Post on 29-Mar-2018
Common Laplace Transform Pairs
Time Domain Function Laplace Domain Function Name Definition*
Unit Impulse )(tδ 1
Unit Step (t)γ † s1
Unit Ramp t 2
1s
Parabola t2 3
2s
Exponential ate− as +1
Asymptotic Exponential )1(1 ate
a−−
)(1
ass +
Dual Exponential ( )btat eeab
−− −−1
))((
1bsas ++
Asymptotic Dual Exponential ( )at bt1 11 be ae
ab a b− − + − −
))((
1bsass ++
Time multiplied Exponential
atte− 2)(1as +
Sine 0sin( t)ω 02 2
0sω+ ω
Cosine 0cos( t)ω 2 20
ss +ω
Decaying Sine atde sin( t)− ω d
2 2d(s a)
ω+ + ω
Decaying Cosine atde cos( t)− ω 2 2
d
s a(s a)
++ +ω
Generic Oscillatory Decay ( ) ( )at
d dd
C aBe Bcos t sin t− −ω + ω ω
( )2 2d
Bs Cs a
+
+ +ω
Prototype Second Order Lowpass, underdamped
( )0t 2002
e sin 1 t1
−ζωωω −ζ
−ζ
20
2 20 0s 2 s
ω+ ζω +ω
Prototype Second Order Lowpass, underdamped
- Step Response
( )0t 202
21
11 e sin 1 t1
1tan
−ζω
−
− ω −ζ + φ−ζ
− ζ φ = ζ
20
2 20 0s(s 2 s )
ω+ ζω +ω
*All time domain functions are implicitly=0 for t<0 (i.e. they are multiplied by unit step, γ(t)).
†u(t) is more commonly used for the step, but is also used for other things. γ(t) is chosen to avoid confusion (and because in the Laplace domain it looks a little like a step function, Γ(s)).
Common Laplace Transform Properties
Name Illustration
Definition of Transform
L
st
0
f (t) F(s)
F(s) f (t)e dt−
∞ −
←→
= ∫
Linearity )()()()( 2121 sBFsAFtBftAf L +→←+
First Derivative )0()()( −−→← fssFdt
tdf L
Second Derivative )0()0()()( 22
2−− −−→← fsfsFs
dttfd L
nth Derivative )0()()( )1(
1
−−
=
−∑−→← in
i
innLn
n
fssFsdt
tfd
Integral )(1)(0
sFs
df Lt→←∫ λλ
Time Multiplication ds
sdFttf L )()( −→←
Time Delay L asf (t a) (t a) e F(s)−− γ − ←→
γ(t) is unit step
Complex Shift )()( asFetf Lat +→←−
Scaling )(asaFatf L→←
Convolution Property )()()(*)( 2121 sFsFtftf L→← Initial Value
(Only if F(s) is strictly proper; order of numerator < order of denominator).
)(lim)(lim0
ssFtfst ∞→→
=+
Final Value (if final value exists;
e.g., decaying exponentials or constants) )(lim)(lim
0ssFtf
st →∞→=