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The Laplace Transform
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The s-Domain
The Laplace Transform of a function, f(t), is defined as;
0
[ ( )] ( ) ( )st
L f t F s f t e dt
The Inverse Laplace Transform is defined by
1 1[ ( )] ( ) ( )
2
j
ts
j
L F s f t F s e dsj
)()( sFtf
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Laplace Transform Pairs
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Cont
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Cont
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Laplace Transform Properties
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Examples
Find laplace transform !
21. 3 cos6 5 sin6
2. sin
cos33.
te t t
t at
d tdt
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2 2 2
2 2
6 3 301. 3 5
36 36 36
3 2 30 3 24
4 402 36
s s
s s s
s s
s ss
22 2
2 2
22.
d a as
ds s a s a
2 2
93. 19 9
sss s
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Background:
There are three cases to consider in doing the partial fraction
expansion of F(s).
Case 1: F(s) has all non repeated simple roots.
1 2( ) . . .
1 2
kk knF s
s p s p s p
n
Case 2: F(s) has complex poles:
*( )1 1 1( ) . . .
( )( )( ) )1
P s k k F s
Q s s j s j s j s j
Case 3: F(s) has repeated poles.
( ) ( )1 11 12 1 1( ) . . . . . .
2 ( )( )( ) ( ) ( )1 11 1 1 1
P s k k k P srF s
r rs p Q sQ s s p s p s p
Inverse Laplace Transforms
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Case 1: Illustration:
Given:
)10()4()1()10)(4)(1(
)2(4)( 321
s
A
s
A
s
A
sss
ssF
274)10)(4)(1(
)2(4)1(|
11 ssss
ssA 94
)10)(4)(1(
)2(4)4(| 42 ssss
ssA
2716)10)(4)(1(
)2(4)10(|
103 ssss
ssA
)()2716()94()274()(104
tueeetf
ttt
Find A1, A2, A3 from Heavyside
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Case 2: Complex Roots:
...)))()((
)()(
*11
1
1
js
K
js
K
jsjssQ
sPsF
F(s) is of the form;
K1 is given by,
jeKKK
jsjssQ
sPjsK
js
||||
))(()(
)()(
111
1
1
1|
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js
eK
js
eK
js
K
js
K jj11
*11
|||
tje
te
je
tje
te
jeK
js
eK
js
eKL
jj
1||
||||111
2
)()(|
1|2
1||
tjetjeateKtj
etej
etj
etejeK
Case 2: Complex Roots:
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)cos(||2|||
1
111 teKjs
eK
js
eKL t
jj
Case 2: Complex Roots:
Therefore:
You should put this in your memory:
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o
jj
j
jss
sK
ss
sA
jsK
jsK
sAsF
jsjss
s
sss
ssF
js
s
10832.0)2)(2(
12
)2(
)1(
5
1
)54(
)1(
22)(
)2)(2(
)1(
)54(
)1()(
|
|
2|1
0|
11
2
2
*
Complex Roots: An Example.
For the given F(s) find f(t)
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Complex Roots: An Example. (continued)
We then have;
jsjsssF
oo
2
10832.0
2
10832.02.0
)(
Recalling the form of the inverse for complex roots;
)(108cos(64.02.0)(2 tutetf ot
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Circuit theory problem:
You are given the circuit shown below.
+_
t = 0 6 k
3 k100 F
+
_v(t)12 V
Use Laplace transforms to find v(t) for t > 0.
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Circuit theory problem:
We see from the circuit,
+_
t = 0 6 k
3 k100 F
+
_v(t)12 V
voltsxv 49
312)0(
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Circuit theory problem:
+
_vc(t) i(t)
3 k
100 F
6 k
05)(
0
)(
0)()(
tv
dt
tdv
RC
tv
dt
tdv
tvdt
tdvRC
c
c
cc
cc
Take the Laplace transform
of this equations including
the initial conditions on vc(t)
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Circuit theory problem:
)(4)(
5
4
)(
0)(54)(
0)(5)(
5tuetv
ssV
sVssV
tvdt
tdv
t
c
c
cc
c
c
Tentukan transformasi laplacenya !
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Tentukan transformasi laplacenya !
Tentukan invers transformasi laplacenya !
Tentukan besarnya i(t) !
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