§ 6-1 The source-free parallel circuit
0
0
)0(
)0(
V
Ii
?
?
?
?
?
02 ??? ststst eLAeRAseCAs
0)(1,0
0
???? ? dtdCtidtLRKCL tt ???
01122 ??? ??? LdtdRdtdC
stAeL et ??
0)/1/( 2 ??? LRsCsAe st
R
)(ti
L C
?
fRe
LC
1)
RC2
1(
RC2
1s
LC
1)
RC2
1(
RC2
1sor 2
2
2
1 ????????
0/1/,2 ??? LRsCsL e t
LCRCRCC
L
C
RRs 1)
2
1
(
2
1
2
4)1(1
2
2
2,1 ????
???
?
tsts eAeA 21 21 ??? ?
A1,A2----two arbitrary constants
s1,s2----real numbers or conjugate complex
Since the s1t and s2t must be dimensionless,s1,s2 ---- (1/s).
0)/1/( 2 ??? LRsCsAe st
-- neper frequency or exponential damping coefficient.
RC2
1??
2
0
22
2
2
0
22
1
1
)
2
1
(
2
1
1
)
2
1
(
2
1
???
???
????????
????????
LCRCRC
s
LCRCRC
s
z e r o
n u m b e ri m a g i n a r y
n u m b e rr e a l
?????
??????
??????
2
0
2
0
2
0
2
0
2
0
2
0
,
,
,
????
????
????
LC
1
0 ??
Let resonant(谐振 ) frequency
1,The over-damped parallel RLC circuit
)12 1(0 LCRC ????? ??
The S1 and S2 are negative real numbers,thus,the response
can be expressed as the sum of two decreasing exponential terms,
both of which approach zero as time increases without limit.
12 ss ?
The term containing s2 ( ) has the more rapid rate of decrease.12 ?? ?
1?
2?
?
t
0202202202 ??????????? ?????????
Example,Find,)(t?
Ai 10)0(
0)0(
?
??
6
42
17
11
0 ?
?
??
LC
?
65.25.32022 ????????? ???s
0?? ??
1525365353 22021 ?????????????,...s ???
tt eAeAt 621)( ?? ???
5.3
42
162
1
2
1 ?
??
?? RC?
Ri ci
?6
)(ti
H7 F
421
?
fRe
tt eAeA
dt
dAi 6
21 610)0(
?? ????? ?
420)0()0()0()0(0 ???????? CiC iiCitdtddtdCi Rcc ??
Veet tt )(84)( 6?? ???
84,84 21 ???? AA
0)6(84,6 ???? ?? tt eedtdl e t ?
te?84
te 684 ?? t
?
mt
At time tm the voltage
becomes maximum,Vst
eee
mm
ttt mmm
9.483 5 8.0
5
6ln
606 56
????
????? ??
?
2100)0( AA ?????
4 2 06 21 ??? AAor
Ri ci
?6
)(ti
H7 F
421
?
fRe
tt eAeAt 621)( ?? ???
V
L1
7H
1
2
C1
0,02 38
R1
6
0
Time
0s 1.0s 2.0s 3.0s 4.0s 5.0s 6.0s
V(R1:2)
0V
25V
50V
2,Critical damping
)65.32112 1(0,2020 ??????????? CLRLCRC????
? ? tteAtA 612 00)0( ?????? ???
6
61
21
0
?????
???
?
??
ss
LC
? ? eee ttt AtAAtAt 626121 )( ??? ???? ??
Vtetet tt 45.26 420420)( ?? ???
16161 0)6( Atdt
deAetA
dt
d tt ?
????
?? ??
1420
)0()0()0()0(
0 AC
i
C
ii
C
i
tdt
d
dt
dCi Rc
c ???
???
???
??
Ri ci
?653.
)(ti
H7 F
421
?
fRe
Vtetet tt 45.26 420420)( ?? ???
V.e.s.t,.mm 1634080420408061 4080452 ?????? ???
?
Vm 1.63??
t
mt0
045.2 1l i m4 2 04 2 0l i m)(l i m 45.245.2 ???
?????? ttttt ee
tt?
0)16(4 2 0,0)( 6 ???? ? tedt td t?
V
C1
0,02 38
0
L1
7H
1
2
R1
8,57 3
Time
0s 1.0s 2.0s 3.0s 4.0s 5.0s 6.0s
V(L1:1) V(L1:1) V(L1:1)
0V
40V
80V
3,The under-damped parallel RLC circuit
)57.8215.1012 1(0,2020 ???????????? CLRLCRC????
By using complex numbers,the exponential response turns
into a sinusoidal response.
djs ?? ???? 2,1
tsts eAeAt 21 21)( ??? 2022,1 ??? ????s
f r e qu e n c yr e s po n s en at u t ald ????? 220 ???
tjtj dd eAeAt )(2)(1)( ????? ???? ??
)1(1,220220202 ???????? jjl e t ??????
tjtj dd eAeAt )(2)(1)( ????? ???? ??
)c o s ()( ??? ? ?? ? tAetor dt
)( 21 tjtjt dd eAeAe ??? ?? ??
]2)(2)[( 2121 j eeAAjeeAAe
tjtjtjtj
t
dddd ????
?
??
? ??????
]s i nco s[ 21 tBtBe ddt ??? ?? ?
246
6
1
2
42/15.102
1
2
1
0
???
??
?
??
??
d
LC
RC
?
?
?
VtBtBet t )2s i n2c o s()( 212 ?? ??
Ri ci
?510.
)(ti
H7 F
421
?
fRe
tVet t 2s in2210)( 2??? ?
100)0( B????
VtBtBet t )2s i n2c o s()( 212 ?? ??
tBet t 2s i n)( 22??? ?
tBetBedt td tt 2s i n22c o s2)( 2222 ?? ???
221 042 0)0(20)( 22 ?????? BCiBtdt td c?
?
t
1mt
2mt
V8.71
V845.0? Vst
Vst
mm
mm
84 5.065 7.2
8.7143 5.0
22
11
????
???
?
?
Ri ci
?510.
)(ti
H7 F
421
?
fRe
V
L1
7H
1
2
0
R1
10, 5
C1
0,02 38
Time
0s 1.0s 2.0s 3.0s 4.0s 5.0s 6.0s
V(C1:2)
-40V
0V
40V
80V
?
t
1mt
2mt
V8.71
V845.0
V1.63
V9.48
d am p edu n de r ?
da m pe dov er ??
da m p in gc ritic al ?
A comparison of the three curves makes these general
conclusions plausible:
1,when the damping is changed by adjusting the size of the
parallel resistance,the maximum magnitude of the response is
greater with smaller damping.
2,The response becomes oscillatory when under-damping is
present,and the minimum setting time is obtained for slight
under-damping.
V
0
R3
1 0, 5
0
C2
0,0 2 3 8
L1
7H
1
2
V
R1
6
L3
7H
1
2
V
C3
0,0 2 3 8
C1
0,0 2 3 8
L2
7H
1
2
R2
8,5 7 3
0
Time
0s 1.0s 2.0s 3.0s 4.0s 5.0s 6.0s
V(C1:2) V(C2:2) V(C3:2)
0V
40V
80V