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HTXDFLRQV GH WHQVLy L FRUUHQW
9 ] = 9 H + 9 H
+ −β O − + βO
R R
, ] = , H + , H
+ −β O − + βO
R R
RQ 9 , 9 , VyQ OHV DPSOLWXGV GH OD WHQVLy L FRUUHQW GH OHV RQHV SURJUHVVLYHV L
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= R SHUPHW UHODFLRQDU OD WHQVLy L HO FRUUHQW GH ORQD TXH
=
R
9 9
= = −
, ,
+ −
R R
+ −
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LPSHGjQFLD / = 6L / = R = ORQD HV WUDQVPHWUj QRUPDOPHQW SHUz VL = / ≠ = R ORQD HV GLYLGLUj
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9 = − =
Γ = =
9
−
R / R
+
R = / + = R
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• (O FLUFXLW WUHEDOOD D OD IUHT qQFLD GH I = *+] → ω = π I = < UDG V
• $VVXPLQW OD LPSHGjQFLD FDUDFWHUtVWLFD GH OD OtQHD = R = Ω
• &RQHL[HP HOV SDUjPHWUHV GHO VXEVWUDW L SHU WDQW SRGHP FDOFXODU
6XEVWUDW
ε
U
µ =
U
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ε = εR ⋅ ε U = Â
− )
×
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µ R = π <
− +
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P
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H[DFWDPHQW
• ε UHII =
• S
•
•
F
Y = = ⋅ P V
ε
UHII
YS
λ = =
I
π
β = =
λ
P
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I I UDG
π
Φ = β ⋅ O = O
λ
/RQJLWXG
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GRQD
ž FP
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ž FP
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ž FP
ž FP
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+tEULG %UDQFK /LQH
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[ 6]
⎛ 6 6 6 6 ⎞
⎜ ⎟
⎜
6 6 6 6
⎟
⎜6 6 6 6 ⎟
⎜ ⎟
⎝6 6 6 6 ⎠
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6LM ≤
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KHUPqWLFD WUDQVSRVDGD L FRQMXJDGD pV LJXDO D OD PDWULX LGHQWLWDW
+
[ 6][ 6 ] = [ , ]
• 5HFtSURFV 6L HO FLUFXLW pV UHFtSURF SRGHP GLU TXH OD VHYD PDWULX GH GLVSHUVLy pV VLPqWULFD
[ ] [ ] W
6 = 6 ⇔ 6 = 6
LM ML
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6 = 6 = 6 = 6 =
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OHV VHJ HQWV UHODFLRQV
6 6 + 6 6 =
6 + 6 + 6 =
6 6 + 6 6 =
6 + 6 + 6 =
6 6 + 6 6 =
6 + 6 + 6 =
6 + 6 + 6 =
6 6 + 6 6 =
6 6 + 6 6 =
6 6 + 6 6 =

8QD GH OHV SRVVLEOHV VROXFLRQV GDTXHVW VLVWHPD FRQVLVWHL[ HQ SUHQGUH 6 = SHO TXH
HV UHGXHL[ D
6 6
6 6
6 6
6 6
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=
=
=
=
6 + 6 =
6 + 6 =
6 + 6 =
6 + 6 =
'H YHLHP TXH 6 = 6 L 6 = 6 6L SUHQHP SHU H[HPSOH 6 = 6 = α
M
6 H θ
β L
M
6 H φ
β OD PDWULX GH GLVSHUVLy TXHGD GH OD VHJ HQW IRUPD
Mθ
⎛ βH α ⎞
⎜ ⎟
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⎜ α βH
⎟
[ 6]
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⎟
