Transcript 3 laplace - WordPress.com

• Ditemukan oleh Piere Simon Maequis de
Laplace tahun (1747-1827) seorang ahli
astronomi dan matematika Prancis
• Menurut; fungsi waktu atau f(t) dapat
ditranspormasi menjadi fungsi komplek
atau F(s)
– Dimana s bilangan komplek dari s = s + j2pf
atau s + j
• s = frekuensi neper = neper/detik
•  = frekuensi radian = radian/detik
• Hasil TL dari f(t) di beri nama F(s)
• Tanda TL diberikan dengan
fungsinya di tulis
£
atau L, dan
F ( s )  L [ f ( t )]
f(t): nilai komplek dari fungsi sebuah fariabel t
F(s): Nilai komplek dari fungsi sebuah fariabel s
Inverse Transformasi Laplace
• Inverse (Bilateral) Transform

L [ f ( t )]  F ( s ) 

f ( t ). e
s t
dt
0
f t   F  s 
L
• Notation
F(s) = L{f(t)}
f(t) = L-1{F(s)}
variable t tersirat untuk L
variable s tersirat untuk L-1
Contoh: Transpormasi Laplace
1. f(t) = A
– Jawab

L [ f ( t )]  L ( A ) 
 A. e
s t
dt
0


A
s
A
[e
[
 st
1
s e

]

0

1
e
0
] 
A
s
( 0  1) 
A
s
Contoh 2. f(t) = At
Jawab

L [ At ] 
 At .e
 st
dt
0

 A  t .e
 st
dt
0
Dibantu dengan formula integral partsiel yaitu
 UV '  UV   U 'V

L [ At ]  A  t .e
 st
dt 
0

A
s
 [ t .e

A
s
 t .de
 st
0

 st

0
]   1 .e
 st
dt

0
A 
0
1
 st 

[ e ]0
(0  ) 
s
1
s
A 
1
A

 0  ( 0  1)  2
s
s
s

• Contoh 3 f(t) = e-at
jawab

L[ e
 at
]
e
 at
.e
 st
dt
0


e
(a  s )t
dt
0



1
 (a  s)
1
 (a  s)
1
as
e
( a  s )t
( 0  1)
0

• Contoh 4 : f(t) = t.e-at

L [ t .e
 at

]
t .e
 at
.e
 st
dt
0

  t .e
(a s)
dt
0


1
 (a  s)
1
 (a  s)
[t .e

(a  s )t 
0
] e
(a  s )t
dt

0
[t .e
(a  s )t 
0
] 
1
 (a  s)
[e
( a  s )t 
0
]

t
Lim (
t 
L [ t .e
e
 at
( a  s )t
]

)0
1
 (a  s)
1
(a  s)
2
( 0  0 ) 
1
(a  s)
( 0  1) 
5.
6.
7.
8.
f(t) = Sin(t)
f(t) = Cos (t)
f(t) = Sin(t+)
f(t) = e-at. Sin(t)
f (t ) 
• Contoh 9;
L[
df

]
dt
df
(
dt
df
)e
 st
dt
dt
0


e
 st
df
0

 [e
 st

f ] 0  s  f .e
 st
dt
0
 { 0  f ( 0  )  s . F ( s )}
 s.F ( s )  f ( 0  )
• f(0+) artinya harga nol untuk fungsi, jika didekati dari
arah positif
• Contoh 10;

f (t ) 
f ( t ) dt

L [  f ( t ) dt   e
 st
{  f ( t )dt } dt
0



 [ e  f (t ) dt ]
s
1
1
s
1
s
 st
[e
 st
[0 

  e

0
f ( t ) dt
0

0
{  f ( t ) dt }] 

 st
f ( t ) dt ] 0  
1
s
1
F (s)
s
F (s)


