*(1(5$725<'5*$ 6,1862,'$/1<&+
:352:$'=(1,(
*HQHUDWRUDPL V XNáDG\ HOHNWURQLF]QH NWyUH Z NRQWURORZDQ\
VSRVyE SU]HWZDU]DM HQHUJL (UyGáD ]DVLODQLD Z GUJDQLD HOHNWU\F]QH
RRNUHORQ\PNV]WDáFLHF] VWRWOLZRFLLPRF\DQDVW SQLHSU]HND]XM
MH GR REFL*HQLD 3RGVWDZRZ\P URG]DMHP GUJD IL]\F]Q\FK
Z\VW SXMF\FK Z SU]\URG]LH V GUJDQLD VLQXVRLGDOQH NWyUH PRJ E\ü
UyZQLH* LQWHUSUHWRZDQH MDNR VNáDGRZH KDUPRQLF]QH ]JRGQLH
]UR]NáDGHP )RXULHUD ZV]HONLFK GUJD IL]\F]Q\FK 'UyGáD GUJD
VLQXVRLGDOQ\FK ]QDMGXM SRZV]HFKQH ]DVWRVRZDQLH Z XU]G]HQLDFK
QDGDZF]R RGELRUF]\FK Z Uy*QHJR URG]DMX DSDUDWXU]H NRQWUROQR
SRPLDURZHMDWDN*HZV\VWHPDFKF\IURZ\FKGRJHQHUDFMLZ]RUFRZ\FK
SU]HG]LDáyZF]DVX
*HQHUDWRU\ GUJD VLQXVRLGDOQ\FK PR*HP\ SRG]LHOLü QD GZLH
JUXS\]DOH*QLHRGVWDZLDQ\FKZ\PDJDFRGRVWDáRFLF] VWRWOLZRFL
LPRF\Z\MFLRZHMJHQHURZDQ\FKGUJD
−
JHQHUDWRU\ F] VWRWOLZRFL WM JHQHUDWRU\ R GX*HM VWDáRFL
F] VWRWOLZRFL
−
JHQHUDWRU\ PRF\ RG NWyU\FK Z\PDJD VL GX*HM PRF\ Z\MFLRZHM
LGX*HM VSUDZQRFL HQHUJHW\F]QHM EH] RSW\PDOL]DFML VWDáRFL
F] VWRWOLZRFL
=HZ]JO GXQDUR]ZL]DQLDXNáDGRZHLVSRVyESUDF\HOHPHQWyZ
DNW\ZQ\FKPR*QDMHSRG]LHOLüQDGZLHJUXS\
−
JHQHUDWRU\ VSU] *HQLRZH Z NWyU\FK HOHPHQW DNW\ZQ\ REM W\ MHVW
S WO GRGDWQLHJR VSU] *HQLD ]ZURWQHJR G]L NL F]HPX X]\VNXMH VL
RGWáXPLHQLHREZRGXJHQHUDF\MQHJR
−
JHQHUDWRU\ GZyMQLNRZH QD]\ZDQH UyZQLH* JHQHUDWRUDPL R XMHPQHM
UH]\VWDQFML Z NWyU\FK HOHPHQW R XMHPQHM UH]\VWDQFML OXE
NRQGXNWDQFML Z\NRU]\VW\ZDQ\ MHVW GR RGWáXPLHQLD REZRGX
JHQHUDF\MQHJR
: SUDNW\FH PDP\ GR F]\QLHQLD QDMF] FLHM ] JHQHUDWRUDPL
VSU] *HQLRZ\PL NWyUH ]H Z]JO GX PD URG]DM Z\NRU]\VW\ZDQ\FK
HOHPHQWyZZREZRG]LHJHQHUDF\MQ\PPR*HP\SRG]LHOLüQD
−
JHQHUDWRU\/&
−
JHQHUDWRU\]HVWDELOL]DFMSLH]RHOHNWU\F]QQSNZDUFRZH
−
JHQHUDWRU\5&
'ODRFHQ\VWDáRFLF] VWRWOLZRFLJHQHURZDQ\FKGUJDGHILQLXMH
VL QDVW SXMFHSDUDPHWU\
−
EH]Z]JO GQQLHVWDáRüF] VWRWOLZRFL
( )
( )
∆
f t
f t
f
=
−
0
JG]LH
f
0
F] VWRWOLZRüQDSRF]WNXREVHUZDFML
( )
f t
F] VWRWOLZRüZFKZLOLWREVHUZDFML
−
Z]JO GQQLHVWDáRüF] VWRWOLZRFL
( )
( )
δ
f t
f t
f
=
∆
0
−
VWDáRü F] VWRWOLZRFL NWyUD MHVW UHGQL Z]JO GQ\FK QLHVWDáRFL
F] VWRWOLZRFL PLHU]RQ\FK SRGF]DV F]DVX REVHUZDFML NWyU\ Z\QRVL
QDMF] FLHMGRE OXEURN
( )
∆
f
f
f
d f dt
T
o
T
0
0
1
=
∫
:VSyáF]HQLHQDMGRNáDGQLHMV]\PLZ]RUFDPLF] VWRWOLZRFLVZ]RUFH
RSDUWH QD DWRPRZ\FK L PROHNXODUQ\FK RGG]LDá\ZDQLDFK
PL
G]\F]VWHF]NRZ\FK NWyUH RVLJDM GRERZH VWDáRFL F]
VWRWOLZRFL
QDSR]LRPLH
(
)
±
−
−
−
10
10
12
14
3R]DW\PLZ]RUFDPLQDMOHSV]VWDáRFL
GáXJRWHUPLQRZ RG]QDF]DM VL JHQHUDWRU\ NZDUFRZH NWyU\FK VWDáRü
∆
I
I
Z\QRVL
(
)
±
−
−
−
D Z XNáDGDFK WHUPRVWDWRZ\FK
(
)
±
−
−
−
QDPLHVLF6WDáRüF] VWRWOLZRFLJHQHUDWRUyZ/&
]DZLHUDVL ZJUDQLFDFK
(
)
±
−
−
−
DVWDáRüJHQHUDWRUyZ5&
Z\QRVLW\SRZR
±
−
−
−
,QQ\PL SDUDPHWUDPL RNUHODMF\PL MDNRü JHQHURZDQHJR
SU]HELHJX V ]DZDUWRü KDUPRQLF]Q\FK Z QRPLQDOQ\FK ZDUXQNDFK
SUDF\]DNUHVSU]HVWUDMDQLDRUD]SR]LRP\LZLGPDIOXNWXDFMLDPSOLWXG\
LID]\
:$581.,*(1(5$&-,'5*$
*HQHUDWRU\VSU] *HQLRZH
$QDOL]D ZDUXQNyZ JHQHUDFML GRW\F]\ HWDSX SUDF\ JHQHUDWRUD WX*
SRZáF]HQLX]DVLODQLDNLHG\DPSOLWXG\V\JQDáyZVPDáHLSR]ZDODM
XNáDG WUDNWRZDü MDNR OLQLRZ\ 'ODWHJR Z SLHUZV]\P HWDSLH SUDF
JHQHUDWRUDPR*QDRSLVDüNRU]\VWDMF]OLQLRZHMWHRULLPDáRV\JQDáRZHM
VSU] *HQLD ]ZURWQHJR SR]ZDODMFHM QD Z\]QDF]HQLH ZDUXQNyZ
SRZVWDQLDGUJDRUD]ZDUWRFLF] VWRWOLZRFLGUJD8Z]JO GQLHQLH
HIHNWyZ QLHOLQLRZ\FK SURZDG]L GR QLHOLQLRZHM ZLHONRV\JQDáRZHM
WHRULL JHQHUDFML RSLVXMFHM VWDQ SU]HMFLRZ\ ]ZL]DQ\ ] QDUDVWDQLHP
LXVWDODQLHPVL GUJD
1D U\V SU]HGVWDZLRQR VFKHPDW EORNRZ\ Z]PDFQLDF]D
]GRGDWQLP VSU] *HQLHP ]ZURWQ\P Z NWyU\P Z]PRFQLHQLH
k
u
Z]PDFQLDF]D EH] VSU] *HQLD ]ZURWQHJR RUD] WUDQVPLWDQFMD
β
u
SDV\ZQHJRF]ZyUQLNDVSU] *HQLD]ZURWQHJRRNUHODM]DOH*QRFL
( )
( )
k
j
U
U
k
j
u
u
u
ω
ϕ
=
=
2
1
exp
( )
( )
β
ω
β
ϕ
β
β
u
u
j
U
U
j
=
=
2
exp
k
u
β
X
Σ
E
g
U
β
U
1
U
2
R
L
5\V6FKHPDWEORNRZ\Z]PDFQLDF]D]GRGDWQLS WOVSU] *HQLD]ZURWQHJR
7UDQVPLWDQFMDXNáDGX]HVSU] *HQLHP]ZURWQ\PZ\QRVL
( )
( )
( ) ( )
k
j
U
E
k
j
k
j
j
f
g
u
u
u
ω
ω
ω β
ω
=
=
−
2
1
8NáDGVWDMHVL QLHVWDELOQ\SU]\VSHáQLHQLXZDUXQNXQD]\ZDQHJR
ZDUXQNLHPJHQHUDFML
( ) ( )
(
)
(
)
(
)
[
]
k
j
j
k
j
k
k
j
u
u
u
u
u
u
u
u
k
ω β
ω
β
β
β
ϕ
ϕ
β
=
+
=
=
+
=
Re
Im
exp
1
:DUXQHNJHQHUDFMLPR*QDSU]HGVWDZLüZ SRVWDFL GZyFK
UyZQDQD]\ZDQ\FKRGSRZLHGQLRZDUXQNLHPDPSOLWXG\
(
)
k
k
u
u
u
u
β
β
= =
=
1
1
Re
RUD]ZDUXQNLHPID]\
(
)
(
)
Im
,
,
, ,
k
n
n
n
u
u
k
β
π
ϕ
ϕ
π
β
=
+
=
=
2
2
0 1
lub
:DUXQHN JHQHUDFML SRZLQLHQ E\ü VSHáQLRQ\ W\ONR GOD MHGQHM
RNUHORQHM F] VWRWOLZRFL =DSHZQLD VL WR ] SHZQ\P SU]\EOL*HQLHP
SU]H] ]DVWRVRZDQLH VHOHNW\ZQ\FK REZRGyZ
LC
OXE
RC
ZH
Z]PDFQLDF]XOXEF]ZyUQLNXVSU] *HQLD]ZURWQHJR
: UR]ZL]DQLDFK SUDNW\F]Q\FK ZDUXQHN SRZLQLHQ E\ü
VSHáQLRQ\]SHZQ\PQDGPLDUHPJG\*PDáH]PQLHMV]HQLHZ]PRFQLHQLD
OXEWUDQVPLWDQFML]ZURWQHMSRZRGRZDáRE\ ]HUZDQLH GUJD 6SHáQLHQLH
ZDUXQNX ] GX*\P QDGPLDUHP PR*H ] NROHL VSRZRGRZDü
SRZVWDQLH VLOQ\FK ]QLHNV]WDáFH QDSL FLD Z\MFLRZHJR ]ZL]DQ\FK
]QLHOLQLRZRFLDPLZ]PDFQLDF]D
3URFHVQDUDVWDQLDGUJDLRVLJDQLDVWDQXXVWDORQHJRJHQHUDWRUD
PR*QD SRJOGRZR SU]HGVWDZLü Z RSDUFLX R FKDUDNWHU\VW\NL
SU]HMFLRZHZ]PDFQLDF]DLWRUXVSU] *HQLD]ZURWQHJR1LHOLQLRZRFL
HOHPHQWyZ DNW\ZQ\FK SRZRGXM *H FKDUDNWHU\VW\NL SU]HMFLRZH
Z]PDFQLDF]\ V WDN*H QLHOLQLRZH L ]DOH*QH RG NODV\ SUDF\
Z]PDFQLDF]D 0R*QD SU]\Mü *H FKDUDNWHU\VW\ND SU]HMFLRZD
F]ZyUQLNDVSU] *HQLD]ZURWQHJRMHVWOLQLRZDMH*HOLMHVWRQ]UHDOL]RZDQ\
]OLQLRZ\FKHOHPHQWyZSDV\ZQ\FK
