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$ At f (t ) = # "0
$ A f (t ) = # "0
for
t ! 0
()
=
)
Ae
' st
dt
=
0
'
)
( ) = ( Ate '
(
st
F s
dt = A t e
0 remember :
F s
t < 0
)
t < 0
(
t ! 0
for
Ae
' st (
s
=
'
Ae
0
'(
s
& Ae ' $$ ' % s
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A =
u
=
t
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=
e
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s
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!
%$=
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=
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v
te
=
'
e
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s
' st )
s
) +
(
e
0 ! t ! t 0
t 0
O
?4 " ! :MM/3:I5.0 X./3 ! '2/,+(.
O
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!"
0+
A s
)
(e
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dt =
0
A s
2
0 ! t ! t 0
$ A t 0 f (t ) = # for " 0
t < 0 and t > t 0
W/.: @+;./ 4@+I1,3+< 4@+I1,3+< ! !
O
=
P@90. ?+M@1
O
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dt !
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O
#
' st
s
0
0
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$ A t 0 f (t ) = # for " 0
dv
u " v # v " du
du
dt
&
F ( s )
=
dt
u
0
! u " dv
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F ( s )
=
" 0
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t < 0 and t > t 0
! A =
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" !
A =
t
=
0
st 0
(1 ! e! ) st 0
& d ' st $ dt (1 ' e 0 A lim $ t (0 d $ st $% dt 0 0
0
0
=
e
st 0
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t 0
! st
=
0
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=
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# ! "
=
A
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C ( s)
O
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C ( s)
C ( s)
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A =
s A
=
s
2
G(s) G(s)
AG( s)
=
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=
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O
A
= A0 +
A1 s
+
A =
s( s + a)
=
A1 s
+
A2 s + a
A = A1 s + A1a + A2 s
(
A = s A1 + A2
C ( s)
1
s( s + a)
=
A 1 a s
(
A !
1
a s + a
"
c(t )
1
s + a
1 =
!
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=
!
+
+
O 6. I:+ @0. .d@:1,3+ 13 .01,F:1. /.0M3+0.
A2
c(t )
A =
a
(1 ! e ! ) at
where A
=
"
and a
!
1
c(t )
=
!
=
& $ %
) $1 ' e
'
t
(
# ! ! "
) + A2 s A 0
=
=
A =
and a
!
=
s + a
A1 s + a
8/3@M I3FF3+ 1./F0
) + A a
A =
! A =
A
" =
A
# "
s + 1 "
B,/01 a/;./J *+,1 E1.M
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1
=
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P:/1,:9 B/:I1,3+ 7YM:+0,3+ s( s + a)
1 "
where A
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C ( s) =
!
1 + " s 1 "
A1a
A1 + A2
A !
a
! A1
a
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e
! at
1,F.
a@1M@1
a
!
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A =
a
(1
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A a
)
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c(t )
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t
(
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c(t )
fge
1
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K $1 ' e
t
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(
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1
A
= A
=
!
1 + " s 1 " where A
C ( s)
C ( s) =
1
" =
A
# "
s + 1 "
and a
=
1 =
!
!
2
!
+
2
+
= A0 +
A1 s
2
+
A2 s
+
A3 s + a
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s + a
A 2
s ( s + a )
"
c(t )
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1 !
=
=
t
=
1
' & ' ( $$1 ' e %
t
# ! ! "
(
( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
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c (t )
=
t
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t
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# ! ! "
G ( s)
=
(
#
!
1 "
1 + " s 1 "
=
A
# "
s + 1 "
=
s + a
!
where A
O a@1M@1J h!
( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
!
A
and a
1 =
!
s + a
O ?+-./0. HKJ C ( s)
( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
C ( s) = 1!
" =
t
A =
$
s + a
c(t )
= Ae
# at
=
"
#
e
!
! ( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
$
!"#!$#%!&'
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B,+:9 U:9@. K5.3/.F
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O K3 4,+; 15. 4,+:9 01.:;2 01:1. -:9@.
"
=
1 !
=
1
t
c(t )
=
lim f (t )
" #! e
t !"
!
O 7Y:FM9.J
X ( s) F ( s )
( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
=
X ( s )
=
x(!)
=
=
lim sF ( s) s!0
G ( s )
=
5 s + 3
; input F(s) =
3 s
5 3 s + 3 s
lim sX ( s ) = 5
( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
K3;:2N0 9.I1@/. O ?+M@10J P@90. R,
[email protected]< E1.M< L:FM O K3 ;.1./F,+. 1,F. /.0M3+0.J 8.1 1/:+04./ 4@+I1,3+ ,+ 0D;3F:,+< :MM92 ,+M@1< 15.+ 83 _:I` 13 1,F. ;3F:,+ @0,+8J T ?+-./0. H:M9:I. 1/:+043/F0 T P:/1,:9 B/:I1,3+ 1.I5+,d@. 15.+ ?HK
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( %!&' *+,-./0,12 34 15. 6.01 34 7+89:+;< =/,0139>
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