ENGS 22 — Systems
Laplace Transform Table
Largely modeled on a table in D’Azzo and Houpis, Linear Control Systems Analysis and Design, 1988
F (s)
1. 1
2.
1
s
3.
1
s2
1
4.
sn
1 −as
5. e
s
1
−as
(
1
−
e
)
6.
s
7.
1
s+a
1
( s + a) n
1
s ( s + a)
f (t) 0 ≤ t
δ (t )
unit impulse at t = 0
1 or u(t )
unit step starting at t = 0
t ⋅ u(t) or t
ramp function
1
t n −1
( n − 1)!
n = positive integer
u (t − a )
unit step starting at t = a
u(t) − u(t − a)
rectangular pulse
e −at
exponential decay
s( s + a)(s + b)
1
12. (s + a)(s + b)
1
t n−1e −at n = positive integer
(n − 1)!
1
(1 − e −at )
a
1
b −at
a −bt
(1 −
e +
e )
ab
b−a
b−a
1
b(α − a) −at a(α − b) −bt
[α −
e +
e ]
ab
b−a
b−a
1
(e − at − e −bt )
b−a
s
13. ( s + a )( s + b)
1
( ae − at − be −bt )
a−b
8.
9.
1
10. s(s + a)(s + b)
s +α
11.
Laplace Table
Page 1
ENGS 22 — Systems
F(s)
s +α
14. ( s + a )( s + b )
1
15. ( s + a)(s + b)(s + c)
s +α
16. (s + a)(s + b)(s + c)
ω
17. 2
s + ω2
s
18. 2
s + ω2
s+α
19. 2
s +ω2
0≤t
f(t)
1
[(α − a)e −at − (α − b)e −bt ]
b−a
e−at
e−bt
e−ct
+
+
(b − a)(c − a) (c − b)(a − b) (a − c)(b − c)
(α − a)e−at
(α − b)e−bt
(α − c)e−ct
+
+
(b − a)(c − a) (c − b)(a − b) (a − c)(b − c)
sin ω t
cos ω t
α 2 + ω2
sin(ωt + φ )
ω
s sin θ + ω cos θ
20.
s2 + ω2
sin(ωt + θ )
1
21. s ( s 2 + ω 2 )
s+α
22. s ( s 2 + ω 2 )
1
1
23. (s + a)(s 2 + ω 2 )
ω2
φ = atan2(ω, α )
(1 − cosωt )
α
α2 +ω2
−
cos( ω t + φ )
φ = atan2(ω,α )
ω2
ω2
e − at
1
+
sin(ωt − φ )
2
2
2
2
a +ω
ω a +ω
φ = atan2(ω, α )
1
24. (s + a) 2 + b 2
1
24a. 2
2
s + 2ζω n s + ω n
s+a
25. ( s + a ) 2 + b 2
Laplace Table
1 − at
e sin(bt )
b
1
ωn 1 − ζ 2
e −ζωnt sin(ωn 1 − ζ 2 t )
e − at cos( bt )
Page 2
ENGS 22 — Systems
F(s)
s +α
26. ( s + a ) 2 + b 2
26a.
(α − a ) 2 + b 2 − at
e sin( bt + φ )
b
(
α
ωn
s +α
s2 + 2ζωns +ωn
0≤t
f(t)
− ζωn
)
1−ζ 2
2
2
+1
⋅ e −ζωnt sin(ω n 1 − ζ 2 t + φ )
φ = atan2(ωn 1 − ζ 2 ,α − ζωn )
1
1
−at
+
e
sin(bt −φ)
2
2
2
2
a +b b a +b
27.
1
s[(s + a)2 + b2 ]
27a.
1
1
s(s 2 + 2ζωn s + ωn
2
ωn
2
−
φ = atan2(b,α − a)
1
ωn 2 1 − ζ 2
φ = atan2( b,− a )
e−ζωnt sin(ωn 1 − ζ 2 t + φ )
φ = cos − 1 ζ
28.
1 (α − a) 2 + b2 −at
+
e sin(bt + φ)
2
2
2
2
a +b b
a +b
φ = atan2( b , α − a ) − atan2( b , − a )
28a.
