Laplace Transformation II

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Laplace Transformation II

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Why use Laplace Transform?

Al b i M i l ti f ODE Algebraic Manipulation of ODE

Solution of ODE is hard Solution of ODE is hard.

Tranform in to a “domain” where it’s easier to solve

Solve in the new domain

f “i ” f

Laplace domain is a frequency domain.

Integration, Differentiation becomes multiplication division

Perform “inverse” transform. becomes multiplication, division.

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Laplace Transformation

f £ [f(t)] f ( )

^st

d F ( )







Laplace Transformation

 Definition: £ [f(t)] =

£

: f(t)  F(s) , s=  +j  (complex variable)

0

( )

^st

( ) f t e dt

^

F s



  Integration from

_0-

f(t) : a time function such that f(t)=0 for t<0

[

^t

] A

Ae s







L   



0



⁰

( ) 1

( )

^{t s}

1

t

t t e



^



  

L L

[1( )] 1

[1( )]

st

t s

t t e





 

L

L [ sin ]

² ²

A t A s

 

  L 

[1( t t

0

)]

 s L

2 2

[ cos ] As

A t

 s

  L 

2

[ ] t 1

 s

L s

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Useful Theorems

Theorem 1. Linearity

[ af t ( )]  aF s ( ) L

Theorem 2. Superposition

[ ( ) f t  f t ( )]  F s ( )  F s ( ) L

Theorem 3. Translation in time.

[ (f )1( )] f( )1( ) ^std





L

1 2 1 2

[ ( ) f t  f t ( )]  F s ( )  F s ( ) L

0

( )

[ ( )1( )] ( )1( )

( )1( ) ( . )

st

s a

f t a t a f t a t a e dt

f

 

e ^ d



let t a







 

    

  



L

⁽^a^⁰⁾

0

( )1( ) ( ) ( ( )1( ) 0 0)

a

s sa sa

f

 

e ^e d



e F s f

 

for







  





 ^  

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Useful Theorems

Theorem 4. Complex differentiations [ ( )] d ( )

tf t F s

  ds L

[1] 1

 s

L

1

₂

[ 1] d ( )

t F s

ds s

   

L

 

2 3

2

1 2

[ ] [ ] [ ( )] ( )

) ( ) ( )

^st

d d

t t t tf t F s

ds s s ds

proof let F s f t e dt

 

 

          

 

 

L L L

0

0 0

) . ( ) ( )

( ) ( )

^st

( )

^st

[ ( )]

proof let F s f t e dt

d F s f t e dt tf t e dt tf t

ds s



 

 



  

       



  L

Th 5 T l ti i th d i

0 0

2

2 2

, [ ( )] ( 1)

2

( ) [ ( )] ( 1) ( )

n

n n

n

d d

similarly t f t F s t f t F s

ds ds

   

L L

Theorem 5. Translation in the s-domain

2 2

( )

[ cos ]

( )

at

s a

e t

s a

 

 

  [ e f t

^at

( )]  F s a (  ) L

L ( )

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Useful Theorems

Theorem

_6.

Real Differentiation

d

( ) d ( )

Df t f t

dt



 [ d ( )] ( ) (0 )

f t s F s f

dt    

L

0

) . [ ( )] ( )

( ) ( ) ( )

st

st st

d d

proof let f t f t e dt

dt dt

f t e f t e dt s





 

 



  



L

0 0

( ) ( ) ( )

(0 ) ( )

( ) (0 )

f t e f t e dt s

f s F s

s F s f





    

   



 

2

, [ ( )] [ ( )] [ ( )]

( ) (0 ) '(0 ) ( ) (0 ) '(0 )

similarly D f t D Df t Df t

s s F s f f s F s s f f

   

            

L L L

 ( ) (0 )  (0 ) ( ) (0 ) (0 )

[

n n

s s F s f f s F s s f f

d f

L dt

¹ ²

( 1)

( )] ( ) (0) '(0 )

(0 )

n n n

n

t s F s s f s f

f

 



      

 f (0 ) 

 



