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1. Lecture 2: The Laplace Transform
• Laplace transform definition
• Laplace transform properties
• Relation between time and Laplace
domains
• Initial and Final Value Theorem
• Introduction to Simulink (during lab.) 1
EPCE-3204,
Lecture
2

2. The Laplace Transform
• The Laplace transform is a mathematical
operation that takes an equation from being a
function of time, t, to being a function of the
Laplace variable, s
• Some mathematical operations become much
simpler in the Laplace domain
• We will never solve this integral, will use tables
0
[ ( )] ( ) ( )
st
f t f t e dt F s


 

L
2
EPCE-3204,
Lecture
2

3. Item No. f(t) F(s)
δ(t)
1(t)
t
tn
e-at
sin (ωt)
cos (ωt)
1.
2.
3.
4.
5.
6.
7.
1
2
1
s
!
n+1
n
s
1
s + a
2 2
ω
s + ω
2 2
s
s + ω
Table of Laplace pairs
on pages 18-19
unit impulse
unit step
unit ramp
t
1
t
t
1
s
3
EPCE-3204,
Lecture
2

4. Properties of the Laplace Transform
1. Linearity
- constants factor out and Laplace operation
distributes over addition and subtraction
- note:
[ ( ) ( )]
[ ( )] [ ( )]
( ) ( )
af t bg t
a f t b g t
aF s bG s
 
 

L
L L
[ ( ) ( )] ( ) ( )
f t g t F s G s
  
L 4
EPCE-3204,
Lecture
2

5. Properties of the Laplace Transform
2. Integration
3. Differentiation
0
0 0
2 2
( )
( )
( )
( ) ( )
( )
( )
t
t t
f t dt
F s
f t dt
s s
f t dt dt f t dt
F s
f t dt dt
s s s

 
 
 
   
 
   
   
    
 


 

L
L
2
2
2
( ) (0)
( ) (0) (0)
df
sF s f
dt
d f
s F s sf f
dt
 
 
 
 
 
  
 
 
 
L
L
These properties
turn differential
equations into
algebraic equations
often zero
5
EPCE-3204,
Lecture
2

6. Properties of the Laplace Transform
4. Multiplication by e-at
- important for damped response
Example:
[ ( )] ( )
at
e f t F s a

 
L
[ cos ]
at
e t


L
Note: roots of
denominator (poles)
in Laplace domain =
roots of characteristic
equation in the time
domain
f(t)
2 2
from Laplace pairs table, ( ) [cos ]
s
F s t
s




= L
2 2
then from prop above, ( )
( )
s a
F s a
s a 

 
 
EPCE-3204,
Lecture
2
6

7. Properties of the Laplace Transform
5. Time shift
- important for analyzing time delays
[ ( )1( )] ( ), 0
as
f t a t a e F s a

   
L
7
EPCE-3204,
Lecture
2

8. Properties of Laplace Transform
6. Multiplication by t
( )
[ ( )]
dF s
tf t
ds
 
L
2
2
2
( )
[ ( )]
d F s
t f t
ds

L
( )
[ ( )] ( 1)
n
n n
n
d F s
t f t
ds
 
L
8
EPCE-3204,
Lecture
2

9. Example
• Find
3
[2 5]
t
te

L
3
2 [ ] 5 [1( )] (by property 1)
t
te t


= L L
1
s
2
1
[ ]
t
s
 
L
3
2
1
[ ] (by property 4)
( 3)
t
te
s

 

L
2
2 5
( 3)
s s


=
0 for t<0

10. Laplace/Time Domain Relationship
• Previously, saw how poles of X(s) relate to x(t)
• Two further relationships between X(s) and x(t):
Initial Value Theorem
Final Value Theorem
(0 ) lim ( ), if the lim exists
s
f sF s



0
( ) lim ( ) lim ( ),
if [poles of ( )] 0
t s
f f t sF s
sF s
 
  

10
EPCE-3204,
Lecture
2

11. Example
• Find the initial value of f(t), where 2
3
( )
( 6 13)
s
F s
s s s


 
(0) lim ( )
s
f sF s

 2
( 3)
lim
( 6 13)
s
s s
s s s



  2
( 3)
lim
( 6 13)
s
s
s s



 
2
1
s

2
1
s

2
2
1 3
lim
6 13
1
s
s s
s s



 
0


12. Example
• Find the final value of f(t), where 2
3
( )
( 6 13)
s
F s
s s s


 
0
( ) lim ( ) lim ( )
t s
f f t sF s
 
   2
0
( 3)
lim
( 6 13)
s
s s
s s s



 
2
0
( 3)
lim
( 6 13)
s
s
s s



 
3
13

poles of ( ) 3 2 ,
sF s j
  
since <0, limit exists