Transformada de laplace

1. LAPLACE TRANSFORM
TABLES
MATHEMATICS CENTRE
©2000

2. DEFINITION
The Laplace transform f ( s ) of a function f(t) is defined by:
f (s) =
∞
∫e
− st
f (t ) dt
0
TRANSFORMS OF STANDARD FUNCTIONS
f(t)
f (s)
1
1
s
e−αt
1
s+α
t
1 −T
e
T
1
1+ sT
1− e −α t
α
s (s + α )
te− α t
1
(s + α )2
e −α t − e − β t
β −α
( s + α )(s + β
t
1
s2
tn
n!
sn +1
e −α t t n
n!
(s + α )n +1
ω
s + ω2
sin ωt
2
s
s + ω2
cosωt
2
2
)

3. f(t)
f (s)
e −α t sin ω t
ω
(s + α )2 + ω 2
e −α t cos ω t
s+α
(s + α )2 + ω 2
ω2
s s2 +ω
1− cosωt
1
2ω
3
(sin ω
(
t − ω t cos ω t )
(s
t
sin ω t
2ω
(s
α


e − α t  cos ω t − sin ω t 
ω


1
2
)
s
2
)
2 2
+ω
)
2 2
+ω
s
(s + α )
2
+ω
2
s sin φ + ω cos φ
s2 +ω 2
sin (ω t + φ )
e −α t +
2
α
sin ω t − cos ω t
ω
α
2
+ω
2
( s + α )(s 2 + ω
sin 2 ωt
(
2ω
2
s s 2 + 4ω
cos2 ωt
s 2 + 2ω
(
s s 2 + 4ω
β
s −β2
sinh βt
2
cosh βt
s
s −β2
2
3
2
2
)
2
2
)
)

4. f(t)
f (s)
e −α t sinh β t
β
( s + α )2 − β
s +α
e −α t cosh β t
( s + α )2 − β
t sinh β t
(s
t cosh β t
2β
3
(β
t cosh β
t − sinh β
2β
2
s
−β
t)
(s
2
−β
−β
)
2 2
δ(t)
1
Unit step :
H(t)
1
s
Ramp:
tH(t)
1
s2
Delayed Unit Impulse: δ(t-T)
e-sT
Delayed Unit Step:
H(t-T)
e − sT
s
Rectangular Pulse:
H(t)-H(t-T)
1− e − sT
s
4
)
2 2
Transforms of Special Functions
Unit impulse :
)
2
1
2
2
2 2
s2 + β
(s
1
2

5. TRANSFORM THEOREMS
f(t)
f (s)
e-αt f(t)
f (s + α )
f(t-T)H(t-T)
e − sT f ( s )
f(kt)
1 s
f 
k k 
Damping:
Delay:
Time scale:
Integral:
∫
t
0
1
f ( s)
s
f (t ) dt
Differentiation
sf ( s ) − f ( 0)
d
f (t )
dt
d2
f (t )
dt 2
dn
f (t )
dt n
Initial Value:
s 2 f ( s) − sf ( 0) − f '( 0)
s n f ( s) − sn −1 f ( 0) − sn −2 f '( 0) −... − f n −1 ( 0)
lim { f (t )} = lim {s f ( s) }
t →0
s →∞
Final Value:
lim { f (t )} = lim {s f ( s) }
t →∞
s →0
Periodic Functions:
If f(t) has period T then:
f ( s) =
1
1 − e − sT
T
∫ f (t )e
− st
dt
0
further, if g(t) is defined as the first cycle of f(t), followed by zero, then f ( s ) =
g ( s)
1 − e − sT
Square Wave:
f (t ) = 1
f (t ) = 0
0< t <
T
<t <T
2
T

2




f (s) =
1  eα T − e−α T
1 +
2s  e α T + e −α T
π 

ω 

π
2π 
<t <
ω
ω

f (t ) = sin ω t
0<t<
Half-Wave Rectified Sine:
f (t ) = 0
Full-Wave Rectified Sine:

 , where

f (t ) = sin ω t
f ( s) =
5
ω
f (s) =
1 + e − sπ / ω
s 2 + ω 2 1 − e − sπ /ω
α =
s
4
ω
1
⋅
2
s + ω 1 − e − sπ / ω
2

6. Saw-Tooth Wave:
t
f (t ) = 0 < t < T
T
6
1
e − sT
f (s) = 2 −
s T s 1 − e − sT
(
)