V(s ) = L ⋅ [s ⋅ I(s ) − i(0 − )] V ( s ) = s ⋅ L ⋅ I (s ) − L ⋅ i ( 0 − ) En “s” I(s) =
V(s) i(0 − ) + sL s
i (0− ) s
Si la condición inicial es nula, es decir i(0-)=0, el modelo de un inductor en el dominio de “s” es una reactancia de valor sL [Ω Ω]. C. MODELO DE UN CAPACITOR
En “t”
i( t ) = C ⋅
dv(t ) dt
I(s ) = C ⋅ [s ⋅ V(s ) − v(0 − )] I( s ) = s ⋅ C ⋅ V ( s ) − C ⋅ v ( 0 − ) En “s” V(s) =
1 v(0 − ) ⋅ I(s) + sC s
Si la condición inicial es nula, v(0-)=0, el modelo en el dominio de “s” de un capacitor, es una reactancia de valor 1/sC [Ω Ω].
CIRCUITOS ELÉCTRICOS II
TRANSFORMADA DE LAPLACE
PARES DE TRANSFORMADAS DE LAPLACE 1. 2.
f(t) = L -1 [F(s)] δ (t ) δ (t − T)
3.
u(t)
4.
u(t-T)
5.
u(t) - u(t-T)
6.
t.u(t)
7.
t n −1 ⋅ u(t ) , n = 1, 2, ..... (n − 1)!
8.
e − α t ⋅ u( t )
9.
1 ⋅ 1 − e − α t ⋅ u( t ) α
10. 11.
(
1 α
2
F(s) = L [ f(t)] 1 e-Ts 1 s 1 −Ts ⋅e s 1 ⋅ 1 − e −Ts s 1
(
)
(
)
⋅ α ⋅ t − 1 + e − α t ⋅ u( t ) t ⋅ e − α t ⋅ u( t )
)
s2 1 sn 1 s+α 1 1 ⋅ s (s + α ) 1 1 ⋅ s 2 (s + α ) 1
(s + α ) 2 1
t n −1 ⋅ e −αt ⋅ u(t ) , n = 1, 2, ..... (n − 1)! 1 ⋅ e −αt − e −βt . u(t ) β−α
1 1 ⋅ (s + α ) (s + β )
14.
e − αt e −β t e − εt . u( t ) + + (β − α ) ⋅ (ε − α ) (α − β ) ⋅ (ε − β) (α − ε ) ⋅ (β − ε )
1 1 1 ⋅ ⋅ ( s + α ) ( s + β ) (s + ε )
15.
sen (ωt ) ⋅ u(t )
ω
16.
cos (ωt ) ⋅ u(t )
17.
sen (ωt + θ) ⋅ u(t )
18.
cos (ωt + θ) ⋅ u(t )
19.
e − α⋅t ⋅ sen (ωt ) ⋅ u(t )
20.
e − α⋅t ⋅ cos (ωt ) ⋅ u(t )
12. 13.
