consulta1 - sergio haffner

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Sistemas de Energia I – Folha de Consulta 1 – SHaffner Versão: 10/9/2007 Página 1 de 2 
 
Expressões gerais 
( ) ( ) jXRI
V
jYjZ +=== ωω
1
 
( ) ( ) jBGV
I
jZjY +=== ωω
1
 
 
jQPjVIVIIVIVS +=+==+−=⋅= θθθθφφ sencos* θ
φ
φ
φ cos
3
3
3 == S
P
FP 
θθφφ cos3cos33 LLL IVIVP == θθφφ sen3sen33 LLL IVIVQ == LLL IVIVS 333 == φφ 
 
Linhas de transmissão 
A
lRCC ρ= 
10
20
12 TT
TT
RR
+
+
= 4
1−
⋅=′ err αα 
 
n
anacaba
M
aMaCaBaA
a DDDr
DDDD
nL
L
L
′
= ln
2pi
µ
 
 
C 
a 
Condutor x
 
b 
c 
n 
A B 
M 
Condutor Y 
bnD bC
D
 
n
bnbcbab
M
bMbCbBbA
b DDDr
DDDD
nL
L
L
′
= ln
2pi
µ
 
s
m
x D
DL ln
2pi
µ
=
 ( )( ) ( )Mn nMnBnAbMbBbAaMaBaAm DDDDDDDDDD LLLL= 
xL fLX pi2= ( )( ) ( )2n nnnbnabnbbbaanabaas DDDDDDDDDD LLLL= 
s
eq
x D
D
L ln
2pi
µ
=
 
3
312312 DDDDeq = Para o vácuo kmH
4
m
H7
0 104 104
−−
== pipiµ 
 
 
abD
a 
b 
c 
αααα 
ββββ 
χχχχ 
bcD
βaD χbD
αaD βbD χcD
caD
αcD
Condutores 
Imagens 
Superfície do solo 
 
 a b 
D 
 
r
D
kCab
ln
pi
=
 
r
D
kCN
ln
2pi
=
 
N
C fCX pi2
1
=
 
 








+
=








−
=
3
3
3
3
lnln
2
lnln
2
αχβ
χβα
χβα
αχβ
pipi
cba
cbaeq
cba
cbaeq
N
DDD
DDD
r
D
k
DDD
DDD
r
D
kC
 
 
Para o vácuo kmF9mF120 1085,8 1085,8 −− ×=×=k 
 
 
k l
lZZ kmkm
⋅
⋅
=
′
γ
γsenh
2
kmY
′
m 
2
2
tanh
22 l
l
YY kmkm
⋅
⋅
=
′
γ
γ
 
yzj ⋅=+= βαγ 
Sistemas de Energia I – Folha de Consulta 1 – SHaffner Versão: 10/9/2007 Página 2 de 2 
 
 
Sistema por unidade 
3
base3
base
φ
φ
S
S = 
3
base 
base 
LVV =φ 
base 
base 3
base 
base 3
base 
base 
base base 3
3
3
LL
YL V
S
V
S
V
S
II φ
φ
φ
φ
====
 
base 
base 3base 
base 33 L
L
V
SI
I φ==∆ 
base 3
2
base 
base 
base 
base 
φ
φ
S
V
I
V
Z L
Y
Y == base base 3 YZZ =∆ ( ) ( )
1 base 3
2 base 3
2
2 base 
1 base 
1 basepu 2 basepu 
φ
φ
S
S
V
VZZ
L
L






=
 
 
Transformadores 
 
mjx 
• 
1I 
Transformador 
Real 
• 
Ideal 
1V 
+ 
– 
2V 
+ 
 
21 : NN 
11 jxr + 22 jxr + 
mr 
 
 
2
1
2
1
N
N
V
V
=
 
1
2
2
1
N
N
I
I
=
 2
2
2
1ref
2 Z
N
NZ 





=
 1
2
1
2ref
1 Z
N
NZ 





=
 
pri
base
1
2sec
base VN
NV =
 
pri
base
2
1
pri
base
1
2
base
sec
base
basesec
base IN
N
V
N
N
S
V
S
I ===
 
 
 
k 
kmI
m 
mkI
kV mVA 
B C 
A, B, C admitâncias 
 
( )
( ) kmkm
kmkmkm
kmkm
yaC
yaaB
yaA
−=
−=
=
1
1
 
 
Expressões gerais dos fluxos de corrente e potência nos ramos 
( ) ( ) mkmjkmkshkmkmkmkm VYeaVjbYaI kmϕ−−++= 2 ( ) ( ) mshkmkmkkmjkmmk VjbYVYeaI km ++−= ϕ 
( ) ( ) ( ) ( )[ ]kmkmkmkmkmkmmkkmkmkkmkm bgVVagVaP ϕθϕθ +++−= sencos2 
( ) ( ) ( ) ( ) ( )[ ]kmkmkmkmkmkmmkkmshkmkmkkmkm bgVVabbVaQ ϕθϕθ +−+−+−= cossen2 
 
Modelo ZIP de carga 
( ) NOM2 kkPkPPk PVcVbaP ++= 1=++ PPP cba ( ) NOM2 kkQkQQk QVcVbaQ ++= 1=++ QQQ cba