:]EXG]DQLD VL GUJD JHQHUDWRUD SU]HGVWDZLRQR QD U\V QD
NWyU\PZW\FKVDP\FKXNáDGDFKZVSyáU] GQ\FK
U
1
L
U
2
SU]HGVWDZLRQR
QLHOLQLRZH FKDUDNWHU\VW\NL Z]PDFQLDF]\ RUD] OLQLRZ FKDUDNWHU\VW\N
XNáDGXVSU] *HQLD]ZURWQHJR:LHONRFL
U
1
L
U
2
R]QDF]DMDPSOLWXG\
OXE ZDUWRFL VNXWHF]QH QDSL ü QD ZHMFLX L QD Z\MFLX Z]PDFQLDF]D
1DOH*\SU]\W\P]DXZD*\ü*HQDSL FLH
U
1
MHVWUyZQLH*QDSL FLHPQD
Z\MFLXF]ZyUQLNDVSU] *HQLD]ZURWQHJR(
)
U
U
β
=
1
1D U\VD SU]HGVWDZLRQR SRJOGRZR SURFHV QDUDVWDQLD GUJD
ZJHQHUDWRU]H Z NWyU\P FKDUDNWHU\VW\ND SU]HMFLRZD Z]PDFQLDF]D
ZSRF]WNRZ\P ]DNUHVLH MHM SU]HELHJX MHVW OLQLRZD FR RGSRZLDGD
SUDF\Z]PDFQLDF]DVHOHNW\ZQHJRZNODVLH$$%OXE%DZDUXQHN
JHQHUDFML VSHáQLRQ\ MHVW ] QDGPLDUHP WDN *H RELH FKDUDNWHU\VW\NL
SU]HFLQDM VL Z SXQNFLH 3 3R ]DáF]HQLX QDSL FLD ]DVLODQLD
LSU]\SDGNRZ\PSREXG]HQLXQSQDSL FLHPV]XPyZDPSOLWXGDGUJD
VWRSQLRZRQDUDVWDRGZDUWRFLSRF]WNRZHM
( )
U 0
GRZDUWRFLXVWDORQHM
( )
U
∞
RGSRZLDGDMFHMSXQNWRZLZNWyU\P
k
u
u
β
=
1
LGDOV]\Z]URVW
DPSOLWXG\ MHVW QLHPR*OLZ\ 7DNL VSRVyE Z]EXG]DQLD VL GUJD MHVW
QD]\ZDQ\Z]EXG]DQLHPPL NNLP
1DU\VESU]HGVWDZLRQRZ]EXG]DQLHVL JHQHUDWRUDZNWyU\P
FKDUDNWHU\VW\ND SU]HMFLRZD Z]PDFQLDF]D QDUDVWD GRSLHUR RG SHZQHM
SURJRZHM ZDUWRFL QDSL FLD
U
1
FR RGSRZLDGD SUDF\ Z]PDFQLDF]D
VHOHNW\ZQHJRZNODVLH&
3U]\ PDá\FK DPSOLWXGDFK ZDUXQHN JHQHUDFML QLH MHVW VSHáQLRQ\
JG\* k
u
u
<
1 /
β
QDWRPLDVWMHVWVSHáQLRQ\GODZL NV]\FKDPSOLWXG
-H*HOL XNáDG ]RVWDQLH VLOQLH SREXG]RQ\ QD VDP\P SRF]WNX QS SU]\
ZáF]HQLX QDSL FLD ]DVLODQLD WR DPSOLWXGD GUJD E G]LH QDUDVWDü RG
ZDUWRFL SRF]WNRZHM
( )
U 0
0
>
GR ZDUWRFL XVWDORQHM
( )
U
∞
RGSRZLDGDMFHM SXQNWRZL 3 7HQ VSRVyE Z]EXG]DQLD VL JHQHUDWRUD
MHVWQD]\ZDQ\Z]EXG]DQLHPWZDUG\P
U
2
U
2
U
2
a)
b)
c)
U
1
U
1
U
1
0
0
0
U ( )
∞
U ( )
∞
U ( )
∞
P
P
P
U
U
u
2
1
=
/
β
U
k U
u
2
1
=
U ( )
0
U ( )
0
U ( )
0
tranzystor zatkany
tranzystor przewodzi
u t
1
( )
t
0
U
BE 0
u
t
BE
( )
t
0
5\V :]EXG]DQLH VL GUJD D Z]EXG]DQLH PL NNLH E Z]EXG]DQLH WZDUGH F
Z]EXG]DQLH Z XNáDG]LH ] DXWRPDW\F]Q SRODU\]DFM REZRGX ZHMFLRZHJR
Z]PDFQLDF]D G SU]HELHJL F]DVRZH QDSL
ü u t
1
( )
RUD]
u
t
BE
( )
Z XNáDG]LH
]DXWRPDW\F]QSRODU\]DFMREZRGXZHMFLRZHJR
1DU\VFSU]HGVWDZLRQRZ]EXG]HQLHVL GUJDZJHQHUDWRU]H
ZNWyU\PZ]PDFQLDF]VHOHNW\ZQ\SUDFXMHZNODVLH&]G\QDPLF]Q
SRODU\]DFM REZRGX ZHMFLRZHJR SDWU] UR]G] 1D SRF]WNX
Z]PDFQLDF] VWDUWXMH Z NODVLH $ G]L NL F]HPX ZXNáDG]LH Z\VW SXMH
Z]EXG]DQLH PL NNLH 3RG ZSá\ZHP QDUDVWDMFHM DPSOLWXG\ GUJD
Z\WZDU]D VL URVQFH QDSL FLH SRODU\]XMFH REZyG ZHMFLRZ\
Z]PDFQLDF]DLSU]HVXZDMFHVWRSQLRZRMHJRSXQNWSUDF\GRNODV\&
U\VG
*HQHUDWRU\/&]HOHPHQWDPLRXMHPQHMUH]\VWDQFML
=DVDGD G]LDáDQLD JHQHUDWRUyZ ] HOHPHQWDPL DNW\ZQ\PL
RXMHPQHMUH]\VWDQFMLRSLHUDVL QDLFK]GROQRFLRGWáXPLDQLDVWUDWQ\FK
REZRGyZUH]RQDQVRZ\FK/&'\QDPLF]QDUH]\VWDQFMDNRQGXNWDQFMD
XMHPQD W\FK HOHPHQWyZ MHVW Z\QLNLHP LFK FKDUDNWHU\VW\N SUGRZR
QDSL FLRZ\FKNWyUHQDSHZQ\PRGFLQNXSRVLDGDMXMHPQHQDFK\OHQLH
,VWQLHMHOHPHQW\RFKDUDNWHU\VW\FHXMHPQHMUH]\VWDQFMLW\SX1
X]DOH*QLRQH QDSL FLRZR MDN QS GLRGD WXQHORZD RUD] HOHPHQW\
RFKDUDNWHU\VW\FHXMHPQHMUH]\VWDQFMLW\SX6X]DOH*QLRQHSUGRZRMDN
QSWUDQ]\VWRUODZLQRZ\OXEW\U\VWRU(OHPHQW\RFKDUDNWHU\VW\FHW\SX
1 V VWRVRZDQH GR RGWáXPLDQLD REZRGyZ UyZQROHJá\FK QDWRPLDVW
HOHPHQW\W\SX6GRRGWáXPLDQLDREZRGyZV]HUHJRZ\FK
1D U\V SU]HGVWDZLRQR SRGVWDZRZH XNáDG\ JHQHUDWRUyZ /&
]HOHPHQWDPLRXMHPQHMUH]\VWDQFMLREXW\SyZ1L6
E
0
L
C
G
R
R
R
=
1 /
N
L
I
0
C
S
R
S
I
I
U
U
g
r
u
u
=
<
1
0
/
A
P
0
B
A
P
0
B
R
S
r
u
<
0
E
0
I
0
0
0
a)
b)
G
R
R
R
=
1 /
5\V3RGVWDZRZHJHQHUDWRU\/&]HOHPHQWDPLRXMHPQHMUH]\VWDQFMLDREZyG
UyZQROHJá\ RGWáXPLDQ\ SU]H] HOHPHQW R FKDUDNWHU\VW\FH W\SX 1 E REZyG
V]HUHJRZ\RGWáXPLDQ\SU]H]HOHPHQWRFKDUDNWHU\VW\FHW\SX6
2GWáXPLHQLH REZRGyZ UH]RQDQVRZ\FK MHVW PR*OLZH MH*HOL
ZSXQNFLH
P
0
Z\]QDF]RQ\PSU]H]QDSL FLH]DVLODMFH
E
0
OXESUG
]DVLODMF\
I
0
Z\SDGNRZD NRQGXNWDQFMD REZRGX UyZQROHJáHJR
U\VD OXE Z\SDGNRZD UH]\VWDQFMD REZRGX V]HUHJRZHJR U\VE
MHVW UyZQD ]HUX 6WUDW\ PRF\ ]ZL]DQH ] NRQGXNWDQFM REZRGX
UyZQROHJáHJR
G
R
R
R
=
1 /
PRJE\üVNRPSHQVRZDQHJG\QDFK\OHQLH
RGFLQNDFKDUDNWHU\VW\NLRGSRZLDGDMFHXMHPQHMNRQGXNWDQFML
g
u
MHVW
QLHFRPQLHMV]HQL*
G
R
U\VD
3RGREQLH VWUDW\ PRF\ ]ZL]DQH ]UH]\VWDQFM REZRGX
V]HUHJRZHJR
R
S
PRJE\üVNRPSHQVRZDQHJG\QDFK\OHQLHRGFLQND
FKDUDNWHU\VW\NLRGSRZLDGDMFHXMHPQHMUH]\VWDQFML
r
u
MHVWQLHPQLHMV]H
QL*
R
S
U\VE
: SUDNW\FH HOHPHQW W\SX 1 ]DZV]H ]DZLHUD QLHZLHON
SDVR*\WQLF]SRMHPQRüUyZQROHJá C
p
]DHOHPHQWW\SX6QLHZLHON
SDVR*\WQLF]LQGXNF\MQRüV]HUHJRZ L
p
1D U\V SU]HGVWDZLRQR OLQHDU\]RZDQH VFKHPDW\ ]DVW SF]H
JHQHUDWRUyZ]HOHPHQWDPLRXMHPQHMUH]\VWDQFML]FKDUDNWHU\VW\NDPL
W\SX1L6
a)
b)
L
L
C
C
p
C
G
R
g
u
R
S
L
p
r
u
5\V/LQHDU\]RZDQHVFKHPDW\]DVW SF]HJHQHUDWRUyZ]HOHPHQWDPLRXMHPQHM
UH]\VWDQFML]FKDUDNWHU\VW\NDPLDW\SX1EW\SX6
1D SRGVWDZLH XNáDGyZ ]OLQHDU\]RZDQ\FK PR*HP\ RNUHOLü
JUDQLF]QHZDUXQNLJHQHUDFMLSU]H]SRUyZQDQLHZ\SDGNRZHMDGPLWDQFML
REZRGX UyZQROHJáHJR OXE Z\SDGNRZHM LPSHGDQFML REZRGX
V]HUHJRZHJRGR]HUD
−
GODXNáDGX]HOHPHQWHPW\SX1
Y
G
g
j
L
j
C
j
C
R
u
p
=
+
+
+
+
=
1
0
ω
ω
ω
D
−
GODXNáDGX]HOHPHQWHPW\SX6
Z
R
r
j
C
j
L
j
L
S
u
p
=
+ +
+
+
=
1
0
ω
ω
ω
E
6WGRWU]\PXMHP\JUDQLF]QHZDUXQNLDPSOLWXG\
G
g
R
r
R
u
S
u
=
=
oraz
RGSRZLHGQLRGODJHQHUDWRUD]HOHPHQWHPW\SX1LW\SX6
'ODSRGWU]\PDQLDGUJDZDUXQHNDPSOLWXGRZ\SRZLQLHQE\üVSHáQLRQ\
]QDGPLDUHPGODWHJR
G
g
R
r
R
u
S
u
<
<
oraz
= SRUyZQDQLD GR ]HUD F] FL XURMRQHM ]DOH*QRFL F]\OL
]ZDUXQNX ID]\ PR*HP\ Z\]QDF]\ü SXOVDFM GUJD
ω
0
W]Z Z]RU\
