TRANSFORMASI TRANSFORMASI LAPLACE INVERS IKRIMA ALFI, S.T., M.Eng.
DEFINISI TRANSFORMASI TRANSFORMASI LAPLACE L APLACE INVERS
Invers Transformasi Laplace adalah suatu metode sederhana untuk mencari fungsi t (f(t)) bila diketahui persamaan fungsi s (F(s)).
Berlaku hubungan timbal balik antara f(t) dan F(s). L −1 Notasi transformasi balik adalah , − 1 L [ F ( s)] = f (t ) sehingga
ontoh ! − 1 a.
c.
L
L −1
1 s + 2
1 s
3
= e −2t
b.
1.2! 1 2 = L −1 = t 3 2! s
2
d.
L −1 L −1
s s
2
+ 16
= cos 4t
4 ( s + 3) + 16 2
= e −3t sin 4t
DEFINISI TRANSFORMASI TRANSFORMASI LAPLACE L APLACE INVERS
Invers Transformasi Laplace adalah suatu metode sederhana untuk mencari fungsi t (f(t)) bila diketahui persamaan fungsi s (F(s)).
Berlaku hubungan timbal balik antara f(t) dan F(s). L −1 Notasi transformasi balik adalah , − 1 L [ F ( s)] = f (t ) sehingga
ontoh ! − 1 a.
c.
L
L −1
1 s + 2
1 s
3
= e −2t
b.
1.2! 1 2 = L −1 = t 3 2! s
2
d.
L −1 L −1
s s
2
+ 16
= cos 4t
4 ( s + 3) + 16 2
= e −3t sin 4t
TABEL T ABEL TRANSFORMASI TRANSFORMASI LAPLACE LAPL ACE INVERS No
1
F(s)
L-1{F(s)}=f(t)
1
1 s
2
t
1 s
3
2 n
1
t
s n +1
n!
4
e
1
at
s − a
5
a s
sin at
+ a2
2
s
6 s
2
7
+ a2 a
s
2
8
− a2
cos at sinh at
s 2
2
cosh at
!t"#!n D$ng!n %$ngg&n!'!n t!($, t$nt&'!n t)!n*+)%!*" !-!$ "n/$)* 0!)" +&ng*"+&ng*" ($)"'&t 6.
1.
2.
7.
3.
8.
4.
.
5.
1.
SIFAT TRANSFORMASI LAPLACE INVERS
SIFAT LINEAR
Cnt#
SIFAT TRANSLASI PERESERAN FREKENSI
"'!
Cnt# "'! %!'!
%!'!
TRANSLASIPERESERAN 9AKT
"'!
Cnt# "'! %!'!
%!'!
SIFAT PENBA:AN SKALA
"'!
Cnt# "'! %!'!
%!'!
TRANSFORMASI LAPLACE INVERS DARI TRNANTRNAN
"'!
Cnt# "'! %!'! !t!&
%!'!
TRANSFORMASI LAPLACE INVERS DARI INTERALINTERAL
"'!
Cnt# K!)$n! %!'!
%!'!
PERKALIAN DENAN *n "'!
0!n F;<=, %!'!
!0" -$)'!"!n 0$ng!n * %$%-&n>!" !'"(!t *$(!g!" menurunkan F(t). "'! F;
0!-!t @&g! 0"!'&'!n
PEMBAIAN DENAN S
"'!
%!'!
!0" -$%(!g"!n 0$ng!n * ;!t!& -$)'!"!n 0$ng!n 1*< %$%-&n>!" !'"(!t mengintegrasikan F(t) dari 0 sampai t Cnt# K!)$n! '"t! -$)$#
P$)&!*!n '$ 0!!% ($nt&' 0!-!t @&g! 0"!'&'!n
SIFAT KONVOLSI
"'!
0!n
%!'!
+g 0"*$(&t 'n/&*" 0!)" + 0!n g, 0!n t$)$%!n>! 0"*$(&t teorema atau sifat konvolusi Cnt# '!)$n! 0!n '"t! -$)$#
PECA:AN PARSIAL BENTK 1 T$nt&'!n
I.
