Post on 07-Jul-2018
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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.