dx(t) dt 2 dx(t) d x(t) ax(t) + b +c dt dt2 ax(t) + b
an
=
f (t)
=
f (t)
dn x(t) dx(t) + a1 + a0 x(t) = F (t) n dt dt
t = 0−
t = 0+ iL (0− ) = iL (0+ )
uC (0− ) = uC (0+ )
f (t) = ax(t) + b
dx(t) dt
5 + 20sen(10t) = 4x(t) + 10 dxdt(t) x(t) = xlibre + x particular
4 = ke− 10 t +
5 4
+
20 sen(10t 42 +(10 10)2
√
∗
−
∗
t = 0 + x(0+ )
i(0− ) = 0
E = Ri + L
arctan( 10410 ))
di = (R + LD)i dt
i = ilibre + iforzada (R + LD)i = 0 R
ilibre = ke− L t
iforzada = E/R
E R
R
i(t) = ke− L t +
i(0− ) = i(0+ ) = 0
i(0+ ) = 0 = k +
i(t) =
e− − E R
R Lt
E R
+
k=
E E = R R
− E R
e− − E R
t τ
τ
τ
5τ
P R P L P E
− Ee−
uR
=
E
uL
=
Ee − L t
R Lt
R
R 2R E 2 (1 2e− L t + e− L t ) R 2R E 2 − R = uL i = (e L t e− L t ) R R E 2 = Ei = (1 e− L t ) R
= uR i =
−
−
−
t2 t = t 0 = Ri(t) + L
− t2
i(t2− ) = i(t2+ ) = E R i(0+ ) = E R
di(t) = (R + LD)i dt R
i(t ) = ke− L t
i(t0+ ) =
i(t ) =
E =k R
E − R e Lt R
E 2 1 2 L( R )
L = L3 + L12 = L3 +
i(t) =
+
uL
= Ee − L t = uL3 + uL12 = L
uL2
i = = =
E R
R Lt
= E
uL1
i2
R Lt
uR
uL3
i1
− Ee−
e− − E R
L1 L2 L1 + L2
di di = (L3 + L12 ) dt dt R di L3 L3 = L3 = uL = Ee − L t dt L3 + L12 L3 + L12 R di L12 L12 = L12 = uL = Ee − L t dt L3 + L12 L3 + L12 = uL1 R
E R 1 L1 1 L2
−
E − R 1 e Lt = R L12 t
L1 + L2 uL1 dt = L1 L2 0+
uL1 dt =
L2 i L1 + L2
uL1 dt =
L1 i L1 + L2
0+ t 0+
t
t
0+
uL1 dt
E = (R1 + LD)i1 + R1 i2 E = R1 i1 + (R1 + R2 )i2
i1 =
i2 =
E R1 E (R1 + R2 ) (R1 + LD) R1 R1 (R1 + R2 )
=
ER 2 (R1 + R2 )LD + R1 R2
(R1 + LD) E R1 E (R1 + LD) R1 R1 (R1 + R2 )
=
0 (R1 + R2 )LD + R1 R2
ER 2
= ((R1 + R2 )LD + R1 R2 )i1
0
= ((R1 + R2 )LD + R1 R2 )i2
i1 i2
= i1libre + i1forzada = i2libre + i2forzada
i1forzada i2forzada
E R1 = 0 =
R1 R2
i1libre
= k1 e− (R1 +R2 )L t
i2libre
= k2 e
1 R2 − (RR1 +R t 2 )L
− (RR+RR )L 1
1
k1
2
2
k2
i1 (0+ ) = 0 t = 0+ di1 di1 )0+ + R1 i2 (0+ ) = L( )0+ + R1 i2 (0+ ) dt dt E = R1 i1 (0+ ) + (R1 + R2 )i2 (0+ ) = (R1 + R2 )i2 (0+ ) E = R1 i1 (0+ ) + L(
k1
i1 (0+ )
= 0
i2 (0+ )
=
E R1 + R2
k2 E + k1 R1 E = k2 R1 + R2
i1 (0+ )
= 0=
i2 (0+ )
=
i1
=
i2
=
E − R1 R2 t (1 e (R1 +R2 )L ) R1 E − R1 R2 t e (R1 +R2 )L R1 + R2
−
1 I = iL + iR = iL (0+ ) + L
t
udt +
0+
u 1 1 =( + )u R LD R
u = ulibre + uforzada
uforzada = 0 R
ulibre = ke− L t
u(0+ ) = k t = 0+
I = iL (0+ ) + iR (0+ ) = 0 +
u(0+ ) R
k = RI R
u = RIe− L t
iR iL
L1
R u = Ie− L t R t 1 = 0+ udt = I L 0+
=
− Ie−
L2
i1
=
i2
=
2E R E R
i1 (0− ) = 2RE = i1 (0+ ) = i2 (0+ ) i2 (0− ) = E R = i2 (0+ ) = i1 (0+ )
R Lt
i(0− ) = 0
√ e(t) = 2Esen(wt + φ)
e = Ri + L
di = (R + LD)i dt
