6=80<:8.à$'$&+(/(.7521,&=1<&+
'5Ï'à$,52'=$-(6=80Ï:
8ZDJLZVW SQH
6]XP\ V HIHNWHP SU]\SDGNRZ\FK IOXNWXDFML HOHNWU\F]Q\FK
SRZVWDMF\FK ZVNXWHN ]LDUQLVWHM VWUXNWXU\ HOHPHQWyZ EG( áDGXQNyZ
HOHNWU\F]Q\FK%H]áDGQ\UXFKáDGXQNyZSRZRGXMH*HFKZLORZDZDUWRü
SUGX OXE QDSL FLD IOXNWXXMH SU]\SDGNRZR ZRNyá UHGQLHM ZDUWRFL
&KDUDNWHULSR]LRPW\FKIOXNWXDFML]DOH*\RGURG]DMXHOHPHQWXVSRVREX
Z\NRQDQLD L ZDUXQNyZ MHJR SUDF\ :\SDGNRZ\ V]XP XNáDGX
HOHNWURQLF]QHJR MHVW Z\QLNLHP VXSHUSR]\FML SRV]F]HJyOQ\FK URG]DMyZ
V]XPyZ SRZVWDMF\FK Z ND*G\P HOHPHQFLH WHJR XNáDGX 1DMZL NV]\
SR]LRP V\JQDáX Z XNáDG]LH Z\]QDF]DM FKDUDNWHU\VW\NL REZRGX
MHGQDN*HQDMPQLHMV]UR]Uy*QLDOQZDUWRüV\JQDáXGHWHUPLQXMHSR]LRP
V]XPyZ 6NXWNL V]XPyZ XMDZQLDM VL Z VWRSQLX Z\MFLRZ\P WRUX
HOHNWURQLF]QHJR QDWRPLDVW (UyGáD V]XPyZ WNZL Z ZHMFLRZHM
QLVNRV]XPQHM F] FL XNáDGX 3RVW S Z PLQLPDOL]RZDQLX V]XPyZ
HOHPHQWyZ L XNáDGyZ HOHNWURQLF]Q\FK MHVW QLHZLHONL Z SRUyZQDQLX ]
SRVW SHP WHFKQRORJLL Z\WZDU]DQLD HOHPHQWyZ L XNáDGyZ L MHVW
RJUDQLF]RQ\ VDP QDWXU V]XPyZ 6]XP MHVW V\JQDáHP FDáNRZLFLH
VWRFKDVW\F]Q\P-HVWRQXWZRU]RQ\]HVNáDGRZ\FKF] VWRWOLZRFLRZ\FK
R SU]\SDGNRZHM DPSOLWXG]LH L ID]LH 0R*HP\ ]PLHU]\ü ZDUWRü
VNXWHF]Q V]XPX MHGQDN*H QLH PR*HP\ SU]HZLG]LHü MHJR ZDUWRFL
FKZLORZHM 0R*OLZ\ MHVW RSLV V]XPX SRM FLDPL WHRULL
SUDZGRSRGRELHVWZD 'OD SU]HELHJyZ VWRFKDVW\F]Q\FK RNUHOD VL
JáyZQHSDUDPHWU\
−
ZDUWRüRF]HNLZDQZDUWRüUHGQL
( )
[ ]
( ) ( )
( L W
L W
7
L W S L W GW
7
7
=
→∞
−
+
∫
OLP
7
−
ZDULDQFM NZDGUDWZDUWRFLVNXWHF]QHMVNáDGRZHM]PLHQQHM
( ) ( )
[
]
( ) ( )
[
]
( )
σ
2
2
2
2
2
1
=
−
=
−
→∞
−
+
∫
E i t
i t
T
i t
i t
p i t dt
T
T
T
lim
,
/
/
: ]DOH*QRFLDFK RUD] V\PEROÄ ´R]QDF]D
XUHGQLHQLH Z F]DVLH D
( )
p i t
,
MHVW IXQNFM J VWRFL
SUDZGRSRGRELHVWZDUR]NáDGXDPSOLWXG\SU]HELHJX
2PyZLRQHZGDOV]HMF] FLURG]DMHV]XPyZ]Z\MWNLHPV]XPX
Z\EXFKRZHJR
PDM
QRUPDOQ\
JDXVVRZVNL
UR]NáDG
SUDZGRSRGRELHVWZDFKZLORZHMDPSOLWXG\SU]HELHJXV]XPRZHJR
( )
p i t
i
,
exp
=
−
1
2
2
2
2
σ
π
σ
5\V LOXVWUXMH SRM FLH UR]NáDGX SUDZGRSRGRELHVWZD
DPSOLWXG\SU]HELHJXV]XPRZHJR
S ,
,
[
]
S L W
,
>
,
W
L W
5\V5R]NáDGSUDZGRSRGRELHVWZDDPSOLWXG\SU]HELHJXV]XPRZHJR
UHGQLD ZDUWRü IOXNWXDFML SUGX V]XPyZ MHVW UyZQD ]HUX =H
Z]JO GX QD PHFKDQL]P SRZVWDZDQLD V]XPyZ QDMRJyOQLHM PR*QD
SRG]LHOLü V]XP\ QD FLHSOQH UXWRZH VWUXNWXUDOQH QD]\ZDQH UyZQLH*
V]XPDPLPLJRWDQLDOXEW\SX
f
1
Z\EXFKRZHLODZLQRZH
6]XP\FLHSOQH
6]XP FLHSOQ\ QD]\ZDQ\ UyZQLH* V]XPHP -RKQVRQD SRZVWDMH
ZVNXWHN SU]\SDGNRZ\FK GUJD FLHSOQ\FK QRQLNyZ áDGXQNX Z
SU]HZRGQLNDFK
:WHPSHUDWXU]HSRZ\*HM]HUDEH]Z]JO GQHJRHOHNWURQ\]QDMGXM
VL Z XVWDZLF]Q\P EH]ZáDGQ\P UXFKX NWyUHJR HQHUJLD ]DOH*\ RG
WHPSHUDWXU\ = ND*G\P HOHNWURQHP SRUXV]D VL áDGXQHN
e
=
⋅
−
1 59 10
19
,
& ]DWHP SRMDZLD VL RJURPQD LORü HOHPHQWDUQ\FK
LPSXOVyZ SUGRZ\FK ]ZL]DQ\FK ] UXFKHP SRMHG\QF]\FK áDGXQNyZ
)OXNWXDFMH WHJR SUGX SRZRGXM SRZVWDZDQLH QD NRFyZNDFK
SU]HZRGQLND Uy*QLF\ SRWHQFMDáyZ R ZDUWRFL UHGQLRNZDGUDWRZHM VLá\
HOHNWURPRWRU\F]QHM e
T
2
OXE UyZQRZD*QHJR (UyGáD SUGX i
T
2
RNUHORQ\FK]DOH*QRFL1\TXLVWD
u
k T R f
T
2
4
=
∆
D
i
k T
R
f
T
2
4
1
=
∆
E
:SRZ\*V]\FK]DOH*QRFLDFKk MHVWVWDá%ROW]PDQD
1D U\V SU]HGVWDZLRQR VFKHPDW ]DVW SF]\ UH]\VWRUD ]
XZ]JO GQLHQLHP ]DVW SF]HJR (UyGáD V]XPRZHJR QDSL FLRZHJR OXE
SUGRZHJR
u
kTR f
T
=
4
∆
R bezszumny
(
)
a)
b)
bezszumny
R
i
kT f
R
T
=
4
∆
5\V6FKHPDW\]DVW SF]HUH]\VWRUDXZ]JO GQLDMFH(UyGáDV]XPyZFLHSOQ\FK
* VWRüZLGPRZDV]XPXFLHSOQHJRMHVWVWDáDD*GRF] VWRWOLZRFL
>+]@ 6]XPFLHSOQ\MHVW]DWHPÄV]XPHPELDá\P´5H]\VWDQFMD
R
]DZDUWD Z UyZQDQLX MHVW U]HF]\ZLVW F] FL LPSHGDQFML
