPL Z
XNáDG]LH
=$6$'$35$&<,2*Ï/1(:à$612&,5(=21$162:<&+
:=0$&1,$&=<02&<./$6<&
2JyOQ\ VFKHPDW EORNRZ\ Z]PDFQLDF]D PRF\ NODV\ &
SU]HGVWDZLRQRQDU\V
+
-
X
L
X
6
( )
X
W
L
Ω
8
%%
<
0
X
2
+
-
&
/
HOHPHQW
Z]PDFQLDM F\
5
/
Ω
X
L
2
L
6
8
&&
>
0
5\V6FKHPDWEORNRZ\Z]PDFQLDF]DNODV\&
$QDOL]DWHJRXNáDGX]RVWDQLHSU]HSURZDG]RQDSU]\QDVWSXMF\FK
]DáR*HQLDFKXSUDV]F]DMF\FK
−
&KDUDNWHU\VW\ND
( )
i u
O
S
F]\OL
( )
i u
C
BE
Z SU]\SDGNX WUDQ]\VWRUD
ELSRODUQHJR MHVW OLQL SRMHG\QF] L PR*H E\ü DSURNV\PRZDQD
RGFLQNDPL SURVWHM 2]QDF]D WR *H REZyG UH]RQDQVRZ\ VWDQRZLF\
REFL*HQLH Z]PDFQLDF]D MHVW GRVWURMRQ\ GR SXOVDFML V\JQDáX
VWHUXMFHJR NWyUD MHVW QD W\OH PDáD *H PR*QD SRPLQü HIHNW\
EH]ZáDGQRFLRZH WUDQ]\VWRUD SRMHPQRFL ZHMFLRZH L Z\MFLRZH
PRJE\üGRáF]RQHGRRGSRZLHGQLFKREZRGyZELHUQ\FKDZSá\Z
UHDNWDQFML ZHZQWU]QHJR VSU]*HQLD ]ZURWQHJR NROHNWRU ED]D MHVW
SRPLMDOQLHPDá\2]QDF]DWRWDN*H*HQDFK\OHQLHFKDUDNWHU\VW\NLMHVW
VWDáH]DWHPWUDQVNRQGXNWDQFMD
J
FRQVW
P
=
−
,PSHGDQFMD ZHMFLRZD QLHOLQLRZD Z]PDFQLDF]D MHVW GX*D Z
SRUyZQDQLX ] LPSHGDQFM JHQHUDWRUD VWHUXMFHJR F]\OL VWHURZDQLH
PR*HP\X]QDü]DQDSLFLRZH
−
,PSHGDQFMD Z\MFLRZD QLHOLQLRZD MHVW GX*D Z VWRVXQNX GR
LPSHGDQFMLREFL*HQLD
−
1DSLFLHZ\MFLRZHMHVWVLQXVRLGDOQHREZyGUH]RQDQVRZ\RG]QDF]D
VLWDNGX*GREURFL Q *HGODSXOVDFMLGRVWURMHQLD
Ω
UHSUH]HQWXMH
UH]\VWDQFM G\QDPLF]Q
R
L
RUD] ]ZDUFLH GOD LQQ\FK SXOVDFML
KDUPRQLF]Q\FK k
Ω
RUD]SRGKDUPRQLF]Q\FK
Ω
/ .
k
1D U\V SU]HGVWDZLRQR FKDUDNWHU\VW\N URERF]
( )
i u
O
S
RUD]
SU]HELHJL QDSLFLD ZHMFLRZHJR
( )
u
t
i
Ω
VWHUXMFHJR
( )
u
t
S
Ω
QDSLFLD
QDREZRG]LHUH]RQDQVRZ\P
( )
u
t
Ω
LSUGXZ\MFLRZHJR
( )
i
t
O
Ω
.
Ω
W
8
RP
( )
X
W
2
Ω
L X
8
7
8
%%
X
6
L
2
J
L
X
P
2
6
=
∂
∂
FRQVW
( )
X
W
L
Ω
8
VP
2
Θ
0
Θ
π
π
2
3
2
π
Ω
W
Ω
W
,
RP
−Θ
0
Θ π
2
π
3
2
π
2
π
( )
X
W
6
Ω
8
7
8
%%
( ) ( )
X
W
X
W 8
6
L
%%
Ω
Ω
=
+
Θ
0
π
π
2
3
2
π
( )
L
W
2
Ω
2
π
8
&&
( )
X W
Ω
0
5\V &KDUDNWHU\VW\ND URERF]D
( )
L X
2
6
L SU]HELHJL F]DVRZH Z LGHDOQ\P
Z]PDFQLDF]XUH]RQDQVRZ\PNODV\&
( )
X
W
L
Ω
QDSLFLHZHMFLRZH
( )
X
W
6
Ω
QDSLFLH VWHUXMFH
( )
L
W
2
Ω
SUG Z\MFLRZ\
( )
X
W
Ω
QDSLFLH QD REZRG]LH
UH]RQDQVRZ\P
( )
X
W
2
Ω
QDSLFLHQDHOHPHQFLHDNW\ZQ\P
3UG Z\MFLRZ\ PD NV]WDáW LPSXOVyZ NRVLQXVRLGDOQ\FK R
DPSOLWXG]LH
I
om
]DOH*QHM RG DPSOLWXG\ QDSLFLD VWHUXMFHJR
U
sm
,
QDSLFLDSRODU\]DFML
U
BB
LQDSLFLDSURJRZHJR
U
T
FKDUDNWHU\VW\NL
(
)
,
8
8
8 J
8
8
8
8
RP
VP
%%
7
P
VP
%%
7
VP
%%
7
=
+
−
≥ −
+
≤ −
+
GOD8
GOD8
0
1DSLFLHVWHUXMFH
( )
u
t
S
Ω
GOD
Ω
Θ
W
=
RVLJDZDUWRü
8
8
8
VP
%%
7
FRV
Θ
= −
+
=]DOH*QRFLRUD]Z\QLND
(
)
,
8
J
RP
VP
P
=
−
1
FRV
Θ
3U]HSá\Z SUGX
( )
i
t
O
Ω
SU]H] REZyG UH]RQDQVRZ\ R SXOVDFML
UH]RQDQVRZHMGRVWURMRQHMGRSXOVDFMLV\JQDáXVWHUXM FHJR
Ω
SRZRGXMH
SRZVWDQLH QD QLP QDSLFLD VLQXVRLGDOQHJR
( )
u
t
Ω
.