1

s
F (s)
s


f ( t )] 0  


1
F (s)
s
f ( t ) dt ] ( 0  )
s
f ( t ) dt ] ( 0  ) artinya h arg a
f ( t ) dt ] ( 0  ) dapat ditulis
int egral
f
1
fungsi
(0  )
pada
0
1
2
3
4
f(t)
L(f)
1
1/s
7
t
1/s2
8
t2
2!/s3
tn
(n=0, 1,…)
n!
n 1
s n!
s
5
ta
(a positive)
6
eat
9
10
f(t)
L(f)
cos t
s
sin t
cosh at
s
sa
2
s 
s
2
s a
2
2
2
a
s a
2
11
eat cos t
2
sa
(s  a)  
2
a 1
1
2
sinh at
n 1
 ( a  1)
s 

12
eat sin
t
2

(s  a)  
2
2
Some useful Laplace transforms
f(t)
F(s)=L[f(t)]
(t)
1
u (t)
1/s
t
2
t
1/ s
n
n! / s
at
1 /( s  a )
( n  1)
e
sin( at )
a /( s  a )
cos( at )
s /( s  a )
2
2
2
2
2
2
2
2
sh ( at )
a /( s  a )
ch ( at )
s /( s  a )
Some useful Laplace transforms
f(t)
(e
e
bt
sin( at )
a /[( s  b )  a ]
e
bt
cos( at )
( s  b ) /[( s  b )  a ]
bt
( be
F(s)=L[f(t)]
bt
2
2
2
2
at
 e ) /( b  a ) 1 /( s  a )( s  b )
a  b
s /( s  a )( s  b )
a  b
 ae
at
) /( b  a )
Laplace Transform Properties
• Linear atau Nonlinear?
f 1 t    F1  s 
L
and
f 2 t    F 2  s 
L
a 1 f 1 t   a 2 f 2 t    a 1 F1  s   a 2 F 2  s 
L
L a 1 f 1 t   a 2 f 2 t  
?

 st






a
f
t

a
f
t
e
dt
1
1
2
2
0 


 st
 st
 a 1  f 1 t e dt  a 2  f 2 t e dt
0
0
 a 1 F1  s   a 2 F 2  s 
f(t)
• Linear operator
L
F(s)
contoh
C
R
• Seperti gambar
disamping, muatan awal
kapasitor = 0. Tentukan
persamaan arusnya;
V
V  RI 
q
C
RI 
1
C
 i dt
 v
• Transpormasi Laplace
1
  RI    
C
RI
S

 i dt    v 
1
 IS
f 0 
V


 
s
s
 C .s

f
1
(0  ) 
 idt
0
 0
 ( t  0 , q  0 )
RI
(S )

I (s)

C .s
V
s
1 
V

I (s)  R 
 
C .s 
s

I (s) 
I (s) 
I (s) 
V
1 

sR 

C .s 

V
1 

R S 

R .C 

V
1
1 
R 
S 

R .C 

• Pembalikan transpormasi laplace

1
1
I (s)  

V
1 
R 
S 

R
.
C


V

1
• Lihat tabel
I (t ) 
R

e
1
1 

S 

R .C 

R
V
1
t
RC
Contoh 2
• Gambar RL seperti
gambar disamping,
jika saklar s di on-kan
maka tentukan
persamaan arunya
• Persamaan rangkaian
L
di
 RI  V
dt
• Transpormasi Laplace
 di 
 L
   RI

 dt 
   V 
 di 
L . 
   RI

 dt 
   V 
L sI ( s )  i ( 0  )   RI
(s)

V
s
,  i(0  )  0, t  0, i  0)
• Transpormasi dari cos t
Laplace transform
Definition of function f(t)
f(t)
• f(t)=0 for t<0
• defined for t>=0
• possibly with discontinuities
• f(t)<Mexp(t)[exponential order]
• s: real or complex
Definition of Laplace transform
t
Examples

L [  ( t )] 
  ( t )e
 st
dt  1
0


L [ f ( t )]  F ( s ) 
 f ( t )e
 st
dt
L [ u ( t )] 
e
 st
dt 
1
s
0

0
L [ tu ( t )] 
 te
0
 st
dt 
1
s
2
Laplace transform

L [ f ( t )]  F ( s ) 
 f ( t )e
 st
dt
Examples
0
Dirac
f(t)