]HVSRORQHM HOHPHQWX : WHPSHUDWXU]H SRNRMRZHM . J VWRü
ZLGPRZDV]XPyZFLHSOQ\FKUH]\VWRUDN
Ω
Z\QRVL
[
]
u
f
V
Hz
T
2
18
2
16 10
∆ = ⋅
−
FR R]QDF]D *H QDSL FLH V]XPyZ WHJR
UH]\VWRUDZSDPLH+]PDZDUWRü Q 9 +]
6]XP\UXWRZH
6]XP\UXWRZHQD]\ZDQHWH*V]XPDPL6FKRWWN\¶HJR]ZL]DQH
V]DZV]H]SU]HSá\ZHPVNáDGRZHMVWDáHMSUGXSU]H]EDULHU SRWHQFMDáX
6]XP\ WH SRZVWDM Z ]áF]RZ\FK SU]\U]GDFK SyáSU]HZRGQLNRZ\FK
ZVNXWHNIOXNWXDFMLG\IX]MLJáyZQLHQRQLNyZPQLHMV]RFLRZ\FKF]\OL
QLHUHJXODUQHJR SU]HFKRG]HQLD QRQLNyZ SU]H] EDULHU\ SRWHQFMDáX
DFLOHMSU]H]REV]DU\áDGXQNXSU]HVWU]HQQHJRQDVW\NXREV]DUyZSQ
SyáSU]HZRGQLND 3RGREQ\ HIHNW Z\ZRáXMH SU]HMFLH QRQLNyZ SU]H]
VWUHI\JUDQLF]QHPHWDOSyáSU]HZRGQLNEDULHU\6FKRWWN\¶HJROXEPHWDO
L]RODWRU
3UGSá\QF\SU]H]]áF]H]DZLHUDGZLHVNáDGRZH]NWyU\FKMHGQD
Z\ZRáDQD MHVW SU]H] QRQLNL PQLHMV]RFLRZH JHQHURZDQH FLHSOQLH
ZZDUVWZLH]DSRURZHMQDWRPLDVWGUXJDVNáDGRZDRGJU\ZDMFDJáyZQ
URO SU]\ SRODU\]DFML Z NLHUXQNX SU]HZRG]HQLD UHSUH]HQWXMH G\IX]M
QRQLNyZ ZL NV]RFLRZ\FK SU]H] EDULHU SRWHQFMDáX QD ]áF]X
REQL*RQ QD VNXWHN GRSURZDG]HQLD GR ]áF]D ]HZQ WU]QHJR QDSL FLD
SRODU\]XMFHJR
2ELH VNáDGRZH SUGX Z\ZRáXM SHáQ\ V]XP UXWRZ\ L FKRü
NLHUXQNLLFKSU]HSá\ZXVUy*QHWRMHGQDNUHGQLRNZDGUDWRZHZDUWRFL
V]XPyZSU]H]QLHZ\WZRU]RQ\FKGRGDMVL
0R*QD Z\ND]Dü *H MH*HOL SUG
( )
i t
]áR*RQ\ MHVW ] VHULL
SU]\SDGNRZ\FK QLH]DOH*Q\FK LPSXOVyZ R UHGQLHM ZDUWRFL I
Q
WR
UHGQLRNZDGUDWRZDZDUWRüV]XPyZUXWRZ\FKZ\QRVL
[ ]
i
q I
f A
s
Q
2
2
2
=
∆
JG]LH
q
áDGXQHNHOHNWURQX
&
I
Q
VNáDGRZDVWDáDSUGX>$@
∆
f
V]HURNRüSDVPD>+]@
=UyZQDQLDZ\QLND*HUHGQLRNZDGUDWRZDZDUWRüSUGX
V]XPyZ UXWRZ\FK MHVW ZSURVW SURSRUFMRQDOQD GR V]HURNRFL SDVPD
∆
f
F] VWRWOLZRFLSRPLDUX
:\QLND VWG *H J VWRü ZLGPRZD PRF\ JHQHUDWRUD SUGRZHJR
V]XPyZ
i
f
q I
s
Q
2
2
∆
=
MHVW VWDáD Z IXQNFML F] VWRWOLZRFL F]\OL V]XP
WHJR W\SX MHVW V]XPHP ELDá\P 6WDáD J VWRü ZLGPRZD Z\VW SXMH Z
SU]HG]LDOH F] VWRWOLZRFL ]QDF]QLH PQLHMV]\FK RG
1
τ
F]\OL RG
F] VWRWOLZRFL Z\QLNDMFHM ] F]DVX SU]HORWX QRQLNyZ
τ
SU]H]
ZDUVWZ ]XER*RQ ]áF]D VWDQRZLF MHG\QH PLHMVFH RGG]LDá\ZDQLD
SROD]HZQ WU]QHJRQDQRQLNLDNW\ZQH
6]XP\VWUXNWXUDOQHV]XP\
f
1
OXEV]XP\PLJRWDQLD
7HUPLQHP W\P RNUHOD VL VNáDGRZ V]XPyZ GRPLQXMF Z
]DNUHVLH PDá\FK F]
VWRWOLZRFL R J
VWRFL ZLGPRZHM PRF\
SURSRUFMRQDOQHM GR 1 f
b
]Z\NOH
b
≈
1
VWG LFK QD]ZD 6]XP WHQ
Z\VW SXMH ]DUyZQR Z SU]\U]GDFK SyáSU]HZRGQLNRZ\FK MDN WH* L Z
HOHPHQWDFK ELHUQ\FK 3RZVWDZDQLH V]XPX
f
1
PR*QD SU]\SLVDü Z
GHF\GXMF\P VWRSQLX HIHNWRP SRZLHU]FKQLRZ\P D JáyZQLH SURFHVRP
SXáDSNRZDQLDVZRERGQ\FKQRQLNyZ]ZL]DQ\FK]SRZROQ\PLVWDQDPL
SRZLHU]FKQLRZ\PL 3HZLHQ ZNáDG Z V]XP\
f
1
GDM FHQWUD
JHQHUDF\MQR UHNRPELQDF\MQH ]QDMGXMFH VL Z REV]DU]H ]XER*RQ\P
SyáSU]HZRGQLND : HOHPHQWDFK HOHNWURQLF]Q\FK V]XP
f
1
PR*H
Z\VW SRZDüMHG\QLHZZDUXQNDFKLFKSRODU\]DFML
UHGQLRNZDGUDWRZ ZDUWRü SUGX V]XPyZ
f
1
Z SDPLH
∆
f
RNUHOD]DOH*QRü
i
K
I
f
f
f
Q
a
b
2
1
=
∆
JG]LH
I
Q
VNáDGRZDVWDáDSUGX
K
1
VWDáDGODGDQHJRSU]\U]GXSyáSU]HZRGQLNRZHJRPR*HE\üUy*QD
QDZHWGODWUDQ]\VWRUyZWHJRVDPHJRW\SX
a
VWDáDZSU]HG]LDOHGR
b
VWDáDUyZQDSUDZLHMHGQRFL
* VWRü ZLGPRZ V]XPyZ I SU]HGVWDZLRQR QD U\V
&KRFLD* J VWRü ZLGPRZD PRF\ V]XPyZ I MHVW QDMZL NV]D SU]\
PDá\FKF] VWRWOLZRFLDFKWRMHGQDNZSU]\U]GDFKZ\ND]XMF\FKGX*\
SR]LRPW\FKV]XPyZPRJRQHE\üGRPLQXMF\PVNáDGQLNLHPV]XPyZ
ZSDVPLHRGPDá\FKF] VWRWOLZRFLD*GR]DNUHVX0+]
skala log
skala log
i
f
f
2
∆
1
f
f
5\V* VWRüZLGPRZD
V]XPyZ I
6]XP\Z\EXFKRZH
7HQ W\S QLVNRF] VWRWOLZRFLRZ\FK V]XPyZ PD V]F]HJyOQH
]QDF]HQLH Z XNáDGDFK VFDORQ\FK JáyZQLH ZH Z]PDFQLDF]DFK
RSHUDF\MQ\FKDOHZ\VW SXMHUyZQLH*ZXNáDGDFKG\VNUHWQ\FK
t
Amplituda szumów
f
s k a l a
log
f
c