3U]HELHJL QDSLFLD
Z\MFLRZHJR
( )
u
t
O
Ω
L QDSLFLD QD REZRG]LH UH]RQDQVRZ\P
( )
u
t
Ω
SU]HGVWDZLRQRQDU\V
=JRGQLH]SU]HGVWDZLRQ\PLZ\NUHVDPLQDSL FLHVWHUXMFH
u
S
RUD]
QDSLFLHZ\MFLRZH
u
O
RSLVXM]DOH*QRFL
(
)
u
U
U
t
U
u
U
U
t
S
BB
sm
BB
O
CC
om
=
+
<
=
−
cos
cos
Ω
Ω
0
1D SRGVWDZLH ]DOH*QRFL PR*HP\ RNUHOLü QDSLFLH
Z\MFLRZH
u
O
ZIXQNFMLQDSLFLDVWHUXMFHJR
u
S
(
)
u
U
U
u
U
U
O
om
sm
S
BB
CC
= −
−
+
:XNáDG]LHZVSyáU]GQ\FK
( )
u
u
O
S
i
i
const
S
0
,
=
W]QZSROXFKDUDNWHU\VW\N
L]RSUGRZ\FK FKDUDNWHU\VW\ND URERF]D VWDQRZL SURVW SU]HFKRG]F
SU]H]SXQNW\
(
)
u
U
u
U
S
BB
O
CC
=
=
,
RUD]
(
u
U
U
S
BB
sm
=
+
)
u
U
U
O
CC
om
=
−
=]DáR*HQLDGZXRGFLQNRZHMDSURNV\PDFMLFKDUDNWHU\VW\NL
( )
i u
O
S
Z\QLND*HLPSXOVSUGX
i
O
PR*QDRSLVDü]DOH*QRFL
L
$ ,
W
2
RP
=
FRV
Ω
%
SU]\F]\PVWDáH$L%Z\]QDF]\P\]ZDUXQNyZ
(
)
(
)
L
W
,
L
W
2
RP
2
Ω
Ω
Θ
=
=
0
VWG
$ ,
% ,
$ ,
%
RP
RP
RP
− =
− =
FRV
Θ
0
=]DOH*QRFLRUD]RWU]\PXMHP\
(
)
L
,
W
W
W
2
RP
=
−
−
− ≤
≤
≤
≤
1
0
FRV
FRV
FRV
Θ
Ω
Θ
Θ Ω
Θ
Θ Ω
GOD
GOD
π
JG]LH
Θ
MHVW NWHP SU]HSá\ZX SUGX Z\MFLRZHJR NWyU\ PR*HP\
REOLF]\ü]]DOH*QRFL
Θ =
−
+
DUFFRV
8
8
8
%%
7
VP
.W\SU]HSá\ZXSUGXZRGSRZLHGQLFKNODVDFKSUDF\Z\QRV]
NODVD$
Θ = π
NODVD$%
π
π
2
< <
Θ
NODVD%
Θ = π
2
NODVD&
0
2
≤ <
Θ π
5H]RQDQVRZ\ REZyG Z\MFLRZ\ GRNRQXMH VHOHNFML ZLGPD
LPSXOVX SUGX Z\MFLRZHJR
i
O
ÄZ\ELHUDMF´ ] MHJR VNáDGRZ\FK
VNáDGRZ SRGVWDZRZ ,QWHUHVXMFH QDV VNáDGRZH SUGX
i
O
PR*HP\
Z\]QDF]\ü QD SRGVWDZLH UR]NáDGX )RXULHUD SU]HELHJX
i
O
6NáDGRZD
VWDáD
,
2
RUD]DPSOLWXG\NW\FKKDUPRQLF]Q\FK
,
RPN
VUyZQH
( )
( )
( )
( )
(
)
(
)
( )
,
,
L
,
,
,
,
,
,
,
N
N
N
N N
,
GOD N
2
&&
R
RP
RP
RP
RP
RP
RPN
RP
RP N
=
= =
=
−
−
=
=
−
−
=
=
−
−
−
=
=
≥
π
α
π
α
π
α
VLQ
FRV
FRV
VLQ FRV
FRV
VLQ
FRV
FRV
VLQ
FRV
Θ Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
1
1
2
1 1
2
0
1
1
2
:VSyáF]\QQLNL
( )
α
N
Θ
QRV] QD]Z ZVSyáF]\QQLNyZ UR]NáDGX LPSXOVX
NRVLQXVRLGDOQHJR:VSyáF]\QQLN
( )
( )
( )
J
1
1
0
Θ
Θ
Θ
=
α
α
QRVL QD]Z ZVSyáF]\QQLND NV]WDáWX =DOH*QRFL ZVSyáF]\QQLNyZ
( )
( )
α
k
g
Θ
Θ
,
1
SU]HGVWDZLRQRQDU\V
-DN ZLGDü ] U\V SUDFD Z NODVLH & ]ZáDV]F]D SU]\ PDá\FK
NWDFK SU]HSá\ZX JáERND NODVD & RG]QDF]D VL ERJDW\P ZLGPHP
KDUPRQLF]Q\FK LPSXOVX SUGX Z\MFLRZHJR
i
O
0LPR WR ]JRGQLH ]
SRF]\QLRQ\PZF]HQLHM]DáR*HQLHPZZ\QLNXVHOHNW\ZQHJRG]LDáDQLD
Z\MFLRZHJR REZRGX UH]RQDQVRZHJR R GX*HM GREURFL QDSLFLH
Z\MFLRZH MHVW ]EOL*RQH GR SU]HELHJX VLQXVRLGDOQHJR R PDNV\PDOQHM
DPSOLWXG]LH
8
RP
Z\QRV]FHM
8
,
5
(
RP
RP
/
&
=
=
1
ξ
JG]LH
ξ =
=
+
U
U
U
U
U
om
CC
om
om
O min
MHVWZVSyáF]\QQLNLHPZ\NRU]\VWDQLDQDSLFLDZ\MFLRZHJR
ξ
]DOH*\P
LQRGREFL*HQLD
R
L
30
60
90
120
( )
α Θ
Θ
150
180
0
α
1
α
2
α
0
α
3
α
4
α
5
α
6
J
1
1
0
=
α
α
0
01
0 5
0 4
0 3
0 2
0
0 4
0 8
−
01
12
2
16
( )
J
1
Θ
&
$%
%
$
5\V:VSyáF]\QQLNLUR]NáDGXLPSXOVXNRVLQXVRLGDOQHJR
=DOH*QRFLHQHUJHW\F]QHZXNáDG]LH
=QDMRPRü NV]WDáWX L SDUDPHWUyZ LPSXOVX SUGX Z\MFLRZHJR
( )
,
RPN
N
α Θ
SDUDPHWUyZ Z\MFLRZHJR REZRGX UH]RQDQVRZHJR L
REFL*HQLD
R
L
RUD] ZDUXQNyZ ]DVLODQLD
U
CC
XPR*OLZLD Z\]QDF]HQLH
PRF\ZXNáDG]LHZ\MFLRZ\P0RFZ\MFLRZRNUHOD]DOH*QRü
( )
3
, 8
,
8
/
RP
RP
RP
&&
=
=
1
2
1
2
1
1
α
ξ
Θ
0RFZ\MFLRZD]DOH*\Z\áF]QLHRGDPSOLWXG\VNáDGRZHMSRGVWDZRZHM
JG\ REZyG UH]RQDQVRZ\ GOD VNáDGRZ\FK KDUPRQLF]Q\FK L
SRGKDUPRQLF]Q\FKVWDQRZL]ZDUFLH
0RF GRVWDUF]RQD ]H (UyGáD ]DVLODQLD
8
&&
GR REZRGX Z\MFLRZHJR
Z\QRVL
( )
3
, 8
,
8
'
&&
&&
RP
&&
=
=
α
0
Θ
JG]LH
I
I
CC
=
0
MHVW UHGQL ZDUWRFL SUGX SREUDQHJR ]H (UyGáD
]DVLODQLD
U
CC
:\NRU]\VWXMF ]DOH*QRFL RUD] PR*HP\
Z\]QDF]\üVSUDZQRüZ]PDFQLDF]D
( )
( )
( )
η
ξ α
α
ξ
=
=
=
3
3
J
/
'
1
2
1
2
1
0
1
Θ
Θ
Θ
= ]DOH*QRFL L REOLF]\P\ PRF VWUDW Z NROHNWRU]H
WUDQ]\VWRUD
(
)
P
P
P
P
P
P
P
C
D
L
D
D
D
L
=
−
=
−
= −
= −
η
η
η
η
1
1
=DOH*QRFLJUDILF]QHPRF\Z\MFLRZHM
P
L
,
PRF\GRVWDUF]RQHM
P
D
L PRF\ WUDFRQHM
P
C
RUD] VSUDZQRFL
η
RG NWD SU]HSá\ZX
Θ
SU]HGVWDZLRQR QD U\V :\NUHV\ WH SU]HGVWDZLRQR Z SRVWDFL
]QRUPDOL]RZDQHMSU]\F]\P
−
PRFZ\MFLRZ
P
L
UHSUH]HQWXMHZ\UD*HQLH
( )
α
1
Θ
( )
2
1
3
, 8
/
RP
&&
ξ
α
=
Θ
−
PRFGRVWDUF]RQUHSUH]HQWXMHZ\UD*HQLH
( )
2
0
α Θ
( )
2
2
0
3
, 8
'
RP
&&
=
α Θ
−
VSUDZQRüZ]PDFQLDF]DUHSUH]HQWXMHZ\UD*HQLH
( )
0 5
1
, g
Θ
( )
η
ξ
=
0 5
1
J
Θ
−
PRF WUDFRQ Z NROHNWRU]H WUDQ]\VWRUD GOD
ξ
UHSUH]HQWXMH
Z\UD*HQLH
(
)
( )
( )
2
2
2
2
2
1
0
1
3
, 8
3
3
, 8
3
, 8
3
, 8
&
RP
&&
'
/
RP
&&
'
RP
&&
/
RP
&&
=
−
+
−
=
α
α
ξ
Θ
Θ
GOD
-H*HOL ]QDQD MHVW FKDUDNWHU\VW\ND SUGX VWHUXMFHJR
( )
i u
S
S
RUD]
NV]WDáW LPSXOVX WHJR SUGX ZUD] ]H ZVSyáF]\QQLNDPL UR]NáDGX WR
DQDORJLF]QH UR]ZD*DQLD SURZDG] GR Z\]QDF]HQLD VNáDGRZ\FK
KDUPRQLF]Q\FKSUGX
i
S
DGDOHMGRPRF\ZHMFLRZHM
P
in
]ZDQHMPRF
Z]EXG]HQLD
P
I
U
in
sm
sm
=
1
2
1
JG]LH
I
sm1
DPSOLWXGD VNáDGRZHM SRGVWDZRZHM LPSXOVX SUGX
ZHMFLRZHJR
U
sm
DPSOLWXGDQDSLFLDVWHUXMFHJR
u
s
30
60
90
120
150
180
0
0
122
π
4
( )
( )
2
2
0
1
3
, 8
&
RP &&
=
−
α
α
Θ
Θ
GOD
ξ
( )
2
2
0
3
, 8
'
RP &&
=
α Θ
( )
2
1
3
, 8
/
RP
&&
ξ
α
=
Θ
( )
η
ξ
=
J
Θ
Θ
5\V=DOH*QRüPRF\LVSUDZQRFLRGNWDSU]HSá\ZX
1DSLFLH
X
6
MHVWQDSLFLHPNRVLQXVRLGDOQ\P
(
)
u
u
U
S
i
BE
= +
RSXOVDFML
Ω
.