7KRPVRQD
(
)
(
)
ω
π
ω
0
0
0
2
1
1
=
=
+
=
+
f
L C
C
L
L
C
p
p
,
RGSRZLHGQLRGODXNáDGX]HOHPHQWHPW\SX1LW\SX6
(OHPHQW R XMHPQHM UH]\VWDQFML W\SX 1 WZRU]\ REZyG
QLHVWDELOQ\ JG\ MHVW REFL*RQ\ PDá NRQGXNWDQFM VWG HOHPHQW WHQ
QD]\ZD VL VWDELOQ\P ]ZDUFLRZR 3U]HFLZQLH HOHPHQW R XMHPQHM
UH]\VWDQFMLW\SX6WZRU]\REZyGQLHVWDELOQ\JG\MHVWREFL*RQ\PDá
UH]\VWDQFMGODWHJRHOHPHQWWHQQD]\ZDVL VWDELOQ\PUR]ZDUFLRZR
/LQHDU\]RZDQH XNáDG\ ]DVW SF]H JHQHUDWRUyZ SU]HGVWDZLRQH QD
U\V SR]ZDODM W\ONR QD Z\]QDF]HQLH JUDQLF]Q\FK ZDUXQNyZ
DPSOLWXGRZ\FKLF] VWRWOLZRFLGUJD-DNMX*ZVSRPQLDQRZF]HQLHM
GOD SRGWU]\PDQLD GUJD ZDUXQHN DPSOLWXGRZ\ JHQHUDFML PXVL E\ü
VSHáQLRQ\]QDGPLDUHPFRR]QDF]D*HSURFHVQDUDVWDQLDGUJDD*GR
RVLJQL FLD VWDQX XVWDORQHJR E G]LH VL ZL]Dá ] FKDUDNWHU\VW\F]Q\PL
GOD JHQHUDWRUyZ HIHNWDPL QLHOLQLRZ\PL =Dáy*P\ GOD SU]\NáDGX *H
HOHPHQWRGWáXPLDMF\PDFKDUDNWHU\VW\N W\SX1:SLHUZV]\PHWDSLH
GUJDQLD V]\ENR QDUDVWDM JG\* XMHPQD NRQGXNWDQFMD GRPLQXMH QDG
NRQGXNWDQFM
G
R
WM g
G
u
R
>
UHSUH]HQWXMF
VWUDW\
HQHUJLL
ZUyZQROHJá\P REZRG]LH UH]RQDQVRZ\P : PLDU VWRSQLRZHJR
QDUDVWDQLD DPSOLWXG\ GUJD FKZLORZ\ SXQNW SUDF\ QD FKDUDNWHU\VW\FH
HOHPHQWX RGWáXPLDMFHJR ZNUDF]D Z REV]DU\ R FRUD] WR PQLHMV]HM
EH]Z]JO GQHM ZDUWRFL XMHPQHM NRQGXNWDQFML D* Z NRFX Z REV]DU
RGRGDWQLHMNRQGXNWDQFML:VWDQLHXVWDORQ\PUHGQLDZDUWRüXMHPQHM
NRQGXNWDQFML HOHPHQWX RGWáXPLDMFHJR VWDMH VL UyZQD FR GR
EH]Z]JO GQHM ZDUWRFL NRQGXNWDQFML VWUDW REZRGX 3U]\NáDGRZR QD
U\VDGODNRQGXNWDQFMLREZRGXZ\QRV]FHM
G
R
FKZLORZ\SXQNW
SUDF\ E G]LH VL SU]HPLHV]F]Dá SR FKDUDNWHU\VW\FH HOHPHQWX
RGWáXPLDMFHJR SRPL G]\ SXQNWDPL
A
L
B
Z\]QDF]DMF\PL UyZQLH*
PL G]\V]F]\WRZZDUWRüJHQHURZDQHJRQDSL FLDQD]DFLVNDFKREZRGX
UH]RQDQVRZHJR
3U]HDQDOL]RZDQLH Uy*Q\FK ]MDZLVN Z\VW SXMF\FK Z XNáDGDFK
JHQHUDF\MQ\FK D WDN*H Z\]QDF]HQLH SRGVWDZRZ\FK SDUDPHWUyZ
JHQHUDWRUD QS DPSOLWXG\ VSUDZQRFL HQHUJHW\F]QHM LWS Z\PDJD
VWRVRZDQLDQLHOLQLRZHMWHRULLJHQHUDFML:UDPDFKWHMWHRULLRSUDFRZDQR
ZLHOH PRGHOL L PHWRG DQDOLW\F]Q\FK ]QDMGXMF\FK ]DVWRVRZDQLH
]DUyZQRZDQDOL]LHMDNLZSURMHNWRZDQLXJHQHUDWRUyZ>@
*(1(5$725</&=(635= )(1,(0=:5271<0
2JyOQDFKDUDNWHU\VW\NDVSU] *HQLRZ\FKJHQHUDWRUyZ/&
(OHPHQWDPL DNW\ZQ\PL VWRVRZDQ\PL Z JHQHUDWRUDFK /& V
QDMF] FLHM WUDQ]\VWRU\ ELSRODUQH OXE XQLSRODUQH DOH WDN*H PR*QD
VSRWNDü UR]ZL]DQLD RSDUWH QD Z]PDFQLDF]DFK RSHUDF\MQ\FK OXE
]OLQHDU\]RZDQ\FK EUDPNDFK F\IURZ\FK 3RáF]HQLH WUyMHOHNWURGRZHJR
HOHPHQWXDNW\ZQHJR]REZRGHPUH]RQDQVRZ\PPR*QD]UHDOL]RZDü]D
SRPRFUy*Q\FKREZRGyZVSU] JDMF\FKMDNWUDQVIRUPDWRU\G]LHOQLNL
SRMHPQRFLRZHOXELQGXNF\MQHSDWU]F],UR]G]
1
2
3
2
3
1
2
3
1
2
1
3
C
2
L
2
L
2
C
2
C
1
C
1
L
1
L
1
L
L
a)
b)
c)
2
3
1
2
1
3
C
C
C
C
5\V =PLHQQRSUGRZH XSURV]F]RQH VFKHPDW\ LGHRZH JHQHUDWRUyZ
WUyMSXQNWRZ\FK/&]WUDQ]\VWRUDPLELSRODUQ\PLL-)(7D&ROSLWWVDE+DUWOH\D
F0HLVVQHUD
1DU\VSU]HGVWDZLRQRWU]\SRGVWDZRZHVWUXNWXU\JHQHUDWRUyZ
/& Uy*QLFH VL VSRVREHP VSU] *HQLD REZRGX UH]RQDQVRZHJR
]HOHPHQWHP DNW\ZQ\P ]QDQH MDNR JHQHUDWRU\ &ROSLWWVD +DUWOH\D
L0HLVVQHUD 1D ED]LH W\FK WU]HFK SRGVWDZRZ\FK XNáDGyZ SRZVWDáR
ZLHOH RGPLDQ XNáDGyZ JHQHUDF\MQ\FK 'R QDMZD*QLHMV]\FK ] QLFK
QDOH*\ JHQHUDWRU &ODSSD VWDQRZLF\ RGPLDQ JHQHUDWRUD &ROSLWWVD
RUD] JHQHUDWRU\ NZDUFRZH QD]\ZDQH XNáDGDPL 3LHUFH¶D
DVWDQRZLF\PLRGPLDQ JHQHUDWRUyZ&ROSLWWVDL+DUWOH\D
: SU]HGVWDZLRQ\FK QD U\V XSURV]F]RQ\FK VWUXNWXUDFK
SRGVWDZRZ\FKJHQHUDWRUyZ/&PR*QDX]LHPLüZ ]HáRWU]\PXMF
XNáDG\ Z NRQILJXUDFML 2% OXE 2* DOER Z ]Há RWU]\PXMF
XNáDG\ZNRQILJXUDFML2(OXE26
:DUXQNLJHQHUDFMLZJHQHUDWRUDFKÄWUyMSXQNWRZ\FK´
2JyOQ\ VFKHPDW JHQHUDWRUyZ ÄWUyMSXQNWRZ\FK´ SU]HGVWDZLRQR
QD U\V : FHOX XSURV]F]HQLD DQDOL]\ UR]SDWU]P\ QDMSLHUZ XNáDG
]HOHPHQWHPDNW\ZQ\PQLHREFL*DMF\PREZRGXUH]RQDQVRZHJRQS
]WUDQ]\VWRUHP-)(7
X
1
X
3
X
2
2
1
3
U
1
U
3
G
0
U
2
5\V2JyOQ\VFKHPDWJHQHUDWRUyZ
WUyMSXQNWRZ\FK]WUDQ]\VWRUHP-)(7
2EZyG UH]RQDQVRZ\ UHSUH]HQWRZDQ\ MHVW WU]HPD UHDNWDQFMDPL
X
X
X
1
2
3
,
,
RUD]G\QDPLF]QNRQGXNWDQFM
G
0
UHSUH]HQWXMFVWUDW\
REZRGX :DUXQHN ID]\ JHQHUDWRUyZ WUyMSXQNWRZ\FK PR*QD QDMáDWZLHM
UR]SDWU]\ü SU]\ ]DáR*HQLX *H XNáDG SUDFXMH Z NRQILJXUDFML 2*
3RQLHZD* XNáDG 2* SRGREQLH MDN 2% QLH RGZUDFD ID]\ QDSL FLD
Z\MFLRZHJRWRG]LHOQLN
X
X
1
2
,
SRZLQLHQVNáDGDüVL ]UHDNWDQFML
WHJR VDPHJR W\SX DE\ ]DSHZQLü SRG]LDá QDSL FLD Z\MFLRZHJR
( )
U
3
EH] ]PLDQ\ ID]\ =DWHP DE\ RWU]\PDü XNáDG UH]RQDQVRZ\ MH*HOL
X
X
1
2
,
VUHDNWDQFMDPLSRMHPQRFLRZ\PLWR
X
3
PXVLE\üUHDNWDQFM
LQGXNF\MQ:WHQVSRVyERWU]\PXMHP\VWUXNWXU JHQHUDWRUD&ROSLWWVD
]G]LHOQLNLHPSRMHPQRFLRZ\P
C C
1
2
,
U\VD
-H*HOL
X
X
1
2
,
VUHDNWDQFMDPLLQGXNF\MQ\PLWR
X
3
PXVLE\ü
UHDNWDQFM SRMHPQRFLRZ GDMF Z HIHNFLH VWUXNWXU JHQHUDWRUD
+DUWOH\D]G]LHOQLNLHPLQGXNF\MQ\P
L L
1
2
,
U\VE
= ZDUXQNX ID]\ PR*HP\ Z\]QDF]\ü F] VWRWOLZRü GUJD
JHQHUDWRUD 6SURZDG]D VL RQ SUDNW\F]QLH GR Z\]QDF]HQLD
F] VWRWOLZRFLUH]RQDQVRZHM
ω
0
REZRGX]ZDUXQNX
X
X
X
1
2
3
0
+
+
=
ERZLHP GOD F] VWRWOLZRFL UH]RQDQVRZHM REZyG UH]RQDQVRZ\
UHSUH]HQWXMHG\QDPLF]QNRQGXNWDQFM
G
0
:DUXQHNDPSOLWXGRZ\Z\PDJDVSHáQLHQLDZDUXQNX
( ) ( )
( )
k
k
X
X
X
u
u
u
0
0
0
0
1
1
2
1
ω β ω
ω
=
+
≥
3RQLHZD*
( )
k
u
ω
0
MHVWGX*HWRSU]\
k
u
>>
1
ZDUXQHN
MHVWáDWZ\GRVSHáQLHQLDQDZHWSU]\
X
X
1
2
<<
WMSU]\
C
C
1
2
>>
GOD
XNáDGX&ROSLWWVDLSU]\
L
L
1
2
<<
GODXNáDGX+DUWOH\D0R*OLZRü
GRERUX
X
X
1
2
<<
MHVW V]F]HJyOQLH ZD*QD GOD JHQHUDWRUyZ
UHDOL]RZDQ\FK QD WUDQ]\VWRUDFK ELSRODUQ\FK ERZLHP ZWHG\
X
1
MHVW
ERF]QLNRZDQHQLHZLHONUH]\VWDQFMZHMFLRZWUDQ]\VWRUD
r
in
.