1
f
i
f
w
2
∆
b)
a)
5\V D 7\SRZ\ RVF\ORJUDP V]XPyZ Z\EXFKRZ\FK E J VWRü ZLGPRZD
V]XPyZZ\EXFKRZ\FK
0HFKDQL]PLFKSRZVWDZDQLDQLHMHVWZSHáQLZ\MDQLRQ\DOHZL*HVL
RQ]REHFQRFL]DQLHF]\V]F]HZSRVWDFLMRQyZFL *NLFKPHWDOL
1D]ZD ÄV]XP Z\EXFKRZ\´ Z\ZRG]L VL VWG *H REVHUZRZDQ\ QD
RVF\ORVNRSLHSU]HELHJV]XPRZ\PDNV]WDáWQLHVWDFMRQDUQ\FK IOXNWXDFML
W\SXLPSXOVRZHJRSRMDZLDMF\FKVL FRSHZLHQF]DV
&] VWRWOLZRüLPSXOVyZV]XPXZ\EXFKRZHJRPLHFLVL Z
SDPLHF] VWRWOLZRFLDNXVW\F]Q\FKDMHJRJ VWRüZLGPRZDPR*HE\ü
RSLVDQD]DOH*QRFL
i
K
I
f
f
f
w
Q
c
c
2
2
2
1
=
+
∆
JG]LH
K
2
VWDáDFKDUDNWHU\VW\F]QDGODGDQHJRSU]\U]GX
I
Q
VNáDGRZDVWDáDSUGXSRODU\]DFML
c
VWDáDZJUDQLFDFKGR
f
c
F] VWRWOLZRü FKDUDNWHU\VW\F]QD ]DOH*QD RG PHFKDQL]PX
SRZVWDZDQLDV]XPyZ
6]XP\ODZLQRZH
3RZVWDZDQLH V]XPyZ ODZLQRZ\FK ZL*H VL ] SU]HFKRG]HQLHP
QRQLNyZSU]H]]áF]DSQVSRODU\]RZDQH]DSRURZR:W\PSU]\SDGNX
]DSRURZHQDSL FLHSRODU\]DFMLPXVLE\üMHGQDNQDW\OHGX*HDE\ SROH
HOHNWU\F]QH Z ZDUVWZLH ]DSRURZHM ]DSHZQLDáR QRQLNRP SU]\
]GHU]HQLX HQHUJL Z\VWDUF]DMF GOD ODZLQRZHJR XZROQLHQLD SDU\
HOHNWURQG]LXUD.D*GH]GHU]HQLHZ\WZDU]DLPSXOVSUGRZ\q /
τ
SU]\
F]\P
τ
MHVW UHGQLP F]DVHP SRPL G]\ ]GHU]HQLDPL 6]XP ]MDZLVND
ODZLQRZHJRPR*HE\üRSLVDQ\]DSRPRFPRGHOX+LQHVDSU]\SRPRF\
NWyUHJRRNUHODVL J VWRüZLGPRZWDNLHJRV]XPX
(
)
i
q I
f
z
2
2
2
2
=
π τ
I
R]QDF]DUHGQLZDUWRüSUGX]áF]DVSRODU\]RZDQHJR]DSRURZR
3U]\MPXMHVL *HUR]NáDGZDUWRFLFKZLORZ\FKSU]HELHJyZV]XPX
ODZLQRZHJRMHVWQRUPDOQ\
6=802:(02'(/(35=<5='Ï:3Ïà35=(:2'1,
.2:<&+
'LRGDSyáSU]HZRGQLNRZD
6]XPRZ\ VFKHPDW ]DVW SF]\ GLRG\ SyáSU]HZRGQLNRZHM MHVW
SRND]DQ\QDU\V
T
K
=
0
i
2
U
6
X
V
T
K
=
0
U
N7
T,
G
'
=
i
D
U
G
5\V6]XPRZ\VFKHPDW]DVW SF]\
GLRG\SyáSU]HZRGQLNRZHM
3U]\ SRODU\]DFML Z NLHUXQNX SU]HZRG]HQLD PDáRV\JQDáRZ\
VFKHPDW ]DVW SF]\ GLRG\ ]DZLHUD UH]\VWDQFM Uy*QLF]NRZ
r
d
L
UH]\VWDQFM V]HUHJRZ
r
S
3RQLHZD*
U
6
MHVW UH]\VWDQFM IL]\F]Q
GODWHJRQDOH*\GRáF]\üGRQLHM(UyGáRQDSL FLRZHV]XPyZWHUPLF]Q\FK
5yZQROHJOH GR
r
d
QDOH*\ GRáF]\ü (UyGáR SUGRZH UHSUH]HQWXMFH
V]XP\UXWRZHLV]XP\
f
1
u
k T r
f
s
s
2
4
=
∆
i
q I
f
K
I
f
f
D
D
a
2
2
=
+
∆
∆
5H]\VWDQFMH
r
S
L
r
d
ZVFKHPDFLH]DVW SF]\PVEH]V]XPQH
7UDQ]\VWRUELSRODUQ\
6]XPRZ\ PRGHO WUDQ]\VWRUD ELSRODUQHJR Z MHJR DNW\ZQ\P
REV]DU]H SUDF\ WZRU]\ VL Z WHQ VSRVyE *H GR VFKHPDWX ]DVW SF]HJR
K\EU\G
π
GRáF]DP\ V\PEROLF]QH (UyGáD V]XPyZ UHSUH]HQWXMFH
RGSRZLHGQLHSURFHV\IL]\F]QHJHQHUXMFHV]XP\U\V
6]XP\ FLHSOQH Z WUDQ]\VWRUDFK V Z\WZDU]DQH SU]H] IL]\F]QH
UH]\VWDQFMHUR]SURV]RQHREV]DUyZHPLWHUDED]\LNROHNWRUD3UDNW\F]QLH
XZ]JO GQLDVL MHG\QLHV]XP\FLHSOQHREV]DUXED]\
r
bb'
u
k T r
f
b
bb
2
4
=
'
∆
r
b e
'
U
b e
'
g U
m
b e
'
&
MF
r
ce
B
'
C
i
c
2
i
b
2
u
b
2
r
bb'
E
B
&
H
5\V6]XPRZ\PRGHOWUDQ]\VWRUDELSRODUQHJR
1RQLNLSUGXZVWU]\NLZDQHSU]H]HPLWHUGRED]\JG]LHVWDM VL
QRQLNDPL PQLHMV]RFLRZ\PL G\IXQGXM Z NLHUXQNX ]áF]D
NROHNWRURZHJRJG]LHVSU]\VSLHV]DQHSU]H]SROHHOHNWU\F]QHZDUVWZ\
]XER*RQHMZNLHUXQNXNROHNWRUD1LHUHJXODUQHSU]HFKRG]HQLHQR QLNyZ
SU]H] REV]DU áDGXQNX SU]HVWU]HQQHJR ]áF]D NROHNWRURZHJR GDMH Z
HIHNFLHV]XPUXWRZ\
i
q I
f
c
C
2
2
=
∆
3RZVWDZDQLH V]XPX UXWRZHJR i
b
2
]ZL]DQH MHVW ] SU]HSá\ZHP
SRV]F]HJyOQ\FK VNáDGRZ\FK SUGX ED]\ : ]áF]X ED]D HPLWHU
SRZVWDMH UyZQLH* V]XP
f
1
RUD] V]XP Z\EXFKRZ\ FR ZL*H VL
JáyZQLH]H]MDZLVNDPLSRZLHU]FKQLRZ\PLL]H]MDZLVNLHPUHNRPELQDFML
ZREM WRFLED]\:\SDGNRZ\V]XPUHSUH]HQWRZDQ\MHVWSU]H]SU GRZH
(UyGáRV]XPyZ
i
q I
f
K
I
f
f
K
I
f
f
f
b
B
B
a
B
c
c
2
1
2
2
2
1
=
+
+
+
∆
∆
∆
szum rutowy szum 1 / f szum wybuchowy
:\SDGNRZ J VWRü V]XPyZ i
b
2
RNUHORQ QD SRGVWDZLH
UyZQDQLDSU]HGVWDZLRQRQDU\V
f
skala log
f
a
i
f
b
2
∆
skala log
2qI