=DWHPKDUPRQLF]QHVUyZQH]HUX
U
U
sm
sm
1
=
U
smk
=
0
'ODWHJR
,
3
, 8
,
8
LQ
VP
VP
VPN
VPN
N
=
+
=
∞
∑
1
2
1
2
0
1
1
2
LZNRQVHNZHQFMLRWU]\PXMHVL]DOH*QRü
0RF GRVWDUF]RQD ]D (UyGáD QDSLFLD
U
BB
Z REZRG]LH ZHMFLRZ\P
Z\QRVL
P
I
U
B
B
B B
=
0
JG]LH
I
B 0
MHVW SUGHP SRELHUDQ\P ]H (UyGáD SRODU\]DFML
U
BB
,
F]\OL
UHGQLZDUWRFLSUGXVWHUXMFHJR
i
S
0RF WUDFRQD QD ZHMFLX
P
Din
MHVW VXP PRF\ Z]EXG]HQLD
P
in
L
PRF\
P
B
GRVWDUF]RQHM]H(UyGáD
U
BB
.
P
P
P
Din
in
B
=
+
1DOH*\SRGNUHOLü*HMH*HOL
U
BB
<
0,
WRPRFGRVWDUF]RQD
P
B
MHVWXMHPQD
SRQLHZD* (UyGáR ]DVLODQLD
8
%%
SRELHUD D QLH GRVWDUF]D PRF\ :WHG\
EDWHULD
U
BB
PR*HE\ü]DVWSLRQDUH]\VWRUHP
R
U
I
B
BB
B
=
0
]ERF]QLNRZDQ\P SRMHPQRFL
C
B
R WDNLHM ZDUWRFL DE\ UHDNWDQFMD
NRQGHQVDWRUD
C
B
GOD VNáDGRZHM SRGVWDZRZHM E\áD GRVWDWHF]QLH PDáD
8NáDG WDNL MHVW QD]\ZDQ\ XNáDGHP SRODU\]DFML G\QDPLF]QHM REZRGX
ZHMFLRZHJR
:QLRVNLZ\QLNDMFH]DQDOL]\SUDF\Z]PDFQLDF]DNODV\&
3U]HSURZDG]RQDDQDOL]DSUDF\Z]PDFQLDF]DNODV\&RSLHUDáDVL
QD V]HUHJX ]DáR*H XSUDV]F]DMF\FK -HGQDN*H Z RSDUFLX R QL PR*QD
SU]HGVWDZLü NLOND ZQLRVNyZ VáXV]Q\FK WDN*H GOD U]HF]\ZLVWHJR
Z]PDFQLDF]DNODV\&
3U]\ VWDáHM ZDUWRFL V]F]\WRZHM LPSXOVX NRVLQXVRLGDOQHJR SUGX
Z\MFLRZHJR
(
)
,
FRQVW
RP
=
RUD] SU]\ VWDá\P ZVSyáF]\QQLNX
ξ
Z\NRU]\VWDQLDQDSLFLD]DVLODMFHJR
U
CC
PRFZ\MFLRZD
P
L
URQLH
JG\ NW SU]HSá\ZX SUGX Z\MFLRZHJR
Θ
URQLH =DOH*QRü
( )
3
/
Θ
MHVW DQDORJLF]QD GR ]DOH*QRFL
( )
α
1
Θ
'OD SU]\MWHM RGFLQNRZR
OLQLRZHM DSURNV\PDFML FKDUDNWHU\VW\NL
( )
i u
O
S
PDNVLPXP PRF\
Z\MFLRZHM SU]\SDGD GOD
Θ
RSW
R
=
122
NODVD $% =PQLHMV]DMF NW
SU]HSá\ZXSUGX
Θ
WDNDE\PRFZ\MFLRZDE\áDVWDáD
(
)
P
const
L
=
,
QDOH*\]ZLNV]\üZDUWRüV]F]\WRZLPSXOVXSUGXZ\MFLRZHJR
I
om
.
'OD X]\VNDQLD GX*HM PRF\ Z\MFLRZHM
P
L
QDOH*\ VWRVRZDü QLH]E\W
PDáHZDUWRFLNWDSU]HSá\ZX
Θ
F]\OLQLH]E\WJáERNNODV&
3U]\VWDáHMZDUWRFLV]F]\WRZHMLPSXOVX
I
om
SUGXZ\MFLRZHJRPRF
GRVWDUF]RQD
P
D
URQLH JG\ NW SU]HSá\ZX
Θ
URQLH =DOH*QRü
( )
3
'
Θ
MHVW DQDORJLF]QD GR ]DOH*QRFL
( )
α
0
Θ
:]URVW PRF\
GRVWDUF]RQHM
P
D
]H Z]URVWHP NWD SU]HSá\ZX
Θ
MHVW V]\EV]\ QL*
Z]URVWPRF\Z\MFLRZHM
P
L
.
3U]\VWDáHMZDUWRFLZVSyáF]\QQLNDZ\NRU]\VWDQLDQDSLFLD]DVLODQLD
(
)
ξ =
const
VSUDZQRüZ]PDFQLDF]D
η
URQLHJG\NWSU]HSá\ZX
Θ
PDOHMH =DOH*QRü
( )
η Θ
MHVW DQDORJLF]QD GR ]DOH*QRFL
ZVSyáF]\QQLND NV]WDáWX
( )
J
1
Θ
: V]F]HJyOQRFL JG\
Θ
→
L
ξ
→
WR
η
η
=
→
=
1 2
1
1
PD[
J
*UDQLF]QD WHRUHW\F]QD VSUDZQRü
Z]PDFQLDF]DZNODVLH&Z\QRVLGODNODV\%
η
max
π
RUD]
GODNODV\$
η
max
MHGQDNPRFZ\MFLRZDPDOHMHZWHG\GR]HUD
= ZQLRVNyZ L Z\QLND *H Z\PDJDQLD RGQRQLH GX*HM PRF\
Z\MFLRZHM
P
L
L GX*HM VSUDZQRFL
η
V VSU]HF]QH L Z ZDUXQNDFK
U]HF]\ZLVW\FKZ\PDJDQ\MHVWNRPSURPLV
:]PDFQLDF] NODV\ & FKDUDNWHU\]XMH VL QLHOLQLRZ ]DOH*QRFL
( )
I
f U
om
sm
1
=
.
