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Page 1: Laplace transformation
Page 2: Laplace transformation

There are techniques for finding the system response of a system described by a differential equation, based on the replacement of functions of a real variable (usually time or distance) by certain frequency-dependent representations, or by functions of a complex variable dependent upon frequency. The equations are converted from the time or space domain to the frequency domain through the use of mathematical transforms.

Page 3: Laplace transformation

Differentialequations

Inputexcitation e(t)

Outputresponse r(t)

Time Domain Frequency Domain

Algebraicequations

Inputexcitation E(s)

Outputresponse R(s)

Laplace Transform

Inverse Laplace Transform

Page 4: Laplace transformation

Let f(t) be a real function of a real variable t (time) defined for t>0. Then

is called the Laplace transform of f(t). The Laplace transform is a function of a complex variable s. Often s is separated into its real and imaginary parts: s=+j , where and are real variables.

Page 5: Laplace transformation

After a solution of the transformed problem has been obtained in terms of s, it is necessary to "invert" this transform to obtain the solution in terms of the time variable, t. This transformation from the s-domain into the t-domain is called the inverse Laplace transform.

Page 6: Laplace transformation

Let F(s) be the Laplace transform of a function f(t), t>0. The contour integral

where c>0 (0 as above) is called the inverse Laplace transform of F(s).

Page 7: Laplace transformation

It is seldom necessary to perform the integration in the Laplace transform or the contour integration in the inverse Laplace transform. Most often, Laplace transforms and inverse Laplace transforms are found using tables of Laplace transform pairs.

Page 8: Laplace transformation

Time Domainf(t), t>0

Frequency DomainF(s)

1. 12. K K/s3. Kt K/s2

4. Ke-at K/(s+a)5. Kte-at K/(s+a)2

6. Ksint K/(s2+2)

7. Kcost Ks/(s2+2)8. Ke-atsint K/((s+a)2+2))

9. Ke-atcost K(s+a)/((s+a)2+2))

Page 9: Laplace transformation

Time Domain f(t), t>0

Frequency Domain F(s)

10. t s 11. f(t) F(s) 12. L-1{F(s)}=f(t) L{f(t)}=F(s) 13. Af1(t) + Bf2(t) AF1(s)+BF2(s)

14. t

df0

)(

F s

s

( )

15. df t

dt

( )

sF s f( ) ( ) 0

Page 10: Laplace transformation

The inverse Laplace transform is usually more difficult than a simple table conversion.

X ss s

s s s( )

( )( )

( )( )

8 3 8

2 4

Page 11: Laplace transformation

If we can break the right-hand side of the equation into a sum of terms and each term is in a table of Laplace transforms, we can get the inverse transform of the equation (partial fraction expansion).

X ss s

s s s

K

s

K

s

K

s( )

( )( )

( )( )

8 3 8

2 4 2 41 2 3

Page 12: Laplace transformation

In general, there will be a term on the right-hand side for each root of the polynomial in the denominator of the left-hand side. Multiple roots for factors such as (s+2)n will have a term for each power of the factor from 1 to n.

Y ss

s

K

s

K

s( )

( )

( ) ( )

8 1

2 2 221 2

2

Page 13: Laplace transformation

Complex roots are common, and they always occur in conjugate pairs. The two constants in the numerator of the complex conjugate terms are also complex conjugates.

Z ss s

K

s j

K

s j( )

.

( ) ( )

*

52

2 5 1 2 1 22

where K* is the complex conjugate of K.

Page 14: Laplace transformation

The solution of each distinct (non-multiple) root, real or complex uses a two step process.

The first step in evaluating the constant is to multiply both sides of the equation by the factor in the denominator of the constant you wish to find.The second step is to replace s on both sides of the equation by the root of the factor by which you multiplied in step 1

Page 15: Laplace transformation

X ss s

s s s

K

s

K

s

K

s( )

( )( )

( )( )

8 3 8

2 4 2 41 2 3

Ks s

s s s

1

0

8 3 8

2 4

8 0 3 0 8

0 2 0 424

( )( )

( )( )

( )( )

( )( )

Ks s

s s s

2

2

8 3 8

4

8 2 3 2 8

2 2 412

( )( )

( )

( )( )

( )

Page 16: Laplace transformation

Ks s

s s s

3

4

8 3 8

2

8 4 3 4 8

4 4 44

( )( )

( )

( )( )

( )

The partial fraction expansion is:

X ss s s

( )

24 12

2

4

4

Page 17: Laplace transformation

The inverse Laplace transform is found from the functional table pairs to be:

x t e et t( ) 24 12 42 4

Page 18: Laplace transformation

Any unrepeated roots are found as before.The constants of the repeated roots (s-a)m are found by first breaking the quotient into a partial fraction expansion with descending powers from m to 0:

B

s a

B

s a

B

s am

m( ) ( ) ( )

2

21

Page 19: Laplace transformation

The constants are found using one of the following:

BP a

Q s s am m

s a

( )

( ) / ( )

1)/()(

)(

)!(

1

1 as

mim

im

i assQ

sP

ds

d

imB

Page 20: Laplace transformation

Y ss

s

K

s

K

s( )

( )

( ) ( )

8 1

2 2 221 2

2

Ks s

ss

ss2

2

22

2

8 1 2

28 1 8

( )( )

( )( )

Page 21: Laplace transformation

The partial fraction expansion yields:

Y ss s

( )( )

8

2

8

2 2

8)2/()2(

)1(8

)!12(

1

222

s

i ss

s

ds

dB

Page 22: Laplace transformation

The inverse Laplace transform derived from the functionaltable pairs yields:

y t e tet t( ) 8 82 2

Page 23: Laplace transformation

Y ss

s

K

s

K

s( )

( )

( ) ( )

8 1

2 2 221 2

2

21 )2()1(8 KsKs

211 288 KKsKs

Equating like terms:

211 288 KKandK

Page 24: Laplace transformation

211 288 KKandK

2828 K

28168 K

Thus

22

8

2

8)(

sssY

tt teety 22 88)(

Page 25: Laplace transformation

221

2 22)2(

)1(8)(

s

K

s

K

s

ssY

Ks s

ss

ss2

2

22

2

8 1 2

28 1 8

( )( )

( )( )

As before, we can solve for K2 in the usual manner.

Page 26: Laplace transformation

2212

22

2

8)2(

2)2(

)2(

)1(8)2(

s

ss

Ks

s

ss

ds

Ksd

ds

sd 82)1(8 1

18 K

22 2

8

2

8

)2(

)1(8)(

sss

ssY

tt teety 22 88)(

Page 27: Laplace transformation

Unrepeated complex roots are solved similar to the process for unrepeated real roots. That is you multiply by one of the denominator terms in the partial fraction and solve for the appropriate constant.Once you have found one of the constants, the other constant is simply the complex conjugate.

Page 28: Laplace transformation

Z ss s

K

s j

K

s j( )

.

( ) ( )

*

52

2 5 1 2 1 22

Ks j

s j s jj

s j

5 2 1 2

1 2 1 213

1 2

. ( )

( )( ).

K j* . 13

Page 29: Laplace transformation

)21(

3.1

)21(

3.1

52

2.5)(

2 js

j

js

j

sssZ

)21()21()(

3.13.1

js

e

js

esZ

jj


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