i = ilibre + iforzada (R + LD)i = 0 R ilibre = ke− L t
iforzada =
√ 2E
R2 + (wL)2
R i(t) = ke− L t +
k=
i(t) =
−
√ 2E
R2 + (wL)2
−
sen(wt + φ
√ 2E
R2 + (wL)2
sen(wt + φ
− arctan(wL/R))
i(0− ) = i(0+ ) = 0
√ 2E
− arctan(wL/R))
R2 + (wL)2
sen(φ
R sen(φ − arctan(wL/R))e− L t +
− arctan(wL/R)) √ 2E
R2 + (wL)2
sen(wt + φ
uC (0− ) = 0
1 E = Ri + uC (0+ ) + C
t
0+
idt = (R +
1 )i CD
− arctan(wL/R))
i = ilibre + iforzada (R + 1/(CD))i = 0 1 ilibre = ke− RC t
iforzada = 0
1
i(t) = ke− RC t i(0+ ) uC (0− ) = uC (0+ ) = 0
E = uC (0+ ) + Ri(0+ ) = Ri(0+ )
i(t) =
1 E − RC E t t e = e− τ R R
τ
τ
5τ
uR
1
Ee − RC t
=
uC = 0+
P R P C P E
1 C
t
idt = E
0+
E 2 − 2R = uR i = e Lt R E 2 − R t = uC i = (e L R E 2 − R = Ei = e Lt R
− Ee−
− e−
1 RC t
2R L t
)
t2 uC (t2− ) = uC (t2+ ) = E U c (o+ ) = E
t = t − t2 1 0 = Ri + uC (0+ ) + C
t
idt = (R +
0+
R
i(t ) = ke− L t
1 )i CD
0 = Ri(0+ ) + uC (0+ ) = Ri(0+ ) + E
k= i(t ) =
− E R
e− − E R
1 RC t
1 2 2 C (E )
C =
C 12 C 3 C 12 +C 3
C 12 = C 1 + C 2
i(t) =
uR uC uC 3 uC 1 uC 2
1 E − RC t e R
1
= Ee − RC t 1 t = idt = E C 0+ =
1 C 3
1 C 12 = uC 1 =
t
idt =
0+ t 0+
idt =
− Ee− RC1 t = uc3 + uc12 = C 3 + C 12
C 12 uC C 3 + C 12 C 3 uC C 3 + C 12
1 E − RC duC 1 duC 1 t e = C 12 = (C 1 + C 2 ) R dt dt duC 1 C 1 = C 1 = i dt C 1 + C 2 duC 2 C 2 = C 2 = i dt C 1 + C 2
i = i1 i2
C 3 C 12
1 )i1 + R1 i2 CD E = R1 i1 + (R1 + R2 )i2 E = (R1 +
t
0+
idt
i1 =
i2 =
E R1 E (R1 + R2 ) 1 (R1 + CD ) R1 R1 (R1 + R2 )
=
ER 2 1 (R1 + R2 ) CD + R1 R2
1 (R1 + CD ) E R1 E 1 (R1 + CD ) R1 R1 (R1 + R2 )
=
1 CD E 1 R2 ) CD +
(R1 +
1 + R1 R2 )i1 CD 1 1 E = ((R1 + R2 ) + R1 R2 )i2 CD CD ER 2
= ((R1 + R2 )
i1
= i1libre + i1forzada
i2
= i2libre + i2forzada
i1forzada
= 0
i2forzada
=
E R1 + R2
i1libre
= k1 e
2 − (RR11 +R R2 )C t
i2libre
= k2 e
2 t − (RR11 +R R2 )C
− (RR R+R)C 1 1
k1
2
2
k2
uC (0+ ) = 0 = (
1 i1 )0+ CD t = 0+
R1 R2
1 i1 )0+ + R1 i2 (0+ ) = R1 i1 (0+ ) + 0 + R1 i2 (0+ ) CD E = R1 i1 (0+ ) + (R1 + R2 )i2 (0+ ) E = R1 i1 (0+ ) + (
i1 (0+ ) i2 (0+ ) k1
E R1 = 0 =
k2 i1 (0+ ) i2 (0+ )
i1
=
i2
=
E = k1 R1 E = 0= + k2 R1 + R2 =
R1 +R2 E − (R t 1 R2 )C e R1 E − R1 +R2 t (1 e (R1 R2 )C ) R1 + R2
−
du u 1 I = iC + iR = C + = (CD + )u dt R R
u = ulibre + uforzada
uforzada = RI 1
ulibre = ke− RC t
u(0+ ) = k + RI
t = 0+ k= u = RI
iR iC
C 1
− RIe −
−RI
1 RC t
1 u = I (1 e− RC t ) R 1 du = C = Ie− RC t dt
=
−
C 2
uC 1 uC 2
= R2I = RI
uC 1 (0− ) = R2I = uC 1 (0+ ) = uC 2 (0+ ) uC 2 (0− ) = RI = uC 2 (0+ ) = uC 1 (0+ )
√ e(t) = 2Esen(wt + φ)
uC (0− ) = 0
t
e = Ri + uC (0+ ) +