B
K
I
f
B
a
1
5\V* VWRüZLGPRZD
V]XPyZSUGXED]\
7UDQ]\VWRUXQLSRODUQ\-)(7
:G]LDáDQLXWUDQ]\VWRUyZXQLSRODUQ\FKELHU]HXG]LDáW\ONRMHGHQ
URG]DM QRQLNyZ VWG PQLHMV]D UROD ]MDZLVN ]ZL]DQ\FK ]
UHNRPELQDFMD]DWHPPQLHMV]DURODV]XPyZUXWRZ\FK=WHJRSRZRGX
GRPLQXMV]XP\FLHSOQHRUD]V]XP\
f
1
6]XPRZ\ VFKHPDW ]DVW SF]\ WUDQ]\VWRUD -)(7 MHVW SRND]DQ\ QD
U\V
C
gs
U
gs
g U
m
gs
C
gd
r
ds
D
i
d
2
i
g
2
S
G
C
gb
5\V6]XPRZ\PRGHOWUDQ]\VWRUD-)(7
=SUGHPXSá\ZQRFLEUDPNL]ZL]DQ\MHVWV]XPUXWRZ\
i
q I
f
g
G
2
2
=
∆
6]XP WHQ PD SUDNW\F]QH ]QDF]HQLH W\ONR Z SU]\SDGNX JG\
UH]\VWDQFMD(UyGáDVWHUXMFHJRMHVWGX*D
*HQHUDWRU i
d
2
UHSUH]HQWXMH V]XP\ WHUPLF]QH SU]HZRG]FHJR
NDQDáX RUD] V]XP\
f
1
NWyUH SRZVWDM JáyZQLH Z Z\QLNX SHZQ\FK
]MDZLVNSRZLHU]FKQLRZ\FKRUD]ZVNXWHNIOXNWXDFMLáDGXQNXZFHQWUDFK
JHQHUDF\MQRUHNRPELQDF\MQ\FKZDUVWZ\]XER*RQHM
i
k T
g
f
K
I
f
f
d
m
D
2
1
2
4
2
3
=
+
∆
∆
szum termiczny szum 1 / f
7UDQ]\VWRUXQLSRODUQ\026)(7
6]XPRZ\VFKHPDW]DVW SF]\WUDQ]\VWRUD026)(7PR*HE\üWDNL
VDPMDNWUDQ]\VWRUD-)(7SU]\LORFLRZ\FKPRG\ILNDFMDFKJHQHUDWRUyZ
V]XPRZ\FK 6]XP\ FLHSOQH Z WUDQ]\VWRUDFK 026)(7 V UyZQLH*
]ZL]DQH ] NRQGXNWDQFM SU]HZRG]FHJR NDQDáX 6]XP\
f
1
V
]GHWHUPLQRZDQH JáyZQLH SU]H] ]MDZLVND SRZLHU]FKQLRZH NWyUH
VWDQRZL UyZQRF]HQLH SRGVWDZ G]LDáDQLD W\FK SU]\U]GyZ
7UDQ]\VWRU\026PDMOHSV]HZáDFLZRFLV]XPRZHZ]DNUHVLHZF]
QL*WUDQ]\VWRU\-)(7FKRFLD*PDM]QDF]QLHZ\*V]\SR]LRPV]XPyZ
Z]DNUHVLHPDá\FKF] VWRWOLZRFL* VWRüZLGPRZDV]XPyZFLHSOQ\FK
SU]HZRG]FHJR NDQDáX WUDQ]\VWRUD 026)(7 PR*H E\ü RNUHORQD
]DOH*QRFL
S
obszar liniowy
obszar nasycenia
IT
=
4 k T
R
4 k T
2
3
g
FET
m
JG]LH
R
FET
MHVW HNZLZDOHQWQ UH]\VWDQFM SU]HZRG]FHJR WUDQ]\VWRUD
ZREV]DU]HOLQLRZ\P
g
m
PDáRV\JQDáRZD WUDQVNRQGXNWDQFMD WUDQ]\VWRUD Z REV]DU]H
QDV\FHQLD
'ODWHPSHUDWXU\.
8
3
1 1 10
20
k T
=
⋅
−
,
V A sec
⋅ ⋅
* VWRü ZLGPRZD V]XPyZ VWUXNWXUDOQ\FK I ]DUyZQR GOD REV]DUX
OLQLRZHJRMDNLQDV\FHQLDZ\QRVL
S
K K I
C
L f
If
f
D
x
=
2
0
2
'
SU]\F]\P
K
f
ZVSyáF]\QQLNV]XPyZPLJRWDQLDW\SRZR K
V F
f
= ⋅
−
3 10
24
2
I
D
SUGGUHQXZVSRF]\QNRZ\PSXQNFLHSUDF\WUDQ]\VWRUD
C
ox
SRMHPQRüEUDPNDNDQDáQDMHGQRVWN SRZLHU]FKQL
K'
SDUDPHWUWUDQVNRQGXNWDQFML
L
:
GáXJRüLV]HURNRüNDQDáX
6802:$1,('5Ï'(à6=802:<&+
=DVW SF]H (UyGáD V]XPyZ ]DZLHUDM ZLHON OLF]E VNáDGRZ\FK
VLQXVRLGDOQ\FK R Uy*Q\FK F] VWRWOLZRFLDFK RUD] SU]\SDGNRZHM
DPSOLWXG]LH L ID]LH *G\ SRáF]\P\ V]HUHJRZR GZD QLH]DOH*QH (UyGáD
V]XPyZU\VWRPRFZ\MFLRZDMHVWVXPQLHVNRUHORZDQ\FKPRF\
Z\MFLRZ\FKRGSRV]F]HJyOQ\FKVNáDGRZ\FKFRSR]ZDODQDZ\UD *HQLH
Z\SDGNRZHMZDUWRFLUHGQLRNZDGUDWRZHMQDSL FLDV]XPyZMDNRVXP\
UHGQLFK NZDGUDWyZ QDSL ü NROHMQ\FK VNáDGRZ\FK 3RZ\*V]H
VWZLHUG]HQLH MHVW ZD*QH UyZQLH* GOD SUGRZ\FK (UyGHá V]XPyZ
SRáF]RQ\FKUyZQROHJOH
R
2
u
1
2
R
1
u
2
2
u
T
2
5\V6XPRZDQLHQLH]DOH*Q\FK(UyGHá
V]XPRZ\FK
5R]ZD*P\ SU]\NáDGRZR WHUPLF]QH QDSL FLH V]XPyZ UH]\VWRUyZ
1
R
L
2
R
SRáF]RQ\FKV]HUHJRZRMDNQDU\V
u
k T R
f
1
2
1
4
=
∆
u
k T R
f
2
2
2
4
=
∆
UHGQLRNZDGUDWRZD ZDUWRü QDSL FLD V]XPyZ R FKZLORZHM ZDUWRFL
]DVW SF]HM
( )
( )
( )
u t
u t
u t
T
=
+
1
2
Z\QRVL
( )
( ) ( )
[
]
( )
( )
( ) ( )
0
2
2
1
2
2
2
1
2
2
1
2
t
u
t
u
t
u
t
u
t
u
t
u
t
u
T
+
+
=
+
=
3RQLHZD* V]XP\ SRFKRG] ] Uy*Q\FK UH]\VWRUyZ V RQH ]DWHP
QLHVNRUHORZDQHLUHGQLDZDUWRü
( ) ( )
u t u t
1
2
PXVLE\üUyZQD]HUX
=DWHP
u
u
u
T
2
1
2
2
2
=
+
3RGVWDZLDMFLGRRWU]\PXMHP\
(
)
u
k T R
R
f
T
2
1
2
4
=
+
∆
: SU]\SDGNX JG\ F] ü V]XPyZ Z REX (UyGáDFK SRFKRG]L RG
SURFHVyZ IL]\F]Q\FK ]ZL]DQ\FK ]H VRE WR Z\SDGNRZD
UHGQLRNZDGUDWRZDZDUWRüQDSL FLDV]XPyZF] FLRZRVNRUHORZDQ\FK
Z\QRVL
u
u
u
C u u
T
r
2
1
2
2
2
1
2
2
=
+
+
JG]LH
C
r