=PLDQDDPSOLWXG\V\JQDáXVWHUXMFHJR
U
sm
SRZRGXMH
]PLDQ\ NWD SU]HSá\ZX
Θ
RUD] ]PLDQ\ V]F]\WRZHM ZDUWRFL SUGX
Z\MFLRZHJR
,
RP
:DUXQHN OLQLRZRFL MHVW VSHáQLRQ\ GRSLHUR SU]\
SUDF\ Z NODVLH % OXE $ ZyZF]DV ERZLHP NW SU]HSá\ZX
Θ
QLH
]DOH*\RG
U
sm
:WHG\]PLDQD
U
sm
SRZRGXMHSURSRUFMRQDOQGRQLHM
]PLDQ
I
om
D ]DWHP UyZQLH*
I
om1
'OD
U
BB
8
7
U
BB
>
0
NW
SU]HSá\ZX
Θ = π
2 Z]PDFQLDF] SUDFXMH Z NODVLH % SU]\ F]\P
DPSOLWXG\SUGyZVUyZQH
I
U
g
I
U g
om
sm
m
om
sm
m
1
2
=
=
'OD
8
8
8
%%
7
VP
=
+
NW SU]HSá\ZX
Θ = π
:WHG\ Z]PDFQLDF]
SUDFXMHZNODVLH$]DPSOLWXGDPLSUGyZ
I
U
g
I
U
g
m
sm
m
om
sm
m
0 1
2
=
=
:]PDFQLDF]H SUDFXMFH SU]\ ]PLHQQHM DPSOLWXG]LH V\JQDáX
X*\WHF]QHJR QS Z]PDFQLDF]H SU]HELHJyZ PRGXORZDQ\FK
DPSOLWXGRZR PXV] ]DWHP SUDFRZDü Z NODVLH % OXE Z NODVLH $
DE\ ]DSHZQLü OLQLRZ ]DOH*QRü QDSLFLD Z\MFLRZHJR RG QDSLFLD
VWHUXMFHJR:W\PSU]\SDGNXQLHMHVWPR*OLZHZ\NRU]\VWDQLHSUDF\
ZNODVLH&LX]\VNDQLHGX*HMVSUDZQRFLHQHUJHW\F]QHM]ZL]DQHM]
NODV&
1D SRGVWDZLH SRZ\*V]\FK XZDJ PR*HP\ RNUHOLü ]DOH*QRFL
HQHUJHW\F]QHZ]PDFQLDF]DZNODVLH%WMGOD
Θ = π
2=DOH*QRFLWH
]RVWDá\ RNUHORQH ZF]HQLHM GOD GRZROQHJR NWD SU]HSá\ZX
Θ
:\NRU]\VWXMF]DOH*QRFLLRWU]\PXMHP\
α
π
π
α
π
ξ
α
π
0
1
1
1
2
1
2
0 5
2
Θ
Θ
Θ
=
=
=
=
=
=
=
=
8
8
,
5
, 5
RP
&&
RP
/
RP
/
6WG
,
8
5
3
,
8
8
5
3
,
8
8
5
J
RP
&&
/
/
RP
&&
&&
/
'
RP
&&
&&
/
=
=
=
=
=
=
=
=
=
=
=
ξ
α
π
α
π
ξ ξ
α
π
π
ξ
η
π ξ π ξ
1
1
2
2
0
2
1
2
1
2
2
2
2
2
1
2
2
4
Θ
Θ
Θ
Θ
: SRGREQ\ VSRVyE PR*HP\ RNUHOLü ]DOH*QRFL HQHUJHW\F]QH GOD
NODV\$WMGOD
Θ = π
:WHG\
(
)
(
)
(
)
α
π
α
π
π
0
1
1
0 5
1
Θ
Θ
Θ
=
=
=
=
=
=
J
:]PDFQLDF] Z NODVLH & PR*H E\ü X*\W\ MDNR NNURWQ\ SRZLHODF]
F]VWRWOLZRFL F]\OL Z]PDFQLDF] NWHM KDUPRQLF]QHM :WHG\
Z\MFLRZ\REZyGUH]RQDQVRZ\MHVWGRVWURMRQ\GR NWHMKDUPRQLF]QHM
V\JQDáX VWHUXMFHJR .W SU]HSá\ZX SU]\ NWyU\P DPSOLWXGD NWHM
KDUPRQLF]QHM SUGX Z\MFLRZHJR
I
omk
RVLJD PDNVLPXP ZWHG\
WDN*H PRF Z\MFLRZD RVLJD PDNVLPXP SU]\ ]DáR*HQLX VWDáHM
ZDUWRFLV]F]\WRZHMSUGXZ\MFLRZHJR
I
om
LVWDá\PZVSyáF]\QQLNX
ξ
Z\NRU]\VWDQLDQDSLFLD]DVLODQLDZ\QRVL
Θ
RSW
N
=
2
3
π
3U]\
Θ
RSW
Z\VWSXM PDNVLPD ZVSyáF]\QQLNyZ UR]NáDGX
NRVLQXVRLGDOQHJR
( )
α
N
Θ
0DNV\PDOQD PRF Z\MFLRZD PDOHMH ]H
Z]URVWHPVWRSQLDSRZLHODQLDN:SUDNW\FHVWRVXMHVL SRZLHODF]HR
VWRSQLXSRZLHODQLDQLHZLNV]\PQL*
2EZyG UH]RQDQVRZ\ R LPSHGDQFML = ]DSHZQLD Z\RGU EQLHQLH
VNáDGRZHM SRGVWDZRZHM L ILOWUDFM SR]RVWDá\FK KDUPRQLF]Q\FK
=DZDUWRüNWHMKDUPRQLF]QHMQDSLFLDZ\MFLRZHJRZ\QRVL
(
)
(
)
( )
( )
8
8
=
N
,
=
,
4 N
N
RPN
RP
R
RPN
R
RP
N
1
1
1
1
1
=
=
=
≈
−
Ω
Ω
Ω Ω
Θ
Θ
α
α
=DWHP ILOWUDFM GDQHM KDUPRQLF]QHM PR*QD SRSUDZLü VWRVXMF
RGSRZLHGQLR GREUDQ ZDUWRü NWD SU]HSá\ZX 1S
Θ = π
2
]DSHZQLDHOLPLQDFMQLHSDU]\VW\FKKDUPRQLF]Q\FK
:3à<: &=6727/,:2&, 6<*1$à8 67(58-&(*2 1$
35$&:=0$&1,$&=$:./$6,(&
$E\ SU]HSURZDG]Lü SHáQ DQDOL] SUDF\ Z]PDFQLDF]D NODV\ &
XZ]JOGQLDMF ZV]\VWNLH LVWRWQH L ]áR*RQH F]\QQLNL ]ZL]DQH ]
QLHOLQLRZRFL
FKDUDNWHU\VW\N
WUDQ]\VWRUD
Uy*QHJR
URG]DMX
RJUDQLF]HQLDPL SUGX QDSLFLD L PRF\ QLHOLQLRZ\PL HIHNWDPL
SDUDPHWU\F]Q\PL Z ]DNUHVLH ZF] LWS QDOH*\ Z\NRU]\VWDü V\PXODFM
NRPSXWHURZ XNáDGX QS Z SURJUDPLH 63,&( $E\ DQDOL]D E\áD
ZLDU\JRGQD PRGHOH HOHPHQWyZ DNW\ZQ\FK Z\NRU]\VW\ZDQ\FK GR
EXGRZ\ Z]PDFQLDF]D PXV] XZ]JOGQLDü ZV]\VWNLH Z\*HM
ZVSRPQLDQHHIHNW\QLHOLQLRZH'DOV]HUR]ZD *DQLDRJUDQLF]\P\MHG\QLH
GR MDNRFLRZHJR Z\MDQLHQLD ]MDZLVN IL]\F]Q\FK EH] LFK LORFLRZHJR
XMFLD
:Sá\Z F]VWRWOLZRFL QD SUDF Z]PDFQLDF]D UR]SDWU]\P\ Z
WU]HFKSU]HG]LDáDFKGOD
Ω
Ω
Ω
<
< <<
<
ω
ω
ω
ω
β
β
7
7
D=DNUHVVWRVXQNRZRPDá\FKF]VWRWOLZRFL
Ω <
ω
β
:W\P]DNUHVLHPR*HP\NRU]\VWDü]FKDUDNWHU\VW\NVWDW\F]Q\FKL
URERF]\FKWUDQ]\VWRUDRUD]]PHWRGJUDILF]Q\FK
=DNáDGDMF *H GREURü REZRGX UH]RQDQVRZHJR MHVW GX*D ]D
UH]\VWDQFMD (UyGáDVWHUXMFHJRMHVWPDáD
(
)
5
U
J
EE
<
PR*HP\SU]\Mü*H
QDSLFLH ZHMFLRZH
X
EH
L Z\MFLRZH
X
FH
V VLQXVRLGDOQH 3U]\SDGHN
QDSLFLRZHJRVWHURZDQLDWUDQ]\VWRUDMHVWU]DGNRVSHáQLRQ\ZSUDNW\FH
SR]ZDOD MHGQDN QD MDNRFLRZH Z\MDQLHQLH ]MDZLVN ]DFKRG]F\FK