idt = (R +
0+
1 )i CD
i = ilibre + iforzada (R + ilibre = ke−
√ 2E
iforzada =
i(t) = ke−
1 RC
1 RC t
R2 + (1/(wC ))2
t
+
e(t) =
i(0+ ) =
k=
R2 + (1/(wC ))2
1/(wC ) )) R
sen(wt + φ + arctan(
1/(wC ) )) R
uC (0− ) = uC (0+ ) = 0
√
√ 2Esen(φ) R
√ 2Esen(φ) R
=0
sen(wt + φ + arctan(
√ 2E
1 CD )i
2Esen(φ) = uC (0+ ) + Ri(0+ ) = Ri(0+ )
=k+
−
√ 2E
R2 + (1/(wC ))2
√ 2E
R2
+ (1/(wC ))2
sen(φ + arctan(
sen(φ + arctan(
1/(wC ) )) R
1/(wC ) )) R
u(t), i(t), di(t)/dt, du(t)/dt iL (0 ), uC (0 )
−
−
iL (0 ) = iL (0+)
−
uC (0 ) = uC (0+)
−
t = 0+ (L didtL )0+
C (C du dt )0+
di 1 E = Ri + L + uC (0+ ) + dt L
t
0+
idt = (R + LD +
1 )i CD
i = ilibre + iforzada iforzada = 0
R + LD +
( 2RL )2 >
1 LC
( 2RL )2 =
1 LC
( 2RL )2 <
1 LC
R + 2L R 2L
D1
=
−
D2
=
− −
1 =0 CD
R 2 ) 2L R ( )2 2L (
1 − LC 1 − LC
−α1 −α2 ilibre = k1 e−α1 t + k2 e−α2 t
−α = − 2RL ilibre = k1 e−αt + k2 te−αt
−α + jw −α − jw ilibre = e−αt (k1 cos(wt) + k2 sen(wt)) = e−αt (Asen(wt + B))
α
R =0 w=
1/(LC )
α =0
i = k1 e−α1 t + k2 e−α2 t i = k1 e−αt + k2 te−αt i = e−αt (k1 cos(wt) + k2 sen(wt)) i(0+ ), ( di dt )0+
i(0− )
uC (0− )
i(0+) = i(0− ) = 0 uC (0+ ) = uC (0− ) = 0
E = Ri(0+) + L(
di di )0+ + uC (0+ ) = L( )0+ dt dt
i(0+) = 0 di E ( )0+ = dt L
k1
k2
uR
= Ri di = L dt
uL uc
1 = uc (0+ ) + C
u 1 I = iR + iL + iC = + iL (0+ ) + R L
t
idt
0
t
du 1 1 udt + C = ( + + CD)u dt R LD 0+
u = ulibre + uforzada uforzada = 0
1 1 + + CD = 0 R LD
1 ( 2RC )2 >
1 LC
1 ( 2RC )2 =
1 LC
1 ( 2RC )2 <
1 LC
D1
=
−
D2
=
−
1 + 2RC 1 2RC
−
1 2 ) 2RC 1 2 ( ) 2RC (
1 − LC 1 − LC
u = k1 e−α1 t + k2 e−α2 t u = k1 e−αt + k2 te−αt u = e−αt (k1 cos(wt) + k2 sen(wt)) u(0+ ), ( du dt )0+
iL (0− )
u(0− )
iL (0+) = iL (0− ) = 0 u(0+ ) = u(0− ) = 0
I =
u(0+) du du + iL (0+ ) + C ( )0+ = C ( )0+ R dt dt
u(0+) = 0 du I ( )0+ = dt C k1
k2
iR
=
u R
du iC = C dt iL
e=
√
1 = iL (0+ ) + L
t
udt
0
di 1 2sen(wt + φ) = Ri + L + uC (0+ ) + dt L
t
0+
idt = (R + LD +
1 )i CD
i = ilibre + iforzada
iforzada =
√ 2E
R2 + (wL
1 2 ) − wC
sen(wt + φ
R + LD +
− arctan
wL
1 − wC )
R
1 =0 CD
√ √
i = 2Isen(wt + ϕ) + k1 e−α1 t + k2 e−α2 t i = 2Isen(wt + ϕ) + k1 e−αt + k2 te−αt i = 2Isen(wt + ϕ) + e−αt (k1 cos(wt) + k2 sen(wt))
√
i(0+ ), ( di dt )0+
i(0− )
uC (0− )
i(0+) = i(0− ) = 0 uC (0+ ) = uC (0− ) = 0
√
2sen(φ) = Ri(0+) + L(
i(0+) di ( )0+ dt
= 0 =
k1
k2
uR
= Ri di = L dt
uL uc
di di )0+ + uC (0+ ) = L( )0+ dt dt
√ 2sen(φ) L
1 = uc (0+ ) + C
t
0
idt
E =
(2R + LD)i1 + (R + LD)i2
E =
(R + LD)i1 + (2R + LD +
i1 i2
= i1libre + i1forzada = i2libre + i2forzada
