Tipos y cálculo de faltas en circuitos trifásicos

TIPOS Y CÁLCULO DE FALTAS
EN CIRCUITOS TRIFÁSICOS.
0.- PRELIMINARES.
I 0 
1 1
 I  = 1 1 a
 1 3
 I 2 
1 a 2
1  I a 
a 2   I b 
a   I c 
 I a  1 1
 I  = 1 a 2
 b 
 I c  1 a
1 I 0 
a   I 1 
a 2   I 2 
V0 
1 1
V  = 1 1 a
 1 3
V2 
1 a 2
1  Va 
a 2  Vb 
a  Vc 
Va  1 1
V  = 1 a 2
 b 
Vc  1 a
1  V0 
a  V1 
a 2  V2 
V1 = VFP − I 1 Z 1

 V2 = − I 2 Z 2
 V = −I Z
0 0
 0
1+ a + a2 = 0
1A.- FALTA MONOFÁSICA FRANCA.
Ib = 0

C.I . I c = 0
V = 0
 a
1 1
I 0 
 I  = 1 1 a
 1 3
1 a 2
 I 2 
I1
V1
V2
1  0 
a 2  Vb 
a  Vc 
1

 V0 = 3 (Vb + Vc )

1
2
V1 = (aVb + a Vc )
3

V = 1 (a 2V + aV )
b
c
 2 3
I 0 = I1 = I 2
I a = 3I 1
S.D.
I2
V0 
1 1
V  = 1 1 a
 1 3
V2 
1 a 2
I0
V0 + V1 + V2 =
S.I.
− I 0 Z 0 + VFP
V0
1  I a 
a 2   0 
a   0 
1

I 0 = 3 I a

1
 I1 = I a
3

I = 1 I
 2 3 a
[
]
1
Vb (1 + a + a 2 ) + Vc (1 + a + a 2 ) = 0
3
− I1 Z1 − I 2 Z 2 = 0
S.H.
I 1 ( Z 0 + Z 1 + Z 2 ) = VFP → I 1 =
VFP
Z 0 + Z1 + Z 2
1B.- FALTA MONOFÁSICA CON RESISTENCIA.
 Ib = 0

C.I . I c = 0
V = I R
a F
 a
RF
I 0 = I1 = I 2
I a = 3I 1
− I 0 Z 0 + VFP
[
]
1
3Va + Vb (1 + a + a 2 ) + Vc (1 + a + a 2 ) = Va
3
− I 1 Z 1 − I 2 Z 2 = I a R F = 3I 1 R F
V0 + V1 + V2 =
I 1 ( Z 0 + Z 1 + Z 2 + 3RF ) = VFP → I 1 =
V FP
Z 0 + Z 1 + Z 2 + 3R F
I1
V1
S.D.
I2
3RF
V2
S.I.
I0
V0
S.H.
Va
2A.- FALTA BIFÁSICA FRANCA.
 Ia = 0

C.I . I b = − I c
V =V
c
 b
I 0 
1 1
 I  = 1 1 a
 1 3
 I 2 
1 a 2
1  0 
a 2   I b 
a  − I b 
I0 = 0
1

I 0 = 3 ( I b − I b ) = 0

1

2
 I 1 = (aI b − a I b )
3

 I = 1 (a 2 I − aI )
b
b
 2 3
I1 = − I 2
I b = − 3I 1 j
1 1
V0 
V  = 1 1 a
 1 3
1 a 2
V2 
1

V0 = (Vb + 2Vb )

3

1
1
2
V1 = (Va + Vb (a + a )) = (Va − Vb )
3
3

1
2
V = (V + V (a + a )) = 1 (V − V )
b
a
b
 2 3 a
3
1  Va 
a 2  Vb 
a  Vb 
V1 = V2
VFP − I 1 Z 1 = − I 2 Z 2 → VFP = I 1 ( Z 1 − (− Z 2 ))
I1 =
V1
VFP
Z1 + Z 2
S.D.
V2
I1
S.I.
V0
I2
S.H.
2B.- FALTA BIFÁSICA CON RESISTENCIA.
Ia = 0