:Sá\Z
WHMUH]\VWDQFMLE G]LHPDá\JG\
X
X
1
2
<<
RUD]
X
r
in
1
<<
=JRGQLH ] ]DOH*QRFL ZDUXQHN DPSOLWXG\ ]DOH*\ RG
Z]PRFQLHQLD
( )
k
u
ω
0
NWyUH]NROHL]DOH*\RGNRQGXNWDQFMLREFL*HQLD
G
L
SRPLQL WHMZGRW\FKF]DVRZ\FKUR]ZD*DQLDFK
5R]SDWU]P\
]DWHP
ZDUXQNL
JHQHUDFML
JHQHUDWRUD
ÄWUyMSXQNWRZHJR´ ] WUDQ]\VWRUHP ZNRQILJXUDFML 26 L REFL*RQHJR
NRQGXNWDQFM
G
L
U\V
G
0
X
3
U
3
U
2
3
2
X
1
1
U
1
X
2
G
L
5\V2JyOQ\VFKHPDWJHQHUDWRUyZ
WUyMSXQNWRZ\FKZXNáDG]LH26]
REFL*HQLHP
G
L
:V]\VWNLHZF]HQLHMV]HUR]ZD*DQLDGRW\F]FHZDUXQNXID]\SR]RVWDM
VáXV]QHGODXNáDGX]U\V3RQLHZD*XNáDG26RGZUDFDID] QDSL FLD
Z\MFLRZHJRR
R
]DWHPG]LHOQLNUHDNWDQF\MQ\
X
X
1
3
,
SRZLQLHQ
ZQRVLüGDOV]HSU]HVXQL FLHID]\R
R
DE\]DSHZQLü]JRGQRüID]
U
1
RUD]
U
2
D W\P VDP\P VSHáQLü ID]RZ\ ZDUXQHN JHQHUDFML -DN
SRND]DQR QD Z\NUHVDFK ZVND]RZ\FK QDSL ü L SUGyZ U\V
]JRGQRüID]
U U
1
2
,
Z\VW SXMHW\ONRZW\PSU]\SDGNXJG\UHDNWDQFMH
X
X
X
1
2
3
,
,
PDMWDNLVDPFKDUDNWHUMDNQDU\V
=JRGQLH ] U\V NRQGXNWDQFM
G
0
UHSUH]HQWXMF VWUDW\
REZRGXPR*QDSU]HWUDQVIRUPRZDüGR]DFLVNyZGRNWyU\FKGRáF]RQD
MHVWNRQGXNWDQFMREFL*HQLD
G
L
SDWU]F],UR]G]
G
G
p
p
X
X
X
0
0
2
2
1
2
'
=
=
+
gdzie
a)
b)
U
U
C2
2
= −
U
U
L2
2
= −
U
U
C1
1
=
U
U
L1
1
=
U
2
U
2
U
L
U
C
I
C2
I
L2
I
I
L
C
1
=
I
I
C
L
1
=
U
U
U
C
C
L
2
1
=
+
U
U
U
L
L
C
2
1
=
+
5\V:\NUHV\ZVND]RZHQDSL üL
SUGyZZXNáDG]LH]U\V
DZJHQHUDWRU]H&ROSLWWVD
EZJHQHUDWRU]H+DUWOH\D
X
1
X
2
X
3
G
0
1
2
3
X
1
X
2
X
3
G
0
'
1
2
3
5\V7UDQVIRUPDFMDNRQGXNWDQFML
*
'OD F] VWRWOLZRFL UH]RQDQVRZHM Z\]QDF]RQHM ] ZDUXQNX
Z]PRFQLHQLH
k
u
(
)
ω
0
RUD]WUDQVPLWDQFMDF]ZyUQLNDVSU] *HQLD
]ZURWQHJR
( )
β ω
u
0
Z\QRV]RGSRZLHGQLR
k
g
g
G
G
p
u
m
ds
L
(
)
/
ω
0
0
2
=
−
+
+
( )
β ω
X
;
;
;
;
;
S
=
+
= −
= −
*UDQLF]Q\ZDUXQHNDPSOLWXG\PR*QDZL FSU]HGVWDZLüZSRVWDFL
( ) ( )
k
g
g
G
G
p
p
u
u
m
ds
L
0
0
0
0
2
1
1
1
ω β ω =
−
+
+
−
=
3RSURVW\FKSU]HNV]WDáFHQLDFKZ\UD*HQLDRWU]\PXMHP\
(
)
p
g
G
g
p g
G
ds
L
m
m
2
0
0
+
+
−
+
=
:DUXQNLHP SRZVWDZDQLD GUJD Z XNáDG]LH MHVW DE\ SLHUZLDVWNL
UyZQDQLDE\á\U]HF]\ZLVWH
∆ ≥
0
(
)
g
G
G
g
g
m
L
ds
m
2
0
4
≥
+
+
7\ONRMHGHQ]SLHUZLDVWNyZUyZQDQLDPDVHQVIL]\F]Q\
(
)
(
)
p
g
g
G
g
G
g
g
G
g
m
m
ds
L
m
ds
L
m
1
2
0
4
2
=
+
−
+
+
+
+
≈
(
)
≈
+
+
−
≈
+
g
g
G
g
G
g
g
g
G
m
ds
L
m
m
m
m
L
0
=]DOH*QRFLLRWU]\PXMHP\
p
X
X
X
g
g
G
m
m
L
1
2
1
2
=
+
≈
+
VNG
X
X
G
g
L
m
1
2
≈
*HQHUDWRU\&ROSLWWVDL+DUWOH\D
:DUXQHN DPSOLWXG\ GOD JHQHUDWRUD &ROSLWWVD SU]\ELHUD
SRVWDü
C
C
G
g
L
m
2
1
=
]DGODJHQHUDWRUD+DUWOH\D
L
L
G
g
L
m
1
2
=
:DUXQHN ID]\ SR]ZDOD QD Z\]QDF]HQLH SXOVDFML GUJD
JHQHUDWRUD D SRQDGWR MHVW GUXJLP UyZQDQLHP ZL*F\P ZDUWRFL
HOHPHQWyZ
X
1
L
X
2
'OD JHQHUDWRUD &ROSLWWVD QD SRGVWDZLH ]DOH*QRFL
RWU]\PXMHP\
1
1
1
2
0
2
C
C
L
+
=
ω
QDWRPLDVWGODJHQHUDWRUD+DUWOH\D
L
L
C
1
2
0
2
1
+
=
ω
: JHQHUDWRU]H &ROSLWWVD LQGXNF\MQRü / PR*QD WHRUHW\F]QLH
GREUDüRWDNLHMZDUWRFLDE\VSHáQLüUyZQDQLDL-HGQDN
ZSUDNW\FH LVWQLHM MHV]F]H LQQH F]\QQLNL ZDUXQNXMFH GREyU ZDUWRFL
ZV]\VWNLFK WU]HFK HOHPHQWyZ REZRGX UH]RQDQVRZHJR SR]D
NRQLHF]QRFL VSHáQLHQLD GZyFK ZVSRPQLDQ\FK UyZQD 3RMHPQRFL
C C
1
2
,
SRZLQQ\ E\ü Z PLDU GX*H DE\ ]PLQLPDOL]RZDü ZSá\Z\
ZV]HONLFK UyZQROHJOH GR QLFK GRáF]RQ\FK SRMHPQRFL
SDVR*\WQLF]\FK Z W\P SRMHPQRFL PL G]\HOHNWURGRZ\FK HOHPHQWX
DNW\ZQHJR :WHG\ GOD RNUHORQHM SXOVDFML
ω
0
]JRGQHM ]H Z]RUHP
QDOH*\ ]PQLHMV]\ü ZDUWRü LQGXNF\MQRFL / FR ZSá\ZD QD
SRJRUV]HQLH GREURFL D W\P VDP\P ]ZL NV]HQLH NRQGXNWDQFML
G\QDPLF]QHM
G
0
REZRGX 0R*H WR ]DEXU]\ü JUDQLF]QH ZDUXQNL
JHQHUDFML Z SURVWHM SRVWDFL : WDNLP SU]\SDGNX SU]\
Z\]QDF]DQLX ZDUXQNX DPSOLWXG\ QLH QDOH*\ VWRVRZDü SU]\EOL*HQLD
Z\NRU]\VW\ZDQHJRZ]DOH*QRFL3RQDGWRDQDOL]DJUDQLF]Q\FK
ZDUXQNyZJHQHUDFML]RVWDáDSU]HSURZDG]RQDSU]\]DáR*HQLX*HHOHPHQW
DNW\ZQ\QLHREFL*DREZRGXUH]RQDQVRZHJR$SU]HFLH*ZSUDNW\FH
QDMF] FLHM VSRW\NDP\ VL ] JHQHUDWRUDPL UHDOL]RZDQ\PL Z RSDUFLX
RWUDQ]\VWRU\ ELSRODUQH NWyU\FK UH]\VWDQFMD ZHMFLRZD MHVW PDáD -DN
MX* ZVSRPQLDQR Z SRSU]HGQLP SRGUR]G]LDOH Z FHOX
]PLQLPDOL]RZDQLD ZSá\ZX WHM UH]\VWDQFML QD ZDUXQNL JHQHUDFML
ZDUWRFL UHDNWDQFML GRELHUD VL WDN DE\
X
X
1
2
<<
FR GOD XNáDGX
&ROSLWWVD R]QDF]D
C
C
1
2
>>
3U]\ DQDOL]LH JUDQLF]Q\FK ZDUXQNyZ
JHQHUDFML JHQHUDWRUD ] WUDQ]\VWRUHP ELSRODUQ\P QDOH*\ UyZQLH*
XZ]JO GQLü ZSá\Z UH]\VWDQFML ZHMFLRZHM WUDQ]\VWRUD QD Z\SDGNRZ
NRQGXNWDQFM G\QDPLF]Q REZRGX UH]RQDQVRZHJR 0R*HP\ WHJR
GRNRQDüZEDUG]RSURVW\VSRVyESRSU]H]WUDQVIRUPDFM NRQGXNWDQFML
ZHMFLRZHMWUDQ]\VWRUD
g
in
GR ]DFLVNyZ Z\MFLRZ\FK Z]PDFQLDF]D
GRNWyU\FKGRáF]RQDMHVWNRQGXNWDQFMDREFL*HQLDU\V
G
0
1
3
2
C
1
C
2
L
G
L
g
in
C
1
C
2
L
G
L
g
in
'
G
0
'
5\V7UDQVPLWDQFMDNRQGXNWDQFMLZHMFLRZHMGR]DFLVNyZZ\MFLRZ\FK
Z]PDFQLDF]D
3U]\ WUDQVIRUPDFML XNáDGX QDOH*\ XZ]JO GQLü DE\ PRFH
Z\G]LHORQHZNRQGXNWDQFMDFK g
g
in
in
i
'
E\á\WDNLHVDPH
=DNáDGDVL SU]\W\P*HSUGG]LHOQLND
C C
1
2
,
MHVWGX*RZL NV]\RG
SUGXSRELHUDQHJR]WHJRG]LHOQLND
=DWHP
G
G
p
0
0
2
'
=
JG]LH p
X
X
X
C
C
C
1
2
1
2
1
1
2
=
+
=
+
D
g
g
X
X
g
C
C
p
in
in
in
'
=
=
=
−
1
2
2
1
2
2
2
1
1
E
8Z]JO GQLDMF g
in
'
QDOH*\ VNRU\JRZDü ]DOH*QRü
RNUHODMFZ]PRFQLHQLHQDSL FLRZHZ]PDFQLDF]D
( )
k
g
g
G
G
p
g
p
u
m
ce
L
in
ω
0
0
2
2
1
1
=
−
+
+
+
−
L SRZWyU]\ü DQDOL] ZDUXQNyZ JHQHUDFML ZJ ZF]HQLHM SRGDQHJR
DOJRU\WPX
:DUWRü LQGXNF\MQRFL / PR*HP\ Z\]QDF]\ü SU]\MPXMF
GRVWDWHF]QLHGX*\PDUJLQHVZDUXQNXQS>@
G
g
g
G
g
m
o
L
m
0
2
0 2
<
+
+
.
SU]\F]\P
g
o
MHVWNRQGXNWDQFMZ\MFLRZWUDQ]\VWRUD
3RGREQH UR]ZD*DQLH PR*HP\ SU]HSURZDG]Lü GOD JHQHUDWRUD
+DUWOH\DZNWyU\PLVWQLHMHSHZLHQVWRSLHVZRERG\SU]\GRELHUDQLX
SRMHPQRFL & : JHQHUDWRU]H +DUWOH\D Z\VW SXM MHGQDN LQQH
GRGDWNRZHXZDUXQNRZDQLDSR]DNRQLHF]QRFLVSHáQLHQLDJUDQLF]Q\FK
ZDUXQNyZ JHQHUDFML SU]\ GRERU]H ZDUWRFL WU]HFK
HOHPHQWyZREZRGXUH]RQDQVRZHJR
:\SDGNRZH SRMHPQRFL SDVR*\WQLF]H EORNX Z]PRFQLHQLD
ZHMFLRZD
C
in
RUD] Z\MFLRZD
C
o
WZRU] REZRG\ UH]RQDQVRZH
]LQGXNF\MQRFLDPL
L L
1
2
,
&] VWRWOLZRFLUH]RQDQVRZHW\FKREZRGyZ
f
L C
f
L C
r
in
r
o
1
1
2
2
1
2
1
2
=
=
π
π
,
SRZLQQ\NLONDNURWQLHSU]HZ\*V]DüF] VWRWOLZRü
f
0
GUJDJHQHUDWRUD
=DVLODQLHJHQHUDWRUyZ&ROSLWWVDL+DUWOH\D
3UDNW\F]QD UHDOL]DFMD JHQHUDWRUyZ PXVL UyZQLH* XZ]JO GQLü
REZRG\ SRODU\]DFML HOHPHQWX DNW\ZQHJR 6SRUyG ZLHOX Uy*Q\FK
VSRVREyZ]DVLODQLDJHQHUDWRUyZ/&QDZ\Uy*QLHQLH]DVáXJXMW\SRZH
XNáDG\]DVLODQLDWDNLHMDNV]HUHJRZHUyZQROHJáHSU]H]GáDZLNZF]
RUD] ]DVLODQLH RG VWURQ\ HPLWHUD OXE (UyGáD WUDQ]\VWRUD 1D U\V
SU]HGVWDZLRQRSU]\NáDG\]DVLODQLDJHQHUDWRUyZ+DUWOH\D>@
a)
b)
R
B 2
R
B1
R
B1
L
1
L
2
C
C
b
L
2
L
1
C
C
b2
C
b1
U
CC
U
CC
L
D
GáDZLN ZF]
R
B 2
C
b1
C
e
R
B
R
E
−
U
EE
DZ
`
L
1
L
2
C
L
1
L
2
C
C
g
R
G 2
R
G1
U
DD
M
c)
d )
5\V=DVLODQLHJHQHUDWRUyZ+DUWOH\DDV]HUHJRZREUyZQROHJOHSU]H]GáDZLN
ZF]FRGVWURQ\HPLWHUDGV]HUHJRZHZXNáDG]LHVSU] *HQLD(UyGáRZHJR]
WUDQ]\VWRUHP9026
8NáDG QD U\VD UHSUH]HQWXMH ]DVLODQLH V]HUHJRZH Z NWyU\P
VNáDGRZD VWDáD SUGX NROHNWRUD Sá\QLH SU]H] FHZN REZRGX
UH]RQDQVRZHJR3RáF]HQLHREZRGXUH]RQDQVRZHJR]ED]WUDQ]\VWRUD
RGG]LHORQH MHVW SU]H] SRMHPQRü VSU] JDMF
C
b
QD NWyUHM SR
Z]EXG]HQLXVL GUJDZ\WZDU]DVL GRGDWNRZHQDSL FLHSU]HVXZDMFH