i1forzada i2forzada
1 )i2 CD
2R + LD R + LD
E 2R = 0 =
R + LD 1 2R + LD + CD
2RLD + (3R2 +
=0
L 2R )+ =0 C CD
i2
E + k1 e−α1 t + k2 e−α2 t 2R = k3 e−α1 t + k4 e−α2 t
i1
=
i1
i2
=
E + k1 e−αt + k2 te−αt 2R = 0 + k3 e−αt + k4 te−αt
i1 i2
E + e−αt (k1 cos( cos(wt) wt) + k2 sen( sen(wt)) wt )) 2R = e−αt (k3 cos( cos(wt) wt) + k4 sen( sen(wt)) wt )) =
di2 1 i1 (0+ ), ( di dt )0+ , i 2 (0+ ), ( dt )0+
i1 (0+) + i2 (0+) = 0 uC (0+ ) = 0
di1 di2 )0+ + Ri2 (0+) + L( )0+ dt dt di1 di2 E = Ri1 (0+) + L( )0+ + 2Ri 2Ri2 (0+) + L( )0+ + uC (0+) dt dt E = 2Ri1 (0+) + L(
t = 0+ 1 i2 − Ri2 − CD i2 RDi1 − RDi 2 − C di1 di1 i2 (0+ ) R( )0+ − R( )0+ − dt dt C
0 = Ri1 0 = 0 =
k1 , k2 , k3 y k 4
di2 1 i1 (0+ ), ( di dt )0+ , i2 (0+ ), ( dt )0+
F ( F (s) =
∞
f ( f (t)dt
0
−
s = σ + jw
A At e−at te−at tn e−at sen( sen(wt + φ) δ (t) − αt 2K e sen( sen(wt φ) df (t) dt
−
t
f (t)dt −∞ f (
F ( F (s) =
F ( F (s) =
A s A s2 1 s+a 1 (s+a)2 1 (s+a)n+1 s[senφ]+w[cosφ] s2 +w 2 Ke jφ s+α+jw
1 +
Ke −jφ s+α jw
− sF ( sF (s) − f (0 f (0− ) 0
F (s) s
−
+
−∞
f (t)dt s
N ( N (s) D(s)
N ( N (s) s(s + a1 )(s )(s + a2 )(s )(s + α + jw)( jw )(ss + α
jw ) − jw)
F ( F (s) =
k1 k2 k3 k4 k5 + + + + s s + a1 s + a2 (s + α + jw) jw ) (s + α jw) jw )
−
s + a1 F ( F (s)(s )(s + a1 ) = [ s=
k1 k3 k4 k5 + + + ](s ](s + a1 ) + k2 s s + a2 (s + α + jw) jw ) (s + α jw) jw )
−
−a1
k1 k2 k3 k4 k5
k2 .
= = = = =
[F ( F (s)]s )]s s=0 [F ( F (s)](s )](s + a1 ) s=−a1 [F ( F (s)](s )](s + a2 ) s=−a2 [F ( F (s)](s )](s + α + jw) jw ) s=−α−jw [F ( F (s)](s )](s + α jw) jw ) s=−α+jw
k1 , k2 , y k3
|
| |
| |
−
k4
k4 = k∗ = Ke jφ
k5
5
F ( F (s) =
k1 k2 k3 K ejφ K e−jφ + + + + s s + a1 s + a2 (s + α + jw) jw ) (s + α jw) jw )
−
f ( f (t) = k1 + k2 e−a1 t + k3 e−a2 t + 2K 2K e−αt sen( sen(wt
− φ) −an
F ( F (s) =
kn kn−1 k2 k1 + + ... + + n n − 1 2 (s + an ) (s + an ) (s + an ) (s + an )
kn kn = F ( F (s)(s )(s + an )n kn−1
|s=−a
n
(s + an )n kn−1 =
d [F ( F (s)(s )(s + an )n ] ds
kn
|s=−a
n
kn−m
=
1 dm [F ( F (s)(s )(s + an )n ] m m! ds
f ( f (t) = k1 e−a1 t + k2 te−a1 t + ... +
kn (n
|s=−a
n
n 1
− 1) t
− e−a1 t
u(t) = Ri( Ri(t)
U (s) = RI (s)
u(t) = L
U (s) = L[sI (s)
di( di(t) dt
− i(0−)] = sLI (s) − Li(0 Li(0− )
I (s) =
U (s) i(0− ) + Ls s
du(t) i(t) = C dt
I (s) = C [sU (s)
− u(0−)] = sCU (s) − Cu(0−)
U (s) =
I (s) u(0− ) + Cs s
ea−a eb−b
=
=
ec−c
ea−a eb−b
==
=
=
ec−c
==
√ √ 2E asen(wt + ϕa ) √ 2E bsen(wt + ϕb) 2E c sen(wt + ϕc )
√ √ 2Esen(wt + ϕ) √ 2Esen(wt + ϕ − 120 ) 2Esen(wt + ϕ − 240 )
ea−a , eb−b , ec−c
ea , eb , ec
