This document provides information about the Laplace transform of periodic functions. It discusses the Laplace transform of periodic square waves, triangular waves, sawtooth waves, and staircase functions. It also discusses the Laplace transform of full-wave and half-wave rectification of a sine wave. Examples are provided to show how to calculate the Laplace transform of each of these periodic and rectified functions.

1. SNPIT & RC
SUBJECT:- ADVANCED ENGINEERING MATHEMATICS
SUBJECT CODE:- 2130002
TOPIC:- LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
1

2.  PANCHAL ABHISHEK -130490109002
 CHANCHAD BHUMIKA -130490109012
 DESAI HALLY -130490109022
 JISHNU NAIR -130490109032
 LAD NEHAL -130490109042
 MISTRY BRIJESH -130490109052
 SURTI KAUSHAL -D2D 002
 CHAUDHARI MEGHAVI -D2D 012
GUIDED BY
-Prof. RASHIK SHAH
-Prof. NEHA BARIA
GROUP MEMBERS
2

3. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
1. PERIODIC SQUARE WAVE
2. PERIODIC TRIANGULAR WAVE
3. PERIODIC SAWTOOTH WAVE
4. STAIRCASE FUNCTION
5. FULL-WAVE RECTIFIER
6. HALF-WAVE RECTIFIER
7. UNIT STEP FUNCTION
8. SHIFTING THEOREM
3

4. 1. PERIODIC SQUARE WAVE
1. Find the Laplace transform of the square wave
function of period 2a
defined as f(t) = k if 0 
t < a
= -k if a < t < 2a
The graph of square wave is shown in figure
4

5.  ANS. :-
since f(t) is a periodic function with period p= 2a
L{f(t)}  
 
2
 
2
0 0
2
2
0
2
2
2
2
1
1
1
1
1
1 2
1
a a
st st
as
a a
st st
as
a
as as as
as
as as
as
ke dt ke dt
e
k e e
e s s
k e e e
e s s s s
k
e e
s e
 

 

  

 

 
    
  
    
     
        
 
     
  
  

5

6.  
2

k 1

e
   
 
s e e
1 1
k e
s e
2 2 2
k e e e
s e e e
k as
s
2 2 2
1
1
tanh
2
{ ( )} tanh
2
as
as as
as
as
as as as
as as as
k as
L f t
s


 
 

 
  
  
  
  
  
    
 
  
 
 
   
 
6

7. 2. PERIODIC TRIANGULAR WAVE
 Find the Laplace transform of the periodic
function shown in figure.
7

8.  ANS.:-The function can be represented as
f(t) = t 0 < t < a
=2a-t a < t < 2a
The function has a period 2a
L{F(t)}
2
2

0
1
( )
1
a
st
as e f t dt
e


 
  
  
8

9. a 2
a
   
2
 
 
  0

st st
 
2
    
2
 
 
  0

2
      
       2
  
     
  
2 2
        
2
1
( ) ( )
1
1
(2 )
1
(1)
1
1
(2 ) ( 1)
as
a
a a
st st
as
a
a
st st
a
as a
st st
a
e f t dt e f t dt
e
e tdt e a t dt
e
e e
t
s s
e
e e
a t
s s

 

 

 
        
     

9

10. as as as 2
as as
 a e e a e e e

1 1
       2 2 2 2 2

e s s s s s s
  
2
  
2 2
2
a e e
2 2
2
1
1
(1 2 )
(1 )
1
{ ( )} (1 )
(1 )
1
{ ( )} tan h
2
as
as as
as
as
as
s e
L f t e
s e
sa
L f t
s
    

 




  

 
   
 
10

11. 3. PERIODIC SAW TOOTH WAVE
 Find the Laplace transform of the saw tooth
wave function given by
f(t)= if 0 < t < p, f(t+p) = f(t)
k
t
p
 ANS.:-
11

12. Since f(t) is a periodic function with period p.
p

0

0


0
2
0
1
( )
1
1
1
p(1 )
1
p(1 )
st
ps
p
st
ps
p
st
ps
p
st st
ps
e f t dt
e
kt
e dt
e p
k
e tdt
e
k e e
t
e s s





 







    
      
      
L{f(t)}
12

13. sp sp
 
 
k pe e
e s s s
k
    2 2

  
 
    
 2
 
2
2
1
p(1 )
e sp e
1 ( )
(1 ) s
s s(1 )
{ ( )}
s s(1 )
ps
sp sp
ps
sp
sp
sp
sp
e p
k ke
p e
k ke
L f t
p e

 





 

  

13

14. 4. STAIRCASE FUNCTION
 Find the Laplace transform of the staircase function
defined as
g(t)=kn for np < t < (n+1)p, where n=0,1,2,….
(Note : This is not a periodic function)
14

15. If h(t) is a saw tooth wave function defined in
example 3 as
kt
p
h(t)= for 0<t<p
and h(t + p) = h(t) for all values of t. It is easy
to observe from the figure that
g(t)= -h(t) for 0 < t <
kt
p

ANS:-
15

16.  
    