VWRSQLRZRSXQNWSUDF\WUDQ]\VWRUDGRSUDF\ZNODVLH$%%OXE&
6WDáD F]DVRZD UR]áDGRZDQLD SRMHPQRFL
C
b
SRZLQQD E\ü GX*D
ZSRUyZQDQLX]RNUHVHPGUJDJHQHUDWRUD
1D U\VE SRND]DQ\ MHVW VFKHPDW LGHRZ\ JHQHUDWRUD +DUWOH\D
]]DVLODQLHP UyZQROHJá\P Z NWyU\P VNáDGRZD VWDáD SUGX NROHNWRUD
Sá\QLHSU]H]GáDZLNZF]]DQLHSá\QLHSU]H]REZyGUH]RQDQVRZ\
2EZyGWHQMHVWGRáF]RQ\GRNROHNWRUDSRSU]H]SRMHPQRüVSU] JDMF
C
b2
]DGRED]\SU]H]SRMHPQRüVSU] JDMF
C
b1
7HQURG]DM]DVLODQLD
MHVW V]F]HJyOQLH SUHIHURZDQ\ Z JHQHUDWRUDFK GX*HM PRF\ SRQLHZD*
SU]H]GáDZLN
L
D
ZF]QLHSá\QLHSUGZLHONLHMF] VWRWOLZRFLSU]H]FR
]DSHZQLRQHMHVWGREUHGRSDVRZDQLH(UyGáD]DVLODMFHJR=DOHW]DVLODQLD
UyZQROHJáHJR SU]H] GáDZLN ZF] V UyZQLH* PDáH VWUDW\ PRF\ Z
REZRG]LH ]DVLODQLD QDWRPLDVW ZDG WUXGQRü Z\NRQDQLD VDPHJR
GáDZLND'áDZLNZF]SRZLQLHQE\üWDNZ\NRQDQ\DE\QLHSRVLDGDá
ZáDVQ\FK UH]RQDQVyZ D MHJR LQGXNF\MQRü
L
D
SRZLQQD E\ü
Z\VWDUF]DMFRGX*D
ω
L
Z
D
o
>>
JG]LH
Z
o
MHVWLPSHGDQFMREZRGX
UH]RQDQVRZHJR 'Rü GáXJL F]DV XVWDODQLD VL DPSOLWXG\ GUJD Z
XNáDG]LH]ZL]DQ\MHVW]áDGRZDQLHPSRMHPQRFL
C
b2
SU]H]GáDZLN
L
D
SR]DáF]HQLX]DVLODQLD
1D U\VF SU]HGVWDZLRQR SU]\NáDG ]DVLODQLD WUDQ]\VWRUD
XMHPQ\P QDSL FLHP
−
U
EE
RG VWURQ\ HPLWHUD %D]D WUDQ]\VWRUD MHVW
X]LHPLRQD GOD VNáDGRZHM ]PLHQQHM F]\OL WUDQ]\VWRU SUDFXMH
ZNRQILJXUDFML 2% =DVLODQLH WUDQ]\VWRUD SRSU]H] UH]\VWDQFM VWRVXMH
VL QDMF] FLHMZJHQHUDWRUDFKPDáHMPRF\
8NáDG QD U\VG MHVW SU]\NáDGHP JHQHUDWRUD ] ]DVLODQLHP
V]HUHJRZ\P -HVW RQ ]UHDOL]RZDQ\ QD WUDQ]\VWRU]H 026)(7 QS
9026 ] NDQDáHP Z]ERJDFDQ\P Z NWyU\P G]LHOQLN
R
R
G
G
1
2
,
]DSHZQLDZVW SQSRODU\]DFM EUDPNLWUDQ]\VWRUD2EZyGUH]RQDQVRZ\
MHVWZáF]RQ\V]HUHJRZR]H(UyGáHPWUDQ]\VWRUD]DWHPZXNáDG]LH
Z\VW SXMHV]HUHJRZHVSU] *HQLH(UyGáRZH3RGREQLHMDNZXNáDG]LHD
QLH PD *DGQHJR HOHPHQWX VHSDUXMFHJR SUG ZLHONLHM F] VWRWOLZRFL
ZREZRG]LHZ\MFLRZ\PWUDQ]\VWRUDRG]DVLODF]DFRMHVWSHZQZDG
UR]ZL]DQLD
,ORü PR*OLZ\FK UR]ZL]D XNáDGyZ ]DVLODQLD JHQHUDWRUyZ
&ROSLWWVDMHVWQLHFRPQLHMV]DQL*JHQHUDWRUyZ+DUWOH\DSRQLHZD*W\ONR
MHGHQHOHPHQWREZRGXUH]RQDQVRZHJRFHZNDSU]HZRG]LSUGVWDá\
3U]\NáDG\ ]DVLODQLD JHQHUDWRUyZ &ROSLWWVD SU]HGVWDZLRQR QD
U\V
a)
c)
R
B 2
L
D
U
CC
b)
U
CC
L
D
R
D
R
B 2
R
B1
R
B1
T
T
T
L
L
L
C
2
C
2
C
1
C
1
C
b2
C
b1
C
b
C
b
R
B
DZ
R
E
C
1
C
2
−
U
EE
'áZF]
'áZF]
5\V=DVLODQLHJHQHUDWRUyZ&ROSLWWVDDUyZQROHJáHSU]H]GáDZLNZF]E
SU]H]GáDZLNZF]LFHZN REZRGXUH]RQDQVRZHJRFRGVWURQ\HPLWHUD
8NáDG ] U\VXQNX D MHVW W\SRZ\P UR]ZL]DQLHP ]DVLODQLD
UyZQROHJáHJRNWyUH]RVWDáRRSLVDQHZF]HQLHMGODJHQHUDWRUD+DUWOH\D
]U\VD
1D U\VE SU]HGVWDZLRQR SHZQ PRG\ILNDFM ]DVLODQLD
UyZQROHJáHJRSU]H]GáDZLNZF]RLQGXNF\MQRFL
L
D
LV]HUHJRZHM
UH]\VWDQFML
R
D
L FHZN REZRGX UH]RQDQVRZHJR >@ 3RQLHZD*
GáDZLNZF]ERF]QLNXMHQLHFDá\REZyGUH]RQDQVRZ\DW\ONRMHJR
QLVNRLPSHGDQF\MQ F] ü
C
C
1
2
>>
GODWHJR Z ]QDF]Q\P VWRSQLX
SRPQLHMV]RQH ]RVWDáR MHJR WáXPLFH G]LDáDQLH QD REZyG UH]RQDQVRZ\
àDJRG]L WR Z\PDJDQLH RGQRQLH ZLHONRFL LPSHGDQFML R
j
L
D
D
+
ω
WDN *H Z REZRG]LH ]DVLODQLD PR*QD ]DVWRVRZDü QDZHW Z\áF]QLH
UH]\VWRU
8NáDGQDU\VXQNXF]DVLODQ\MHVW]H(UyGáDXMHPQHJRQDSL FLD
−
U
EE
RG VWURQ\ HPLWHUD L MHVW SRGREQ\ GR RSLVDQHJR ZF]HQLHM
XNáDGX]U\VF
*HQHUDWRU0HLVVQHUD
:JHQHUDWRU]H0HLVVQHUDVSU] *HQLHREZRGXNROHNWRUDGUHQX
]REZRGHPED]\EUDPNLUHDOL]RZDQHMHVWSRSU]H]WUDQVIRUPDWRU1D
U\V SU]HGVWDZLRQR ]PLHQQRSUGRZ\ VFKHPDW LGHRZ\ JHQHUDWRUD
0HLVVQHUD RUD] MHJR OLQLRZ\ PRGHO SU]\ UR]FL WHM S WOL VSU] *HQLD
]ZURWQHJRQDZL]XMF\GRVFKHPDWXEORNRZHJRQDU\V
a)
U
1
'
U
1
A
p
1
p
2
U
L
R
L
C U
2
n
1
n
2
n
T
p
U
U
n
n
p
U
U
n
n
L
1
1
2
1
2
2
2
=
=
=
=
n
−
OLF]EDX]ZRMH
b)
g U
m
1
'
p g
1
2
11
p g
2
2
11
G
0
U
2
U
1
1
1
: p
5\V*HQHUDWRU0HLVVQHUD
DVFKHPDWLGHRZ\
EPRGHOOLQLRZ\]UR]FL WS WO
ZREZRG]LHED]\ZSXQNFLH$
5R]FL FLH S WOL VSU] *HQLD ]ZURWQHJR SR]ZDOD QD RNUHOHQLH
Z]PRFQLHQLD S WOL
k
u
u
β
=DNáDGDMF *H F] VWRWOLZRü JUDQLF]QD
WUDQ]\VWRUDMHVWGX*RZL NV]DQL*F] VWRWOLZRüJHQHUDFMLPR*QDSU]\Mü
*HDGPLWDQFMDZHMFLRZDWUDQ]\VWRUDPDFKDUDNWHUU]HF]\ZLVW\2EZyG
/&GODF] VWRWOLZRFLUH]RQDQVRZHM
f
L C
0
1
2
=
π
PR*HE\ü]DVWSLRQ\NRQGXNWDQFMG\QDPLF]Q
G
0
G
C
Q
0
0
=
ω
SU]\F]\PQ MHVWGREURFLREZRGX
:\SDGNRZDNRQGXNWDQFMDREZRGXGODF]
VWRWOLZRFLUH]RQDQVRZHM
f
0
SR SU]HWUDQVIRUPRZDQLX NRQGXNWDQFML ZHMFLRZHM
g
11
L REFL*HQLD
G
L
Z\QRVL
G
G
p g
p G
R
L
=
+
+
0
1
2
11
2
2
JG]LH
p
1
L
p
2
VWDQRZL SU]HNáDGQLH WUDQVIRUPDWRUD ]GHILQLRZDQH QD
U\VD3RQLHZD*XNáDGZNRQILJXUDFML2(RGZUDFDID] R
R
WR
GODVSHáQLHQLDZDUXQNyZJHQHUDFMLGRGDWNRZHSU]HVXQL FLHID]\R
R
PXVL]DSHZQLüWUDQVIRUPDWRUSU]\RGSRZLHGQLPSRáF]HQLXX]ZRMHQLD
ZREZRG]LHED]\:DUXQHNDPSOLWXG\RWU]\PXMHP\]SU]\UyZQDQLD
Z]PRFQLHQLDS WOLGRMHGQRFL
( ) ( )
(
)
k
g
G
p g
p G
p
u
u
m
L
ω β ω
0
0
0
1
2
11
2
2
1
1
=
−
+
+
−
=
: RSDUFLX R JUDQLF]Q\ ZDUXQHN DPSOLWXG\ PR*HP\
Z\]QDF]\üSU]HNáDGQL WUDQVIRUPDWRUD
3UDNW\F]QH XNáDG\ JHQHUDWRUyZ 0HLVVQHUD ]DZLHUDMFH REZRG\
]DVLODQLDSU]HGVWDZLRQRQDU\V
8NáDG SU]HGVWDZLRQ\ QD U\VD ]DVLODQ\ MHVW V]HUHJRZR SU]H]
JáyZQHX]ZRMHQLHWUDQVIRUPDWRUD
.RQGHQVDWRU
C
b
]DSHZQLDPDáLPSHGDQFM ZREZRG]LHVWHUXMF\P
GODSU]HELHJyZZLHONLHMF] VWRWOLZRFL]ZLHUDUH]\VWDQFM Z\MFLRZ
G]LHOQLND
R
R
B
B
1
2
,
DSRQDGWRMHVWHOHPHQWHPDXWRPDW\F]QHMSRODU\]DFML
ED]\
T
L
n
1
n
2
n
C
b
R
B1
R
B2
R
L
C
U
CC
C
e
R
E
R
B
L
D
'áZF]
a)
b)
U
CC
R
L
C
L
C
b2
C
b1
5\V*HQHUDWRU0HLVVQHUDD]]DVLODQLHPV]HUHJRZ\PE]]DVLODQLHP
UyZQROHJá\P
: PLDU QDUDVWDQLD DPSOLWXG\ GUJD QD SRMHPQRFL
C
b
VWRSQLRZR
QDUDVWDXMHPQHQDSL FLHE GFHHIHNWHPMHMáDGRZDQLDLPSXOVDPLSUGX
ED]\']L NLWHPXJHQHUDWRUSRVLDGDZ]EXG]HQLHPL NNLHW]QZ]EXG]D
VL SU]\SUDF\WUDQ]\VWRUDZNODVLH$SRODU\]DFMDZVW SQD]DSRPRF
G]LHOQLND
R
R
B
B
1
2
,
DZPLDU QDUDVWDQLDDPSOLWXG\GUJDMHJRSXQNW
SUDF\ SU]HVXZD VL GR NODV\ $% % OXE & 7DNL VSRVyE SUDF\
Z\ND]XMHZáDFLZRFLVWDELOL]XMFHDPSOLWXG JHQHURZDQHJRSU]HELHJX
8NáDGSU]HGVWDZLRQ\QDU\VEMHVW]DVLODQ\UyZQROHJOHSU]H]
GáDZLN ZF] L Z\PDJD VWRVRZDQLD ZL NV]HM LORFL HOHPHQWyZ QL*
SU]\]DVLODQLXV]HUHJRZ\P.RQGHQVDWRU
C
b1
VSHáQLD SRGREQ URO
MDN ZXNáDG]LH QD U\VD ]D NRQGHQVDWRU
C
b2
VSU] JD REZyG
UH]RQDQVRZ\]NROHNWRUHPWUDQ]\VWRUD
*HQHUDWRU\&ODSSDL.KQD+XWKD
,VWQLHMH ZLHOH RGPLDQ JHQHUDWRUyZ WUyMSXQNWRZ\FK VSRUyG
NWyU\FKQDZ\Uy*QLHQLH]DVáXJXMJHQHUDWRU\&ODSSDL.KQD+XWKD
U\V
:XNáDG]LH&ODSSDVWDQRZLF\PRGPLDQ JHQHUDWRUD&ROSLWWVD
JHQHUDFMD MHVW PR*OLZD MHG\QLH GOD F] VWRWOLZRFL ZL NV]\FK RG
F] VWRWOLZRFLUH]RQDQVRZHMV]HUHJRZHJRREZRGX
L C
,
.
3
a)
b)
T
T
L
L
z
L
z1
L
z 2
C
3
C
1
C
2
C
1
C
2
C
L
1
L
2
5\V2GPLDQ\JHQHUDWRUyZ/&DJHQHUDWRU&ODSSDEJHQHUDWRU.QKQD+XWKD
: W\P ]DNUHVLH F] VWRWOLZRFL REZyG WHQ PR*H VSHáQLDü URO
LQGXNF\MQRFL]DVW SF]HM
L
L
C
z
= −
1
2
3
ω
R ZDUWRFL VLOQLH URVQFHM ]H Z]URVWHP F] VWRWOLZRFL ']L NL WHPX
X]\VNXMH VL ]ZL NV]HQLH VWDáRFL F] VWRWOLZRFL JHQHURZDQ\FK GUJD
-H*HOL SRMHPQRFL
C
1
L
C
2
]RVWDQ WDN GREUDQH *H E G Z\UD(QLH
ZL NV]H FR QDMPQLHM NLONDNURWQLH RG
C
3
WDN DE\ Z\SDGNRZD
SRMHPQRü REZRGX Z\QLNDMFD ] V]HUHJRZHJR SRáF]HQLD
C C C
1
2
3
,
,
E\áD QLHZLHOH PQLHMV]D RG
C
3
WR XNáDG JHQHUXMH GUJDQLD
]F] VWRWOLZRFL QLH]QDF]QLH ZL NV] RG F] VWRWOLZRFL UH]RQDQVX
V]HUHJRZHJR
L C
,
.