(t)
t
L [  ( t )] 
0
f(t)
(t  t 0 )
t
t0

L [  ( t  t 0 )] 
  ( t  t 0 )e
0
  ( t )e
 st
dt  e
 st 0
 st
dt  1
Laplace transform

 f ( t )e
L [ f ( t )]  F ( s ) 
 st
Examples
dt
0
Heaviside
f(t)
u ( t )]
t

 u ( t )e
L [ u ( t )] 
0

 st
dt 
e
0
 st
 st

e 
1
dt  
 
s
  s 0
f(t)
u ( t  t 0 )]
t
t0

 st
e 
e 0
L [ u ( t  t 0 )]  
 
s
  s  t0
 st
Laplace transform

L [ f ( t )]  F ( s ) 
 f ( t )e
 st
Examples
dt
0
Ramp
f(t)
r ( t )  at , t  0
t

  ate
 st
L [ r ( t )]   ate dt  
s

0
 st


 
0


0
a
s
e
 st
dt 
a
s
2
•Linearity
F1 ( s )  L [ f 1 ( t )]
Laplace transform properties
c 1 , c 2  Cons tan ts
F2 ( s )  L [ f 2 ( t )]
L [ c 1 .f 1 ( t )  c 2 .f 2 ( t )] 
c 1 .L [ f 1 ( t )]  c 2 .L [ f 2 ( t )] 
c 1 .F1 ( s )  c 2 .F2 ( s )
Laplace transform properties
• Translation
a) if
L[ e f ( t )]  F ( s  a )
at
F(s)=L[f(t)]

L [ e f ( t )] 
at

 [ e f ( t ) ]e
at
 st
dt 
0
Example
t
 f ( t )e
 (s a ) t
dt  F ( s  a )
0
L [ Cos ( 2 t )] 
L [ e Cos ( 2 t )] 
s 1
( s  1)  4
2
s
s 4
2

s 1
s  2s  5
2
Laplace transform properties
g(t)
f(t)
• Translation
b) if g(t) = f(t-a) for t>a
= 0 for t<a
L [ g ( t )]  e
 as
t
F (s )
a

L [ g ( t )] 

 f ( t  a ) ]e
 st
dt 
0

 f ( u )e
s(u a )
du  e
 as
0
0
L[ t ] 
3
Example
3!
s
g (t)  (t  2) , t  2
4

6
s
4
3
g ( t )  0, t  2
 f ( u )e
L [ g ( t )] 
6e
s
2s
4
 su
du
Laplace transform properties
•Change of time scale
L [ f ( a .t )] 

L [ f ( a .t )] 

 f ( a .t ) ]e
0
1
 st
dt 
 f ( u )e

su
a
0
du

s
F( )
a
a
1
s
a
F( )
a
a
1
1
Example
L [ Sin ( t )] 
1
s 1
2
L [ Sin ( 3 t )] 
2
3 s
3 1
 

3
s2  9
Laplace transform properties
• Derivatives
L [ f ' ( t )]  L [
df


]  L [ f ( t )]  s .F ( s )  f ( 0 )
dt

L [ f ' ( t )] 

0
e
 st


f dt  e
 st
f (t)

    se

0
 st
f ( t ) dt  sF ( s )  f ( 0
0

L [ f ' ( t )]  s .F ( s )  f ( 0 )
)
Laplace transform properties
• Derivatives
L [ f ' ( t )]  L [
df