ZWUDQ]\VWRU]H
&KDUDNWHU\VW\NURERF]XNáDGX2(]JRGQLH]UyZQDQLHP
RNUHOD]DOH*QRü
(
)
u
U
U
U
u
U
CE
CC
cem
bem
BE
BB
−
= −
+
=DOH*QRüWSU]HGVWDZLRQRJUDILF]QLHZSROXFKDUDNWHU\VW\NWUDQ]\VWRUD
( )
i
f u
C
CE
=
SU]\
u
const
BE
=
QDU\V
X
&(
L
&
Ω
W
Θ
8
&&
Θ
Θ
L L
&
%
L
&
L
%
L
%
L
%
L
&
L
&
6
6
6
6
6
3
*
8
&( PLQ
4
6
X
%(
8
8
8
%(
EHP
%%
PD[
=
−
Ω
W
5\V&KDUDNWHU\VW\NDURERF]DZSROXFKDUDNWHU\VW\NNROHNWRURZ\FKWUDQ]\VWRUD
*UDQLF]QD OLQLD * VWDQX QDV\FHQLD QD U\V UR]G]LHOD VWDQ\
SUDF\ QLHGRZ]EXG]RQHM L SU]HZ]EXG]RQHM WUDQ]\VWRUD 3XQNWRZL
V]F]\WRZHPX
S
QDFKDUDNWHU\VW\FHURERF]HMRGSRZLDGDVWDQNU\W\F]Q\
6WDQRZL SU]HZ]EXG]RQHPX RGSRZLDGD SUDFD Z REV]DU]H QDV\FHQLD
SXQNW
6
]D VWDQRZL QLHGRZ]EXG]RQHPX RGSRZLDGD SUDFD Z
REV]DU]HDNW\ZQ\PSXQNW
S'
: VWDQLH SU]HZ]EXG]RQ\P Z RWRF]HQLX SXQNWX V]F]\WRZHJR
Z\VWSXMH FKDUDNWHU\VW\F]QH RGNV]WDáFHQLH LPSXOVX SUGX NROHNWRUD
6SRZRGRZDQH MHVW WR ]QDF]Q\P Z]URVWHP SUGX ED]\ ]áF]D ED]D
HPLWHU L ED]D NROHNWRU ]RVWDM VLOQLH VSRODU\]RZDQH Z NLHUXQNX
SU]HZRG]HQLD]D (UyGáDQDSLFLRZHJRNRV]WHPSUGXNROHNWRUD3UG
HPLWHUD
i
i
i
E
C
B
= +
MHVWZ\PXV]RQ\SU]H]REZyGUH]RQDQVRZ\]DWHP
JG\
i
B
JZDáWRZQLH URQLH WR
i
C
PDOHMH 3UDFD Z VWDQLH VLOQLH
SU]HZ]EXG]RQ\P QLH MHVW NRU]\VWQD SRQLHZD * SRZRGXMH ]PQLHMV]HQLH
PRF\Z\MFLRZHMLVSUDZQRFLRUD]Z]URVWPRF\ZHMFLRZHM
5yZQLH* SUDFD Z VWDQLH QLHGRZ]EXG]RQ\P FKDUDNWHU\]XMH VL
]PQLHMV]HQLHP PRF\ Z\MFLRZHM L VSUDZQRFL XNáDGX SRQLHZD*
JZDáWRZQHPX ]PQLHMV]HQLX XOHJD ZVSyáF]\QQLN Z\NRU]\VWDQLD
QDSLFLRZHJR
ξ
E=DNUHVUHGQLFKF]VWRWOLZRFL
ω
ω
β
< <<
Ω
7
: UR]SDWU\ZDQ\P ]DNUHVLH F]VWRWOLZRFL RJUDQLF]HQLH
PLQLPDOQHJR QDSLFLD
U
CE min
VWDQRZL W]Z SDUDPHWU\F]QD OLQLD
JUDQLF]QD3U\V:REV]DU]HÄSDUDPHWU\F]Q\P´F]\OLQDOHZRRG
FKDUDNWHU\VW\NL JUDQLF]QHM * F]VWRWOLZRü JUDQLF]QD WUDQ]\VWRUD
ω
T
PDOHMH C
jc
URQLHSRQDGWRSDUDPHWU\WHVZVLOQ\PVWRSQLXQLHOLQLRZH
L X]DOH*QLRQH ]DUyZQR RG SUGX MDN L RG QDSLFLD NROHNWRUD RSUyF]
WHJRVSRGF]DVSUDF\IXQNFMDPLF]DVX
: REV]DU]H DNW\ZQHM SUDF\ WUDQ]\VWRUD ]DF]\QD RGJU\ZDü URO
VNRF]RQ\ F]DV SU]HORWX QRQLNyZ SUGX SU]H] ED] (IHNW\ZQH
QDSLFLH HIHNW\ZQLH VWDUXMFH SUDF WUDQ]\VWRUD MHVW RSy(QLRQH
Z]JOGHP ]HZQWU]QHJR QDSLFLD VWHUXMFHJR RUD] MHVW ]QLHNV]WDáFRQH
ZVNXWHN FDáNXMFHJR G]LDáDQLD XUHGQLRQHM SRMHPQRFL G\IX]\MQHM
(IHNWHPW\FK]MDZLVNMHVW
−
RSy(QLHQLH SUGX NROHNWRUD
i
C
L QDSLFLD
u
B E
'
QD ]áF]X ED]D
ZHZQWU]QDHPLWHUWUDQ]\VWRUDZVWRVXQNXGRQDSLFLDVWHUXMFHJR
u
BE
−
]ZLNV]HQLHNWDSU]HSá\ZX
Θ
SUGXNROHNWRUDZVWRVXQNXGRMHJR
ZDUWRFL
Θ
SU]\PF]
−
]PQLHMV]HQLH ZDUWRFL V]F]\WRZHM
I
cm
LPSXOVX SUGX NROHNWRUD Z
SRUyZQDQLXGRMHJRZDUWRFLSU]\PF]
−
Z]URVWDPSOLWXG\LF]DVXWUZDQLDLPSXOVXSUGXED]\
3XOVDFMDJUDQLF]QD
ω
g
SRZ\*HMNWyUHMF]DVSU]HORWXQRQLNyZRGJU\ZD
LVWRWQ URO MHVW Z SU]\EOL*HQLX RNUHORQD VWDá F]DVRZ REZRGX
ZHMFLRZHJRWUDQ]\VWRUDZVWDQLHDNW\ZQ\PW]Q
ω
ω
g
bb
e
T
eb
bb
r C
r
r
≈
=
1
'
'
'
Ω
W
Θ
Θ Θ
+
RS
Θ
RS
Θ
L
&
X
L
%(
&
J
Ω
ω
=
L
&
J
Ω
ω
=
L
&
J
Ω
ω
=
L
&
J
Ω
ω
=
Ω
W
Θ
Θ
L
%
X
%(
L
%
J
Ω
ω
≥
L
%
J
Ω
ω
=
L
%
J
Ω
ω
=
L
%
J
Ω
ω
≈
L
%
J
Ω
ω
=
5\V3U]HELHJLF]DVRZHQDSLüLSUGyZWUDQ]\VWRUDSUDFXMFHJRZNODVLH&SU]\
Uy*Q\FKF]VWRWOLZRFLDFKURERF]\FK
1DU\VSU]HGVWDZLRQRSRJOGRZRSU]HELHJL
( )
X
W
%(
Ω
( )
L
W
&
Ω
RUD]
( )
L
%
W
Ω
GODNLONXZDUWRFLSDUDPHWUX
Ω
ω
J
F=DNUHVZLHONLFKF]VWRWOLZRFL
Ω
<
ω
7
: W\P ]DNUHVLH RERN Z\*HM Z\PLHQLRQ\FK HIHNWyZ ]DF]\QDM
RGJU\ZDü URO LQQH QS LQGXNF\MQRü HPLWHUD 3RZRGXMH WR GDOV]H
]PQLHMV]HQLHZDUWRFLV]F]\WRZHM
I
om
SUGXNROHNWRUDRUD]]ZLNV]HQLH
NWD SU]HSá\ZX Z NRQVHNZHQFML ]PQLHMV]HQLH PRF\ Z\M FLRZHM
VSUDZQRFLRUD]Z]URVWPRF\ZHMFLRZHM
52=:,=$1,$8.à$'2:(:=0$&1,$&=<./$6<&
1DU\VXQNXSU]HGVWDZLRQRSU]\NáDG\UR]ZL]DXNáDGRZ\FK
UH]RQDQVRZ\FKZ]PDFQLDF]\PRF\NODV\&
5
%
+
8
&&
a)
&
%
5
/
/
&
&
E
'
Z F]
á
5
%
+
8
&&
&
%
5
/
/
&
&
E
'
Z F]
á
b)
5
/
&
%
5
%
&
%
5
%
&
/
'
Z F]
á
&
E
+
8
&&
c)
5\V 5H]RQDQVRZH Z]PDFQLDF]H NODV\ & D Z]PDFQLDF] Z NRQILJXUDFML 2(
EZ]PDFQLDF]ZNRQILJXUDFML2%FZ]PDFQLDF]SU]HFLZVREQ\