3
:XNáDG]LH&ODSSDZDUWRFLHOHPHQWyZ
C C L
1
2
,
,
PRJE\ü]QDF]QLHZL NV]HRGVZRLFKRGSRZLHGQLNyZZJHQHUDWRU]H
&ROSLWWVD :áDFLZRü WD MHVW V]F]HJyOQLH ZD*QD SU]\ SURMHNWRZDQLX
JHQHUDWRUyZGX*\FKF] VWRWOLZRFLJG\ZDUWRFL
C C L
1
2
,
,
SRWU]HEQH
GR ]DSHZQLHQLD Z\PDJDQHM F] VWRWOLZRFL JHQHUDFML VWDM VL
SRUyZQ\ZDOQH]SRMHPQRFLDPLLLQGXNF\MQRFLDPLSDVR*\WQLF]\PL
XNáDGX
:JHQHUDWRU]H.KQD+XWKDVWDQRZLF\PRGPLDQ JHQHUDWRUD
+DUWOH\D U\VE IXQNFMH LQGXNF\MQRFL
L
L
z
z
1
2
,
VSHáQLDM
RGSRZLHGQLRREZRG\
L C
1
1
,
L
L C
2
2
,
3DWU]FQDVWUXNWXU JHQHUDWRUD
.KQD+XWKD áDWZR PR*QD VRELH XZLDGRPLü GODF]HJR
ZZVNRSDVPRZ\P Z]PDFQLDF]X VHOHNW\ZQ\P ] SDU REZRGyZ
UH]RQDQVRZ\FK QD ZHMFLX L Z\MFLX PR*H GRMü GR XWUDW\ MHJR
VWDELOQRFLMH*HOLZ]PRFQLHQLHLSRMHPQRüED]DNROHNWRUEUDPND
GUHQV]E\WGX*H
3RSUDZNDOLQLRZDLSRSUDZNDQLHOLQLRZDF] VWRWOLZRFL
/LQLRZ SRSUDZN F] VWRWOLZRFL QD]\ZD VL RGFK\OHQLH
JHQHURZDQHM F] VWRWOLZRFL
f
0
VSRZRGRZDQH RGG]LDá\ZDQLHP
REFL*HQLD VWUDWDPL HOHPHQWyZ RUD] ZSá\ZHP HOHPHQWyZ
SDVR*\WQLF]\FK ]DEXU]DMF\FK ELODQV PRF\ XURMRQHM Z XNáDG]LH RUD]
]PLDQ SU]HVXQL FLD ID]\ V\JQDáX ]ZURWQHJR ZREHF Z\MFLRZHJR
3RSUDZNDWDZ\VW SXMHZHZV]\VWNLFKW\SDFKJHQHUDWRUyZ
1DZHW Z SU]\SDGNX QDMSURVWV]HJR VWUDWQHJR REZRGX
UH]RQDQVRZHJRMHJRF] VWRWOLZRüUH]RQDQVRZD
f
r
MHVWPQLHMV]DQL*
F] VWRWOLZRüUH]RQDQVRZD
f
0
REZRGXEH]VWUDWQHJR>SRUF],UR]G]
UyZQ@
(
)
f
f
r
f
L
r
L
=
−
0
2
0
2
2
1
2
π
D
JG]LH
f
L C
0
1
2
=
π
E
:SU]\SDGNXJHQHUDWRUD&ROSLWWVDZSá\ZUH]\VWDQFMLVWUDWFHZNL
r
L
QDF] VWRWOLZRüJHQHUDFMLMHVWQDVW SXMF\
(
)
f
f
C
C
C
r
g
G
g
L
o
L
=
+
+
+
0
1
1
2
1
D
JG]LH
f
L C
C
0
1
2
1
2
1
1
1
=
+
π
E
MHVWF] VWRWOLZRFLUH]RQDQVRZEH]VWUDWQHJRREZRGX
/LQLRZHSRSUDZNLF] VWRWOLZRFLPRJE\üGRGDWQLHOXEXMHPQH
DLFKEH]Z]JO GQHZDUWRFLVPDáHJG\GREURFL]DVWRVRZDQ\FKFHZHN
LNRQGHQVDWRUyZVGX*H
1LHOLQLRZD SRSUDZND F] VWRWOLZRFL Z\QLND ] ]DVDG\
UyZQRZDJL PRF\ ELHUQHM K DUPRQLF]Q\FK Z SU]HELHJX Z\MFLRZ\P
E GF\FK HIHNWHP QLHOLQLRZRFL FKDUDNWHU\VW\NL SU]HMFLRZHM
Z]PDFQLDF]D(OHPHQWDNW\ZQ\ GRVWDUF]D W\ONR PRF F]\QQ D ]DWHP
PRFH ELHUQH PXV] VL UyZQRZD*\ü Z REZRG]LH :SURZDG]HQLH
KDUPRQLF]Q\FK VSRZRGRZDQH QLHOLQLRZRFLDPL QDUXV]D ELODQV PRF\
ELHUQ\FK
-H*HOLSRMHG\QF]\REZyGUH]RQDQVRZ\RDGPLWDQFML
( )
Y
G
j
C
j
L
ω
ω
ω
= +
+
1
MHVWGRVWURMRQ\GRSXOVDFMLSRGVWDZRZHM
ω ω
=
0
W]Q
( )
Y
G
j
C
j
L
G
L C
ω
ω
ω
ω
0
0
0
0
2
1
1
= +
+
=
=
,
WRPRFELHUQDKDUPRQLF]Q\FK]PDJD]\QRZDQDZSROXPDJQHW\F]Q\P
FHZNL MHVW PQLHMV]D QL* PRF ELHUQD ]PDJD]\QRZDQD Z SROX
HOHNWU\F]Q\P NRQGHQVDWRUD SRQLHZD* GOD K DUPRQLF]Q\FK REZyG
UH]RQDQVRZ\PDFKDUDNWHUSRMHPQRFLRZ\
(
)
Y k
G
j k
C
j k
L
G
j
k
k
L
ω
ω
ω
ω
0
0
0
2
0
1
1
1
0
= +
+
= +
−
>
$E\ SU]\ZUyFLü QDUXV]RQ ZVNXWHN REHFQRFL KDUPRQLF]Q\FK
UyZQRZDJ F] VWRWOLZRü SRGVWDZRZD PXVL ]PDOHü WDN DE\ GOD
VNáDGRZHMSRGVWDZRZHMPRFELHUQDZFHZFHE\áDZL NV]DRGPRF\
ELHUQHMZNRQGHQVDWRU]HWMDE\GODVNáDGRZHMSRGVWDZRZHMREZyGPLDá
FKDUDNWHU LQGXNF\MQ\ =DWHP QLHOLQLRZRFL Z]PDFQLDF]D SRZRGXM
]PQLHMV]HQLH JHQHURZDQHM F] VWRWOLZRFL R QLHOLQLRZ SRSUDZN
F] VWRWOLZRFL
3U]\NáDGRZR GOD JHQHUDWRUD &ROSLWWVD QLHOLQLRZD SRSUDZND
F] VWRWOLZRFLZ\QRVL>@
ω
ω
ω
g
k
k
Q
k
h
−
≅ −
−
=
∞
∑
0
0
2
2
2
2
1
2
1
1
*HQHUDWRU\NZDUFRZH
3HZQRGPLDQVSU] *HQLRZ\FKJHQHUDWRUyZ/&VJHQHUDWRU\
NZDUFRZH Z\NRU]\VWXMFH UH]RQDWRU NZDUFRZ\ RSLVDQ\ Z F],
UR]G] *HQHUDWRU\ NZDUFRZH SR]ZDODM QD RVLJQL FLH GX*HM
VWDáRFL F] VWRWOLZRFL QLHRVLJDOQHM SU]\ X*\FLX NRQZHQFMRQDOQ\FK
REZRGyZ /& =DVWRVRZDQLD UH]RQDWRUD NZDUFRZHJR Z XNáDGDFK
JHQHUDF\MQ\FKZ\QLNDM]FKDUDNWHUX]PLDQMHJRLPSHGDQFMLZIXQNFML
F] VWRWOLZRFL&KDUDNWHU\VW\NL]DVW SF]HMUH]\VWDQFML
( )
R
z
ω
L
( )
X
z
ω
U]HF]\ZLVWHJR UH]RQDWRUD NZDUFRZHJR RSLVDQR Z F], UR]G]
L]RVWDá\SRZWyU]RQHQDU\V
ω
R
z
R
z
X
z
r
k
X
z
ω
s
ω
r
ω
m
1
ω
C
o
≈ ω
s
k
Q
2
5\V,PSHGDQFMDUH]RQDWRUD
NZDUFRZHJR
0R*OLZHVGZLHJUXS\XNáDGRZHJHQHUDWRUyZ
−
JHQHUDWRU\ Z NWyU\FK UH]RQDWRU Z\NRU]\VWDQ\ MHVW MDNR VHOHNW\ZQ\
HOHPHQWVSU] JDMF\RPDáHMUH]\VWDQFMLSUDFDSU]\SXOVDFML
ω
s
−
JHQHUDWRU\ZNWyU\FKUH]RQDWRUSUDFXMHMDNR]DVW SF]DLQGXNF\MQRü
L
z
R ZDUWRFL V]\ENR URVQFHM ] F] VWRWOLZRFL SUDFD Z
SU]HG]LDOHSXOVDFML
ω
ω
s
m
−
U\V
3LHUZV]D ] Z\PLHQLRQ\FK JUXS JHQHUDWRUyZ QD]\ZDQD MHVW
JHQHUDWRUDPL %XWOHUD *HQHUDWRU\ GUXJLHM JUXS\ Z\ZRG] VL
EH]SRUHGQLR]SRGVWDZRZ\FKVWUXNWXU&ROSLWWVDL+DUWOH\DLQD]\ZD
VL MHJHQHUDWRUDPL3LHUFH¶D
6FKHPDW\LGHRZHSRGVWDZRZ\FKXNáDGyZ%XWOHUDSU]HGVWDZLRQR
QD U\V : XNáDGDFK W\FK Z]PRFQLHQLH S WOL RVLJD EDUG]R RVWUH
PDNVLPXPZRWRF]HQLXSXOVDFML
ω
s
SU]\NWyUHMUH]RQDWRUSUDFXMH
MDNR HOHPHQW VSU]
JDMF\ *HQHUDWRU\ %XWOHUD SR]ZDODM X]\VNDü
VWDáRüF]
VWRWOLZRFLU]
GX10
10
6
8
−
−
−
.
