6 Transformasi Laplace Invers

8/19/2019 6 Transformasi Laplace Invers

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TRANSFORMASILAPLACE INVERS

IKRIMA ALFI, S.T., M.Eng.


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DEFINISI TRANSFORMASI LAPLACE INVERS

Invers Transformasi Laplace adalah suatumetode sederhana untuk mencari fungsi t (f(t))bila diketahui persamaan fungsi s (F(s)).

Berlaku hubungan timbal balik antara f(t) danF(s).

Notasi transformasi balik adalah ,sehingga

ontoh !

1−L

[ ] )()(1 t f s F =−L

t e

s

d. t

s s

c.

t s

s b. e

sa.

t

t

4sin

16)3(

41

2

1

!2

!2.1111

4cos16

1

2

11

3

2

2

33

2

2

−

−

=

++

−=−=−

=+

−=+

−

LLL

LL


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TABEL TRANSFORMASI LAPLACE INVERSNo F(s) L-1{F(s)}=f(t)

1 1

2 t

3

4

5

6

7

8

s

1

2

1

s

11+n s

a s −

1

22a s

a

+

22a s

s

+

22a s

a

−

22a s

s

−

!nt

n

at e

at sin

at cos

at sinh

at cosh


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!t"#!n

D$ng!n %$ngg&n!'!n t!($, t$nt&'!n t)!n*+)%!*" !-!$ "n/$)*0!)" +&ng*"+&ng*" ($)"'&t

1. 6.

2. 7.

3. 8.

4. .

5. 1.


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SIFAT TRANSFORMASI LAPLACE INVERS

SIFAT LINEAR

Cnt#


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SIFAT TRANSLASI PERESERANFREKENSI

"'! %!'!

Cnt#

"'!

%!'!


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TRANSLASIPERESERAN9AKT "'! %!'!

Cnt#

"'!

%!'!


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SIFAT PENBA:AN SKALA

"'! %!'!

Cnt#

"'!

%!'!


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TRANSFORMASI LAPLACE INVERS DARI TRNANTRNAN

"'! %!'!

Cnt#

"'!

%!'!

!t!&


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TRANSFORMASI LAPLACE INVERS DARIINTERALINTERAL

"'! %!'!

Cnt#

K!)$n!

%!'!


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PERKALIAN DENAN *n

"'! 0!n F;


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PEMBAIAN DENAN S

"'! %!'!

!0" -$%(!g"!n 0$ng!n * ;!t!& -$)'!"!n 0$ng!n1*< %$%-&n>!" !'"(!t mengintegrasikan F(t) dari 0sampai t

Cnt#

K!)$n! '"t! -$)$#

P$)&!*!n '$ 0!!% ($nt&'0!-!t @&g! 0"!'&'!n


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SIFAT KONVOLSI

"'! 0!n %!'!

+g 0"*$(&t 'n/&*" 0!)" + 0!n g, 0!nt$)$%!n>! 0"*$(&t teorema atau sifatkonvolusi

Cnt#'!)$n! 0!n

'"t! -$)$#


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PECA:AN PARSIAL BENTK1 T$nt&'!n

I. K!"'!n!# '$0&! )&!* 0$ng!n -$n>$(&t >!"t& ;*2<;*1! ($)#!)g! n

.A%(" ;*2< = , !)t"n>! *&(*t"t&*"'!n * =2

4=3A %!'!

A=43


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Cnt#

A%(" ;*1< = , !)t"n>! *&(*t"t&*"'!n *=1

2 =12B, %!'!

B=212 =16

A%(" ;*3


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L!t"#!n

T$nt&'!n t)!n*+)%!*" L!-!$ "n/$)*($)"'&t

1.

2.

3.

++

+− s s s

s L

23

2723

1

( ) ( ) ( )

+++

++−321

872

1

s s s

s s L

( ) ( )

( ) ( )

++

++−42

311

s s s

s s L


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KASS 2

B$nt&' -$!#!n -!)*"!n>! !0!!#

". B$)*"#'!n -$n>$(&t 0$ng!n %$ng!"'!n '$0&!)&!*n>! 0$ng!n ;*2


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KASS 2 ....

""". M$n>!%!'!n '$*"$n ;'!)$n! t"0!' !0! '&)&ng !"n 0!!%-$)*!%!!n "n"<

*2 1 = A ;* 2!ng !"n 0!n '"t! *!%!'!n '$*"$n-!ng'!t t$)$n0!#, >!"t& *&'& 'n*t!nt! -!0! '$0&! )&!*n>!

1 = 4A 2B C 1 = 4 ;1< 2B 5

8 = 2B

B = 4

M!'!

( ) ( ) ( ) ( )323

2

2

5

2

4

2

1

2

1

+

+

+

−+

+

=

+

+

s s s

s

s


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K!*&* 2

*$#"ngg!