⎜ β H α
⎟
⎜ Mφ
α β H
⎟
⎝ ⎠
3HU ~OWLP SHU D TXH [ ]
6 VLJXL XQLWjULD VKDXUDQ GH FRPSOLU OHV VHJ HQWV UHODFLRQV
α + β =
− Mθ Mφ
− βαH = αβH ⇒ θ + φ = ( Q + ) π
([LVWHL[HQ GXHV SRVVLEOHV VROXFLRQV SHU D OHV HTXDFLRQV
$FREODGRU VLPqWULF
θ = φ =
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π
π
⎛ Mβ
α ⎞
⎜ ⎟
α Mβ
[ 6]
= ⎜ ⎟
⎜ Mβ
α ⎟
⎜ ⎟
⎝ α Mβ
⎠

⎛ β α ⎞
⎜ ⎟
α −β
[ 6]
= ⎜ ⎟
⎜ β α ⎟
⎜ ⎟
⎝α −β
⎠
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HOOV
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FRP D ILJXUHV GH PqULW
• $FREODPHQW & = ( 3 3 ) = − β ( )
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' = ORJ 3 3 =
β
ORJ
6
G%
• 'LUHFWLYLWDW ( ) ( )
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8QD UHODFLy GLUHFWH HQWUH HOOHV pV , & '
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ORJ ORJ G%
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6 pV
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θ = φ = DPE α = β = /D VHYD PDWULX GH SDUjPHWUHV
[ 6]
=
⎛ M ⎞
⎜ ⎟
⎜
M
⎟
⎜ M ⎟
⎜ ⎟
⎝ M ⎠
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• 3DVVLX L VHQVH SqUGXHV >6@>6@ >,@
• 5HFtSURF 6LM = 6 ML
• 7RWV HOV SRUWV DGDSWDWV 6 LL =
• 'RV SDUHOOV GH SRUWV DwOODWV HQWUH HOOV 6 6 L 6 6
• ,QWURGXHL[ XQ GHVIDVDPHQW GH ž HQWUH HOV SRUWV GLUHFWH L DFREODW

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VRUWLGHV HQ IXQFLy GH OHQWUDGD
[ E] = [ 6][ D]
⎡E ⎤ ⎛ M ⎞⎛ D ⎞
⎢
E
⎥ ⎜ ⎟⎜⎜ ⎟
⎢ ⎥
M
= ⎜ ⎟ ⎟
⎢E ⎥ ⎜ M ⎟⎜ ⎟
⎢ ⎥ ⎜ ⎟⎜
⎟
⎣E ⎦ ⎝ M ⎠⎝ ⎠
E = M
D
D
E =
9HLHP FRP SHOV SRUW L VXUW OD PHLWDW GH OD SRWqQFLD GH VHQ\DO TXH HQWUD SHU
D 3
3 = 3 = E = =
3HO TXH ID DO IXQFLRQDPHQW FRP D FRPELQDGRU GH SRWqQFLHV HQWUHP VHQ\DO SHOV SRUWV L
L D OHV VRUWLGHV WHQLP OHV VHJ HQWV HTXDFLRQV
⎡E ⎤ ⎛ M ⎞⎛ D ⎞
⎢
E
⎥ ⎜ ⎟⎜⎜ ⎟
⎢ ⎥
M D
= ⎜ ⎟ ⎟
⎢E ⎥ ⎜ M ⎟⎜ ⎟
⎢ ⎥ ⎜ ⎟⎜
⎟
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D D
E = M + = +
( MD D )
D D
E = + M = +
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( D MD )
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SHUz HV YHXUj GHVIDVDW HQ IXQFLy GH OD GLVWjQFLD D OD TXDO HO FXUWFLUFXLW HV VLWXw 0pV H[SOLFDFLy
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$QDOLW]HP HO FRPSRUWDPHQW GXQ KtEULG DPE GRV GHOV VHXV SRUWV FDUUHJDWV DPE XQD LPSHGjQFLD
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FDUUHJDWV DPE XQD LPSHGjQFLD TXDOVHYRO
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⎡E ⎤ ⎛ M ⎞⎛ D ⎞
⎢
E
⎥ ⎜ ⎟⎜⎜ ⎟
⎢ ⎥
M
= ⎜ ⎟ ⎟
⎢E ⎥ ⎜ M ⎟⎜ D ⎟
⎢ ⎥ ⎜ ⎟⎜
⎟
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E = ( MD + D ) E ( D MD )
= + E = M D L E = D
/HV RQHV LQFLGHQWV L UHIOHFWLGHV HV SRGHQ UHODFLRQDU DPE HO FRHILFLHQW GH UHIOH[Ly
, D OHV VRUWLGHV WHQLP
D = Γ ,1 E L D = Γ,1E
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⎛
