EEM 3117 Laplace Transformation Dr. Sezai Taskin Department of - - PowerPoint PPT Presentation

Automatic Control EEM 3117

Laplace Transformation

• Dr. Sezai Taskin

Department of Electrical&Electronics Engineering Faculty of Engineering, Manisa Celal Bayar University

Laplace Transformation

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by: L F(s)

Slides 2 2

[ ( )] ( ) ( )

st

f t F s f t e dt

 

   L

Why s-domain?

• We can transform an ordinary differential

equation into an algebraic equation which is easy to solve.

• It is also easy to analyze and design

interconnected (series, feedback etc.) systems.

Slides 2 3

Unit step function

Slides 2 4

1 ( ) ( ) t f t u t t       

[1] 1 1

st st

e dt e s s s s

   

             



L

• 1

[1] s  L

Exponential function

Slides 2 5

( ) ( )

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

at at at st s a t s a t

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

       

                   

 

L

1 [ ]

at

e s a   L

Frequency shift

Slides 2 6

1 [1] s  L

1 [ 1]

at

e s a    L

[ ( )] ( )

at

e f t F s a   L

Sine and cosine functions

Slides 2 7

2 2

[cos ] s t s     L

2 2

[sin ] t s      L

Impulse function

Slides 2 8

( ) f t  

[ ( )] ( ) 1

st st t

t t e dt e  

    

  



L

Unit ramp

Slides 2 9

( ) ( ) t t f t u t t       

udv uv vdu  

 

(integration by parts)

2

[ ] 1

st st st st

t e t dt e t e e t d dt s s s s

       

             

  

L

1

! [ ]

n n

n t s   L

similarly

Differentiated function

Slides 2 10

[ ( )] ( ) f t F s  L

( ) ( ) (0) df t sF s f dt         L

1 2 ( 1)

( ) ( ) (0) (0)..... (0)

n n n n n n

d f t s F s s f dt s f f

  

           L

Integrated function

Slides 2 11

( ) ( )

t

F s f t dt s       



L

2nd shifting theorem

 

( ) ( )

as

f t a e F s



  L

Solution of differential equations using Laplace Transformation

Slides 2 12

Differential Equation Transformed Equation Transformed Solution Solution

Laplace Transformation Algebraic Manipulation Inverse Laplace Particular Integral Complementary function

Example

Slides 2 13

2 2 2

2 5 5 ( ) (0) 2 ( ) 5 ( ) ( 2) 5 ( ) 5 2 2.5(1 )

t t t t t

dx x dt sx s x x s s x s s s x t e dt e e

  

               



Transformed Solution Transformed Equation

Convolution integral

Slides 2 14

1 2 1 2

( ) ( ). ( ) ( ) ( ) ( ) F s F s F s f t f t f t     Ex:

2 2 1 2 2( ) 2 2 2

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

t t t t t t t t t t t

F s s s s s f t e e f f t d e e d e e d e e e e

  

    

        

                 

  