PPT: Laplace Transform & Its Applications | Signals and Systems - Electrical Engineering (EE) PDF Download

 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