Laplace Transformation I
Why use Laplace Transform?
Algebraic Manipulation of ODE
Definition:
£ [f(t)]
=£: f(t) F(s) , s= +j (complex variable)
f(t) : a time function such that f(t)=0 for t<0
Inverse Laplace Transformation
L
1[ ( )]F s f t( )0
( )
st( )
f t e dt F s
Laplace Transformation
f(t) Laplace – transformable
if i) f(t) piecewise-continuous
ii) f(t) of exponential order as t approaches infinity
- bounded, exist.
- or approaches zero as t approaches infinity.
If
the : the abscissa of convergence
Existence of Laplace Transformation
lim
t( ) 0
ct c
e f t for
for
ct ( ) e f t )
(t et f
2
0 ( )
0 0, T< t
e
tfor t T
f t
for t
0 0
lim sin
0
t t
e t t if
if
lim
t ct0
t
if c
e te
if c
c
0
c
c
, , sin , .
ct ct ct
e
te
e
t c const , sin , sin ...
t t t t
example : 1)
the abscissa of convergence
2)
the abscissa of convergence
2 2
t
,
te te
does not possess L. T.L[f(t)] exists.
- The signals that can be physically generated always have corresponding Laplace transforms
Existence of Laplace Transformation
0 0
( ) t 0
for t f t Ae for t
0
( )
0
( ) 0
( )
[ ]
1 1
[ ]
1 1
[ ]
t t st
s t
s
j
Ae Ae e dt
Ae dt
A e e
s s
A e
s s
L
Laplace Transformation of simple function
Exponential function
A,
a
: constants
(s j)
j
step function
( ) 0
0 0
f t A t
t
unit step input function
0 0
1 1
[ ( )] [ ]
[0 1 ] Re[ ] 0
st s s
f t Ae dt A e e
s s
A if s
s
L
Laplace Transformation of simple function
0 0
0
1( ) 1
0
1 0
1( ) 0 0
[1( )] 1
t t t t
t t t t
t
t s
L
Sinusoidal Functions
cos 1 ( )
2
sin 1 ( )
2
j t j t
j t j t
t e e
t e e
j
0
2 2
2 2
[ sin ] ( )
2
1 1 1
( )
2
[ cos ]
j t j t st
A t A e e e dt
j
j s j s j
A s A t As
s
L
L
Ramp function
Laplace Transformation of simple function
[ ]
0 stst st
At A te dt
e e
A t dt
L
sin 0
( ) 0 0
A t t
f t t
( ) 0
0 0
At t
f t t
0 0
0
( ) 0
0 0, < t
A for t t
f t t
for t t
0
0 0
( ) A 1( ) A 1( )
f t t t t
t t
0 0
0
0
0 0
0 0
0 0
[ ( )] lim 1(1 )
(1 )
lim
( )
t s t
t s
t
f t A e
t s
d A e
dt A s
d s A
dt t s
L
Pulse function
Impulse function
0
0 0 0
0
lim 0
( )
0 0,
t
A for t t
f t t
for t t t
0
0
0 0
0
[ ( )] 1( ) 1( )
1 (1
t s)
A A
f t t t t
t t
A e
t s
L L L
Laplace Transformation of simple function
0
0 0 0
0
0
0 0
lim 1 0
( )
0 0,
( ) 0 ( ) ( ) ( ) 1
0 0
t
st
for t t
f t t
for t t t
t t t t e dt t dt
t
L
The unit-impulse function occuring at
0 0 00
0
0 0 0
0 0
( ) ( ) ( ) ( ) 1
0
t
st t s
st t
t t
t t t t t e dt t t e dt e
t t
L
t t
0Laplace Transformation of simple function
Unit impulse function ; impulse function of magnitude 1
