Summary New Laplace Transform Table

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) f (t) 0 ≤ t
1. 1 δ (t ) unit impulse at t = 0
1 1 or u(t ) unit step starting at t = 0
2.
s
1 t ⋅ u(t) or t ramp function
3.
s2
1 1
4. t n −1 n = positive integer
sn ( n − 1)!
1 −as u (t − a ) unit step starting at t = a
5. e
s
1 −as
u(t) − u(t − a) rectangular pulse
6. (1 − e )
s
7.
1 e −at exponential decay
s+a
1 1
8. t n−1e −at n = positive integer
( s + a) n (n − 1)!
1 1
9. (1 − e −at )
s ( s + a) a
1 1 b −at a −bt
10. s(s + a)(s + b) (1 − e + e )
ab b−a b−a
s +α 1 b(α − a) −at a(α − b) −bt
11. [α − e + e ]
s( s + a)(s + b) ab b−a b−a
1 1
12. (s + a)(s + b) (e − at − e −bt )
b−a
s 1
( ae − at − be −bt )
13. ( s + a )( s + b) a−b




Laplace Table Page 1

, ENGS 22 — Systems



F(s) f(t) 0≤t
s +α 1
14. ( s + a )( s + b ) [(α − a)e −at − (α − b)e −bt ]
b−a
1 e−at e−bt e−ct
15. ( s + a)(s + b)(s + c) + +
(b − a)(c − a) (c − b)(a − b) (a − c)(b − c)
s +α (α − a)e−at (α − b)e−bt (α − c)e−ct
16. (s + a)(s + b)(s + c) + +
(b − a)(c − a) (c − b)(a − b) (a − c)(b − c)
ω sin ω t
17. 2
s + ω2
s cos ω t
s + ω2
18. 2
s+α α 2 + ω2
19. 2
s +ω2 sin(ωt + φ ) φ = atan2(ω, α )
ω
s sin θ + ω cos θ sin(ωt + θ )
20.
s2 + ω2
1 1
21. s ( s 2 + ω 2 ) (1 − cosωt )
ω2
s+α α α2 +ω2
22. s ( s 2 + ω 2 ) − cos( ω t + φ ) φ = atan2(ω,α )
ω2 ω2
1 e − at 1
23. (s + a)(s 2 + ω 2 ) + sin(ωt − φ )
a +ω
2 2
ω a +ω
2 2

φ = atan2(ω, α )
1 1 − at
24. (s + a) 2 + b 2 e sin(bt )
b
1 1
24a. 2 e −ζωnt sin(ωn 1 − ζ 2 t )
s + 2ζω n s + ω n
2
ωn 1 − ζ 2
s+a e − at cos( bt )
25. ( s + a ) 2 + b 2
Laplace Table Page 2