⎜ M ⋅ MΓ ,1
⎝
D + Γ ,1
⎞
D ⎟ =
⎠
=
⎛
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⎝
D − Γ ,1
⎞
D ⎟ =
⎠
E = ( D + MD ) = ( Γ ,1 E + MΓ ,1 E ) =
⎛
⎜ MΓ ,1
⎝
D + MΓ ,1 D
⎞
⎟ =
⎠
=
⎛
⎜ M
⎝
Γ ,1 D ⎞
⎟ = M
⎠
Γ ,1 D
= MΓ ,1 D
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−
+
= −
D OD VRUWLGD HO VHQ\DO GHVIDVDW ž
E = − MD L E =
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Γ = Γ ⋅
− Mβ O
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OD VRUWLGD GH OKtEULG VHUj LJXDO D
E = MΓ D = MD Γ ⋅ H = − MD ⋅ H
,1 /
− Mβ O − Mβ O
− Mβ O
Γ ,1 = −H L
− Mβ O
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OHV OtQLHV
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IXQFLRQL GH FRPPXWDGRU SHU D VHQ\DOV 5) $TXHVW pV HO GtRGH 3,1 VHJRQV HV SRODULW]D HQ
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FRQQHFWDW D PDVVD HQ OHV GLVWjQFLHV DGHTXDGHV GH OHV OtQLHV GH WUDQVPLVVLy GH FjUUHJD GH
O KtEULG SHUPHWUDQ YDULDU HO SXQW GH UHIOH[Ly GHO VHQ\DO L SHU WDQW HO GHVIDVDPHQW HQWUH OHQWUDGD
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ILJXUD RQ VREVHUYD FRP OHV OtQLHV DFDEDGHV HQ FXUWFLUFXLW SRGHQ SUHQGUH ORQJLWXGV
GLIHUHQWV HQ IXQFLy GH OD SURJUDPDFLy GHO GHVIDVDGRU D WUDYpV GHOV FRUUHQWV DSOLFDWV DOV GtRGHV
9 L 9
3/$17(-$0(17 *(1(5$/
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3URFHGLP GH OD VHJ HQW PDQHUD
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• &jOFXO GHOV GHVIDVDPHQWV QHFHVVDULV SHU DFRQVHJXLU HOV WUHV HVWDWV GHO 93'
• (VWXGL GHO GHVIDVDGRU TXH FRPSOHL[ HOV WUHV GHVIDVDPHQWV
• (VWXGL GH OHVWUXFWXUD GHO 93' ILQDO KtEULG GHVIDVDGRU KtEULG

(VWUXFWXUD EDVH KtEULG KtEULG
(V UHDOLW]D OHVWXGL HQ DTXHVW DSDUWDW GH OHVWUXFWXUD GH OD ILJXUD SHU XQ VHQ\DO GHQWUDGD DO
SRUW GH OKtEULG + D L VXSRVDQW TXH HO GHVIDVDGRU YDULDEOH LQWURGXHL[ XQ GHVIDVDPHQW
M
TXDOVHYRO ( H ) φ
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⎛ M ⎞
⎜ ⎟
⎡ M
6
⎤ ⎡ ⎜ ⎟
6
⎤
⎢ = =
⎣ ⎥⎦ ⎢⎣ ⎥⎦
⎜
M
⎟
⎜ ⎟
⎜ M ⎟
⎝ ⎠
SHU XQ VHQ\DO LQFLGHQW DO SRUW GH + WHQLP D OD VRUWLGD GH OKtEULG
E = M
D
E =
(OV VHQ\DOV GH VRUWLGD E L E VyQ GHVIDVDWV SHU HO GHVIDVDGRU
M
H ψ
D
M
H φ
UHVSHFWLYDPHQW GH PDQHUD TXH D OHQWUDGD GH OKtEULG + WHQLP
Mφ
Mφ
D H
D = E H = M
Mψ
Mψ
D H
D = E H =
L SHU OD OtQLD GH WUDQVPLVVLy
, HQ HOV SRUWV GH VRUWLGD GH OKtEULG + WUREHP OD FRPELQDFLy GH WRWV GRV VHQ\DOV GH OD VHJ HQW
PDQHUD
⎡E ⎤ =
⎡
6
⎤ ⎡D ⎤
⎣ ⎦ ⎣⎢ ⎦⎥
⎣ ⎦
⎡ ⎤
⎢E ⎥ ⎛ M ⎞ ⎛ ⎞
⎢ ⎥ ⎜ ⎟ ⎜ D ⎟
⎢
E
⎥ ⎜ M ⎟ ⎜ ⎟
= D
⎢E ⎥ ⎜
M
⎟ ⎜ ⎟
⎢ ⎥ ⎜ ⎟ ⎜ ⎟
⎜
E
M ⎟ ⎜ ⎟
⎢ ⎥ ⎝ ⎠ ⎝ ⎠
⎣ ⎦
⎛
D H
Mφ D H
Mψ
⎞
D
E = MD + D =
⎜
M M
⎟
⎜
⋅ +
⎟
= ⋅ − D H + D H = H −H
⎜ ⎟
⎝ ⎠
Mφ Mψ Mψ Mφ
( ) ( )
⎛
D H
Mφ D H
Mψ
⎞
D
E = D + MD =
⎜
M M
⎟
⎜
+
⎟
= ⋅ M D H + D H = M H + H
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