1

SUBJECT:- ADVANCED ENGINEERING MATHEMATICSSUBJECT CODE:- 2130002TOPIC:- LAPLACE TRANSFORM OF PERIODIC FUNCTIONS

SNPIT & RC

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

3

1. PERIODIC SQUARE WAVE2. PERIODIC TRIANGULAR WAVE3. PERIODIC SAWTOOTH WAVE4. STAIRCASE FUNCTION5. FULL-WAVE RECTIFIER6. HALF-WAVE RECTIFIER7. UNIT STEP FUNCTION8. SHIFTING THEOREM

LAPLACE TRANSFORM OF PERIODIC FUNCTIONS

4

1. PERIODIC SQUARE WAVE1. 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

5

ANS. :- since f(t) is a periodic function with period p= 2a

2

20 0

2

20

2

2

22

11

1

11

1 21

a ast st

as

a ast st

asa

as as as

as

as asas

ke dt ke dte

k e ee s s

k e e ee s s s sk e e

s e

L{f(t)}

6

2

2 2 2

2 2 2

1

1 1

11

tanh2

{ ( )} tanh2

as

as as

as

as

as as as

as as as

k e

s e e

k es e

k e e es e e e

k ass

k asL f ts

7

Find the Laplace transform of the periodic function shown in figure.

2. PERIODIC TRIANGULAR WAVE

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

20

1 ( )1

ast

as e f t dte

9

2

20

2

20

2

2

2 2

2

1 ( ) ( )1

1 (2 )1

(1)1

1(2 ) ( 1)

a ast st

asa

a ast st

asa

ast st

aas ast st

a

e f t dt e f t dte

e tdt e a t dte

e ets s

e e ea ts s

10

2

2 2 2 2 2

22 2

22 2

2

1 11

1 (1 2 )(1 )

1{ ( )} (1 )(1 )

1{ ( )} tan h2

as as as as as

as

as asas

asas

a e e a e e ee s s s s s s

a e es e

L f t es e

saL f ts

11

Find the Laplace transform of the saw tooth wave function given by

f(t)= if 0 < t < p, f(t+p) = f(t)

3. PERIODIC SAW TOOTH WAVE

k tp

ANS.:-

12

Since f(t) is a periodic function with period p.

0

0

0

20

1 ( )1

11

p(1 )

1p(1 )

pst

ps

pst

ps

pst

ps

pst st

ps

e f t dte

kte dte p

k e tdte

k e ete s s

L{f(t)}

13

2 2

2

2

2

1p(1 )

1 ( )(1 )s

s s(1 )

{ ( )}s s(1 )

sp sp

ps

sp spps

sp

sp

sp

sp

k pe ee s s sk e sp e

e p

k kep e

k keL f tp e

14

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)

4. STAIRCASE FUNCTION

15

If h(t) is a saw tooth wave function defined in example 3 as

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 <

ktp

ktp

ANS:-

16

2 2

[g(t)] { ( )}

[g(t)]1

[g(t)]1

sp

sp

sp

sp

ktL L L h tp

k k keLps s p s e

keLs e

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 0t t

sin sint t

This function is called the full sine-wave rectifier function with period

18

We may write the definition of f(t) as follows: f(t)= for and for all t.

sin t 0 t

( )f t f t

19

0

2 20 0

2 20

2 20

1[ ( )] sin1

sin ( .sin cos )

1sin

sin 1

sts

stst

sst

sst

L f t e tdte

eNow e tdt s t ts

e tdt es

e tdt es

20

2 2

2

2 22

2 2

1[ ( )] . 11

1[ ( )] .1

[ ( )] coth2

s

s

s s

s s

L f t es

e

e eL f ts

e esL f t

s

21

Find the Laplace transform of half-wave rectification of sinωt defined by

6. HALF-WAVE RECTIFICATION

sin 0( )

0 2

2 ( )

wt if t wf t

if w t w

nf t f tw

Where For all integer n.

22

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

23

2

20

20

2 ( ) for all

1

11 = sin .....(1)

1

wst

sw

wst

sw

f t w f t t

L f t e f t dte

e wtdte

24

2 20 0

2 2

2 2

Now sin sin cos

1 =

= 1

ww stst

sw

sw

ee w t d t s wt w wts w

we ws w

w es w

25

2 2 2

1

2 21 1

2 2

1 11

=

=1

sw

s sw w

sw

sw

e

e e

sw

wL f t es w

e

ws w

w

s w e

(1) becomes

26

The unit step function u(t - a) is defined as u(t – a) =0 if t < a (a ≥ 0) =1 if t ≥ a figure.

7. UNIT STEP FUNCTION (OR HEAVISIDE’S FUNCTION

27

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

28

0

0

0

{u( )} . ( )

(0) (1)

1

st

st st

a

stst as

a

L t a e u t a dt

e dt e dt

ee dt es s

Laplace transform of unit step function By definition of Laplace transform,

29

1

1

1 1{ ( )} { ( )}

in particular, if a 01 1{ ( )} ( )

as asL u t a e L e u t as s

L u t L u ts s

30

If

Second shifting theorem

0

0

{f( )} ( ),

{f(t a) u(t a)} (s)

{ ( ) u(t a)} . ( ) ( )

(u ) (0)dt ( ) (1)

as

st

ast 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

Proof : by definition of Laplace transform, we have,

31

( )

0

0

1

(u )) ,

(r)

(r) ( )

{ ( ) u(t a)} ( ) [ ( )]

[ ( )] ( ) ( )

st

a

s a r

as sr as

as as

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

32

.1: { ( ) ( )} [ ( )]

.2 : [ ( )] [1. ( )] (1)

.3 : [ ( ) ( )]

.4 : [ ( ) { ( ) ( )] { ( )} { ( )}

as

asas

as bs

as bs

Cor L f t u t a e L f t a

eCor L u t a L u t a e Ls

e eCor L u t a u t bs

Cor L f t u t a u t b e L f t a e L f t b

(alternative form of second shifting theorem)

33

REFERENCE BOOK :- DR K.R.KACHOT

34