C
1
C
2
2
3
1
2
1
3
C
L
2
L
1
ω
S
ω
S
T
T
ω
S
a)
b)
c)
C
1
C
2
L
R
L
U
CC
U
EE
5\V3RGVWDZRZHXNáDG\%XWOHUDD]F]ZyUQLNLHPVSU] JDMF\P&ROSLWWVDE
]F]ZyUQLNLHPVSU] JDMF\P+DUWOH\DFSUDNW\F]QDUHDOL]DFMD]F]ZyUQLNLHP
VSU] JDMF\P&ROSLWWVD
1DU\VSU]HGVWDZLRQRVFKHPDW\LGHRZHJHQHUDWRUyZ
3LHUFH¶D
a)
b)
c)
d )
( )
L
( )
L
(
)
L
1
(
)
L
1
L
2
L
2
L
2
C
1
C
2
C
1
C
2
U
CC
U
CC
C
2
C
1
2
3
1
1
2
3
C
C
C
2
C
U
CC
U
CC
L
L
z
=
L
L
z
=
5\V*HQHUDWRU\D&ROSLWWVD3LHUFH¶D]GZRPDSRMHPQRFLDPLE&ROSLWWVD
3LHUFHD]REZRGHPUH]RQDQVRZ\PF+DUWOH\D3LHUFHD]LQGXNF\MQRFLG
+DUWOH\D3LHUFHD]REZRGHPUH]RQDQVRZ\P
2EZRG\ UH]RQDQVRZH R UHDNWDQFML SRMHPQRFLRZHM U\VE L
UHDNWDQFML LQGXNF\MQHM U\VG Z ]DNUHVLH JHQHURZDQHM
F] VWRWOLZRFLXáDWZLDMGREyUVSU] *HQLD]ZURWQHJR
3U]HGVWDZLRQH UR]ZL]DQLD SR]ZDODM X]\VNDü VWDáRü F]
VWRWOLZRFL
U]
GX
−
−
−
*áyZQ\ ZSá\Z QD QLHVWDáRü F] VWRWOLZRFL Z JHQHUDWRUDFK
NZDUFRZ\FK PD ]PLDQD WHPSHUDWXU\ : FHOX X]\VNDQLD ZL NV]HM
VWDáRFL F] VWRWOLZRFL NRQLHF]QH MHVW VWRVRZDQLH XNáDGyZ
]NRPSHQVDFMZSá\ZyZWHPSHUDWXU\OXEXNáDGyZ]WHUPRVWDWHP
:WHG\ PR*OLZH MHVW X]\VNDQLH VWDáRFL F] VWRWOLZRFL U] GX
(
)
±
−
−
−
10
10
8
10
QDPLHVLFRNUHORQHM]DOH*QRFL
*(1(5$725<5&=(635= )(1,(0=:5271<0
2JyOQDFKDUDNWHU\VW\NDVSU] *RQ\FKJHQHUDWRUyZ5&
*HQHUDWRU\5&]QDOD]á\V]HURNLH]DVWRVRZDQLHGRZ\WZDU]DQLD
SU]HELHJyZ VLQXVRLGDOQ\FK Z ]DNUHVLH PDá\FK F] VWRWOLZRFL
SRQLHZD*ZJHQHUDWRUDFK/&ZDUWRFLSRMHPQRFLLLQGXNF\MQRFL
VWDMVL ]E\WGX*HGREURüREZRGXUH]RQDQVRZHJRMHVWPDáDLHOHPHQW\
WHQLHQDGDMVL GRPLQLDWXU\]DFML*HQHUDWRU\WHZ\NRU]\VW\ZDQHV
QDMF] FLHMZ]DNUHVLHNLONX+]GR0+]
:JHQHUDWRUDFK/&REZyGVSU] *HQLD]ZURWQHJRWZRU]
HOHPHQW\ QDWRPLDVW Z JHQHUDWRUDFK 5& QDMF] FLHM MHVW WR
HOHPHQWyZ FR SR]ZDOD QD ]ZL NV]HQLH LORFL PR*OLZ\FK UR]ZL]D
DWDN*H VWZDU]D GX*H PR*OLZRFL RSW\PDOL]DFML QS SRG NWHP
ZUD*OLZRFL SU]HVWUDMDQLD ]QLHNV]WDáFH F]\ ZUD*OLZRFL QD ]PLDQ\
LPSHGDQFMLZHMFLRZHMLZ\MFLRZHMZ]PDFQLDF]D
6WRVRZDQH V F]ZyUQLNL VHOHNW\ZQH W\SX PRVWHN :LHQD
SRGZyMQH77ERF]QLNRZDQHLLQQHRUD]SU]HVXZQLNLID]RZH5&
:SRUyZQDQLX]JHQHUDWRUDPL/&JHQHUDWRU\5&PDMJRUV]VWDáRü
F] VWRWOLZRFL MHGQDN*H JHQHUXM V\JQDá R EDUG]R PDá\FK
]QLHNV]WDáFHQLDFK L XPR*OLZLDM SU]HVWUDMDQLH F] VWRWOLZRFL
ZVWRVXQNX QD MHGQ\P SRG]DNUHVLH 7DN GX*\ ]DNUHV
SU]HVWUDMDQLDSU]\ZVSyáELH*QHMUHJXODFMLGZyFKSRMHPQRFLOXEGZyFK
UH]\VWDQFMLMHVWPR*OLZ\SRQLHZD*F] VWRWOLZRüZW\FKJHQHUDWRUDFK
MHVW RGZURWQLH SURSRUFMRQDOQD GR LORF]\QX
RC
f
RC
0
1 2
=
/
π
ZJHQHUDWRUDFK
LC
F] VWRWOLZRü MHVW RGZURWQLH SURSRUFMRQDOQD GR
SLHUZLDVWND]LORF]\QX
LC
= WHJR Z]JO GX JHQHUDWRU\
RC
V
SRZV]HFKQLHVWRVRZDQHMDNRJHQHUDWRU\VHUZLVRZHLODERUDWRU\MQH
6HOHNW\ZQHF]ZyUQLNL5&
1DMF] FLHM
VWRVRZQ\P
F]ZyUQLNLHP
VHOHNW\ZQ\P
ZJHQHUDWRUDFK5&MHVWPRVWHN:LHQDSU]HGVWDZLRQ\QDU\V
:PRVWNX:LHQDZ\Uy*QLDVL JDá(VHOHNW\ZQRWUDQVPLWDQFML
β
β
+
+
=
U
U
/
2
SyáPRVWHN:LHQD]HOHPHQWDPL
R C
1
1
,
L
R C
2
2
,
RUD]
JDá( DSHULRG\F]Q R WUDQVPLWDQFML
β
β
−
−
=
U
U
/
2
G]LHOQLN
UH]\VWDQF\MQ\
R R
a
b
,
U
2
R
1
R
2
R
b
R
a
C
1
C
2
U
β
+
U
β
−
U
1
5\V0RVWHN:LHQD
7UDQVPLWDQFM JDá ]LVHOHNW\ZQHM
β
+
RSLVXMH]DOH*QRü>@
β
β
ν
β
+
+
+
+
=
=
+
8
8
M 4
JG]LH
−
RGVWURMHQLH
ν ω
ω
ω
ω
=
−
0
0
D
−
SXOVDFMDSU]\NWyUHM
β
+
RVLJDPDNVLPXP
β
0
+
ω
0
2
1
1
2
2
1
=
C R C R
E
−
PDNVLPXPPRGXáXWUDQVPLWDQFML
β
+
β
0
2
1
1
2
1
1
+
=
+
+
C
C
R
R
F
−
GREURüJDá ]L
Q
d
d
C R
C R
+
+
= −
=
ω
ϕ
ω
β
β
ω
0
0
2
1
1
2
2
0
G
7UDQVPLWDQFMDJDá ]LDSHULRG\F]QHMRNUHORQDMHVW]DOH*QRFL
β
β
−
−
=
=
+
U
U
R
R
R
b
a
b
2
1DSL FLHZ\MFLRZH
U
1
]SU]HNWQHMPRVWNDZ\QRVL
(
)
U
U
U
U
U
U
U
1
2
2
2
2
=
−
=
−
=
−
=
+
−
+
−
+
−
β
β
β
β
β
β
β
JG]LH
β
=
U
U
1
2
/
MHVWWUDQVPLWDQFMQDSL FLRZPRVWND
:SUDNW\F]Q\FK]DVWRVRZDQLDFKQDMF] FLHM
& &
& 5
5
5
=
=
=
=
FRR]QDF]D
ω
β
0
0
1
1 3
1 3
=
=
=
+
+
/
,
/ ,
/
RC
Q
RUD]
(
)
R
R
a
b
= +
2
ε
SU]\F]\P
ε <<
1
'ODSRZ\*V]\FK]DáR*HWUDQVPLWDQFMHJDá ]LRNUHODM]DOH*QRFL
β
ν
β
ε
+
−
=
+
=
+
1
3
1
3
j
3RGVWDZLDMFGRRWU]\PXMHP\
β
ν
ε
ε
ν
ε
ν
ε
ε
ν
+
=
+
−
+
=
−
+
+
1
3
1
3
9
1
1
1
j
j
j
=DNáDGDMF
ν <<
3,
FRMHVWSUDNW\F]QLHVSHáQLRQHGOD
ω ω
=
0
ν =
0
Z]DNUHVLH
ε <<
3
RWU]\PXMHP\
β ε
ν
ε
β
ν
ε
≈
−
=
−
9
1
1
0
j
j
JG]LH
β
ε
0
9
=
/
D
:VSyáF]\QQLN
ε
MHVWPLDUQLH]UyZQRZD*HQLDPRVWNDGODSXOVDFML
ω
0
.
1DU\VSRND]DQRZ\NUHVELHJXQRZ\WUDQVPLWDQFML
β
RUD]
FKDUDNWHU\VW\N DPSOLWXGRZLID]RZPRVWNDSU]\F]\P
ϕ
β
ε
ν
ε
=
≈
−
arg
arg
9
1
j
1D SRGVWDZLH DQDOL]\ WUDQVPLWDQFML PRVWND :LHQD L ZDUXQNyZ
JHQHUDFMLRWU]\PXMHP\SXOVDFM GUJDJHQHUDWRUD
ω
0
1
=
R C
'REURüPRVWND:LHQDZ\]QDF]RQD]QDFK\OHQLDFKDUDNWHU\VW\NL
ID]RZHMZ\QRVL
Q
=
−
=
=
ω
ε ω
ε
β
0
0
0
2
2
1
1
9
6HOHNW\ZQ\]DSRURZRSU]HELHJPRGXáXWUDQVPLWDQFML
β
Z\QLND
W\ONR]VHOHNW\ZQHJRSU]HSXVWRZRSU]HELHJXPRGXáX
β
+
SRQLHZD*
PRGXá
β
−
QLH]DOH*\RGF] VWRWOLZRFL
ν
|
|
β
1 3
/
ε <
0
ε >
0
ε =
0
ε
9
Im
β
Re
β
a)
b)
ε>
0
ε =
0
ε<
0
ω =∞
ω =
0
−
1 3
/
β
0
0
<
β
0
0
=
β
0
0
>
ω
β
ω
o
ω
o
ω
o
arg
β
arg
β
ε =
0
ε >
0
π
−
π
π
/ 2
−
π
/ 2
ε
ν
ν
0
0
π
ε <
0
ε
c)
d )
5\V&KDUDNWHU\VW\NLWUDQVPLWDQFML
( )
β ν
GODPRVWND:LHQDDDPSOLWXGRZR
ID]RZD
( ) ( )
5H
,P
β
β
EPRGXáX
β
FID]\
( )
DUJ
β
= ZDUXQNX DPSOLWXG\
( ) ( )
k
u
ω β ω
0
0
1
≥
SU]\ Z\NRU]\VWDQLX
]DOH*QRFL D RWU]\PXMH VL ZDUWRü
ε
QLH]E GQ GOD
]DSHZQLHQLDJHQHUDFML
( )
ε
ω
≥
9
0
k
u
JG]LH
( )
k
u
ω
0
R]QDF]D Z]PRFQLHQLH F]áRQX Z]PDFQLDMFHJR
JHQHUDWRUD
)D]\ WUDQVPLWDQFML
β
β
i
+
V ]JRGQH 3U]\
ε ≥
PRVWHN QLH
RGZUDFD ID]\ L ZDUXQHN ID]\ MHVW VSHáQLRQ\ GOD SXOVDFML
ω
0
QLH]DOH*QLH RG VWRSQLD QLH]UyZQRZD*HQLD PRVWND 5H]\VWRU\
R R
a
b
,
SUDFXM]Z\NOHZJDá ]LDXWRPDW\F]QHMUHJXODFMLZ]PRFQLHQLD$5:
JHQHUDWRUD
2SUyF]PRVWND:LHQDF] VWRVWRVRZDQ\PXNáDGHPVHOHNW\ZQ\P
Z JHQHUDWRUDFK 5& MHVW F]ZyUQLN SRGZyMQH 7 SU]HGVWDZLRQ\ QD
U\V
C
C
C n
/
R
R
nR
U
2
U
β
5\V&]ZyUQLN77
7UDQVPLWDQFMDQDSL FLRZDF]ZyUQLNDSRGZyMQH7Z\QRVL
( )
( )
(
)
(
)
β ν
β ν
ν
ν
β
u
U
U
n
j
n
n
j
=
=
=
− +
+ +
+
+
2
2
1
2
1 1 /
JG]LH
ν
ω
ω
ω
ω
ω
=
−
=
0
0
0
1
,
R C
)XQNFMDWUDQVPLWDQFMLSRVLDGDGZD]HUDLGZDELHJXQ\:\NUHV\
ELHJXQRZHRUD]FKDUDNWHU\VW\NLDPSOLWXGRZHLID]RZHF]ZyUQLNDGOD
Uy*Q\FK ZDUWRFL n SU]HGVWDZLRQR QD U\V 3U]HELHJ
FKDUDNWHU\VW\N F] VWRWOLZRFLRZ\FK F]ZyUQLNyZ SRGZyMQH 7
ZVND]XMH QD LFK SU]\GDWQRü Z XNáDGDFK JHQHUDF\MQ\FK MHG\QLH
ZyZF]DVJG\n
<
0 5
,
6WRSLH QLH]UyZQRZD*HQLD F]ZyUQLND SRGZyMQH 7 ]DOH*\ RG
ZDUWRFL n SU]\ F]\P GOD n
=
0 5
,
XNáDG MHVW ]UyZQRZD*RQ\
=DNáDGDMFQLHZLHONLHUR]UyZQRZD*HQLHF]ZyUQLNDSU]\n
=
0 5
,
(
)
n
=
+
<<
0 5 1
1
,
'
'
ε
ε
dla
WR WUDQVPLWDQFMD QDSL FLRZD
β
u