[g(t)] { ( )}
 
     
 
 
2 2
    
 
 
[g(t)]
1
[g(t)]
1
sp
sp
sp
sp
kt
L L L h t
p
k k ke
L
ps s p s e
ke
L
s e





16

17.  Find the Laplace transform of the full wave
rectification of f(t)=
Ans:-The graph of the function f(t)is shown in figure.
Observe that
for any t.
5. FULL-WAVE RECTIFIER
sin t t  0

 
   
 
sin t sin t
 

This function is called the full sine-wave rectifier
function with period


17

18. We may write the definition of f(t) as follows:
f(t)= for
sin t 0 t


 


 
   
 
and f t f (t)
for all t.
18

19. 
L f t e tdt
 

0
st
st

   
2 2
s
st
0 0
2 2
0
2 2
0
1
[ ( )] sin
1
sin ( .sin cos )
1
sin
sin 1
s
st
s
st
e
e
Now e tdt s t t
s
e tdt e
s
e tdt e
s




 
 

 


 


  












 
    
  
 
   
  
 
   
  



19

20. 

 
L f t e
 [ ( )]  .  1
 
2 2
   
2
1


 
2 2
2


2 2
1
1

[ ( )] .
1
[ ( )] coth
2
s
s

s s
s s
s
e
e e
L f t
s

e e
s
L f t
s




 
 
 
 


  

 
   
  
20

21. 6. HALF-WAVE RECTIFICATION
Find the Laplace transform of half-wave rectification
of sinωt defined by
wt if t w
 sin 0
 
 
  
( )
0 2
2
( )
f t
if w t w
n
f t f t
w

 

 
   
 
Where For all integer n.
21

22. Ans. The graph of function F(t) is shown in the
figure
22

23.  
    
f t  2 w 
f ( t ) for all
t
2
L f t e f t dt
 
s
w
2
0
2
w
0
1
1
1
w
st
st
= sin .....(1)
s
1
w
e
e wtdt
e













23

24. w w st
 
 
      2  2

 
Now sin sin cos
0 0
1
 
  
2 2
  
 
  
2 2
=
= 1
st
s
w
s
w
e
e w t d t s wt w wt
s w
we w
s w
w
e
s w






  

24

25.   
2 2 2
1
 
1
2 2
1 1
2 2
1
1
=
=
1
s
w
s s
w w
s
w
s
w
e
e e
s
w
w
L f t e
s w
e
w
s w
w
s w e

 




 


 
  
 
 
  
    
  
  

 
   
   

 
   
 
(1) becomes 
25

26. 7. UNIT STEP FUNCTION (OR
HEAVISIDE’S FUNCTION
 The unit step function u(t - a) is defined as
u(t – a) =0 if t < a (a ≥ 0)
=1 if t ≥ a figure.
26

27. The unit step function is also called the Heaviside
function. In particular if a = 0, we have
u(t) = 0 if t < 0
= 1 if t ≥ 0
27

28.  Laplace transform of unit step function
By definition of Laplace transform,
   
{u( )} . ( )
0
 
0
0
(0) (1)
 
1
st
st st
a
st
st as
a
L t a e u t a dt
e dt e dt
e
e dt e
s s


 
 
  
 
    
  

 

28

29. 1 1
 
L u t a e  as L  1
e 
as u t a
       
{ ( )} { ( )}
s s

1
in particular, if a 0
 

1 1
 
L u t L u t
     
{ ( )} ( )
s s
 
29

30. If
Second shifting theorem
{f( )} 
( ),
{f(t  a) u(t  a)} 

as
(s)
Proof : by definition of Laplace transform, we have,
{ (  ) u(t  a)}  . (  ) ( 
)

0
0
(u ) (0) dt ( ) (1)
st
a
st st
a
L t f s then
L e f
L f t a e f t a u t a dt
e f a e f t a dt



 
   
 
30

31. st
s a r
 
( )


a


0


as sr as
  

0
1
(u )) ,
(r)
(r) ( )
as as
 
{ ( ) u(t a)} ( ) [ ( )]
[ as
( )] ( ) ( )
e f a dt Substituting t a r
e f dr
e e f dr e f s
L f t a e f s e L f t
L  e 
f s f t a u t a
   

 
    
   
31

32. as

Cor L f t u t a e L f t a
.1: { ( ) ( )} [ ( )]
(alternative form of second shifting theorem)
as
  

Cor L u t a L u t a e L
.2 : [ ( )] [1. ( )] (1)
    
Cor L u t a u t b
.3 : [ ( ) ( )]
as
as bs
e

s
 
e e
as bs

s
 
   
Cor L f t u t a u t b e L f t a e L f t b
.4 : [ ( ) { (  )  (  )]  { (  )}  { ( 
)}
32

33. REFERENCE BOOK :- DR K.R.KACHOT
33

34. 34