Page 13 - Servo Motors and Industrial Control Theory

1.4 Laplace Transform 5

Table 1.1 Laplace transform of some common functions
f( t) F( s)
A A
s
At; At n A An!
;
s 2 s n 1+
Ae− αt A
s α+
dft() ( s) − f(0)
dt
2
d ft() 2 d f 0 ()
dt 2 sF s() − sf 0() − t d

d n n
ft () with zero initials F( s)
dt 2
t 1 1  t 
∫ f ( )dζζ = f ( 1)− ( )t Fs f ζζ
( ) +  ∫
( )d 
−∞ s s  −∞  t= 0
n
f (− )( t) zero initials
f (− )( t) fs()
n
s n
−at
e f(t) F( s + a)
te −at 1
(s + ) a 2
t e n!
n −at
s ( + a) n+1
f(t − t ), t > t ;0,t < t d e − st Fs()
d
d
d
δ( t) unit impulse 1
c f ( t) + c f ( t) c F ( s)+c F ( s)
2 2
1 1
1 1
2 2
Aω
A sin (ωt)
s + ω 2
2
( (
 1  ω 2 n
−
2

t
1+   ·e − ξω n t ·cos ω n . . 1 ξ +Φ ) ξ < 1 ( ss + 2ξω s ω 2 )
+
2
−
2
 1 ξ  n n
Π 1 ξ− 2
−
(θ =+ ) φ = tan ( )
φ
1
2 ξ
represent step input; the ramp and acceleration inputs are other parameters that are
often used in analysis to determine the performance.
The above table gives the Laplace Transform of the most useful function. Also
note that the Laplace Transform is the only function of s only. This enables us to
treat differential equation like algebraic function. Some examples will clarify this
point, and they show how differential equation is converted to s domain and how the
solution can be obtained from the Laplace Transform.