Z\]QDF]RQD ] ]DOH*QRFL
LRNUHORQDMHVW]DOH*QRFL
β
ε
ν
ν
ε
ν
ε
u
j
j
j
'
'
'
'
= +
+
≈
+
4
4
1
'OD
ε
'
>
0
F]ZyUQLNQLHRGZUDFDID]\]DGOD
ε
'
<
0
RGZUDFDID]
R
R
L
β
ε
0
4
'
'
/
=
6WRVRZDQLH WDNLHJR QLH]UyZQRZD*RQHJR F]ZyUQLND SRGZyMQH 7 Z
REZRG]LHVSU] *HQLD]ZURWQHJRJHQHUDWRUD5&FKRFLD* WHRUHW\F]QLH
PR*OLZHQLHMHVWGRJRGQHZ SUDNW\FH SRQLHZD* UHJXODFMD VWRSQLD
QLH]UyZQRZD*HQLD
( )
ε
'
Z\PDJDáDE\ MHGQRF]HVQHM ]PLDQ\ GZyFK
HOHPHQWyZ UH]\VWDQFML LSRMHPQRFL U\V 'ODWHJR F]ZyUQLN
SRGZyMQH7 VWRVXMH VL
QDMF]
FLHM MDNR VHOHNW\ZQ\ XNáDG
]UyZQRZD*RQ\ W]Q SU]\ n
=
=
=
0 5
0
0
0
, ,
,
'
'
ε
β
ZáF]RQ\ Z
JDá(XMHPQHJRVSU] *HQLD]ZURWQHJR]DGRGDWQLHVSU] *HQLH]ZURWQH
SRWU]HEQH GR VSHáQLHQLD ZDUXQNX JHQHUDFML ]DSHZQLD VL SU]H]
]DVWRVRZDQLHGRGDWNRZHMJDá ]LUH]\VWDQF\MQHM
ν
|
|
β
1
n
=
0 5
,
n
≠
0 5
,
0
ω
0
1
=
/ RC
2
1
2
1 1
n
n
n
−
+ +
/
( )
Im
β
u
( )
Re
β
u
β
u0
n
=
0 5
,
n
>
0 5
,
n
<
0 5
,
1
ω = ∞
ω
=
0
ω
a)
b)
arg
β
u
ν
ν
arg
β
u
π
/ 2
−
π
/ 2
0
n
=
0 5
,
n
>
0 5
,
c)
n
<
0 5
,
n
→
0 5
,
0
−
π
−
2
π
5\V&KDUDNWHU\VW\NLWUDQVPLWDQFML
β
β
X
=
+
GODF]ZyUQLND77
DDPSOLWXGRZRID]RZD
( ) ( )
5H
,P
β
β
X
X
EPRGXáX
β
X
FID]\
( )
DUJ
β
X
7UDQVPLWDQFMD QDSL FLRZD
β
u
WDN SRZVWDáHJR PRVWND
]áR*RQHJR ] F]ZyUQLND SRGZyMQH 7 L G]LHOQLND UH]\VWDQF\MQHJR
R R
a
b
,
MHVWUyZQD
(
)
( )
−
= −
−
= −
+
+
+
−
=
β
β
β
β ν
u
u
n
b
a
b
R
R
R
0 5
,
-H*HOL
R
R
R
b
a
b
+
= ε
4
WR]]DOH*QRFLLRWU]\PXMHP\
−
= −
+
≈ −
+
≈
−
β
ε
ν
ν
ε
ν
ν
ε
ν
ε
u
j
j
j
j
j
4
4
4
4
1
= SRUyZQDQLD ]DOH*QRFL L Z\QLND *H XNáDG
SRGZyMQH7PDSRGREQHZáDFLZRFLMDNPRVWHN:LHQD]WUy*QLF
*HID]D
β
u
MHVWSU]HFLZQDRUD]LQQ\MHVWZVSyáF]\QQLN
β
0
ZPRVWNX
:LHQD
β
ε
0
9
=
/
ZF]ZyUQLNXSRGZyMQH7
β
ε
0
4
= −
/
6WUXNWXU\JHQHUDWRUyZ5&
: JHQHUDWRU]H 5& VNáDGDMF\P VL ]H Z]PDFQLDF]D
RZ]PRFQLHQLX k
k
j
u
u
k
=
exp
ϕ
RUD] VHOHNW\ZQHJR F]ZyUQLND
VSU] *HQLD ]ZURWQHJR R WUDQVPLWDQFML
β β
ϕ
β
=
uu
j
exp
SRZVWDM
GUJDQLD JG\ VSHáQLRQ\ MHVW ZDUXQHN DPSOLWXG\ k
u
u
β =
1
RUD]
ZDUXQHNID]\
ϕ
ϕ
π
β
k
n
+
=
2
1DMSURVWV]DVWUXNWXUDJHQHUDWRUD5&
VNáDGD VL
]H Z]PDFQLDF]D QLHRGZUDFDMFHJR L MHGQHM JDá
]L
VHOHNW\ZQHJR VSU]
*HQLD ]ZURWQHJR
β
+
QS F]ZyUQLN 5& W\SX
SRáRZDPRVWND:LHQD
6FKHPDW EORNRZ\ WDNLHJR XNáDGX RUD] Z\NUHV ELHJXQRZ\
]DVWRVRZDQHJR F]ZyUQLND VSU] *HQLD ]ZURWQHJR SU]HGVWDZLRQR QD
U\VDE 2 VWDáRFL F]
VWRWOLZRFL GUJD JHQHUDWRUD Z JáyZQHM
PLHU]H GHF\GXMH GREURü ID]RZD Q
+
JDá ]L VHOHNW\ZQHM ]DOH*QRü
G ,P ZL NV]D MHVW GREURü Q
+
W\P OHSV]D MHVW VWDáRü
F]
VWRWOLZRFL-HGQDNVHOHNW\ZQHF]ZyUQLNL5&PDMPDáHZDUWRFL
GREURFLQ
+
QDMF] FLHMQ
+
k
u
>
β
+
Im(
)
β
+
Re(
)
β
+
ϕ
β
1/ k
u
ω
ω
0
ω
=
0
ω
= ∞
a)
b)
d )
c)
k
u
>
β
−
Re(
)
β
+
Im(
)
β
+
ϕ
β
A
'
A
0
ω
0
1 / k
u
β
−
β
+
β
+
1 /
'
k
u
β
+
5\V6FKHPDW\EORNRZHJHQHUDWRUyZ5&D]MHGQS WOF]GZRPDS WODPLL
LFKFKDUDNWHU\VW\NLDPSOLWXGRZRID]RZHRGSRZLHGQLRELG
'REURüID]RZ Q JHQHUDWRUD5&PR*QD]QDF]QLH]ZL NV]\ü
SU]H] SRZL NV]HQLH Z]PRFQLHQLD
k
u
F]áRQX Z]PDFQLDMFHJR
JHQHUDWRUD GR ZDUWRFL k
u
'
SU]\ UyZQRF]HVQ\P ZSURZDG]HQLX
GRGDWNRZHMJDá ]L
β
−
DSHULRG\F]QHJRXMHPQHJRVSU] *HQLD]ZURWQHJR
6FKHPDW EORNRZ\ XNáDGX SU]HGVWDZLRQR QD U\VF SU]\ F]\P
WUDQVPLWDQFMH JDá ]L VSU] *HQLD ]ZURWQHJR PDM WDN SRVWDü MDN GOD
UR]ZD*DQHJR ZF]HQLHM PRVWND :LHQD *UDQLF]Q\ DPSOLWXGRZ\
ZDUXQHNJHQHUDFMLRNUHORQ\MHVW]DOH*QRFL
( )
[
]
(
)
k
k
k
u
u
u
'
'
'
β ω
β
β
β
β
+
−
−
−
=
−
=
=
0
0
1
JG]LH
( )
( )
β β ω
β
β
β ω
=
−
=
+
−
+
0
0
0
,
3RUyZQXMF]DOH*QRFLGLZLG]LP\*H]ZL NV]HQLH
Z]PRFQLHQLDSU]\UyZQRF]HVQ\PZSURZDG]HQLXJDá
]LDSHULRG\F]QHJR
XMHPQHJR VSU]
*HQLD ]ZURWQHJR
β
−
SRZRGXMH SRZL NV]HQLH GREURFL
FDáHJRXNáDGXGRZDUWRFL
Q
k
k
Q
u
u
=
+
'
:SURZDG]HQLH JDá ]L
β
−
Z LVWRWQ\ VSRVyE SROHSV]D
ZáDFLZRFLJHQHUDWRUDL]WHJRSRZRGXJHQHUDWRU\5&VQDMF] FLHM
UHDOL]RZDQH]GZLHPDJDá ]LDPLVSU] *HQLD]ZURWQHJR
1D U\V SU]HGVWDZLRQR VFKHPDW EORNRZ\ JHQHUDWRUD 5&
]V\PHWU\F]Q\PPRVWNLHPSRGZyMQH7:JDá ]LXMHPQHJRVSU] *HQLD
]ZURWQHJR]QDMGXMHVL F]ZyUQLNSRGZyMQH7]DZJDá ]LGRGDWQLHJR
VSU] *HQLD ]ZURWQHJR ]DVWRVRZDQR DSHULRG\F]Q\ G]LHOQLN
UH]\VWDQF\MQ\
β
+
k
u
>
β
−
β
+
Re(
)
β
+
Im(
)
β
+
ω = ∞
ω =
0
ω
ω
0
β
0
ω =
0
ω = ∞
Im(
)
β
Re(
)
β
ω
0
β
a)
b)
c)
1
5\V 6FKHPDW EORNRZ\ JHQHUDWRUD ] F]ZyUQLNLHP SRGZyMQH 77 L MHJR
FKDUDNWHU\VW\NLDPSOLWXGRZRID]RZH
*UDQLF]Q\ DPSOLWXGRZ\ ZDUXQHN JHQHUDFML RNUHORQ\ MHVW
]DOH*QRFL
( )
[
]
(
)
k
k
k
u
u
u
β
β ω
β
β
β
−
+
−
−
=
−
=
=
0
0
1
*HQHUDWRU\ R WHM VWUXNWXU]H G]LDáDM ZVNXWHN ]DQLNX XMHPQHJR
VSU] *HQLD ]ZURWQHJR UHDOL]RZDQHJR SU]H] VHOHNW\ZQ\ F]ZyUQLN
URGNRZRSU]HSXVWRZ\ZSREOL*XJHQHURZDQHMSXOVDFML
ω
0
LSU]HZDJL
QDGQLPGRGDWQLHJRVSU] *HQLD]ZURWQHJRUHDOL]RZDQHJRSU]H]G]LHOQLN
UH]\VWDQF\MQ\
5R]ZL]DQLDXNáDGRZHJHQHUDWRUyZ5&
1D U\V SU]HGVWDZLRQR SU]\NáDG\ UR]ZL]D XNáDGRZ\FK
JHQHUDWRUyZ 5& ] PRVWNLHP :LHQD L ] PRVWNLHP SRGZyMQH 7
ZNWyU\FKZS
WOL
β
−
]DVWRVRZDQR XNáDG\ DXWRPDW\F]QHM UHJXODFML
DPSOLWXG\GUJDJHQHUDWRUDS WODWDQRVLQD]Z $5:$XWRPDW\F]QD
5HJXODFMD:]PRFQLHQLD
: SU]HGVWDZLRQ\FK SU]\NáDGDFK XNáDG\ $5: ]RVWDá\
]UHDOL]RZDQH Z SRVWDFL VWHURZDQ\FK HOHNWU\F]QLH G]LHOQLNyZ
UH]\VWDQF\MQ\FK )XQNFM UHJXORZDQ\FK HOHNWU\F]QLH UH]\VWDQFML
VSHáQLDM WUDQ]\VWRU\ -)(7 SUDFXMFH Z REV]DU]H OLQLRZ\P
WULRGRZ\P Z NWyU\FK ]DVWRVRZDQR OLQHDU\]DFM FKDUDNWHU\VW\N
Z\MFLRZ\FKSU]\SRPRF\XMHPQHJRVSU] *HQLD]ZURWQHJRUH]\VWRU\
R R
2
7
,
a)
b)
R
R
R
R
R / 2
C
C
C
1
C
2
U
o
R
4
R
3
R
2
R
7
R
G
R
6
−
U
GG
−
U
GS
JFET
JFET
D
1
D
2
U
C2
P
2
C
R
4
P
2
R
3
R
2
R
7
R
6
C
2
D
1
U
C 2
C
1
D
2
R
5
R
5
U
GS
U
o
C
C
5\V*HQHUDWRU\5&D]PRVWNLHP:LHQDE]PRVWNLHPSRGZyMQH7
=HZ]URVWHPDPSOLWXG\QDSL FLDZ\MFLRZHJR
U
o
URQLHEH]Z]JO GQD
ZDUWRü QDSL FLD VWDáHJR QD NRQGHQVDWRU]H
C
2
RWU]\P\ZDQHJR
]GZXSRáyZNRZHJR GHWHNWRUD ZDUWRFL V]F]\WRZHM WHJR QDSL FLD
]GLRGDPL
D D
1
2
,
:JHQHUDWRU]H]PRVWNLHP:LHQDQDSL FLHQDNRQGHQVDWRU]H
C
2
MHVW XMHPQH L EH]SRUHGQLR VWHUXMH EUDPN WUDQ]\VWRUD -)(7
=DWHP JG\ DPSOLWXGD QDSL FLD Z\MFLRZHJR URQLH WR URQLH XMHPQH
QDSL FLHQDEUDPFH-)(7URQLHMHJRUH]\VWDQFMDVWDW\F]QD
r
DS
URQLH
XMHPQH VSU] *HQLH ]ZURWQH L Z HIHNFLH PDOHMH DPSOLWXGD QDSL FLD
Z\MFLRZHJR
:JHQHUDWRU]H]PRVWNLHPSRGZyMQH7WUDQ]\VWRU-)(7MHVW
SRODU\]RZDQ\ZVW SQLHXMHPQ\PQDSL FLHP
−
U
GG
DGZXSRáyZNRZ\
GHWHNWRUV]F]\WRZ\Z\WZDU]DQDSL FLHGRGDWQLH=DWHPJG\DPSOLWXGD
QDSL FLD Z\MFLRZHJR URQLH WR PDOHMH Z\SDGNRZH XMHPQH QDSL FLH
QD EUDPFH WUDQ]\VWRUD -)(7 PDOHMH MHJR UH]\VWDQFMD VWDW\F]QD
r
DS
PDOHMHGRGDWQLHVSU] *HQLH]ZURWQHZXNáDG]LHLZHIHNFLHPDOHMH
DPSOLWXGDQDSL FLDZ\MFLRZHJR