II.
K!"'!n!# '$0&! )&!* 0$ng!n -$n>$(&t >!"t& ;*2< ;*1<;*3<, *$#"ngg! %$n@!0"
P""#!# #!)g! * >!ng !'!n %$%(&!t *!!# *!t& '&)&ngn>! ($)#!)g! n . A%(" ;*2< = , !)t"n>! *&(*t"t&*"'!n * =2 4=3A %!'! A=43
Cnt# A%(" ;*1< = , !)t"n>! *&(*t"t&*"'!n * =1 2 =12B, %!'! B=212 =16 A%(" ;*3<=, !)t"n>! *&(*t"t&*"'!n *=3 14=4C, %!'! C=72 S$t$!# *$$*!", '$%(!"'!n '$ -$)%"*!!n. S$#"ngg!
L!t"#!n T$nt&'!n t)!n*+)%!*" L!-!$ "n/$)* ($)"'&t −1
L
1. −1
L
7 s + 2 s 3 + 3 s 2 + 2 s
s 2 + 7 s + 8 ( 1) ( 2) ( 3) s + s + s +
2. −1
L
3.
( s + 1) ( s + 3) s ( s + 2) ( s + 4)
KASS 2 2 +1 s −1 L 3 ( s + 2)
B$nt&' -$!#!n -!)*"!n>! !0!!# A B C +1 = + + 3 2 ( s + 2) ( s + 2) ( s + 2) ( s + 2) 3 s
".
"".
2
B$)*"#'!n -$n>$(&t 0$ng!n %$ng!"'!n '$0&! )&!*n>! 0$ng!n ;*2< 3 %!'! 0"-$)$# *2 1 = A ;* 2< 2 B ;*2< C A%(" ;*2< = , *$#"ngg! *&(*t"t&*"'!n * = 2 ;2<2 1 = A ;< B ;< C C=5
KASS 2 .... """.
M$n>!%!'!n '$*"$n ;'!)$n! t"0!' !0! '&)&ng !"n 0!!% -$)*!%!!n "n"< *2 1 = A ;* 2<2 B ;*2< C *2 1 = A ;*2 4* 4< B* 2B C *2 1 =A *2 4A * 4A B * 2B C *2 1 = A *2 ;4A B<* 4A 2B C . M&!" %$n>!%!'!n '$*"$n 0!)" -!ng'!t t$)t"ngg", >!"t& * 2 1=A . K$%&0"!n '"t! t"n@!& $'*t)"% >!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 = 24 s + 1 1 5 −4 M!'! = + + ( s + 2 ) 3 ( s + 2 ) ( s + 2 ) 2 ( s + 2 ) 3
K!*&* 2
*$#"ngg! −1
L
s 2 + 1 1 5 −4 −1 = L + + 3 2 3 ( ) 2 s + ( ) ( ) ( ) + + + 2 2 2 s s s = e − 2t − 4te − 2t +
5 2
2 − 2 t
t e
L!t"#!n ;'!*&* 2< 1.
2.
3.
4.
−1
L
s 2 ( ) s − 2
2 s 2 ( ) ( ) s + 1 s − 1
−1
L
−1
L
−1
L
s + 4 2 2 s 1 s 3 + + ( ) ( )
s 2 2 s s 1 s 2 + + ( ) ( )
KASS III 2 P$n>$(&t ($)($nt&'as + bs + c
D"%!n! as
2
c + bs + t"0!' 0!-!t 0"+!'t)'!n,
%!'! ($nt&' -$!#!n -!)*"!n>!
As + B
as
Cnt# −1
L
( s − 1) s ( s 2 + 1)( s 2 + 2)
2
+ bs + c
KASS III −1
L
( s − 1) s ( s 2 + 1)( s 2 + 2)
B$nt&' -$!#!n -!)*"!n>! !0!!#
( s − 1)
(
s s I.
