Laplace transform of the periodic function

1. CIRCUIT THEORY
LAPLACE TRANSFORM OF PERIODIC
FUNCTIONS
SYED HASAN SAEED
Thursday, August 29, 2019 1syed hasan saeed

2. REFERENCE BOOKS
• Introductory Circuit Analysis, Robert L. Boylested, Pearson Education,
Prentice Hall.
• Networks And Systems, Ashfaq Husain, Khanna Book Publishing Co (P)
Ltd. Delhi.
• Networks And Systems, A Sudhakr, Shyammohan S Palli, Tata McGraw
Hill, New Delhi.
• Network Analysis, M.E. Van Valkenburg, PHI Learning Private limited,
New Delhi.
• Circuit Analysis Principle and Applications, Allan H. Robbins &Wilhelm
C. Miller, DELMAR CENGAGE Learning, Indian Reprint.
Thursday, August 29, 2019 syed hasan saeed 2

3. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 3
Let f(t) be a periodic function which satisfies the condition f(t) = f(t + T) for
all t > 0 where T is the period of the function.
ℒ [ f(t)] =
ℒ [ f(t)] 

  
 





T
0
st-
sT-
T
0
st-nsT-2sT-sT-
T
0
T
0
T
0
nsT-st-sT-st-st-
2T
T
T1)(n
nT
st-st-
0
st-
dtef(t)
e-1
1
dtef(t)....)e......ee1(
dteef(t).....dteef(t)dtef(t)
ef(t).......dtef(t)dtef(t)
T


4. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 4
EXAMPLE 1 : Find the transform of the following waveform.
f(t)
T 2T 3T 4T
A
-A
t

5. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 5
ℒ [f(t)] =
 
   
 


























































sT-
sT-
2sT-
sT-sT-
sT-sT-2sT-
2sT-
sT-2sT-sT-
2sT-
2T
T
st-
T
0
st-
2sT-
2T
T
st-
0
st-
2sT-
2T
0
st-
2sT-
e1
e-1
s
A
)e-1(
s
A
)e1()e-1(
1
1e-e-e
)e-1(s
A
e-e
s
A
1-e
s
A-
)e-1(
1
e
s
A
e
s
A-
)e-1(
1
dteA-dteA
)e-1(
1
dtef(t)
)e-1(
1
T

6. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 6
EXAMPLE 2 : Find the transform of the following waveform.
f(t)
T
K
0
Slope = K / T
t
t
T
K
f(t)
x
T
K
y
cmxy


STEP 1: Find the function

7. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 7
 
 
 
  T)-u(tK-T)-t(uT-t
T
K
-u(t)t
T
K
T)-u(tT
T
K
-T)-t(uT-t
T
K
-u(t)t
T
K
T)-t(uTT-t
T
K
-u(t)t
T
K
T)-t(ut
T
K
-u(t)t
T
K
T)-u(t-u(t)t
T
K
(t)Gt
T
K
f(t) o






STEP 2: Apply the Gate function Go

8. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 8
 
  Ts-
2
Ts-Ts-
2
Ts-
2
Ts-
2
eTs1-1
sT
K
eTs-e-1
sT
K
e
s
1
K-
s
1
e
T
K
-
s
1
T
K



F(s) = ℒ f(t)

9. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 9
EXAMPLE 3 : Find the transform of the following waveform
f(t)
A
0 T/2
t

10. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 10
Ꞷ = 2πf
Ꞷ = 2π/T
f(t) = A sin Ꞷt [ u(t) – u(t-T/2)]
f(t) = A sin Ꞷt u(t) – A sin Ꞷt u(t-T/2)
Let f1 = A sin Ꞷt u(t) and f2 = A sin Ꞷt u(t-T/2)
F(s) = F1(s) – F2(s)
F1(s) = ℒ f1(t) = ℒ A sin Ꞷt u(t)
=ℒ A sin (2π /T) t u(t)
= A (2π /T) / [s2+ (2π /T)2 ]
ℒ f2(t) = A sin Ꞷt u(t-T/2)
= A sin 2π/T ( t-T/2 + T/2) u (t-T/2)
= A sin [2π/T ( t-T/2) + π] u (t-T/2)
= - A sin [2π/T ( t-T/2)] u (t-T/2) sin(180+Ө)= - sin Ө

11. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 11
ℒ f2(t) = - A sin 2π/T (t-T/2) u(t-T/2)
ℒ f2(t) = -A ℒ sin 2π/T (t-T/2) u(t-T/2)
ℒ f2(t) = F2(s) = -A [ e-(T/2)s (2π/T)/ [s2 + (2π /T)2 ] ]
F(s) = F1(s) + F2(s)
F(s) = A (2π /T) / [s2+ (2π /T)2 ] -A [ e-(T/2)s (2π/T)/ [s2(2π /T)2 ] ]
F(s) = A (2π /T) / [s2+ (2π /T)2 ] (1+ e-(T/2)s )

12. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 12
EXAMPLE 4 : Find the transform of the following waveform
First find the functions f1(t) and f2(t). Find the slope of both functions by
using
1 20
f(t)
K
( 1, K )
( 2, 0 )
A
B
f1(t) f2(t)
)x-(x
x-x
y-y
y-y 1
12
12
1 

13. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 13
For f1(t): For f2(t):
Ky0y
1x0x
21
21


0yKy
2x1x
21
21


We get,
y = K x
f1(t) = Kt
We get,
y = K (x -1)
f1(t) = Kt
f(t)= f1(t) + f2(t)
f(t) = Kt [u(t) – u(t-1)] – K (t-2) [u(t-1) – u (t-2)]
f(t) = Kt u(t) – 2K (t-1) u(t-1) + K (t-2) u(t-2)
Take the Laplace:
2s-
2
2s-s-
2
2s-
2
s-
22
)e-(1
s
K
F(s)
)e2e-(1
s
K
F(s)
e
s
1
Ke
s
1
2K-
s
1
KF(s)




14. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 14
EXAMPLE 5 : Find the transform of the following waveform
2aa0
K
f(t)
t
2a)-(tuK2a)]-u(t-a)-[u(ta)-(t
a
K
f(t) 

15. LAPLACE TRANSFORM OF PERIODIC FUNCTIONS
Thursday, August 29, 2019 syed hasan saeed 15
Contd.
2a)]-u(t2a)-(t
a
K
-a)-u(ta)-(t
a
K
f(t)
2a)-(tuK2a)-(tuK2a)]-u(t2a)-(t
a
K
-a)-u(ta)-(t
a
K
f(t)
2a)-(tuK2a)]-u(t2a)]-a[(t
a
K
-a)-u(ta)-(t
a
K
f(t)
2a)-(tuK2a)]-u(t-a)-[u(ta)-(t
a
K
f(t)




Take the Laplace of above equation
)e-(1
as
Ke
F(s)
e
s
1
a
K
-e
s
1
a
K
F(s)
as-
2
as-
2as-
2
as-
2



16. THANK YOU
shasansaeed@yolasite.com
hasansaeed872726549.wordpress.com
saeed.moodlecloud.com
syedhasansaeed.gnomio.com
Thursday, August 29, 2019 syed hasan saeed 16