C.I . I b = − I c
V = V + I R
c
b F
 b
RF
I0 = 0
Vb
I1 = − I 2
Vc
I b = − 3I 1 j
1   Va

2 
a  Vc + I b RF 

a   Vb
1 1
V0 
V  = 1 1 a
 1 3
1 a 2
V2 
[
[
VFP = I 1 ( Z 1 + Z 2 + RF ) →
V1
[
[
] 
1
Va + (a + a 2 )Vc + aI b RF
3
1
V2 = Va + (a + a 2 )Vc + a 2 I b RF
3
VFP − I 1 Z 1 − (− I 2 Z 2 ) = I 1 RF
V1 =
1

V0 = (Vb + 2Vc + I b RF )

3

1
2
V1 = Va + a (Vc + I b RF ) + a Vc
3

V = 1 V + a 2 (V + I R ) + aV
c
b F
c
 2 3 a
S.D.
V2
I1
1
2
 V1 − V2 = I b (a − a ) RF = I 1 RF
3


]
I1 =
VFP
Z1 + Z 2 + RF
S.I.
V0
I2
RF
S.H.
]
]
2C.- FALTA BIFÁSICA FRANCA A TIERRA.
 Ia = 0

C.I .
V = V = 0
c
 b
I 0 
1 1
 I  = 1 1 a
 1 3
 I 2 
1 a 2
1  0 
a 2   I b 
a   I c 
[
1

 I 0 = 3 (I b + I c )

1
2
 I 1 = (aI b + a I c )
3

 I = 1 (a 2 I + aI )
b
c
 2 3
]
1
I b (1 + a + a 2 ) + I c (1 + a + a 2 ) = 0
3
 I b + I c = 3I 0
I b = 3I 0 − I c



2
2
aI b + a I c = 3I 1 a(3I 0 − I c ) + a I c = 3I 1
I 0 + I1 + I 2 =
3aI 0 + I c (a 2 − a) = 3I 1
Ic =
3( I 1 − aI 0 ) 3( I 1 − aI 0 )
=
= 3 j ( I 1 − aI 0 )
a2 − a
− 3j
I b = 3I 0 − 3 j ( I 1 − aI 0 ) = I 0 (3 + 3 ja ) − 3 jI 1
1 1
V0 
V  = 1 1 a
 1 3
1 a 2
V2 
1  Va 
a 2   0 
a   0 
1

V0 = 3 Va

1
V1 = Va → V0 = V1 = V2
3

V = 1 V
 2 3 a
VFP − I 1 Z 1 = ( I 1 + I 0 ) Z 2 (*)


VFP − I 1 Z 1 = − I 2 Z 2 = − I 0 Z 0 
− I1Z 2
( I 1 + I 0 ) Z 2 = − I 0 Z 0 → I 0 = Z + Z
2
0

2


I1 Z 2
Z2


(*) VFP − I 1 Z 1 = ( I 1 −
) Z 2 → VFP = I 1  Z 1 + Z 2 −

+
Z2 + Z0
Z
Z
2
0 

I1 =
VFP
Z Z
Z1 + 2 0
Z2 + Z0
V1
S.D.
S.I.
V2
V0
I1
S.H.
I2
I0
2D.- FALTA BIFÁSICA CON RESISTENCIA A TIERRA.
Ia = 0


C.I .
V = V = ( I + I ) R
c
b
c
F
 b
RF
I 0 + I1 + I 2 = 0
I b + I c = 3I 0
1 1
V0 
V  = 1 1 a
 1 3
1 a 2
V2 
1 
Va

2 
a  ( I b + I c ) RF 
a  ( I b + I c ) RF 
V1 = V2

V0 − V2 = ( I b + I c ) RF = 3I 0 RF
I1 =
V FP
Z ( Z + 3R F )
Z1 + 2 0
Z 0 + Z 2 + 3R F
I b = I 0 (3 + 3 ja ) − 3 jI 1
1

V0 = [Va + 2( I b + I c ) R F ]