R]QDF]D ZVSyáF]\QQLN NRUHODFML L PR*H SU]\ELHUDü ZDUWRü ]
SU]HG]LDáX
[
]
− +
1 1
;
&$à.2:,7< 6=80 =$67 3&=< 2'1,(6,21< '2
:(-&,$:=0$&1,$&=$
=DVW SF]H(UyGáDV]XPyZRGQLHVLRQHGRZHMFLD
Z]PDFQLDF]D
'ODGRZROQHJRF]ZyUQLNDLVWQLHMHXQLZHUVDOQ\VFKHPDW]DVW SF]\
V]XPyZSU]HGVWDZLRQ\QDU\V
i
i
2
u
i
2
5
J
6LHü
EH]V]XPQD
5
J
6LHü
] V]XPDPL
5\V=DVW SF]H(UyGáDV]XPyZRGQLHVLRQHGRZHMFLDF]ZyUQLND
0R*QDZ\ND]DüNRQLHF]QRüX*\FLDGZyFK(UyGHá]DVW SF]\FKQD
ZHMFLX Z]PDFQLDF]D SUGRZHJR L QDSL FLRZHJR NWyUH UHSUH]HQWXM
VRE ZV]\VWNLH PR*OLZH ZNáDG\ V]XPRZH SRFKRG]FH RG HOHPHQWyZ
F]ZyUQLNDDQDOL]XMF]DFKRZDQLHVL
XNáDGXGODJUDQLF]Q\FKZDUWRFL
UH]\VWDQFML R
g
*G\ R
g
=
0
(UyGáR i
i
2
QD U\V MHVW ]ZDUWH L GOD
UHSUH]HQWDFML V]XPyZ QD Z\MFLX XNáDGX QDOH*\ ]DVWRVRZDü (UyGáR
QDSL
FLRZH u
i
2
3RGREQLH JG\ R
g
= ∞
(UyGáR QDSL FLRZH u
i
2
QLH Z\WZDU]D
V]XPyZ QD Z\MFLX XNáDGX L GOD LFK UHSUH]HQWDFML QD ZHMFLX QDOH*\
]DVWRVRZDüSUGRZH(UyGáRV]XPyZi
i
2
=DVW SF]H(UyGáDV]XPyZWUDQ]\VWRUDELSRODUQHJR
=DVW SF]H (UyGáD V]XPyZ QD ZHMFLX WUDQ]\VWRUD ELSRODUQHJR
PR*HP\ Z\]QDF]\ü QD SRGVWDZLH MHJR V]XPRZHJR VFKHPDWX
]DVW SF]HJR SU]HGVWDZLRQHJR QD U\VD 6]XP\ REZRGX
Z\MFLRZHJR V DQDOL]RZDQH SU]\ ]ZDUW\P REFL*HQLX L SRPLQL FLX
SRMHPQRFL
jc
C
8NáDG]DVW SF]\]U\VEUHSUH]HQWXMHV]XP\WUDQ]\VWRUD Z
SRVWDFLUyZQRZD*Q\FKJHQHUDWRUyZV]XPyZQDZHMFLXGDMFWHVDPH
V]XP\ QD Z\MFLX SU]\ GRZROQHM LPSHGDQFML (UyGáD VWHUXMFHJR
=ZLHUDMF ZHMFLD REX XNáDGyZ ]DVW SF]\FK U\V Z\]QDF]\P\
ZDUWRü]DVW SF]HJR(UyGáDQDSL FLRZHJRV]XPyZ u
i
2
r
b e
'
U
b e
'
g U
m
b e
'
&
H
r
ce
i
c
2
i
b
2
X
EE
2
U
EE
L
R
r
b e
'
U
b e
'
g U
m
b e
'
&
H
r
ce
L
L
2
X
L
2
U
EE
L
R
a)
b)
5\V D 6]XPRZ\ VFKHPDW ]DVW SF]\ WUDQ]\VWRUD ELSRODUQHJR E XNáDG
UyZQRZD*Q\]]DVW SF]\PL(UyGáDPLV]XPRZ\PLQD]DFLVNDFKZHMFLRZ\FK
3U]\MPXMHP\ *H SRV]F]HJyOQH (UyGáD V]XPyZ V ]H VRE
QLHVNRUHORZDQH RUD] SRPLMDP\ LFK ]QDNL SRQLHZD * V WR V\JQDá\
VWRFKDVW\F]QH=DNáDGDMF
r
r
bb
b e
<<
'
RWU]\PXMHP\
g u
i
g u
m
bb
c
m
i
+ =
VWG
u u
i
g
i
bb
c
m
=
+
RUD]
u
u
i
g
i
bb
c
m
2
2
2
2
=
+
3RGVWDZLDMF]DOH*QRFLLGRUyZQDQLD
RWU]\PXMHP\
u
k T r
f
q I
f
g
i
bb
C
m
2
2
4
2
=
+
∆
∆
3RQLHZD* g
I
q I
k T
m
C
T
C
=
=
ϕ
WR]UyZQDQLDRWU]\PXMHP\
u
f
k T r
g
i
bb
m
2
4
1
2
∆
=
+
* VWRüZLGPRZ]DVW SF]HJR(UyGáDSUGRZHJRV]XPyZ i
f
i
2
∆
Z\]QDF]D VL SU]\ UR]ZDUFLX ZHMü REX XNáDGyZ ]DVW SF]\FK
U\V3RUyZQXMF]HVRESUG\Z\MFLRZHRWU]\PXMHP\
( )
( )
β
ω
β
ω
j
i
i
j
i
i
c
b
= +
VWG
( )
i
i
i
j
i
b
c
= +
β ω
=DNáDGDMF *H (UyGáD V]XPyZ
i
b
RUD]
i
c
V QLHVNRUHORZDQH
RWU]\PXMHP\
( )
i
i
i
j
i
b
c
2
2
2
2
= +
β ω
JG]LH
( )
β ω
β
ω
ω
β
j
j
=
+
0
1
3RGVWDZLDMF]DOH*QRFLLGRUyZQDQLD
RWU]\PXMHP\
( )
i
f
q I
K
I
f
I
j
i
B
B
a
C
2
1
2
2
∆
=
+
+
'
β ω
JG]LH
K
K
q
1
1
2
'
=
'ODXSURV]F]HQLDSRPLQL WRV]XP\Z\EXFKRZH
8Z]JO GQLDMF QD ZHMFLX UH]\VWDQFM (UyGáD VWHUXMFHJR L ]ZL]DQH ]
QLV]XP\WHUPLF]QHOLF]E ]DVW SF]\FK(UyGHáV]XPRZ\FKQDZHMFLX
Z]PDFQLDF]D ]UHGXNRZDQR GR WU]HFK G]L NL ]DVWRVRZDQLX (UyGHá
i
u
i
i
2
2
oraz
U\VDSU]\F]\PWUDQ]\VWRU7PR*HE\üXZD*DQ\]D
EH]V]XPQ\
L
L
2
X
L
2
a)
X
J
2
5
J
7
5
/W
X
L1
2
b)
5
J
7
5
/W
5\V D =DVW SF]H (UyGáD V]XPRZH RGQLHVLRQH GR ZHMFLD Z]PDFQLDF]D E
FDáNRZLWH]DVW SF]H(UyGáRV]XPRZHQDZHMFLXZ]PDFQLDF]D
-DNSRND]DQRQDU\VE]DVW SF]H(UyGáRV]XPyZQDZHMFLX
u
iN
2
UHSUH]HQWXMHVXPDU\F]Q\ZSá\ZZV]\VWNLFKWU]HFK (UyGHáV]XPyZ
Z PLHMVFX Z NWyU\P XPLHV]F]RQH MHVW (UyGáR V\JQDáX ']L NL WHPX
RNUHOHQLHVWRVXQNX
N
S
V\JQDáV]XPMHVWEDUG]RXáDWZLRQH
=DNáDGDMF *H VNáDGRZH (UyGáD V]XPyZ V QLHVNRUHORZDQH
RWU]\PXMHP\
u
u
u
i R
iN
g
i
i
g
=
+ +
u
u
u
i R
iN
g
i
i
g
2
2
2
2
2
=
+
+