ea (t) + eb (t) + ec (t) = 0 o´ E a + E b + E c = 0
U ab U bc U ca
= E a = E b = E c
U ab
=
U bc
=
U ca
=
− E b − E c − E a
√ √ 3E a 1∠30 √ 3E b 1∠30 3E c 1∠30
U ab ,
U bc = U ab 1∠
U ab
U ab ,
− 120 ,
U bc = U ab 1∠120
U ab
=
U bc
=
U ca
=
U ca = U ab 1∠120
U ca = U ab 1∠
− 120
√ √ 3E a 1∠ − 30 √ 3E b 1∠ − 30 3E c 1∠ − 30
U bc = U ab 1∠120 ,
E a = E b = E c = U F
U ca = U ab 1∠
− 120
U ab = U bc = U ca = U L
Secuencia directa :
U L =
√
Secuencia inversa :
U L =
√
π
3U F 1ej 6
π
3U F 1e−j 6
√ 3
U ab U bc U ca
= E ab = E bc = E ca
Z Y , Z Z Y =
Z 3
Z = 3Z Y
U AB , U BC , U CA
U ab , U ab
U bc = U ab 1∠
U bc ,
− 120
I ab = I ab
I a
I bc = I ab 1∠
est n equilibradas
U ca = U ab 1∠ + 120
U ab Z
− 120
I bc =
I b
=
I c
=
− 120
I c = I a 1∠ + 120
I L =
I b = I a 1∠
− 120
I a =
U aO Z Y
I ca =
est n equilibradas
U ca Z est n equilibradas
√ − I ca = √ 3I ab 1∠ − 30 I bc − I ab = 3I bc 1∠ − 30 √ I ca − I bc = 3I ca 1∠ − 30
= I ab
I a = I b = I c = I L
I a
U bc Z
I ca = I ab 1∠ + 120
I a
I b = I a 1∠
U ca
I ab = I bc = I ca = I F
√
π
3I F 1e−j 6
I c = I a 1∠ + 120
est n equilibradas
I b =
U bO Z Y
I c =
est n equilibradas
U cO Z Y
U aO = Z Y I a
U bO = Z Y I b
U aO ,
U ab = U aO U ab =
√
3U aO 1∠30
U bO ,
− U bO ,
U bc = U ab 1∠
U cO = Z Y I c
U cO
est n equilibradas
U bc = U bO
− 120
− U cO ,
U ca = U cO
U ca = U ab 1∠ + 120
U aO = U bO = U cO = U F U L =
− U aO
est n equilibradas
U ab = U bc = U ca = U L
√
π
3U F 1ej 6
−
Z C E a E b = E a 1∠
− 120 Z G Z L
E c = E a 1∠120
Z G , Z L , Z C
Z G , y Z L
Z Y = Z G + Z L + Z C U OO = 0
Z N
I N = 0
Z Y
I a
=
I b
=
I c
=
E a E a = Z Y Z G + Z L + Z C E b E a 1∠ 120 = = I a 1∠ 120 Z Y Z G + Z L + Z C E c E a 1∠120 = = I a 1∠120 Z Y Z G + Z L + Z C
−
−
I a + I b + I c = 0 = I n
I A =
E a Z G + Z L + Z C
U a O = I A Z C
U aO = E a
− I AZ G
(corriente de l nea)
(tensi n fase carga)
(tensi n fase fuente real)
Z C
U L = U ab = U a b =
√
3U aO 1∠30
√
π
3U F 1ej 6
U bc = U ab 1∠
3U a O 1∠30
√
U b c = U a b 1∠
(secuencia directa)
− 120
− 120
U ca = U ab 1∠ + 120
(fuente)
U c a = U a b 1∠ + 120
(carga)
Z Y =
Z 3
U L/Z Z G
E ab , E bc , y E ca
Impedancias fuente estrella E A E B E C
F.e.m. f uente estrella
I L
Y
− Y, Y − , − Y, −
√ = 3I
F
Z G =
E
Z G 3
√ ab3 1∠ 30 = = E A 1∠ 120 = E A 1∠ + 120
π 1e−j 6
I A √ 1∠30 3 I 1 1∠ − 120
I 1
=
I 2 I 3
= = I 1 1∠ + 120
− −
Z Al
E AN
−
I A
+
Z Ac
E BN
Z Bl
I B
− +
−
+
E AN
Z Bc
Z Cc Z Cl I C
Z N
I N Z N
Z A = Z Al + Z Ac
Z B = Z Bl + Z Bc
Z C = Z Cl + Z Cc
U AN ; U BN ; U CN
I A =