3

1
1
2
V1 = Va + ( I b + I c ) RF (a + a ) = [Va − ( I b + I c ) RF ]
3
3

1
2
V = V + ( I + I ) R (a + a ) = 1 [V − ( I + I ) R ]
b
c
F
a
b
c
F
 2 3 a
3
[
[
 VPF − I 1 Z 1 = − I 2 Z 2

− I 0 Z 0 − ( − I 2 Z 2 ) = 3 I 0 R F
(*) I 0 (− Z 0 − Z 2 − 3RF ) = I 1 Z 2 → I 0 =
(**) VFP = I 1 Z 1 + I 1 Z 2 −
I c = 3 j ( I 1 − aI 0 )
]
]
 VPF − I 1 Z 1 = ( I 0 + I 1 ) Z 2 (**)

− I 0 Z 0 − ( I 0 + I 1 ) Z 2 = 3I 0 RF (*)
− I1 Z 2
Z 0 + Z 2 + 3R F
2
2
2

Z Z + Z 2 + Z 2 (3RF ) − Z 2
I1 Z 2
= I 1  Z 1 + 0 2
Z 0 + Z 2 + 3R F
Z 0 + Z 2 + 3R F





V0
V1
S.D.
V2
S.H.
S.I.
RF
I1
I2
I0
3A.- FALTA TRIFÁSICA FRANCA.
C.I . ⇒ Va = Vb = Vc = 0
1 1
V0 
V  = 1 1 a
 1 3
1 a 2
V2 
V0 = 0

V1 = 0
V = 0
 2
1  0 
a 2  0
a  0
 V0 = − I 0 Z 0 = 0 → I 0 = 0

VFP

V1 = VFP − I 1 Z 1 = 0 → I 1 =
Z1

 V2 = − I 2 Z 2 = 0 → I 2 = 0
1

1 1 1   I a 
0
 0 = 3 ( I a + I b + I c ) (1)
 I  = 1 1 a a 2   I 

1
 b 
 1 3 
I 1 = ( I a + aI b + a 2 I c ) (2)

2
1 a
 0 
a   I c 
3

1
 0 = ( I + a 2 I + aI ) (3)
a
b
c

3
(1) − (3) → 0 = I b (1 − a 2 ) + I c (1 − a ) = I b (1 − a )(1 + a ) + I c (1 − a ) = I b (1 + a ) + I c → I c = − I b (1 + a )
(1) → I c = − I a − I b = − I b (1 + a ) → I b =
Ia
= a 2 I a ; Ic = −a 2 I a (1 + a ) = − a 2 I a (−a 2 ) = aI a
a
1
(2) → I 1 = ( I a + a·a 2 I a + a 2 ·aI a ) = I a
3
 I a = I1

2
I b = a I 1
 I = aI
1
 c
Z1
I1
3B.- FALTA TRIFÁSICA CON RESISTENCIA.
Va = I a RF

C.I .Vb = I b RF
V = I R
c F
 c
RF
V0 
1 1
V  = 1 1 a
 1 3
V2 
1 a 2
1 1
I 0 
 I  = 1 1 a
 1 3
1 a 2
 I 2 
1  Va 
a 2  Vb 
a  Vc 
1

 V0 = 3 (Va + Vb + Vc )

1
2
V1 = Va + aVb + a Vc
3

V = 1 V + a 2V + aV
2
a
b
c
3

1  I a 
a 2   I b 
a   I c 
1

 I 0 = 3 (I a + I b + I c )

1
2
 I 1 = ( I a + aI b + a I c ) → I a = I 1
3

 I = 1 ( I + a 2 I + aI )
b
c
 2 3 a
(
(
)
)
1
R F ( I a + I b + I c ) = R F I 0 = − Z 0 I 0 → I 0 (R F + Z 0 ) = 0 → I 0 = 0
3
VFP
1
V1 = ( I a + aI b + a 2 I c ) = RF I 1 = VFP − Z 1 I 1 → I 1 =
3
RF + Z1
1
I 2 = ( I a + a 2 I b + aI c ) = RF I 2 = − Z 2 I 2 → I 2 (RF + Z 2 ) = 0 → I 2 = 0
3
V0 =
 I a = I1

2
I b = a I 1
 I = aI
1
 c
Z1
RF
I1