Common Laplace Transform Pairs
Content
Common Laplace Transform Pairs Time Domain Function Definition* (t ) Unit Impulse
Laplace Domain Function 1 1 s 1 s2 2 s3 1 sa 1 s( s a) 1 ( s a )( s b) 1 s ( s a )( s b) 1 (s a) 2 0 2 s 02 s 2 s 02 0 (s a) 2 d2 sa (s a) 2 d2
Name
Unit Step
(t)
Unit Ramp
t
Parbola
t2
Exponential
e at
†
1 (1 e at ) a
Asymptotic Exponential
1 e at e bt ba 1 1 1 be at ae bt ab a b
Dual Exponential Asymptotic Dual Exponential Time multiplied Exponential
te at
Sine
sin(0 t)
Cosine
cos(0 t)
Decaying Sine
e at sin(d t)
Decaying Cosine
e at cos(d t)
Generic Oscillatory Decay Prototype Second Order Lowpass, underdamped Prototype Second Order Lowpass, underdamped Step Response
C aB e at Bcos d t sin d t d 0 e 0 t sin 0 1 2 t 2 1
1
1 1
2
e 0 t sin 0 1 2 t
1 tan 1
2
Bs C
s a
2
d2
02 s 2 20s 02
02 s(s 2 20s 02 )
*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 Definition of Transform
Illustration L f (t) F(s)
F(s) f (t)e st dt 0
Linearity First Derivative Second Derivative nth Derivative Integral Time Multiplication Time Delay Complex Shift
L Af1 (t ) Bf 2 (t ) AF1 ( s ) BF2 ( s ) df (t ) L sF ( s ) f (0 ) dt 2 d f (t ) L s 2 F ( s ) sf (0 ) f (0 ) dt 2 n d n f (t ) L n s F ( s ) s ni f (i 1) (0 ) n dt i 1 t 1 L 0 f ( )d s F (s) dF ( s ) L tf (t ) ds L f (t a) (t a) e as F(s)
γ(t) is unit step
f (t )e
at
L F (s a)
Initial Value
t L f aF (as ) a L f1 (t ) * f 2 (t ) F1 ( s ) F2 ( s ) lim f (t ) lim sF ( s )
Final Value
lim f (t ) lim sF ( s)
Scaling Convolution Property
(if final value exists)
t 0
t
s s 0
Laplace Domain Function 1 1 s 1 s2 2 s3 1 sa 1 s( s a) 1 ( s a )( s b) 1 s ( s a )( s b) 1 (s a) 2 0 2 s 02 s 2 s 02 0 (s a) 2 d2 sa (s a) 2 d2
Name
Unit Step
(t)
Unit Ramp
t
Parbola
t2
Exponential
e at
†
1 (1 e at ) a
Asymptotic Exponential
1 e at e bt ba 1 1 1 be at ae bt ab a b
Dual Exponential Asymptotic Dual Exponential Time multiplied Exponential
te at
Sine
sin(0 t)
Cosine
cos(0 t)
Decaying Sine
e at sin(d t)
Decaying Cosine
e at cos(d t)
Generic Oscillatory Decay Prototype Second Order Lowpass, underdamped Prototype Second Order Lowpass, underdamped Step Response
C aB e at Bcos d t sin d t d 0 e 0 t sin 0 1 2 t 2 1
1
1 1
2
e 0 t sin 0 1 2 t
1 tan 1
2
Bs C
s a
2
d2
02 s 2 20s 02
02 s(s 2 20s 02 )
*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 Definition of Transform
Illustration L f (t) F(s)
F(s) f (t)e st dt 0
Linearity First Derivative Second Derivative nth Derivative Integral Time Multiplication Time Delay Complex Shift
L Af1 (t ) Bf 2 (t ) AF1 ( s ) BF2 ( s ) df (t ) L sF ( s ) f (0 ) dt 2 d f (t ) L s 2 F ( s ) sf (0 ) f (0 ) dt 2 n d n f (t ) L n s F ( s ) s ni f (i 1) (0 ) n dt i 1 t 1 L 0 f ( )d s F (s) dF ( s ) L tf (t ) ds L f (t a) (t a) e as F(s)
γ(t) is unit step
f (t )e
at
L F (s a)
Initial Value
t L f aF (as ) a L f1 (t ) * f 2 (t ) F1 ( s ) F2 ( s ) lim f (t ) lim sF ( s )
Final Value
lim f (t ) lim sF ( s)
Scaling Convolution Property
(if final value exists)
t 0
t
s s 0