( ) ( ) ( ) ( )

t t t et tee

s s s L

s

s L

2222

32

1

3

21

254

2

5

2

4

2

1

2

1

−−−

−−

+−=

++

+

−+

+=

+

+


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L!t"#!n ;'!*&* 2<

1.

2.

3.

4.

( )

−

−

2

1

2 s

s L

( ) ( )

++

−

2122

1

s s s

s L

( ) ( )

++

+−22

1

31

4

s s

s L

( ) ( )

−+

−

2

2

1

11 s s

s L


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KASS III

cbsas

B As

+++2

D"%!n! cbsas ++2

t"0!' 0!-!t 0"+!'t)'!n,

P$n>$(&t ($)($nt&' cbsas ++2

%!'! ($nt&' -$!#!n -!)*"!n>!

Cnt#

( )

( )( )

++

−−21

122

1

s s s

s

L


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KASS III

B$nt&' -$!#!n -!)*"!n>! !0!!#

I. :"!ng'!n -$n>$(&tn>! 0$ng!n %$ng!"'!n

II. nt&' *=

( )( )( )

++

−−21

122

1

s s s

s L

( )( )( ) ( ) ( )2121

12222 +

++

+

++=

++

−

s

E Ds

s

C Bs

s

A

s s s

s

( )( )21 22 ++ s s s( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )12211 2222 ++++++++=− s s E Ds s sC Bs s s A s

2

1

21

−=

=−

A

A


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KASS III

III. M$n>!%!'!n '$*"$n

. ABD= 1< BD=12 ..6<

. CE= .2<

. 3A2BD= ...3< 2BD=32..7<

. 2CE=1....4<

. 2A=1 .....5<

( )

( ) ( ) ( ) ( ) ( ) A s E C s D B A s E C s s

Es Es Ds DsCsCs Bs Bs A As As s

2223DBA1

22231

234

32432424

++++++++++=−

++++++++++=−


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KASS III

P$)*!%!!n 5<

A=12

P$)*!%!!n 2< 4<

2CE=1 CE=

C =1 E= 1

P$)*!%!!n 6< 7<

2BD=32

BD=12

B =1 D= 12


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KASS III

!0"

M!'!

( )( )( ) ( ) ( )

( ) ( ) ( ) ( )21

2

21

1

1

1

21

2

12

1

1

121

21

1

2222

2222

+−

+−

++

++

−=

+

−−+

+

++

−=

++

−

s s

s

s s

s

s

s

s

s

s

s s s s

s

( ) ( ) ( ) ( )

t t t t

s s

s

s s

s

s L

2sin2

12cos

2

1sincos

2

1

2

1

2

21

1

1

1

21

2222

1

−−++−=

+−+−++++

−−


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LATI:AN

T$nt&'!n t)!n*+)%!*" L!-!$ "n/$)*+&ng*" ($)"'&t

1. 2.

3. ( ) ( )11

132 +−

+

s s

s


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K!"0!# P$!#!n P!)*"!

K!"0!# P$!#!n P!)*"! !0!!# *$(!g!" ($)"'&t P$%("!ng 0!)" +&ng*" >!ng 0"($)"'!n #!)&* $("# )$n0!# 0$)!@!tn>!

0!)"-!0! 0$)!@!t -$n>$(&tn>!.

"'! t"0!' 0$%"'"!n, %!'! '"t! #!)&* %$%(!g"n>! 0!#&& 0$ng!n-$%(!g"!n ("!*!.

F!'t)'!n!# -$n>$(&tn>! %$n@!0" +!'t)+!'t) -)"%!n>! F!'t) "n$!) ;!G(< %$%($)"'!n -$!#!n -!)*"! >!ng ($)($nt&'

F!'t) ;!G (


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P$n>$$*!"!n P$)*!%!!n D"H$)$n*"!

S$$*!"'!n!# >;t< >;t;J;$$*!"!n

1. T)!n*+)%!*"'!n L!-!$ '$0&! )&!* -$)*!%!!n 0"H$)$n*"! 0"!t!*

%"*! =;*;t


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3. T)!n*+)%!*"'!n L!-!$ "n/$)*-$)*!%!!n t*(

$n>$$*!"!n P$)*!%!!n D"H$)$n*"!


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Cnt# 2

S$$*!"'!n

1. A%(" t)!n*+)%!*" !-!$ 0!)" -$)*!%!!n0"+$)$n*"! 0" !t!*

2. &n!'!n!# >;J;


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3. D$ng!n %$ngg&n!'!n t)!n*+)%!*"L!-!$ "n/$)*, '"t! %$%-$)$#-$n>$$*!"!n >!ng 0""ng"n'!n

Cnt# 2


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LATI:AN

S$$*!"'!n t"!- -$)*!%!!n 0"+$)$n*"!($)"'&t

1.

2.

3.

6 Transformasi Laplace Invers

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