U AN E AN U N N U BN E BN U N N U CN E CN U N N = I B = = I C = = Z A Z A Z B Z B Z C Z C
−
−
−
I N =
I N =
U N N = I A + I B + I C Z N
U N N E AN U N N E BN U N N E CN U N N = + + Z N Z A Z B Z C
−
U N N =
U AB U AN
E A
E B U CA U CN
E C
U BN
U BC
−
E AN E BN E CN Z A + Z B + Z C 1 1 1 1 Z A + Z B + Z C + Z N
−
ea (t), e b (t), y ec (t) ia (t), ib (t), ic (t), in (t)
p(t) = ea (t)ia (t) + eb (t)ib (t) + ec (t)ic (t)
P = E a I a cos(ϕE a ϕI a ) + E b I b cos(ϕE b Q = E a I a sen(ϕE a ϕI a ) + E b I b sen(ϕE b S = E a I ∗a + E b I ∗b + E c I ∗c = P + jQ
− −
S =
|S | =
P 2 + Q2
− ϕI ) + E c I c cos(ϕE − ϕI ) − ϕI ) + E cI csen(ϕE − ϕI ) b
b
c
c
c
c
uan (t), ubn (t), y ucn (t) ia (t), ib (t), ic (t), in (t) uan (t), ubn (t), y ucn (t)
U ab , U bc , U ca I a , I b , I c
U ab + U bc + U ca = 0 U ab y U bc U ab y U bc
p(t) = uab (t)ia (t) + ubc (t)( ic (t)) = uab (t)ia (t)
−
p(t) = ubc (t)ib (t)
− uca(t)ia (t)
− ubc(t)ic (t)
p(t) = uca (t)ic (t)
P = Q = S =
U ab I a cos(ϕU ab U ab I a sen(ϕU ab U ab I ∗a U bc I ∗c
S =
|S | =
U an ,
−
− ϕI ) − U bcI ccos(ϕU − ϕI ) − ϕI ) − U bc I c sen(ϕU − ϕI ) a
bc
a
bc
c
c
P 2 + Q2
U bn = U an 1∠
I a ,
− uab(t)ib(t)
I b = I a 1∠
− 120 ,
U cn = U an 1∠120
− 120 ,
I c = I a 1∠120
P = U an I a cos(ϕU an ϕI a ) + U bn I b cos(ϕU bn ϕI b ) + U cn I c cos(ϕU cn = 3U an I a cosϕ = 3U bn I b cosϕ = 3U cn I c cosϕ
−
−
ϕ
ϕ = ϕU an
− ϕI ) = c
− ϕI
a
U an = U bn = U cn = U F I a = I b = I c = I L P = 3U F I L cosϕ
Q = 3U F I L senϕ S = 3U an I ∗a S = 3U F I L
U ab =
√
3U an 1∠
− 30
U bc = U ab 1∠
(4 hilos)
(4 hilos)
− 120
U ab = U bc = U ca = U L
U ca = U ab 1∠ + 120
√ √ √
P = 3U F I L cosϕ = 3U L I L cosϕ Q = 3U F I L senϕ = 3U L I L senϕ S = P 2 + Q2 = 3U L I L S = 3U L I L ∠ϕ
√
U AN ,
U BN = U AN 1∠ I A ,
I B = I A 1∠
− 120 ,
− 120 ,
U CN = U AN 1∠120 I C = I A 1∠120
U AB = U BC = U CA = U L I A = I B = I C = I L
U
U AN U BN U CN
F uente estrella
AB √ = 1∠ 30 3 = E AN 1∠ 120 = E AN 1∠ + 120
− −
U AN = U BN = U CN = U F
√ √ √
P = 3U F I L cosϕ = 3U L I L cosϕ Q = 3U F I L senϕ = 3U L I L senϕ S = P 2 + Q2 = 3U L I L S = 3U L I L ∠ϕ
√
ϕ = ϕU an
− ϕI
a
cos(ϕ = cos(ϕU an I a
U an
p(t) = + +
uan (t)ia (t) + ubn (t)ib (t) + ucn (t)ic (t) 2U F cos(wt) 2I F cos(wt ϕ)+ 2U F cos(wt 120 ) 2I F cos(wt 120 ϕ)+ 2U F cos(wt + 120) 2I F cos(wt + 120 ϕ)
√ √
√
−
√ √ √
− −
− −
− ϕI ) a
p(t) = P = 3U F I F cosϕ
U L P L ρ
I L = P/(U L cosφ)
P mo = 2Rmo I L2 = 2Rmo
P 2 U L2 cos2 (φ)
√
I L = I a = I b = I c = P L /( 3U L cos(φ))
P tr =
3Rtr I a2
