MODELAGEM TENSORIAL DE SVC E TCSC NO DOMÍNIO s PARA ...

2 T
I1 = ∫ i cos
T
0
( t) ( ωt)
dt
(4.4.23)
Para tensão senoidal ideal, os dois semiciclos da corrente apresentam simetria ímpar, de
tal forma que:
( ωt) −V
cos( ωt
)
4 toff
4ω
toff V cos
on
I1 = ∫ i( t) cos( ωt)
dt = ∫ −
dt
T ton
2π
ton ωL
(4.4.24)
tcr
I
1
V
=
ω L
2 toff
2
tcr
∫
ton
[ − cos ( ωt) + cos( ωt
) cos( ωt)
]
on
dt
(4.4.25)
I
= 2V
⎡ 1
− sin
sin
1
⎤
( ) ( ) ( ) ( ) toff
ωt
cos ωt
− ωt
+ cos ωt
ωt
1 on
L ⎢
⎥
(4.4.26)
πω
tcr ⎣ 2
2
⎦ ton
1
( ωt
) cos( ωt
) + sin( ωt
) cos( ωt
)
⎡ 1
⎤
⎢−
sin
off
off
on
on
2V
2
2
I
⎢
⎢ ( ) ( ) ( ) ( ) ( ) ⎥ ⎥⎥ 1
= ⋅
πω Ltcr
1
(4.4.27)
− ω toff
− ton
+ cos ωton
sin ωtoff
− cos ωton
sin ωton
⎢⎣
2
⎥ ⎦
Utilizando
π − ωt
= ωt
− π ⇒ ωt
= 2 π − ωt
, tem-se:
on
off
off
on
( ωt
) cos( ω )
cos
off
= t on
. (4.4.28)
sin
( ωt
) −sin( ω )
= . (4.4.29)
off
t on
Substituindo:
2V
⎡
1
⎤
I1 =
⎢
− sin( ωton
) cos( ωton
) − ( 2π − 2ωton
) − 2cos( ωton
) sin( ωton
)
πω L ⎣
2
⎥⎦
, (4.4.30)
tcr
e fazendo
t
on
α
= ⇒ α = ωton
tem-se:
ω
112