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Useful Theorems

Theorem 7. Real Integration

) 0 ( )

( )

(

¹ ¹

0^t

f t dt  D

^

f t  D

^

f



₀

0 0 0

( ) ( )

1 1

t t

st

t

f t dt

^

f   d e dt

^

 

 

 

 

   

L

¹

0

( ) (0)

( )

t

F s f

f t dt

s s

 



 

 

   L

0 0 0

1

1 1

( ) ( )

1 1 ( ) (0)

( ) ( )

st st

t

st

e f d e f t dt

s s

F s f

f d e f t dt

 

 

 



   

   

 

0 0 0

( ) ( )

t

f d e f t dt

s   s s s



     

Theorem 8. Complex Integration

( )

 f t 

^

( ) 

f t t

  

 

 

L ( )

s^

F s ds



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Useful Theorems

Theorem 9. Final value Theorem

0

( ) ( )

lim lim

t s

f t sF s

 



Theorem 10. Initial value Theorem

0

( ) ( )

lim lim

t s

f t sF s

 



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Inverse Laplace Transformation

1

: ( ) ( ) ( ) ( ) f t F s

F f

L  L

p

 

-1

: ( ) ( )

( ) ( ) 1 ( )

c j

st

F s f t

f t F s F s e ds







  

L

L  

_{( :}

c real constant

₎

( ) ( ) ( )

2

_{c j}

f t F s F s e ds

j

_



_



L

_{( :}

c real constant

₎

Inverse Laplace Transformation by Partial Fraction Method

*

1

1 0

1

1 1 0

( ) ( )

( )

m m

n n

n

a s a s a

F s P s

Q s s b s b s b



  

 

  



⁽ⁿ^^m⁾

1

1 1 0

2

1 2 1 2

real complex conjugate, a , b : real num.

( )( ) ( )

n n

s b s

n

b s b

s c s c s d s d







   

    



1 2 1 2

2

1 2 1 2

( ) s

F s s c s c s d s d

    

     

     

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Examples of Inverse Laplace Transformation

2 2

) ( ) 1

( 2) ( 3) ( 2) ( 2) ( 3)

a b c

ex F s

s s s s s

   

    

p p

2

d ds

( 2)( 3) ( 3) ( 2) 1

2, 1, 3, 1

2 1 ( 3) ( 2) 1

( 2)

a s s b s c s

let s than b let s than c

s s s

a s b c a c

      

     

    

         ₂

( 2)

3 3 3 ( 3)

1, 1

a s b c a c

s s s s

a b

        

   

    1 1 ₂ 1

, 1, ( )

( 2) ( 2) ( 3)

c F s

s s s

    

  

2 2 3

( )

^t ^t ^t

f t

^

t

^ ^

Inverse Laplace Transformation

=>

2 2 3

( ) f t   e

^t

 te

^t

 e

^t

partial fraction method

( )

2 2 2 2 2

10 1 4

10 4 10 4

) ( )

ex F s

 ₂   ₂ ₂  ₂ ₂

3

) ( )

6 25 ( 3) 4 4 ( 3) 4

( ) 10sin 4 4

t

s s s s

f t te^

     

 

4

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Solution of Differential Equation by Laplace Transformation

 

2

2 4 1 (0) 0 , (0) 2

L T: ( ) (0) (0) 2 ( ) (0) 4 ( ) 1

y y y y y

s Y s sy y sY s y Y s

       

     

2

L.T: ( ) (0) (0) 2 ( ) (0) 4 ( )

1 2 1

( 2 4) ( ) 2

s Y s sy y sY s y Y s

s s s Y s s

     

     

( s 2 s 4) ( ) Y s 2

s s

  

 

²

2 2

2 1 1 1 1 1

( )

( 2 4) 4 4

s s

Y s s s s s

  

  

 

²

 

( 2 4) 4 4 ( 1) 3

1 1 1

( ) cos 3 sin 3

4 4 4 3

t t

s s s s s

y t te

^

te

^

   

   

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Laplace Transform Table

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Laplace Transform Theorems

^{Nise Ch}^.2