:\NRU]\VWXMFUyZQDQLDRUD]RWU]\PXMHP\
( )
u
f
k T R
k T r
g
R
q I
I
j
iN
g
bb
m
g
B
C
2
2
2
4
4
1
2
2
∆
=
+
+
+
+
β ω
5yZQDQLHMHVWVáXV]QHSU]\PDá\FKZDUWR FLDFK
r
bb
=DVW SF]H(UyGáDV]XPyZWUDQ]\VWRUyZSRORZ\FK
=DVW SF]H (UyGáD V]XPyZ QD ZHMFLX WUDQ]\VWRUD SRORZHJR
Z\]QDF]\P\ Z SRGREQ\ VSRVyE MDN WR XF]\QLRQR GOD WUDQ]\VWRUD
ELSRODUQHJR 1D U\VD SU]HGVWDZLRQR V]XPRZ\ VFKHPDW
]DVW SF]\ WUDQ]\VWRUD -)(7 D QD U\VXQNX E MHJR UyZQRZD *Q\
XNáDG]]DVW SF]\PL(UyGáDPLV]XPRZ\PLQD]DFLVNDFKZHMFLRZ\FK
$QDOL]D ]RVWDQLH SU]HSURZDG]RQD SU]\ ]ZDUW\P Z\M FLX
LSRPLQL FLX SRMHPQRFL C
gd
3U]\ ]ZDUW\P ZHMFLX REX XNáDGyZ
SRUyZQXMFSUG\
L
R
RWU]\PXMHP\
i
g u
d
m i
=
6WG
u
i
g
i
d
m
2
2
2
=
3RGVWDZLDMFGRRWU]\PXMHP\
u
f
k T
g
K
I
g f
i
m
D
m
2
1
2
2
4
2
3
1
∆
=
+
8
JV
J 8
P
JV
&
JV
U
GV
L
G
2
L
J
2
L
R
8
JV
J 8
P
JV
&
JV
U
GV
L
L
2
X
L
2
L
R
a)
b)
'
6
*
5\VD6]XPRZ\VFKHPDW]DVW SF]\WUDQ]\VWRUD-)(7EXNáDGUyZQRZD*Q\]
]DVW SF]\PL(UyGáDPLV]XPRZ\PLQDZHMFLX
1D U\V SU]HGVWDZLRQR ]DOH*QRü J VWRFL ZLGPRZHM
QDSL FLD V]XPyZ ]DVW SF]HJR (UyGáD ZHMFLRZHJR u
f
i
2
∆
Z IXQNFML
F] VWRWOLZRFL
f
1
I
10
5
10
6
10
7
10
8
+]
10
4
10
3
10
2
10
10
18
−
10
17
−
10
16
−
10
15
−
10
14
−
9 +]
2
X
I
L
2
∆
5\V* VWRüZLGPRZDQDSL FLDV]XPyZ]DVW SF]HJR(UyGáDZHMFLRZHJR
X
I
L
∆
GODWUDQ]\VWRUD-)(7
: SU]HFLZLHVWZLH GR WUDQ]\VWRUD ELSRODUQHJR ]DVW SF]\
JHQHUDWRUQDSL FLRZ\V]XPyZWUDQ]\VWRUD-)(7RGQLHVLRQ\GRZHM FLD
XNáDGX ]DZLHUD VNáDGRZ V]XPX PLJRWDQLD
f
1
UR]FLJDMF VL Z
QLHW\SRZ\VSRVyED*GR]DNUHVXF] VWRWOLZRFLU] GX0+]
6]XP\ PLJRWDQLD PDM V]F]HJyOQH ]QDF]HQLH Z WUDQ]\VWRUDFK
026)(7NWyUHZL*VL ]HQHUJHW\F]Q\PLVWDQDPLSRZLHU]FKQLRZ\PL
QD JUDQLF\ 6L 6L2
: WUDQ]\VWRUDFK 026 GOD F] VWRWOLZRFL
PQLHMV]\FK RG GR N+] V]XP\ PLJRWDQLD V ZL NV]H RG V]XPyZ
WHUPLF]Q\FK QDMF] FLHM QLH]DOH*QLH RG SUGyZ SRODU\]XMF\FK
LJHRPHWULLWUDQ]\VWRUyZ
: ZL NV]RFL SU]\SDGNyZ DPSOLWXGD ]DVW SF]HJR (UyGáD
QDSL FLRZHJRV]XPyZWUDQ]\VWRUD026MHVWRGZURWQLHSURSRUFMRQDOQD
GR MHJR DNW\ZQHM SRZLHU]FKQL EUDPNL *G\ SRZLHU]FKQLD EUDPNL MHVW
GX*D ZL NV]Rü GHIHNWyZ SRZLHU]FKQLRZ\FK L ]ZL]DQ\FK ] QLPL
HQHUJHW\F]Q\FK VWDQyZ SRZLHU]FKQLRZ\FK ]QDMGXMH VL SRG EUDPN
SRZRGXMFRJyOQH]PQLHMV]HQLHV]XPyZ
* VWRü ZLGPRZD ]DVW SF]HJR (UyGáD V]XPyZ RGQLHVLRQHJR GR
ZHMFLDWUDQ]\VWRUD026PR*HE\üRSLVDQD]DOH*QRFL
u
f
k T
g
K
W L C
f
i
m
f
x
2
0
4
2
3
1
∆
=
+
* VWRü ZLGPRZ ]DVW SF]HJR (UyGáD SUGRZHJR V]XPyZ
Z\]QDF]DP\SU]\UR]ZDUFLXZHMüREXXNáDGyZ]DVW SF]\FKU\V
3RUyZQXMF VNXWHF]QH ZDUWRFL Z\MFLRZ\FK SUGyZ V]XPyZ
i
o
REX
XNáDGyZRWU]\PXMHP\
i
g
j
C
i
g
j
C
i
i
m
gs
g
m
gs
d
ω
ω
=
+
6WG
i
i
j
C
g
i
i
g
gs
m
d
=
+
ω
3RQLHZD* i
g
RUD]
i
d
V QLH]DOH*QH WR ] UyZQDQLD
RWU]\PXMHP\
i
i
C
g
i
i
g
gs
m
d
2
2
2
2
2
2
= +
ω
3RGVWDZLDMF]DOH*QRFLLGRRWU]\PXMHP\
i
f
q I
C
g
k T
g
K
I
f
i
G
gs
m
m
D
2
2
2
2
1
2
2
4
2
3
∆
=
+
+
ω
:UyZQDQLXZLHONRü
K
g
C
i
m
gs
=
ω
R]QDF]D Z]PRFQLHQLH SUGRZH 6WG V]XP\ JHQHURZDQH QD Z\MFLX
SU]HQRV]RQHVQDZHMFLH]HZVSyáF]\QQLNLHP1
2
K
i
: ]DNUHVLH PDá\FK F] VWRWOLZRFL ]DVW SF]H (UyGáR SUGRZH V]XPyZ
MHVWGHWHUPLQRZDQHSU]H]SUGXSá\ZQRFLEUDPNL
I
G
NWyU\MHVWEDUG]R
PDá\
I
G
$ : WUDQ]\VWRUDFK SRORZ\FK 026)(7 VNáDGRZD
V]XPyZ GHWHUPLQRZDQD SU]H] SUG XSá\ZQRFL EUDPNL 2q I
G
MHVW
SRPLMDOQLHPDáD :SU]\SDGNX(UyGáDVWHUXMFHJRRGX*HMLPSHGDQFML
JG\ ZRJyOQ\P ELODQVLH FDáNRZLWHJR ]DVW SF]HJR V]XPX ZHMFLRZHJR
GRPLQXMF URO RGJU\ZD (UyGáR SUGRZH WUDQ]\VWRU\ -)(7
]DSHZQLDM PQLHMV]\ SR]LRP V]XPyZ QD Z\MFLX QL* WUDQ]\VWRU\
ELSRODUQH 3URSRUFMH WH PRJ E\ü F]DVDPL RGZUyFRQH QD NRU]\ü
WUDQ]\VWRUyZ ELSRODUQ\FK Z SU]\SDGNX (UyGáD VWHUXMFHJR R PDáHM