: SU]HGVWDZLRQ\FK XNáDGDFK ]DVWRVRZDQR SRODU\]DFM
G\QDPLF]Q UHDOL]RZDQ SU]H] HOHPHQW\
R
C
B
B
,
:]PDFQLDF] Z
NRQILJXUDFML 2% SRVLDGD PQLHMV]H ]DVWRVRZDQLH ]H Z]JOGX QD MHJR
PQLHMV]H Z]PRFQLHQLH PRF\ -HGQDN*H XNáDG 2% SRVLDGD OHSV]H
ZáDFLZRFLF]VWRWOLZRFLRZHLSR]ZDODQDX]\VNDQLHQLHFRZLNV]HM
PRF\Z\MFLRZHMZ]DNUHVLHUHGQLFKF]VWRWOLZRFL
5(=21$162:( :=0$&1,$&=( 02&< :&= ./$6< '
,./$6<(
2JyOQHZáDFLZRFLZ]PDFQLDF]\ZF]NODV\'LNODV\(
:]PDFQLDF]H PRF\ NODV\ ' L NODV\ ( Z F] Z\NRU]\VWXM
LPSXOVRZ SUDF HOHPHQWyZ DNW\ZQ\FK SU]HáF]DQ\FK MDNR NOXF]H ]
GX*F]VWRWOLZRFLRGSHáQHJR]DáF]HQLDGRZ\áF]HQLD7DNLURG]DM
SUDF\ HOHPHQWyZ DNW\ZQ\FK SR]ZDOD QD X]\VNDQLH GX*HM VSUDZQRFL
HQHUJHW\F]QHMZ]PDFQLDF]\NODV\'L( NWyUD Z SU]\SDGNX LGHDOQ\FK
NOXF]\ Z\QRVL 0DNV\PDOQD PRF REFL*HQLD MHVW OLPLWRZDQD
SU]H] JUDQLF]QH ZDUWRFL SUGyZ L QDSLü ]DVWRVRZDQ\FK HOHPHQWyZ
DNW\ZQ\FKDQLHSU]H]LFKGRSXV]F]DOQPRFVWUDW6WUDW\PRF\ZWHJR
W\SXZ]PDFQLDF]DFKVNáDGDMVLJáyZQLH]GZyFKVNáDGQLNyZ
−
VWUDW\ PRF\ ]ZL]DQH ] QDSLFLHP QD ]DáF]RQ\P WUDQ]\VWRU]H
WUDQ]\VWRU QLH MHVW LGHDOQ\P NOXF]HP L Z VWDQLH ]DáF]HQLD
Z\VWSXMHQDQLPQLHZLHONLHQDSLFLH
−
VWUDW\PRF\]ZL]DQH]SURFHVHPSU]HáF]DQLDWUDQ]\VWRUDJG\MHJR
SXQNW SUDF\ SU]HPLHV]F]D VL SU]H] REV]DU DNW\ZQ\ L MHGQRF]HQLH
PRJZ\VWSLüGX*HFKZLORZHZDUWRFLQDSLFLDLSUGXWUDQ]\VWRUD
2 HIHNW\ZQRFL W\FK XNáDGyZ Z GX*\P VWRSQLX GHF\GXMH
ZáDFLZ\ GREyU HOHPHQWyZ ]H Z]JOGX QD V]\ENRü LFK G]LDáDQLD MDN
UyZQLH* ]DSURMHNWRZDQLH RGSRZLHGQLFK XNáDGyZ VWHURZDQLD
WUDQ]\VWRUDPL Z FHOX ]PLQLPDOL]RZDQLD VWUDW PRF\ ]ZL]DQ\FK ]
SURFHVDPL SU]HáF]DQLD : SU]HFLZQ\P UD]LH VNáDGQLN WHQ E\áE\
GRPLQXMF\ZRJyOQ\PELODQVLHVWUDWPRF\
5H]RQDQVRZHZ]PDFQLDF]HNODV\'LNODV\(RPRF\Z\MFLRZHM
GRNLONXVHW>:@LF]VWRWOLZRFLNLONXQDVWX>0+]@VUHDOL]RZDQHSU]\
X*\FLX ELSRODUQ\FK WUDQ]\VWRUyZ PRF\ ZF] D Z V]F]HJyOQRFL SU]\
X*\FLX SRORZ\FK WUDQ]\VWRUyZ PRF\ W\SX 9026 *áyZQLH G]LNL
ZáDFLZRFLRPWUDQ]\VWRUyZ9026SDWU]UR]G]LDáZ]PDFQLDF]H
NODV\'L(]QDOD]á\EDUG]RV]HURNLH]DVWRVRZDQLHZSUDNW\FHLVWDá\VL
JáyZQ\P VSRVREHP UHDOL]DFML UH]RQDQVRZ\FK Z]PDFQLDF]\ PRF\ ZH
ZVSRPQLDQ\FKZ\*HM]DNUHVDFKPRF\LF]VWRWOLZRFL
:SRUyZQDQLXGRZ]PDFQLDF]\NODV\&Z]PDFQLDF]HNODV\'L
NODV\(FKDUDNWHU\]XMVLQLHW\ONRZLNV]VSUDZQRFLHQHUJHW\F]QL
OHSV]\P Z\NRU]\VWDQLHP QDSLFLRZ\P L SUGRZ\P HOHPHQWyZ
DNW\ZQ\FK SR]ZDOD WR QD X]\VNDQLH GX*R ZLNV]HM PRF\ Z\MFLRZHM
SU]\W\PVDP\PHOHPHQFLHDNW\ZQ\PDOHWDN*HPQLHMV]]DZDUWRFL
Z\*V]\FKKDUPRQLF]Q\FKZV\JQDOHZ\MFLRZ\P
,VWQLHMHZLHOHRGPLDQUR]ZL]DXNáDGRZ\FKZ]PDFQLDF]\NODV\
'L(ZF]1DV]F]HJyOQHZ\Uy*QLHQLH]DVáXJXMUR]ZL]DQLDW\SX=96
DQJ ]HUR YROWDJH VZLWFKLQJ : XNáDGDFK W\FK SU]HáF]DQLH
HOHPHQWyZDNW\ZQ\FKRGE\ZDVLSU]\]HURZ\PQDSLFLX3R]ZDODWR
QD PLQLPDOL]DFM VWUDW PRF\ Z SURFHVDFK SU]HáF]DQLD 1DV]H
UR]ZD*DQLD RJUDQLF]\P\ MHG\QLH GR NUyWNLHJR SU]HGVWDZLHQLD RJyOQHM
]DVDG\ G]LDáDQLD Z]PDFQLDF]\ NODV\ ' ] NRPXWDFM QDSLFLD OXE ]
NRPXWDFM SUGX >@ RUD] QLHV\PDWU\F]QHJR Z]PDFQLDF]D NODV\ (
>@
5H]RQDQVRZHZ]PDFQLDF]HPRF\ZF]NODV\'
,VWQLHM GZD SRGVWDZRZH XNáDG\ Z]PDFQLDF]\ PRF\ NODV\ '
ZF]6WRXNáDG\]NRPXWDFMQDSLFLDOXE]NRPXWDFMSUGX
5
/V
X
LQ
Ω
W
7 2
L
'1
X
'61
0
0
0
7
Ω
W
8
'6 PLQ
Ω
W
L
'2
X
'6 2
0
0
Ω
W
Ω
W
,
P
8
''
0
Ω
W
8
'6 PLQ
L
/
&
/
X
'61
X
'6 2
+
8
''
L
'1
L
'2
X
LQ
X
/V
8
'6 PLQ
,
P
,
P
8
'6 PLQ
8
''
X
LQ
M
1
M
2
L
/
b)
a)
5\V :]PDFQLDF] NODV\ ' ] NRPXWDFM QDSLFLD D VWRSLH NRFRZ\
ESU]HELHJLF]DVRZHZXNáDG]LH
8SURV]F]RQ\VFKHPDWLGHRZ\VWRSQLDNRFRZHJRZ]PDFQLDF]DNODV\'
]NRPXWDFMQDSLFLDSU]HGVWDZLRQRQDU\VDQDWRPLDVWLGHDOL]RZDQH
SU]HELHJL F]DVRZH QDSLü L SUGyZ Z W\P XNáDG]LH SU]\ SRPLQLFLX
HIHNWyZ ]ZL]DQ\FK ] SURFHVDPL SU]HáF]DQLD WUDQ]\VWRUyZ
SU]HGVWDZLRQRQDU\VE
: F]DVLH WUZDQLD XMHPQHJR QDSLFLD
u
in
] XNáDGX VWHUXMFHJR
WUDQ]\VWRU 0
]RVWDMH ]DáF]RQ\ D WUDQ]\VWRU 0
Z\áF]RQ\ 1DSLFLH
U
DD
SRPQLHMV]RQH R QLHZLHONL VSDGHN QDSL FLD
U
DS min
QD
SU]HZRG]F\P WUDQ]\VWRU]H 0
]RVWDMH SRGDQH QD V]HUHJRZ\ REZyG
UH]RQDQVRZ\
R C L
Ls
.