P 2 P 2 = 3Rtr 2 2 = Rtr 2 2 3U L cos (φ) U L cos (φ) U L
P L
P mo 2Rmo = P tr Rtr 2 Rmo = ρl/(πr mo )
2 Rtr = ρl/(πr tr )
rmo
rtr
P mo 2r 2 = 2tr P tr rmo r 2 = 2r2
2 2 Material para monof a ´sico 2(πr mo l) 2rmo 2 = = = (2) = 1, 33 2 2 M aterial para trif a ´sico 3(πrtr l) 3rtr 3
W = U.I.cosφU ,I
W 1
I a W 2
I b W 3
I c
P = U a .I a cosφU a ,I a + U b .I b cosφU b ,I b + U c .I c cosφU c ,I c = W 1 + W 2 + W 3
W 1
I a W 2
I b I c
P = U ac .I a cosφU ac ,I a + U bc .I b cosφU bc ,I b = W 1 + W 2
Z Q = RQ + jX Q U nQ
2 U nQ Z Q = 1, 1 S kQ
S kQ RQ
RQ = 0, 1X Q
X Q = 0, 995Z Q
X Q
N 1
N 2 rt = N 1 /N 2
R1
jX L2 .rt2
jX L1
R2 .rt2
I 2 /rt
I 1 jX m
U 1
R1 /rt2
Rm
jX L1 /rt2
U 2 .rt
jX L2
I 1 .rt
I 2 jX m /rt2
U 1 /rt
R1
R2
jX L1
jX L2 .rt2
Rm /rt2
U 2
R2 .rt2 Z cc I 2 /rt
I 1 U 1
I 1 U 2 .rt
U cc
I n =
√ S εcc εcc U 2 U cc = U Z cc = 3 = 100 I n 100 S 3U
√
S I n
εcc
Z cc referidaprimario = Z cc referidasecundario .rt2 zT
Z cc referidaprimario = zT
RL
U 2 S
Z cc referidasecundario = Z cc referidaprimario/rt2
jX L
I 1
I 2 I 1
I 2
Π RL
jX L
I 1 U 1
Z I 2
− jX c
I 1
− jX U c 2
I = I 2
− j Y 2 U 2
Y U 1 = (1 + jZ )U 2 + ZI 2 2
−
I
U 1
U 1 = U 2 + ZI
I 1 = jY (1
−
I 2
− jY /2
I 2 = I j
−
− jY /2
Y U 2 2
ZY )U 2 + (1 − j )I 2 − j ZY 4 2
U 2
Z
I
I 1
I 2
U 1
U 2
− jY /2 − jY /2
Z = Z c sinh(γl) γ
Y = Y
tanh(γl/2) γl/2
Z c
γ =
(r + jwL)(g + jwC )
Z c =
(r + jwL) (g + jwC )
Φex Φex E o = kΦext w E o = f (I ex)
Φr Φt = Φex + Φr
E c
Φt E c = U + RI + jX σ I
Φt
⇒ E c
Φo
⇒ E o
Φr
⇒ E r
E c E c = E o + E r = E o
− jX r I
X r
E o = E c
− E r = U + RI + jX σ I + jX r I E o = U + RI + jX s I
X s
U, I
E o , U
I carga
I carga
Rs U carga
jX p
R p
U carga jX s
Rg + jX g
E
−
+
R1 + jX 1
jX m1
R2 + jX 2
Rm1
Rl + jX l
R3 + jX 3
jX m2
R4 + jX 4
Rm1
Rc + jX c
V ariable pu =
V alor real variable V alor base variable
√ 3
U pu =
U U base
I pu =
I I base
Z pu =
Z Z base
S pu =
U base
S S base S base
S =
√
3UI
I base =
S base =
√
3U base I base
S base (M V A) √ (kA) 3U (kV ) base
I base
S pu .S base =
√
3U pu U base I pu I base
S pu = U pu I pu
Z base
√
U base (kV )/ 3 U 2 (kV 2 ) Z base (Ω) = = base I base (kA) S base (M V A)
N 1 I 1
I 2
Z
U 1 U 2
N 1 N 2
N 2
U base1 U base2 U 1 pu =
U 1 U base 1
I 1 N 2 = I 2 N 1
U 2 pu =
N 1 N 2
U 2 U base2
I base1 N 2 = I base2 N 1
I 1 = I 1 pu I base1
⇒
⇒
U 1 pu = U 2 pu
I 1 pu = I 2 pu
I 2 = I 2 pu I base2
U 2 = ZI 2
U 2 pu = Z pu I 2 pu
U 1 = Z 1 I 1
U 1 pu = Z 1 pu I 1 pu
Z l
Z
S gn
U gn
X gpu
S T 1n
U 1n , U 2n
X T 1 pu
S T 2n
U 3n , U 4n
X T 2 pu
Z l Z
jX 1
E
−
+
jX 2
Z 3