LPSHGDQFML JG\ Z ELODQVLH ]DVW SF]HJR V]XPX ZHMFLRZHJR
GRPLQXMFURO RGJU\ZD(UyGáRQDSL FLRZH']LHMHVL WDNGODWHJR*H
GOD GDQHJR SUGX SRODU\]DFML WUDQ]\VWRU\ ELSRODUQH PDM ZL NV]
ZDUWRüWUDQVNRQGXNWDQFML
g
m
:3à<: 635= )(1,$ =:5271(*2 1$ 3$5$0(75<
6=802:(:=0$&1,$&=$
6WRVRZDQLH GZyFK ]DVW SF]\FK (UyGHá V]XPRZ\FK QD ]DFLVNDFK
ZHMFLRZ\FK WUDQ]\VWRUD SU]\ XPRZQ\P XF]\QLHQLX UHV]W\ VFKHPDWX
EH]V]XPQHM MHVW V]F]HJyOQLH SU]\GDWQH SU]\ UR]ZD*DQLX ZSá\ZX
VSU] *HQLD]ZURWQHJRQDZáDFLZRFLV]XPRZHZ]PDFQLDF]D
1D U\VD SU]HGVWDZLRQR VFKHPDW EORNRZ\ Z]PDFQLDF]D ]H
VSU] *HQLHP ]ZURWQ\P QDSL FLRZ\P V]HUHJRZ\P Z NWyU\P
]DVWRVRZDQRXQLODWHUDOQ\LGHDOQ\F]ZyUQLNVSU] *HQLD]ZURWQHJR
=DVW SF]H JHQHUDWRU\ V]XPyZ u
ik
2
RUD] i
ik
2
UHSUH]HQWXM V]XP\
F]ZyUQLND Z]PDFQLDMFHJR N ]D u
i
2
RUD] i
i
2
U\VE V
]DVW
SF]\PL JHQHUDWRUDPL V]XPyZ Z]PDFQLDF]D ]H VSU]
*HQLHP
]ZURWQ\P :DUWRü u
i
2
PR*HP\ Z\]QDF]\ü ]ZLHUDMF ZHMFLD REX
XNáDGyZQDU\V
L
LN
2
X
LN
2
8
R
N
8
LQ
β
8
R
a)
L
L
2
X
L
2
8
R
N
8
LQ
β
8
R
b)
5\V D 6FKHPDW EORNRZ\ Z]PDFQLDF]D ] LGHDOQ\P VSU] *HQLHP ]ZURWQ\P
QDSL FLRZ\PV]HUHJRZ\P ] XZ]JO GQLHQLHP V]XPyZ E XNáDG ] ]DVW SF]\PL
(UyGáDPLV]XPyZ
3RQLHZD* UH]\VWDQFMD Z\MFLRZD F]ZyUQLND VSU] *HQLD ]ZURWQHJR MHVW
UyZQD]HUXZL F
u
u
i
ik
2
2
=
:DUWRü i
i
2
Z\]QDF]DP\ SU]\ UR]ZDUW\FK ZHMFLDFK REX
XNáDGyZ 3U]\ UR]ZDUW\P ZHMFLX XNáDGX U\VE V\JQDá ]ZURWQ\
QLHRGG]LDá\ZDQDZHMFLHZ]PDFQLDF]D]DWHPGODUyZQRFLV]XPyZQD
Z\MFLXRWU]\PXMHP\
i
i
i
ik
2
2
=
=UyZQRFLLZ\QLND*HQDZHMFLXREXXNáDGyZEH]
VSU] *HQLD]ZURWQHJRMDNUyZQLH*]HVSU] *HQLHP]ZURWQ\PVWRVXQHN
V\JQDáX GR V]XPX
S N
SR]RVWDMH EH] ]PLDQ\ : XNáDG]LH ]H
VSU] *HQLHP ]ZURWQ\P Z\VWSL ]PQLHMV]HQLH Z]PRFQLHQLD ]DUyZQR
GOD V\JQDáX X*\WHF]QHJR MDN L GOD V]XPX ]DWHP VWRVXQHN V\JQDáX GR
V]XPXQDZ\MFLXZ]PDFQLDF]DSR]RVWDQLHWDNLVDPMDNZXNáDG]LHEH]
VSU] *HQLD]ZURWQHJR
1D U\VD SRGDQR SUDNW\F]Q\ VSRVyE UHDOL]DFML XMHPQHJR
VSU] *HQLD ]ZURWQHJR QDSL FLRZHJRV]HUHJRZHJR 6\JQDá ]ZURWQ\
RWU]\P\ZDQ\ MHVW ] G]LHOQLND UH]\VWDQF\MQHJR
R
R
E
F
,
SU]\ F]\P GOD
FHOyZ DQDOL]\ ZáDFLZRFL V]XPRZ\FK XNáDGX Z V]HUHJ ] ND*G
UH]\VWDQFM ZáF]RQR (UyGáR QDSL FLRZH UHSUH]HQWXMFH MHJR V]XP\
WHUPLF]QH
u
k T R
f
u
k T R
f
e
E
f
F
2
2
4
4
=
=
∆
∆
X
LN
2
N
a)
5
)
5
(
X
H
2
X
I
2
L
LN
2
X
L
2
N
b)
5
)
5
(
L
L
2
5\VDZ]PDFQLDF]D]XMHPQ\PVSU] *HQLHP]ZURWQ\PQDSL FLRZ\P
V]HUHJRZ\PXZ]JO GQLDMF\(UyGáDV]XPyZEXNáDG]]DVW SF]\PL(UyGáDPL
V]XPyZ
=DVW SF]H JHQHUDWRU\ V]XPyZ u
i
ik
ik
2
2
,
UHSUH]HQWXM V]XP\
Z]PDFQLDF]DEH]VSU] *HQLD]ZURWQHJR
: XNáDG]LH SRND]DQ\P QD U\VE ]DVW SF]H (UyGáD V]XPyZ
u
i
2
RUD] i
i
2
UHSUH]HQWXM FDáNRZLW\ ]DVW SF]\ V]XP ZHMFLRZ\
Z]PDFQLDF]D]XMHPQ\PVSU] *HQLHP]ZURWQ\P'RZ\]QDF]HQLD u
i
2
]ZLHUDP\ ZHMFLD REX XNáDGyZ ] U\V L SRUyZQXMHP\ V]XP\
Z\MFLRZH=Dáy*P\SU]\W\P*H UH]\VWDQFMD Z\MFLRZD Z]PDFQLDF]D
R
R
o
F
<<
u
u
i R
R
R
R
u
R
R
R
u
i
ik
ik
F
F
E
e
E
F
E
f
=
+
+
+
+
+
SU]\F]\P
R
R
R
F
E
=
||
=DNáDGDMF*HSRV]F]HJyOQHVNáDGRZHV]XPyZVQLH]DOH*QHRUD]
Z\NRU]\VWXMF]DOH*QRüRWU]\PXMHP\
u
u
i R
k T R f
i
ik
ik
2
2
2
2
4
=
+
+
∆
$QDOL]XMFVWDQUR]ZDUFLDREZRGyZZHMFLRZ\FKREXXNáDGyZ]
U\VRWU]\PXMHP\
i
i
i
ik
2
2
=
=UyZQDQLDZ\QLND*H]ZL NV]DVL UHGQLRNZDGUDWRZD
ZDUWRü QDSL FLD V]XPyZ ]DVW SF]HJR (UyGáD QDSL FLRZHJR ]DWHP
]ZL NV]\VL VWRVXQHNV\JQDáXGRV]XPX
S N
QDZHMFLXXNáDGX]H
VSU] *HQLHP]ZURWQ\P:WDNLPVDP\PVWRSQLXZ]URQLHWHQVWRVXQHN
QD Z\MFLX XNáDGX 3RGREQH UR]ZD*DQLD PR*HP\ SU]HSURZDG]Lü GOD
Z]PDFQLDF]D]XMHPQ\PVSU] *HQLHP]ZURWQ\PLQQHJRW\SX
2JyOQLH PR*QD VWZLHUG]Lü *H ZH Z]PDFQLDF]X ] XMHPQ\P
VSU] *HQLHP]ZURWQ\PVWRVXQHNV\JQDáXGRV]XPX
S N
QDZHMFLX