]  L [ f ( t )]  s .F ( s )  f ( 0 )
dt


2
L [ f " ( t )]  L [ f ( t ) ]  s .F ( s )  s .f ( 0 )  f ' ( 0  )
(n )
n
L [ f ( t )]  s F ( s )  s
n 1
f (0)  s
n
(n )
n
L[ f ( t ) ]  s F (s ) 
s
n i
n2
( n 1)
(1 )
f ( 0 ) .....  f ( 0 )
( i 1)
. f (0)
i 1
•If discontinuity in a


f (a )  f (a )
L[ f ' ( t )]  s.F ( s )  f ( 0 )  e
 as


[ f ( a )  f ( a )]
Laplace transform properties
• Derivatives examples

L [ Sin (  t )] 
d [sin(  t )]
s 
2
L [ Cos (  t )] 
Cos (  t ) 
dt
L [ Cos (  t )] 

d [ Cos (  t )]

L [ Sin (  t )] 
Sin ( 0 )
   Sin (  t )
L [ Sin (  t )] 

2



dt

s
 (s   )
Sin (  t ) 
2
2
Cos ( 0 )


s
s 
2
 1 d [ Cos (  t )]

dt

L [ Cos (  t )] 
2
1 d [ Sin (  t )]
dt
s
s 
2
  Cos (  t )
s
s


(s   )
2
2
2
Remarques sur la dérivation
Deux cas à prévoir
a)
L[ u ( t )
df ( t )

]
dt
L[ u ( t )
df ( t )
e
 st
0
]  [f
dt
u (t)
df ( t )
dt
L[ f ( t )
dt
dt
En intégrant par parties
dt

 st 
( t )e ]0


 se
 st
f ( t ) dt
L[ u ( t )
df ( t )

]  sF ( s )  f ( 0 )
dt
0
 f (t)
du ( t )
dt
du ( t )
d [ u ( t ) f ( t )]
dt
b)
d [ u ( t ) f ( t )]
df ( t )
 u (t)
df ( t )
dt


]  L [ f ( t )  ( t )]  L [ f ( 0 )]  f ( 0 )
dt
Si f(t) et toutes ses dérivées sont nulles pour t<0, alors on
peut ne pas tenir compte des valeurs initiales
pour étudier le comportement
L[
d [ u ( t ) f ( t )]
dt
]  sF ( s )
Laplace transform properties
• Integral
t
g(t) 

F ( s )  L[ f ( t )]
f ( u ) du ]
0

g(t)  f (t)


L [ g ( t )]  sL [ g ( t )]  g ( 0 )  F ( s )
t

L [ f ( u ) du ] 
0
F (s )
s
Laplace transform properties
Multiplication by t
dF ( s )
d
'
 F (s ) 
ds
dF ( s )
ds
ds



 s


[ e
 st
Leibnitz’s rule
f ( t ) dt
0

[e
 st
f ( t ) dt ] 
0

e
 st
[  tf ( t ) ]dt   L [ tf ( t )]
0
'
L [ tf ( t )]   F ( s )
More general
n
n
L [ t f ( t )]  (  1)
n
d F (s )
ds
Laplace transform properties
Division by t
g(t) 
f (t)
f ( t )  tg ( t )
t
L [ f ( t )]  
dL [ g ( t )]

dG ( s )
ds
 F (s )
ds
s


 f ( u ) du
G ( s )   f ( u ) du 

L[
s
f (t)
t

]
 f ( u ) du
s
Laplace transform properties
• Periodic function
T
L [ f ( t )]  F ( s ) 
2T
e
 st
3T
e
f ( t ) dt 
0
 st
 st
f ( t ) dt .......
2T
e
 st
f ( t ) dt 
T
e
s(u T )
f ( u  T ) du 
0
e
T
e
 st
f ( t ) dt  e
0
f ( u  2 T ) du .......
 sT
e
T
 su
0

s(u  2T )
0
T
L [ f ( t )]  F ( s ) 
e
T
0
L [ f ( t )]  F ( s ) 
f ( t ) dt 
T
T
L [ f ( t )]  F ( s ) 
 t, k
f ( t  kT )  f ( t )
f ( u ) du  e
 2 sT
e
 su
f ( u ) du .......
0
T
e
 nsT
n 0
[ e
 st
f ( t ) dt ]
0
T

