§ 15-7 The complete response
The first is the more fundamental,for it involves
writing the differential equations for the network and
then taking the Laplace transform of those equations.
)0(1 ??Let =10Vand =25V,
find,
2?
)0(2 ??
When initial energy is present in a circuit,the Laplace
transform method may be used to obtain the complete
response.
1?
Vttu )(2co s50
?
?
2?
??
?
?
F481
F24130
20
24
2
1
Solution:
at the node 1:
)1(22048124 '112'121 ?????? ????? or
0241302420 )(2c o s50 '22122 ?????? ????? ttu
)2()(2c o s603 2'21 ttuor ??? ???
at the node 2:
Identifying as the desired response,we eliminate '11 ?? and2?
).(/)( tdttdu ??by taking the derivative of (2),remembering that
)3()(60)(2s i n6023 '2''2'1 tttu ???? ?????
)0(1 ??Let =10Vand =25V,
find,
2?
)0(2 ??
1?
Vttu )(2co s50
?
?
2?
??
?
?
F481
F24130
20
24
2
1
substituting (2) and (3) into (1),
?22 2 ?? ? ??? )(2c o s603 2'2 ttu?? )(60)(2s in6023 '
2''2 tttu ??? ????
)(60)(2s i n1 2 0)(2c o s1 2 045 2'2''2 tttuttuor ???? ?????
We now take the Laplace transform,
604 21 2 01 2 0)(4)0(5)(5)0()0()( 2222'2222 ?? ???????? ??? sssVssVssVs ???
604 2120120)0(5)0()0()()45( 22'2222 ?? ???????? ??? ssssVssor ???
)1(22 '112 ??? ??
)2()(2c o s603 2'21 ttu??? ???
)3()(60)(2s i n6023 '2''2'1 tttu ???? ?????
604 21 2 01 2 0)0(5)0()0()()45( 22'2222 ?? ???????? ??? ssssVss ???
25)0(2 ???L e t
604 21 2 01 2 0)0(1 2 525)()45( 2'222 ?? ???????? ? ssssVss ?
And need a value for, This we may obtain from (1) and
(2) by evaluating each term at t=0-,Actually,we need use only
(2) in this problem:
)0('2 ??
0)0(3)0()0( 2'21 ??? ??? ??? 65)0('2 ????a n d
604 21 2 01 2 0651 2 525)()45( 222 ?? ???????? ssssVss
4
21 2 01 2 01 2 025
2 ?
?????
s
ss
)4)(1(
)4/()240120(12025)( 2
2 ??
?????
ss
ssssV
)2()(2c o s603 2'21 ttu??? ???
)4)(1(
)4/()240120(12025)( 2
2 ??
?????
ss
ssssV
4
66
4
66
4
3/16
1
3/23
22 ?
??
?
??
???? s
j
s
j
ss
? ? Vtuteet tt )()452c o s (212316323)( 42 ?? ???? ??
4
2412
4
316
1
323
2 ?
??
???? s
s
s
/
s
/
The second technique requires each initial capacitor
voltage or inductor current to be replaced by an
equivalent dc source.
The frequency-domain equivalent of an inductor L:
dt
tdiLt )()( ?? )0()()( ??? Liss L IsV
s
i
sL
sVsI )0()()( ???
)t(i
L
??
?
?
t?
)0( ?i
)a(
?
?
)s(I
sL
??
?
?
sV
)b(
)0( ?Li
)s(I
sL??
?
?
sV
)c(
s
i )0( ?
The frequency-domain equivalent of a capacitor C:
ssC
sIsV )0()()( ??? ?
dt
tdCti )()( ?? )0()()( ??? ?Css C VsI
)t(i C
??
?
?
t?
)0( ??
)a(
?
?
)s(I
??
?
?
sV
)b(
s
)0( ??
sC/1 )s(I
??
?
?
sV
)c(
)0( ??C
sC/1
0
/24
25)(
30
)(
/4824
10)(
20
4
50)(
22222
?
?
??
?
?
??
?
s
s
sVsV
s
s
sV
s
ssV
4
2412
4
3/16
1
3/23
2 ?
??
???? s
s
ss
Solution:
Vtutteet tt )(]2s i n122c o s12316323[)( 42 ???? ???
)4)(1(
)4/()240120(12025)( 2
2 ??
?????
ss
ssssV
?? 4502?s s s
48 30
20 2
1
?????
?
sV1
s24
s10 s
25??
??
?
?
sV2
24
1?
Vttu )(2cos50
?
?
2?
??
?
?
F481
F24130
20
24
2
1
Vtuteetor tt )(]452c o s (212316323[)( 42 ????? ???
Example 3:
.0),()( ?f o r tta n dtiF i n d c?
0
/1
)(
/1
3/4)(
/22
3/8/4)(
??
?
?
?
??
s
sVc
s
ssVc
s
sssVc
251
15280
.s
/
s
.)s(Vc
?
???
Vc 3/215/25/4)0( ?????
]0)0([ ??c?
s
sVcsI
/1
)()( ?
Atuetti t )(61)(32)( 25.1??? ?
Solution:
Vtuet tc )()15/25/4()( 25.1?????
?
)(sVC
2 )(sI
??
s3/8
s/1
?
?
s3/4?
?
s/4
s/2
s/1
2 )(ti
?
?
)(tC??
?
V4 F1 F1
F5.0
电荷守恒 1F2/3+ 1F2/3 =1F4/3
25.1
1
6
1
3
2
??? s25.1
)15/2(
5
4
??? s
s
DP,2
(a)Draw the frequency domain circuit that is valid for
the circuit shown in Fig.
(b)Find i(t),and,for t>0.)(t
C? )(tL?
Atu )(?
?
?
)(tC?
)t(i
10
0?t
F321
2040
??
V30
H2
?
?
)(tL?
Solution:
02410323020 ?????? )s(sIs/)s(sI/s/)s(I
1610
102
2 ??
???
ss
s)s(I A)t(u)ee()t(i tt 82 ?? ???
8
1
2
1
?
??
?
??
ss
sss
s
ss)s(Is)s(V C
10
1610
102321032
2 ???
??????
V)t(u)ee()t( ttC 82 41630 ?? ?????
41610 102242 2 ??? ???????? ss ss)s(sI)s(V L
V)t(u)ee()t( ttL 82 164 ?? ????
8
4
2
1630
????
??
sss
8
16
2
4
?
??
?
??
ss
)(sI
s/32
20
??
s/30
s2
s/10? ??
??
4 ?
?
)(sVC
?
?
)(sVLAtu )(?
?
?
)(tC?
)(ti
10
0?t
F321
2040
H2
?
?
)(tL?
??
V30