jX 4
Z 5
S base
U 1base
U 2base =
U 1base
U 3base = U 1base .(
U 3n U 4n
U 1n ) U 2n
Z 1base =
U 12base S base
Z lpu =
Z l = Z 3 Z 1base
Z 2base =
U 22base S base
Z cpu =
Z c = Z 5 Z 2base
X T 2 pu
X T 2 punueva = X T 2 pu .(
U 42n )/(U 22base /S base ) = X 4 S T 2n
X T 1 pu
X T 1 punueva = X T 1 pu .(
U 22n )/(U 12base /S base ) = X 2 S T 1n X gpu
X gpunueva = X gpu .(
U 12n )/(U 32base /S base ) = X 1 S gn
E pu =
U gn U 3base
U m
I k
I k
i p idc
I k
I k
I k (Rk /X k ) = (Rb /X b )
τ d
τ d
τ d
X d
X d
X d
I k
i p Z K i p
idc
i p
X g
Z g = Rg + jX g U ng
2 x U ng X g = 100 S ng
S ng
x
2 U nm 1 U nm Z m = = I arr /I nm S nm 3I arr
√
Z m = Rm + jX m U nm
I arr
I nm
S nm Rm /X m = 0, 1 con X m = 0, 995
≥
Rm /X m = 0, 15 con X m = 0, 989
<
Rm /X m = 0, 3 con X m = 0, 958
E cc = c
U n √ 3
I k =
n √ cU 3Z
k
Z 1
=
√ 3
cU n R2k + X k2
I k3 = I 1 =
n √ cU 3Z
1
=
√ 3
cU n
R12 + X 12
κ
κ
√
i p = κ 2I k
donde κ = 1, 02 + 0, 98e−3R/X κ
t p
κ κ i p κ
κ
κ
Rk /X K Rk /X K
f c = 20 Hz κ
R/X = (Rc .f c)/(X c ,50) f c
S rg = 100 MV A U rg = 10 kV xd = 11 % rg = 0, 11 % S T 1 = 100 M V A U T 1H = 220 kV U T 1L = 10 kV εccx = 11 % εccr = 0, 11 % S T 1 = 31, 5 MV A U T 1H = 220 kV U T 1L = 10 kV εccx = 6 % εccr = 0, 75 % RL = 0, 231 Ω/km X L = 0, 104 Ω/km RR = 0, 00458 Ω/km X R = 0, 458 Ω/km
2
2
10 X g = 0, 11 10 100 == 0, 11 rg = 0, 011 100 == 0, 011
2
2
10 X T 1 = 0, 11 10 100 == 0, 11 rT 1 = 0, 011 100 == 0, 011
2
2
10 10 X T 2 = 0, 06 31 ,5 == 0, 1904 rT 1 = 0, 075 31,5 == 0, 0238
RL = 0, 3465 X L = 0, 156 RR = 0, 00458 X R = 0, 0458
Z I = Z T 1 + Z L Z I = 0, 00513 + j0, 513 RI /X I = 0, 01 Z I = 0, 51303 Z II = (Z T 2 //Z G ) + Z R Z II = 0, 371 + j0, 345 RII /X II = 1, 0754 Z II = 0, 5066 Z I /Z II = 1, 0127
Rc + jX c
Z K = 0, 11276 + j0,25349 Z K = 0, 27744 Rk /X k = 0, 445 R X minima
Rk Xk
=
RI Xi
= 0, 01 κ = 1, 971
κ = 1, 55
Rk > 0, 3X k
= 0, 4448 κ = 1, 278 1,15 = 1, 4697
∗
Z c = Rc + jX c
R X
=
Rc 20 Xc . 50
= 0, 1732 κ = 1, 6029
I b = I k
I b = µI k µ
µ
I K /I n
µ
µ = 0, 84 + 0, 26e−0,26I kG /I rG para tmin = 0, 02 s
µ = 0, 71 + 0, 51e−0,30I kG /I rG para tmin = 0, 05 s µ = 0, 62 + 0, 72e−0,32I kG /I rG para tmin = 0, 01 s
µ = 0, 56 + 0, 94e−0,38I kG /I rG para tmin
I b = µqI k tmin
≥ 0, 25 s
q = 1, 03 + 0, 12ln(P rM /p) q = 0, 79 + 0, 12ln(P rM /p) q = 0, 57 + 0, 12ln(P rM /p) q = 0, 26 + 0, 12ln(P rM /p)
para para para para
tmin = 0, 02 s tmin = 0, 05 s tmin = 0, 01 s tmin 0, 25 s
P rM
V a V b V c
= = =
V a1 + V a2 + V a0 V b1 + V b2 + V b0 V c1 + V c2 + V c0
≥