e
n 0
 nsT

 f ( t )e
1
1 e
 sT
L [ f ( t )]  F ( s ) 
 st
0
1 e
 sT
dt
Hint

e
 nsT

n 0
n
1
sn 
 sT
1 e
2
3
4
n
sn 
p
q0
q


3
4
n 1
1 p
n
ps n  p  p  p  p  .......... .p  p
s n (1  p )  1  p
q
q0
s n  1  p  p  p  p  .......... .p
2
p
1 p
n
n 1
n 1
1 p

n 1
1 p
p 1
s 
p
q0
q

1
1 p
Laplace transform properties
Sine and cosine are periodic functions
e
j t
 Cos (  t )  jSin (  t )

L[e
j t
]  L [ Cos (  t )]  jL [ Sin (  t )] 

e
j t
e
 st
dt 
0
e
( j  s ) t
dt
0
T
L[e
T
e
( j  s ) t
dt 
0
L[ e
1
j  s
j t
]
e
j t
e
]

( j  s ) t T
0
1
s  j

( j  s ) t
dt
0
1 e

 sT
1
j  s
[e
j T
e
 sT
s  j
( s  j  )( s  j  )

 1] 
1
j  s
s  j
s 
2
2
[e
 sT
 1]
Laplace transform properties
Example
f(t)
1
t
0
-1
F (s ) 
1
s
th ( )
s
2
1
2
3
Laplace transform properties
Periodic function
T
1
 f ( t )e
 st
0
dt 
1
e
 st
e
dt 
0
 st
dt
0
1
T
2
e
 1  st 
 1  st 
 st
 f ( t ) e dt  [   s e   [   s e  ] 
0
1
0
s
F (s ) 
(1  e )
s (1  e
s

)
F (s ) 
s
s
s
s (1  e )( 1  e )
s
e
2
s
se
2
s
s
 1  e (e
s

s
(e 2  e
2
)
s
2
)

1
s
s (1  e )
s
th ( )
s
2
 1)
s

(1  e )
s
(1  e )
s
(e 2  e
s
s
(1  e )( 1  e )
2
2s
s
2
Laplace transform properties
Example
1
t
0
F (s ) 
1
s
2

e
s
s
s (1  e )
1
2
3
Laplace transform properties
1
 te
 st
0
  te
dt  
 s
1
 te
0
 st
dt 
e
 st
s

s
1

 
0
1
1
se
s
2
dt 
e
s
0
s
e
 st
1

s
2
s

e
s

s
1
s
2
 st
 e
 2
 s
s
(1  e )
1
F (s ) 
 te
 st
dt
0
1 e
s

1
s
2

e
s
s
s (1  e )
1


0
Laplace transform properties
•Limit behaviour
Initial value


L [ f ( t )]  sF ( s )  f ( 0 )

Lim [  e
Exponential order
 st

f ( t ) dt ]  0
0
s 


Lim [ sF ( s )]  f ( 0 )
Lim [ f ( t )]  f ( 0 )
s 
t 0
Lim [ f ( t )]{ t  0}  lim[ sF ( s )]{ s   }
Laplace transform properties
•Limit behaviour
Final value


Lim [ s  0 ] e
0
 st

f ( t ) dt 


L [ f ( t )]  sF ( s )  f ( 0 )




f ( t ) dt  lim[ p   ][ f ( p )  f ( 0 )]
0
Lim [ f ( t )]{ t   }  lim[ sF ( s )]{ s  0}
Laplace transform applications
RC circuit
Equation describing the circuit
R
e ( t )  RC
e0.(t)
C
Laplace transform
dv
 v(t)
dt
v(t)
v (0)  0
E ( s )  RCsV ( s )  V ( s )  V ( s )[ 1  RCs ]
V (s ) 
E (s )
1  RCs
V (s ) 
Laplace transform applications
E (s )
1  RCs
Impulse function
e( t )  e 0( t )
E (s )  e 0
1
(s 
)
RC
e0
RC
RC

e
sV ( s ) 
1  RCs
Impulse response
e0
1
V (s ) 
V (s ) 
e0
e0
v(t) 
t
e CR
RC
t
t
CR
s
se 0
( RCs  1)
RC
s 0
Laplace transform applications
V (s ) 
Step function
V (s ) 
s