:F]DVLHWUZDQLDGRGDWQLHJRSyáRNUHVXQDSLFLD
VWHUXMFHJR
u
in
QDVWSXMH Z\áF]HQLH WUDQ]\VWRUD 0
RUD] ]DáF]HQLH
WUDQ]\VWRUD0
:Z\QLNXWDNLHMSUDF\WUDQ]\VWRUyZV]HUHJRZ\REZyG
UH]RQDQVRZ\]RVWDMHSREXG]RQ\QDSLFLHPZNV]WDáFLHV\PHWU\F]QHMIDOL
SURVWRNWQHM R ZDUWRFL PLG]\V]F]\WRZHM
U
U
DD
DS
−
2
min
6]HUHJRZ\
REZyGUH]RQDQVRZ\GRVWURMRQ\GRF]VWRWOLZRFLV\JQDáXVWHUXMFHJR
Z\G]LHOD SLHUZV] KDUPRQLF]Q ] SU]HELHJX SURVWRNWQHJR 3U]H]
WUDQ]\VWRU\0
L0
SU]HSá\ZDSUGRNV]WDáFLHSRáyZHNVLQXVRLG\RUD]
R ZDUWRFL PDNV\PDOQHM RJUDQLF]RQHM UH]\VWDQFM REFL*HQLD
R
Ls
1DSLFLD SUG\ L PRFH Z XNáDG]LH SU]\ SRPLQLFLX HIHNWyZ
SU]HáF]DQLD RNUHORQH V Z SU]\EOL*HQLX SURVW\PL ]DOH*QRFLDPL
Z\QLNDMF\PL]UR]NáDGXSU]HELHJXSURVWRNWQHJRZV]HUHJ)RXULHUD
−
1DSLFLHLSUGZREFL*HQLX
(
)
X
8
8
W
/
''
'6
=
−
2
2
PLQ
π
VLQ
Ω
(
)
L
8
8
5
W
/
''
'6
/V
=
−
2
2
PLQ
π
VLQ
Ω
−
0RFX*\WHF]QD
(
)
P
U
U
R
L
DD
DS
Ls
=
−
2
2
2
2
min
π
−
UHGQLDZDUWRüSUGXGRVWDUF]RQHJR]]DVLODF]D
(
)
I
U
U
R
DD
DD
DS
Ls
=
−
2
2
2
min
π
−
0RFGRVWDUF]RQD]]DVLODF]D
(
)
P
U
U
U
R
D
DD
DS
DD
LS
=
−
2
2
2
min
π
−
6SUDZQRüZ]PDFQLDF]D
η =
= −
P
P
U
U
L
D
DS
DD
1
2
min
=]DOH*QRFLZ\QLND*HVSUDZQRüZ]PDFQLDF]DNODV\']
NRPXWDFM QDSLFLD ]DOH*\ W\ONR RG VWRVXQNX QDSLFLD
U
DS min
GR
QDSLFLD]DVLODQLD
U
DD
1DSLFLH
U
DS min
QDSU]HZRG]F\PWUDQ]\VWRU]H
Z REV]DU]H OLQLRZ\P MHVW GOD ZVSyáF]HVQ\FK WUDQ]\VWRUyZ 9026
EDUG]R PDáH GODWHJR VSUDZQRü HQHUJHW\F]QD Z]PDFQLDF]D PR*H E\ü
GX*D -HGQDN SU]\ GX*\FK F]VWRWOLZRFLDFK SU]HáF]DQLD QLH PR*QD
SRPLQü HIHNWyZ SU]HáF]DQLD WUDQ]\VWRUyZ NWyUH WR HIHNW\ SRZRGXM
GRGDWNRZHVWUDW\HQHUJLLREQL*DMFHVSUDZQRüZ]PDFQLDF]D
8SURV]F]RQ\ VFKHPDW LGHRZ\ VWRSQLD NRFRZHJR Z]PDFQLDF]D
NODV\ ' Z F] ] NRPXWDFM SUGX SU]HGVWDZLRQR QD U\VD ]D
LGHDOL]RZDQH SU]HELHJL F]DVRZH QDSLü L SUGyZ Z XNáDG]LH SU]\
SRPLQLFLXHIHNWyZSU]HáF]DQLDQDU\VE
5
/U
X
LQ
Ω
W
7 2
L
'1
X
'61
0
0
0
7
Ω
W
8
'6 PLQ
(
)
π
8
8
''
'6
−
PLQ
Ω
W
L
'2
X
'62
0
0
Ω
W
(
)
π
8
8
''
'6
−
PLQ
Ω
W
,
''
,
''
0
Ω
W
8
'6 PLQ
(
)
π
8
8
''
'6
−
PLQ
X
/
&
/
/
1
X
'61
X
'6 2
+
8
''
,
''
L
'1
L
'2
X
LQ
X
LQ
X
/
M
1
M
2
D
E
5\V :]PDFQLDF] NODV\ ' ZF] ] NRPXWDFM SUGX D VWRSLH NRFRZ\
ESU]HELHJLF]DVRZHZXNáDG]LH
:F]DVLHWUZDQLDGRGDWQLHJRSyáRNUHVXQDSLFLDVWHUXMFHJR
u
in
]RVWDMH ]DáF]RQ\ WUDQ]\VWRU 0
D Z\áF]RQ\ 0
3U]H] ]DáF]RQ\
WUDQ]\VWRU 0
Sá\QLH SUG VWDá\ Z\PXV]RQ\ SU]H] GáDZLN /
ZLHONLHM
F]VWRWOLZRFL R GX*HM LQGXNF\MQRFL : F]DVLH WUZDQLD XMHPQHJR
SyáRNUHVX QDSLFLD VWHUXMFHJR WUDQ]\VWRU 0
]RVWDMH Z\áF]RQ\ D
WUDQ]\VWRU 0
]RVWDMH ]DáF]RQ\ SU]HMPXMF SU]HZRG]HQLH SUGX ]
]DVLODF]D Sá\QFHJR SU]H] LQGXNF\MQRü
L
1
: Z\QLNX WDNLHJR
VWHURZDQLDUyZQROHJá\REZyGUH]RQDQVRZ\]RVWDMHSREXG]RQ\SU GHP
R NV]WDáFLH V\PHWU\F]QHM IDOL SURVWRNWQHM L QD UH]\VWDQFML REFL*HQLD
RWU]\PXMH VL SU]HELHJ VLQXVRLGDOQ\ 1DSLFLH GUHQ (UyGáR
u
DS
WUDQ]\VWRUDRGFLWHJRQLHMHVWVWDáHOHF]PDNV]WDáWSRáyZNLVLQXVRLG\R
DPSOLWXG]LH
(
)
π
U
U
DD
DS
−
min
UHGQLD ZDUWRü QDSLFLD QD
LQGXNF\MQRFL
L
1
]D MHGHQ RNUHV Z\QRVL ]HUR ]D FKZLORZD ZDUWRü
QDSLFLD QD LQGXNF\MQRFL
L
1
MHVW Uy*QLF PRGXáX QDSLFLD
VLQXVRLGDOQHJRLVWDáHJRQDSLFLD]DVLODMFHJR
U
DD
1DSLFLD SUG\ L PRFH Z XNáDG]LH SU]\ SRPLQLFLX HIHNWyZ
SU]HáF]DQLDRNUHODMQDVWSXMFH]DOH*QRFL
−
1DSLFLHLSUGZUH]\VWDQFMLREFL*HQLD
(
)
X
8
8
W
/
''
'6
=
−
π
PLQ
VLQ
Ω
(
)
L
8
8
5
W
/
''
'6
/U
=
−
π
PLQ
VLQ
Ω
−
0RFX*\WHF]QD
(
)
P
U
U
R
L
DD
DS
Lr
=
−
π
2
2
2
min
−
UHGQLSUGGRVWDUF]RQ\]]DVLODF]DSUGGáDZLNDZF]
/
1
(
)
I
U
U
R
DD
DD
DS
Lr
=
−
π
2
2
min
−
0RFGRVWDUF]RQD]]DVLODF]D
(
)
P
U
U
U
R
D
DD
DD
DS
Lr
=
−
π
2
2
min
−
1DSLFLHQDGáDZLNXZF]
L
1
(
)
u
U
U
t
DD
DS
1
2
1
=
−
−
min
sin
π
Ω
−
6SUDZQRüZ]PDFQLDF]D
η =
= −
P
P
U
U
L
D
DS
DD
1
min
=SRUyZQDQLD]DOH*QRFLLZ\QLND*HSU]\SRPLQLFLX
SURFHVyZSU]HáF]DQLDVSUDZQRüZ]PDFQLDF]D]NRPXWDFMSUGXMHVW
QLHFR ZLNV]D QL* Z]PDFQLDF]D ] NRPXWDFM QDSLFLD : XNáDG]LH ]
NRPXWDFMSUGXQDRGFLW\PWUDQ]\VWRU]HZ\VWSXMHGRüGX*DZDUWRü
V]F]\WRZDQDSLFLDUyZQD
(
)
π
U
U
DD
DS
−
min
1LHV\PHWU\F]Q\Z]PDFQLDF]NODV\(ZF]
: SUDNW\FH VSUDZQRü HQHUJHW\F]QD RSLVDQ\FK ZF]HQLHM
Z]PDFQLDF]\ NODV\ ' VLJD QLHFR SRZ\*HM SU]\ PDNV\PDOQHM
F]VWRWOLZRFL SUDF\ QLH SU]HNUDF]DMFHM QD RJyá NLONXQDVWX 0+]
'DOV]H]ZLNV]HQLHVSUDZQRFLHQHUJHW\F]QHMRUD]UR]V]HU]HQLH]DNUHVX
F]VWRWOLZRFLRZHJRSUDF\X]\VNDQRZHZ]PDFQLDF]DFKNODV\(ZF]
G]LNL Z\HOLPLQRZDQLX VWUDW PRF\ ]ZL]DQ\FK ] SURFHVDPL ]DáF]DQLD
WUDQ]\VWRUyZ Z]PDFQLDF]H NODV\ ( ] UyZQROHJá\P L V]HUHJRZ\P
