Courses

# PPT - Laplace Transform and its Applications Electrical Engineering (EE) Notes | EduRev

## Electrical Engineering (EE) : PPT - Laplace Transform and its Applications Electrical Engineering (EE) Notes | EduRev

``` Page 1

Laplace Transform And Its
Applications
Page 2

Laplace Transform And Its
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Page 3

Laplace Transform And Its
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Definition of Laplace Transform
? Let f(t) be a given function of t defined for all
then the Laplace Transform ot f(t) denoted by L{f(t)}
or         or F(s) or          is defined as
provided the integral exists,where s is a parameter real
or complex.
0 ? t
) (s f ) (s ?
dt t f e s s F s f t f L
st
) ( ) ( ) ( ) ( )} ( {
0
?
?
?
? ? ? ? ?
Page 4

Laplace Transform And Its
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Definition of Laplace Transform
? Let f(t) be a given function of t defined for all
then the Laplace Transform ot f(t) denoted by L{f(t)}
or         or F(s) or          is defined as
provided the integral exists,where s is a parameter real
or complex.
0 ? t
) (s f ) (s ?
dt t f e s s F s f t f L
st
) ( ) ( ) ( ) ( )} ( {
0
?
?
?
? ? ? ? ?
Linearity of the Laplace Transform
? If  L{f(t)}=        and                            then for any
constants a and b
) (s f
) ( )] ( [ s g t g L ?
)] ( [ )] ( [ )] ( ) ( [ t g bL t f aL t bg t af L ? ? ?
)] ( [ )] ( [ )} ( ) ( {
) ( ) (
)] ( ) ( [ )} ( ) ( {
Definition -By : Proof
0 0
0
t g bL t f aL t bg t af L
dt t g e b dt t f e a
dt t bg t af e t bg t af L
st st
st
? ? ?
? ?
? ? ?
? ?
?
?
?
?
?
?
?
Page 5

Laplace Transform And Its
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Definition of Laplace Transform
? Let f(t) be a given function of t defined for all
then the Laplace Transform ot f(t) denoted by L{f(t)}
or         or F(s) or          is defined as
provided the integral exists,where s is a parameter real
or complex.
0 ? t
) (s f ) (s ?
dt t f e s s F s f t f L
st
) ( ) ( ) ( ) ( )} ( {
0
?
?
?
? ? ? ? ?
Linearity of the Laplace Transform
? If  L{f(t)}=        and                            then for any
constants a and b
) (s f
) ( )] ( [ s g t g L ?
)] ( [ )] ( [ )] ( ) ( [ t g bL t f aL t bg t af L ? ? ?
)] ( [ )] ( [ )} ( ) ( {
) ( ) (
)] ( ) ( [ )} ( ) ( {
Definition -By : Proof
0 0
0
t g bL t f aL t bg t af L
dt t g e b dt t f e a
dt t bg t af e t bg t af L
st st
st
? ? ?
? ?
? ? ?
? ?
?
?
?
?
?
?
?
Laplace Transform of some Elementary
Functions
a s if
a - s
1
) (

e . ) e (
Definition -By : Proof
a - s
1
) L(e  (2)
) 0 ( ,
s
1
1 . ) 1 (
Definition -By : Proof
s
1
L(1)  (1)
0
) (
0
) (
0
at at
at
0
0
? ?
?
?
?
?
?
?
? ?
?
? ?
?
? ?
?
?
?
?
?
?
?
? ?
?
?
? ?
?
? ?
?
?
?
?
?
?
? ?
?
a s
e
dt e dt e L
s
s
e
dt e L
t a s
t a s st
st
st
```
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

## Signals and Systems

40 videos|40 docs|25 tests

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;