II.
2
+ 1)( s + 2) 2
=
A s
+
Bs + C Ds + E
( s
2
+ 1)
+
( s
2
+ 2)
(
)(
s s + 1 s + 2 %$ng!"'!n 2
2
:"!ng'!n -$n>$(&tn>! 0$ng!n ( s − 1) = A( s 2 + 1)( s 2 + 2) + ( Bs + C ) ( s ) ( s 2 + 2) + ( Ds + E ) ( s ) ( s 2 + 1) nt&' *= − 1 = 2 A A =
−
1 2
)
KASS III M$n>!%!'!n '$*"$n
III.
( s − 1) = As 4 + 3 As 2 + 2 A + Bs 4 + 2 Bs 2 + Cs 3 + 2Cs + Ds 4 + Ds 2 + Es 3 + Es ( s − 1) = ( A + B + D ) s 4 + ( C + E ) s 3 + ( 3 A + 2 B + D ) s 2 + ( 2C + E ) s + 2 A .
ABD= 1<
.
CE=
.
3A2BD= ...3< ..7<
.
2CE=1....4<
.
2A=1 .....5<
BD=12 ..6<
.2< 2BD=32
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
KASS III
!0"
( s − 1)
(
s s
2
+ 1)( s + 2) 2
= =
−1 s
2+
−1 s
s + 1
( s
2+
2
+ 1)
+
s
( s
2
+ 1)
−1
+
( s
s − 1
2
2
+ 2)
1
( s
2
+ 1)
−
1
( s
2
s
2
+ 2)
−
1
( s
2
M!'! 1 s − 1 s 1 1 2+ 2 − L−1 + − 2 2 2 2 s + 1) ( s + 1) ( s + 2 ) ( s + 2) ( s 1
1
2
2
= − + cos t + sin t − cos 2t −
1 2
sin 2t
+ 2)
LATI:AN T$nt&'!n t)!n*+)%!*" L!-!$ "n/$)* +&ng*" ($)"'&t 1. 2.
3 s + 1
( s − 1) ( s 2 + 1)
3.
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&'
A
F!'t) ;!G (<2 !'!n %$%($)"'!n -$!#!n -!)*"!
F!'t) ;!G (<3 !'!n %$%($)"'!n -$!#!n -!)*"! A
ax + b
ax + b
+
+
B
( ax + b ) 2 B
+
C
( ax + b ) 2 ( ax + b ) 3
F!'t) '&!0)!t ;!G 2(G< !'!n %$%($)"'!n -$!# -!)*"!
P$n>$$*!"!n P$)*!%!!n D"H$)$n*"!
S$$*!"'!n!# >;t< >;t<=1, >;<=1, 0!n >J;<= P$n>$$*!"!n 1. T)!n*+)%!*"'!n L!-!$ '$0&! )&!* -$)*!%!!n 0"H$)$n*"! 0" !t!* %"*! =;*<=L>;t<, %!'! L >;t< L>;t<=L1 !t!&
2.
K!)$n! >;<=1, >J;<= %!'!
$n>$$*!"!n P$)*!%!!n D"H$)$n*"! 3.
T)!n*+)%!*"'!n L!-!$ "n/$)* -$)*!%!!n t*(
Cnt# 2 S$$*!"'!n 1.
2.
A%(" t)!n*+)%!*" !-!$ 0!)" -$)*!%!!n 0"+$)$n*"! 0" !t!* &n!'!n!# >;<=2 0!n >J;<=1, *$$*!"'!n -$)*!%!!n !@!(!) "n" &nt&' %$n$nt&'!n .
D$ng!n %$ngg&n!'!n -$!#!n -!)*"!, 0"0!-!t
Cnt# 2 3.
D$ng!n %$ngg&n!'!n t)!n*+)%!*" L!-!$ "n/$)*, '"t! %$%-$)$# -$n>$$*!"!n >!ng 0""ng"n'!n