α
1
α
−ζω t
2
2
2
+
(
−
ζ
)
+
(
1
−
ζ
)
⋅
e
sin(
ω
1
−
ζ
t +φ)
n
2
2
ωn ωn 1−ζ ωn
α
s +α
s[(s + a)2 + b2 ]
s +α
n
s(s2 + 2ζωn s + ωn )
2
29.
1
(s + c)[(s + a)2 + b2 ]
Laplace Table
φ = atan2(ωn 1 − ζ 2 ,α − ω nζ ) − atan2( 1 − ζ 2 ,−ζ )
e −ct
e −at sin(bt − φ )
+
(c − a) 2 + b 2 b (c − a) 2 + b 2
φ = atan2(b, c − a)
Page 3
ENGS 22 — Systems
F(s)
30.
1
s(s + c)[(s + a)2 + b2 ]
31.
s +α
s(s + c)[(s + a)2 + b2 ]
0 ≤1
f(t)
1
e−ct
e−at sin(bt − φ)
−
+
c(a2 + b2 ) c[(c − a)2 + b2 ] b a2 + b2 (c − a)2 + b2
φ = atan2(b,−a) + atan2(b, c − a)
(c − α )e −ct
α
+
c(a 2 + b 2 ) c[(c − a) 2 + b 2 ]
(α − a) 2 + b 2
+
e −at sin(bt + φ )
b a 2 + b 2 (c − a) 2 + b 2
φ = atan2(b, α − a) − atan2(b,−a) − atan2(b, c − a)
1
32. s 2 ( s + a )
1
33. s(s + a)2
s +α
34. s(s + a)2
s 2 + α1s + α 0
35. s(s + a)(s + b)
s 2 + α1s + α 0
36. s[(s + a) 2 + b 2 ]
1
(at−1+ e−at )
2
a
1
(1− e−at − ate−at )
2
a
1
−at
−at
[
α
−
α
e
+
a
(
a
−
α
)
te
]
2
a
α0 a2 −α1a + α0
+
e
−at
b2 −α1b + α0 −bt
e
−
b(a − b)
ab
a(a − b)
α0 1 2 2
2
+
[(
a
−
b
−
α
a
+
α
)
1
0
c 2 bc
1
2 2
+ b (α1 − 2a) ] e−at sin(bt + φ)
φ = atan2[ b(α 1 − 2a ), a 2 − b 2 − α 1 a + α 0 ] − atan2( b,− a )
2
c2 = a2 +b2
Laplace Table
Page 4
ENGS 22 — Systems
0 ≤1
F(s)
f(t)
37.
(1 / ω ) sin(ωt + φ1 ) + (1 / b)e − at sin(bt + φ 2 )
1
(s2 +ω2 )[(s + a)2 +b2 ]
38.
s +α
(s2 +ω2 )[(s + a)2 +b2 ]
s +α
39. s2[(s + a)2 + b2 ]
s 2 + α1s + α0
40. s 2 (s + a)(s + b)
2
2
2
φ1 = atan2(−2aω, a2 + b2 − ω2 )
φ2 = atan2(2ab, a 2 − b 2 + ω 2 )
1 α 2 + ω2 2
(
) sin(ωt + φ1 )
ω
c
1
1 (α − a)2 + b2 2 −at
+ [
] e sin(bt +φ2 )
b
c
c = (2aω)2 + (a2 + b2 −ω2 )2
1
φ1 = atan2(ω , α ) − atan2( 2aω , a 2 + b 2 + ω 2 )
φ2 = atan2(b,α − a) + atan2(2ab, a 2 − b 2 − ω 2 )
1
2
1
2α a
[b + (α − a ) ] − at
(α t + 1 −
)+
e sin( bt + φ )
c
bc
c
2
2
c = a 2 + b2
φ = 2atan2(b, a) + atan2(b,α − a)
α1 +α0t α0 (a + b) 1
α1 α0
ab
−
Laplace Table
1
2 2 2
[ 4a ω + ( a + b − ω ) ]
2
−
(ab)
2
−
a −b
(1−
a
+
2
a
)e−at
α α
1
(1− 1 + 20 )e−bt
b−a
b b
Page 5