1  RCs
E (s ) 
e( t )  e 0 u ( t )
e0
E (s )
e0
V (s ) 
s
e0
(s 
1
e0
s (1  RCs )
t
v ( t )  e 0 [1  e cr ]
)
RC
e0
0 , 63 e 0
t
t
v ( t )  e 0  e 0 e CR  e 0 [1  e CR ]
RC
Laplace transform applications
Step function and initial conditions v(0)  0
E ( s )  RC [ sV ( s )  v ( 0 )]  V ( s )  V ( s )[ 1  RCs ]  RCv ( 0 )
V (s ) 
V (s ) 
e0
s (1  RCs )
e0
s


RCv ( 0 )
1  RCs
t
v ( t )  e 0  [ v ( 0 )  e 0 ]e cr
v (0)  e 0
(s 
1
e0
)
RC
v (0)
t
v ( t )  e 0  [ v ( 0 )  e 0 ]e
CR
sV ( s )  e 0 
RCs [ v ( 0 )  e 0 ]
RCs  1
Laplace transform applications
V (s ) 
Ramp function
E (s ) 
e(t)  r(t)  t
V (s ) 
1
s
2

RC
s
1
s
V (s ) 
2
E (s )
1  RCs
1
s (1  RCs )
2
RC

(s 
1
RC
)
v(t)
t  CR
t
v ( t )  t  RC  RCe
dv
dt
t
 1  e CR
CR
RC
sV ( s ) 
1
s
2
 RC 
( RC ) s
( RCs  1)
Laplace transform properties
e(t)  E 0 u (t  a )
E0
(Heaviside)
E (s ) 
E 0e
t
a
 as
s
V (s ) 
E (s )
RC ( s 
1

)
v0
s
RC
V (s )  E 0 e
V (s ) 
1
1
[ 
s
 as
sRC ( s 
RC
 as
E 0e
1
RC
1
1
s
]
v0
s
RC
v ( t )  u ( t  a ) E 0 [1  e

RC
ta
cr
1
]  v 0e

t
cr

)
v0
s
1
RC
Laplace transform properties
v ( t )  u ( t  a ) E 0 [1  e
v(t)  v 0e
ta
ta


ta
cr
]  v 0e
t

cr
t
cr
v ( t )  E 0 [1  e

ta
cr
]  v 0e
t

cr
E0
v0
a
t
Laplace transform properties
Limits
sV ( s ) 
E 0e
 as
RC ( s 
1
RC
Initial value
s
sV ( s )  v 0  v ( 0 )
Final value
s 0
sV ( s )  E 0  v (  )

)
sv 0
s
1
RC
Laplace transform properties
Harmonic analysis
e(t)
E(s)
R
v(t)
V(s)
C
RC ( s 
1

)
aE ( s )
sa
A
sa
E (s )  e 0

Bs  C
s 
2
2

ae 0 
a 
2
2
1
RC
s 
2
V (s ) 
2
ae 0 
( s  a )( s  
2
A 
)
B
V (s ) 
,a 
RC
e ( t )  e 0 sin(  t )
V ( s )  ae 0  (
E (s )
V (s ) 
(
1
sa

a
s 
2
2

s
s 
2
2
)
C
1
a 
2
2
1
a 
2
2
a
a 
2
2
2)
Laplace transform properties
V (s ) 
v(t) 
ae 0 
a 
2
2
a 
sa

t
ae 0 
2
(
1
2
[ e CR 
a
s 
2
1
RC 
tg (  )   RC 
2

s
s 
2
2
)
sin(  t )  cos(  t )]
Cos (  ) 
1
1  ( RC  )
2
t
v ( t )  e 0 Cos (  )[sin(  t   )  sin(  ) e CR ]
Forced
Transient