This document discusses the Laplace transform of periodic functions. It begins by defining a periodic function f(t) and derives the Laplace transform of f(t) as an infinite sum. Several examples are then worked through to demonstrate finding the Laplace transform of square waves, triangular waves, and other periodic functions. Transform properties and techniques like using gate functions are employed in the examples. Reference books on circuit analysis and Laplace transforms are also listed.

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
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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
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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
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 
  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
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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