XNáDGXMHVWZL NV]\QL*ZXNáDG]LHEH]VSU] *HQLD]ZURWQHJRZVNXWHN
SHZQHM GRGDWNRZHM SRUFML V]XPX FLHSOQHJR JHQHURZDQHJR
ZHOHPHQWDFKVSU] *HQLD]ZURWQHJR
3$5$0(75<6=802:(8.à$'Ï:(/(.7521,&=1<&+
1DMF] FLHM VWRVRZDQ\P SDUDPHWUHP RSLVXMF\P ZáDFLZRFL
V]XPRZH HOHPHQWX DNW\ZQHJR OXE XNáDGX HOHNWURQLF]QHJR MHVW
ZVSyáF]\QQLN V]XPyZ ) DQJ QRLVH IDFWRU -HVW RQ GHILQLRZDQ\ MDNR
VWRVXQHNFDáNRZLWHMPRF\V]XPyZQDZ\MFLXXNáDGXZSDPLH
∆
f
GR
PRF\ V]XPyZ QD Z\MFLX SRFKRG]F\FK ] V]XPyZ WHUPLF]Q\FK
UH]\VWDQFML(UyGáDV\JQDáX
F
P
k T
f k
oN
ps
=
≥
∆
1
JG]LH k
ps
MHVWVNXWHF]Q\PZ]PRFQLHQLHPPRF\
3R SRG]LHOHQLX OLF]QLND L PLDQRZQLND Z\UD*HQLD SU]H] PRF
G\VSRQRZDQV\JQDáX P
E
R
s dysp
g
g
=
2
4
RWU]\PXMHP\
F
P
k T
f
k
P
P
S
N
S
N
s dysp
ps
s dysp
oN
in
in
o
o
=
=
∆
:SRZ\*V]HM]DOH*QRFL
S
N
in
in
RUD]
S
N
o
o
VVWRVXQNLHPV\JQDáXGR
V]XPyZRGSRZLHGQLRQDZHMFLXLQDZ\MFLXF]ZyUQLNDU\V
6 1
R
R
6
1
LQ
LQ
5\V0RFV\JQDáXLV]XPXQD
ZHMFLXLZ\MFLXF]ZyUQLND
:VSyáF]\QQLN V]XPyZ MHVW ZL NV]\ RG MHGQRFL L MHVW PLDU
SRJRUV]HQLD VL VWRVXQNX V\JQDáX GR V]XPX VSRZRGRZDQHJR SU]H]
V]XP\Z]PDFQLDF]D
: F]ZyUQLNX SU]HGVWDZLRQ\P QD U\V ZHMFLRZD PRF
V]XPyZ
N
in
SRFKRG]LRGUH]\VWDQFML(UyGáDVWHUXMFHJR]DZ\MFLRZD
PRF V]XPyZ
N
o
UHSUH]HQWXMH ZV]\VWNLH (UyGáD V]XPyZ F]ZyUQLND
ZUD] ] PRF V]XPyZ UH]\VWDQFML (UyGáD VWHUXMFHJR SU]HQLHVLRQ QD
Z\MFLHXNáDGX
:VSyáF]\QQLN V]XPyZ PR*H E\ü WDN*H Z\UD*RQ\ ]D SRPRF
ZLHONRFL u
i
2
RUD] i
i
2
WM UHGQLRNZDGUDWRZ\FK ZDUWRFL QDSL FLD
LSUGX ]DVW SF]\FK (UyGHá V]XPRZ\FK QD ZHMFLX F]ZyUQLND *G\
ZV]\VWNLH(UyGáDV]XPyZVSU]HQLHVLRQHGRREZRGXZHMFLRZHJRWR
ZVSyáF]\QQLNV]XPyZPR*HE\üRNUHORQ\ MDNR VWRVXQHN FDáNRZLWHJR
UHGQLRNZDGUDWRZHJR ]DVW SF]HJR V]XPX QD ZHMFLX GR
UHGQLRNZDGUDWRZHMZDUWRFLV]XPyZWHUPLF]Q\FK(UyGáDV\JQDáX
:\NRU]\VWXMF]DOH*QRFLLRWU]\PXMHP\
F
u
u
u
u
i R
u
u
k T R
f
i
k T
R
f
iN
gT
gT
i
i
g
gT
i
g
i
g
=
=
+
+
= +
+
2
2
2
2
2
2
2
2
2
1
4
4
1
∆
∆
JG]LH
u
gT
2
UHGQLRNZDGUDWRZD ZDUWRü QDSL FLD V]XPyZ WHUPLF]Q\FK
UH]\VWDQFML R
g
(UyGáDVWHUXMFHJR
0LQLPDOQD ZDUWRü ZVSyáF]\QQLND V]XPyZ ) Z\VW SXMH GOD
RSW\PDOQHMUH]\VWDQFMLJHQHUDWRUD
R
g opt
2EOLF]DMFSRFKRGQ dF dR
g
]UyZQDQLDRWU]\PXMHP\
R
u
i
g opt
i
i
2
2
2
=
3RGVWDZLDMF
R
g opt
GRUyZQDQLDRUD]SU]\MPXMF
∆
I +]
RWU]\PXMHP\PLQLPDOQZDUWRüZVSyáF]\QQLNDV]XPyZ)
PLQ
F
u i
k T
i
i
min
= +
1
2
2
2
1DMPQLHMV]\ ZVSyáF]\QQLN V]XPyZ Z\VW SXMH GOD LPSHGDQFML
(UyGáDNWyUDMHVW]Z\NOHUy*QDRGLPSHGDQFMLZHMFLRZHMVSU] *RQHM
DZL FSU]\QLHGRSDVRZDQLXHQHUJHW\F]Q\PQDZHMFLXXNáDGX=WHJR
Z]JO GX SURMHNWRZDQLH PDáRV\JQDáRZHJR Z]PDFQLDF]D Z\PDJD
SHZQHJR NRPSURPLVX SRPL G]\ MHJR ZáDFLZRFLDPL V]XPRZ\PL D
Z]PDFQLDMF\PL
3U]\DQDOL]LHZ]PDFQLDF]\RPDá\PSR]LRPLHV]XPyZGHILQLXMH
VL ]DVW SF]UH]\VWDQFM V]XPyZL]DVW SF]WHPSHUDWXU V]XPyZ
=DVW SF]DUH]\VWDQFMDV]XPyZ5
Q
MHVWWRWDNDZDUWRüUH]\VWDQFML
GOD NWyUHM V]XP FLHSOQ\ MHVW UyZQ\ FR GR ZLHONRFL ]DVW SF]HPX
ZHMFLRZHPX V]XPRZL Z]PDFQLDF]D 3RUyZQXMF V]XP FLHSOQ\ ]
V]XPDPLZ]PDFQLDF]DU\VRWU]\PXMHP\
4
2
2
2
2
k T R
f
u
u
i R
n
iN
i
i
g
∆ =
=
+
6WG
f
T
k
R
i
u
R
g
i
i
n
∆
4
2
2
2
+
=
=DVW SF]DUH]\VWDQFMDV]XPyZQLHMHVW]ZL]DQDDQL]UH]\VWDQFM
ZHMFLRZZ]PDFQLDF]DDQLWH*]UH]\VWDQFM(UyGáDV\JQDáX
=DVW SF] WHPSHUDWXU V]XPyZ
T
n
QD]\ZDP\ WDN ZDUWRü
WHPSHUDWXU\UH]\VWDQFML(UyGáDSU]\NWyUHMUH]\VWDQFMDWDJHQHURZDáDE\
V]XP FLHSOQ\ R ZDUWRFL UyZQHM ]DVW SF]HPX ZHMFLRZHPX V]XPRZL
Z]PDFQLDF]D
3RUyZQXMFREDV]XP\SRUZ]yURWU]\PXMHP\
4
2
2
2
2
k T R
f
u
u
i R
n
n
iN
i
i
g
∆ =
=
+
6WG]DVW SF]DWHPSHUDWXUDV]XPyZ
T =
u
i R
k R f
n
i
i
g
g
2
2
2
4
+
∆
-H*HOL V]XP\ Z]PDFQLDF]D ]RVWDQ RNUHORQH Z WHPSHUDWXU]H
T
]Z\NOH.WR
(
)
T
T
F
n
=
−
1