NRQGHQVDWRUHP OXE Z\áF]DQLD WUDQ]\VWRUD Z]PDFQLDF]H NODV\ ( ]
V]HUHJRZLQGXNF\MQRFL:DGQLHV\PHWU\F]Q\FKZ]PDFQLDF]\NODV\
(MHVWGRüGX*D]DZDUWRüKDUPRQLF]Q\FKZV\JQDOHZ\MFLRZ\PJG\
GREURü UH]RQDQVRZHJR REZRGX REFL*DMFHJR MHVW PDáD FR MHVW
SRZRGRZDQH QLHV\PHWU\F]Q\P SU]HáDGRZ\ZDQLHP WHJR REZRGX :
FHOX Z\HOLPLQRZDQLD WHM ZDG\ UHDOL]RZDQH V UyZQLH* SU]HFLZVREQH
Z]PDFQLDF]HNODV\(ZF]
1DV]H
UR]ZD*DQLD
RJUDQLF]\P\
GR
QLHV\PHWU\F]QHJR
Z]PDFQLDF]D NODV\ ( ] UyZQROHJá\P NRQGHQVDWRUHP NWyUHJR
XSURV]F]RQ\ VFKHPDW LGHRZ\ ZUD] ] MHJR PRGHOHP SU]HGVWDZLRQR QD
U\VD1DU\VESU]HGVWDZLRQRLGHDOL]RZDQHSU]HELHJLF]DVRZH
ZXNáDG]LHRGSRZLDGDMFHRSW\PDOQ\PZDUXQNRPSUDF\Z]PDFQLDF]D
SU]\NWyU\FK]RVWDMFDáNRZLFLHZ\HOLPLQRZDQHVWUDW\PRF\]ZL]DQH]
]DáF]DQLHPWUDQ]\VWRUD :HZ]PDFQLDF]XW\PVWUDW\PRF\]ZL]DQH]
]DáF]DQLHP WUDQ]\VWRUD ]RVWDá\ Z\HOLPLQRZDQH G]LNL RGSRZLHGQLR
GREUDQHPX REZRGRZL GUHQRZHPX NWyU\ ]DSHZQLD *H SU]HMFLH
WUDQ]\VWRUD ]H VWDQX RGFLFLD GR VWDQX ]DáF]HQLD RGE\ZD VL ]
SRPLQLFLHPVWDQXDNW\ZQHJR6SURZDG]HQLHW\FKVWUDWGR]HUDZ\PDJD
MHGQRF]HVQHJRVSHáQLHQLDZDUXQNyZDE\QDSLFLHGUHQX
u
DS
LSRFKRGQD
WHJRQDSLFLDZFKZLOLZáF]DQLDWUDQ]\VWRUDE\á\UyZQH]HUX>@
( )
(
)
( )
( )
X
W
8
Q
GX
W
G
W
'6
'6
'6
W
Q
ω
π
ω
ω
π
=
=
=
=
2
0
0
2
Q
Ω
-DN SRND]DQR QD U\VF VSHáQLHQLH SRZ\*V]\FK ZDUXQNyZ
SRZRGXMH *H Z FKZLOL ]DáF]DQLD WUDQ]\VWRUD SUG GUHQX ]DF]\QD
áDJRGQLHQDUDVWDüRG]HUDSU]\QDSLFLX
u
DS
=
0
FRZHIHNFLHSURZDG]L
GR Z\HOLPLQRZDQLD SURFHVX SU]HMFLRZHJR $E\ X]\VNDü QDMZLNV]
PRFZ\MFLRZZ]PDFQLDF]DZVSyáF]\QQLNZ\SHáQLHQLDSURVWRNWQHJR
SU]HELHJXVWHUXMFHJRSRZLQLHQZ\QRVLü
X
'6
X
LQ
,
''
L
6
L
'
&
1
&
2
/
2
5
/
/
G á
' Z F]
á
+
8
''
X
5
X
;
X
6
&
1
5
/
X
5
X
;
X
6
X
'6
8
&&
L
'
,
''
/
G á
L
6
&
2
/
2
;
/
=
ω
π
2
π
X
LQ
.OXF]
]ZDUW\
.OXF]
UR]ZDUW\
ω
W
π
2
π
ω
W
,
''
π
2
π
ω
W
π
2
π
ω
W
L
6
L
'
π
2
π
ω
W
π
2
π
ω
W
8
'6 PD[
X
'6
0
0
0
0
0
0
,
' PD[
L
&1
a)
b)
c)
L
&1
L
&1
U
'6
5\V 1LHV\PHWU\F]Q\ Z]PDFQLDF] NODV\ ( ] UyZQROHJá\P NRQGHQVDWRUHP
DVWRSLHNRFRZ\EXSURV]F]RQ\PRGHOZ]PDFQLDF]DFLGHDOL]RZDQH
SU]HELHJLF]DVRZHZXNáDG]LH
3U]HSURZDG]RQD DQDOL]D WHRUHW\F]QD XNáDGX Z SUDFDFK > @
SURZDG]L GR QDVWSXMF\FK ]ZL]NyZ PLG]\ ZDUWRFLDPL HOHPHQWyZ
REZRGXGUHQRZHJRSU]\NWyU\FKVSHáQLRQHVZDUXQNL
&
5
4
4
/
/
/
1
2
1
5 447
1
0 81
4
=
+
+
ω
C
Q
R
Q
L
L
L
2
1
1
111
1 7879
=
+
−
ω
,
,
L
Q R
L
2
=
ω
JG]LH Q
L
R
L
= ω
2
GREURüV]HUHJRZHJRREZRGXUH]RQDQVRZHJR
-DN ZLGDü ] SRZ\*V]\FK ]DOH*QRFL REZyG NROHNWRURZ\
GUHQRZ\GODF]VWRWOLZRFLSUDF\SRVLDGDFKDUDNWHULQGXNF\MQ\WM
X
L
X
C
L
C
=
>
=
ω
ω
2
2
1
: FHOX RNUHOHQLD SROD FKDUDNWHU\VW\N HOHPHQWX DNW\ZQHJR
SUDFXMFHJRMDNRNOXF]SU]\' PR*HP\Z\NRU]\VWDüQDVWSXMFH
]ZL]NL
−
PDNV\PDOQ\SUGWUDQ]\VWRUD
I
I
I
D
DD
DD
max
,
=
+ +
≈
π
2
4
2
1
2 8611
−
PDNV\PDOQHQDSLFLHQDWUDQ]\VWRU]H
U
U
U
DS
DD
DS
max
min
,
,
=
−
3 562
2 562
JG]LH
I
DD
SUG]DVLODQLD
U
DS min
VSDGHNQDSLFLDQD]DáF]RQ\PWUDQ]\VWRU]H
3DUDPHWU\Z\MFLRZHZ]PDFQLDF]DRNUHODM]DOH*QRFL
−
DPSOLWXGDQDSLFLDZ\MFLRZHJR
(
)
U
U
U
Lm
DD
DS
=
+
−
4
4
2
π
min
−
DPSOLWXGDSUGXZ\MFLRZHJR
I
U
R
U
U
R
Lm
Lm
L
DD
DS
L
=
=
+
−
4
4
2
π
min
−
PRFZ\MFLRZD
P
U
I
U
U
R
L
Lm
Lm
DD
DS
L
=
=
+
−
2
8
4
2
2
π
(
)
min
−
SUG]DVLODQLD
I
U
U
R U
DD
DD
DS
L
DD
=
−
+
8
4
2
2
(
)
(
)
min
π
η
−
VSUDZQRüZ]PDFQLDF]D
( )
( )
( )
η
π
π
π
=
−
−
+ −
−
1
2
3
1
2
3
1
2
2
2 3
A
U
U
A
A
A
DS
DD
min
/
JG]LH
A
Q
f t
f
= +
+
1
0 82
1 1525
,
,
D
t
f
F]DVRSDGDQLDSUGXGUHQXSU]\Z\áF]DQLXWUDQ]\VWRUD
−
PRF]DVLODQLD
P
P
U
U
R
D
L
DD
DS
L
=
=
+
−
η π
η
8
4
2
2
(
)
min
2JUDQLF]HQLDF]VWRWOLZRFLRZHZ]PDFQLDF]DNODV\(ZL*VL]
SRMHPQRFL Z\MFLRZ WUDQ]\VWRUD NWyUD áF]QLH ] SRMHPQRFL
GáDZLND ]DVLODMFHJR L SRMHPQRFL PRQWD*X WZRU]\ SHZQ PLQLPDOQ
SRMHPQRü
C
1min
SU]\ NWyUHM PR*OLZH MHVW MHV]F]H VNRQVWUXRZDQLH
REZRGX GUHQRZHJR R RSW\PDOQ\FK ZDUWRFLDFK HOHPHQWyZ
0DNV\PDOQD F]VWRWOLZRü SUDF\ Z]PDFQLDF]D NODV\ ( MHVW RNUHORQD
Z]RUHP>@
(
)
f
R C
R C
L
L
max
min
min
,
=
+
≈
4
4
0 02922
2
2
1
1
π π
: SUDNW\FH VSUDZQRü Z]PDFQLDF]D NODV\ ( Z F] PR*H RVLJDü
ZDUWRü QDZHW ']LNL VZ\P ZáDFLZRFLRP UH]RQDQVRZH
Z\VRNRVSUDZQH Z]PDFQLDF]H NODV\ ' L ( Z F] V FRUD] V]HU]HM
Z\NRU]\VW\ZDQHZUy*QRURGQ\FKXU]G]HQLDFKGRSU]HWZDU]DQLDHQHUJLL
Z\VRNRVSUDZQH UH]RQDQVRZH SU]HWZRUQLFH QDSL FLD VWDáHJR
Z]PDFQLDF]H UH]RQDQVRZH Z F] SURVWRZQLNL UH]RQDQVRZH ZF]
VWHURZDQHSURVWRZQLNLV\QFKURQLF]QHZF]DZDU\MQH]DVLODF]HQDSL FLD
SU]HPLHQQHJR LWS FR SR]ZDOD QD PLQLDWXU\]DFM L ]ZLNV]HQLH
QLH]DZRGQRFLW\FKXU]G]H