(
)
(s + α ) n
s 2 + ω2 s s 2 + ω2 s ⋅ sen (θ) + ω ⋅ cos(θ) s 2 + ω2 s ⋅ cos (θ) − ω ⋅ sen(θ) s 2 + ω2 ω (s + α ) 2 + ω 2 s+α (s + α ) 2 + ω 2
CIRCUITOS ELÉCTRICOS II
TRANSFORMADA DE LAPLACE
OPERACIONES CON LA TRANSFORMADA DE LAPLACE
Operación Suma Multiplicación Escalar
f(t) f 1 (t ) ± f 2 (t )
F(s) F1 (s ) ± F2 (s )
k ⋅ f (t )
k ⋅ F (s )
df (t ) dt 2 d f (t )
Derivada respecto al tiempo
2
s 2 ⋅ F (s ) − s ⋅ f ( 0 − ) − f ' ( 0 − )
3
s 3 ⋅ F (s ) − s 2 ⋅ f ( 0 − ) − s ⋅ f ' ( 0 − ) − f ' ' ( 0 − )
dt d 3 f (t ) dt
s ⋅ F (s ) − f ( 0 − )
Convolución
f 1 (t ) * f 2 (t )
1 ⋅ F (s ) s 1 1 0− ⋅ F (s ) + ⋅ ∫ f (t ) dt s s −∞ F1 (s ) ⋅ F2 (s )
Cambio de Tiempo Cambio de Frecuencia Derivada de la Frecuencia Integral de la frecuencia
f ( t − a ) ⋅ u( t − a ) ; a ≥ 0
e −as ⋅ F(s )
e −at ⋅ f (t )
F (s + a )
− t ⋅ f (t )
dF(s ) ds
Cambio de Escala
f (at ) ; a ≥ 0
Valor Inicial
f (0 + )
Valor Final
f (∞ )
t
Integral respecto al tiempo
∫0
-
f (t ) dt
t
∫− ∞ f (t ) dt
f (t ) t
∞
∫s
1 s ⋅ F a a lim [s ⋅ F(s )]
s→∞
lim [s ⋅ F(s )]
s→0
1
Periodicidad
f (t ) = f (t + nT) ; n = 1, 2,…
F(s ) ds
1 − e −Ts
⋅ F1 (s )
donde F1 (s ) = ∫
T 0
-
f (t ) ⋅ e −st dt
FRACCIONES PARCIALES 1 1 ⋅ s+α s+β
A B + s+α s+β
(β − α ) (s + α ) ⋅ (s + β )
A B + s+α s+β
1 s ⋅ (s + α )
A B + s s+α
α s ⋅ (s + α )
A B + s s+α
A=
1 (β − α )
B=
−1 (β − α )
A = 1 B = −1
A=
1 α
B=−
1 α
A = 1 B = −1
CIRCUITOS ELÉCTRICOS II 1
A B C + + 2 s s s+α
s 2 ⋅ (s + α ) 1 1 s ⋅ (s + α ) 2 +α 1
α α
s ⋅ (β − α ) (s + α ) ⋅ (s + β )
A B + s+α s+β
s (s + α ) ⋅ (s + β )
A B + s+α s+β
s (s + α )
Se realiza el artificio matemático
s2 (s + α ) 2 1 s ⋅ (s + α ) 3 ⋅
+α 1
s 2 ⋅ (s + α ) 3
s−α+
C=
C=
1 α
α
1
α2 D=−
α2
2 α
D=
3
1 α3
1 α
C=−
C=
2
1
1 α2
A = −α B = β
A=− s+α−α (s + α )
α (β − α )
B=
y se calcula 1 −
( s 2 + 2α s + α 2 ) − 2α s − α 2 (s + α )
β (β − α )
α (s + α )
y se calcula
para
α2 (s + α )
Se completa 1−
α
1
B=
3
2
B=−
α2
2
A=−
1 α
1
1 α
B=
2
B=−
3
A=
A B C D + + + 2 s s s + α (s + α ) 2
s2 (s + α )
1
A=
s 2 ⋅ (s + α ) 2
Se completa cuadrado
1
A=−
A B C D + + + 2 3 s s s+α s A B C + + s s + α (s + α ) 2
s 3 ⋅ (s + α ) ⋅
TRANSFORMADA DE LAPLACE
cuadrado
( s 2 + 2α s + α 2 ) − 2α s − α 2
y se calcula para
(s + α ) 2
2α α2 + ( s + α ) (s + α ) 2 A B C D + + + s s + α (s + α ) 2 (s + α ) 3
A B C D E + + + + s s 2 (s + α ) (s + α )2 (s + α ) 3
1
A=
α
B=−
3
α
3
A=−
1
B=
α4 α
1 α3
α
C=
α
1
C=−
3
3
D=
α4
D=−
2
α
2
1 α
E=
α3 α
1 α2
RAÍCES COMPLEJAS α > 0 1 s ⋅ (s 2 + α ) 1 s 2 ⋅ (s 2 + α ) 1 ( s + β ) ⋅ (s 2 + α )
1 ( s + β ) 2 ⋅ (s 2 + α )
A Bs + C + s (s 2 + α )
A=
A B Cs + D + + s s 2 (s 2 + α )
A=0 B=
A Bs + C + s + β (s 2 + α ) A B Cs + D + + s + β (s + β ) 2 (s 2 + α )
