TRANSFORMADA DE LAPLACE

Tablas

A. MODELO DE UN RESISTOR

En “t”

v( t ) = R ⋅ i( t )

En “s”

V (s ) = R ⋅ I (s )

B. MODELO DE UN INDUCTOR

En “t”

v( t ) = L ⋅

di(t ) dt

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 )

A= A= C=

1 α

1

B=− 1 α

B=−

2

(α + β ) 2β 2

2

4

( 2αβ + α + β ) − 2β s ( 2αβ 2 + α 2 + β4 )

1 α

C=0

C=0 D=− 1 2

(α + β ) B= D=

C=

1 α β (α + β2 )

1 (α + β 2 ) − α + β2 ( 2αβ 2 + α 2 + β4 )