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8/6/2019 LRC
1/18
II
7: LRC
1 18
1 3020 2980
1
::
, .
::
::
1, LRC . 1,
0
Q . , 0t s= ,
2
L . , , RC, ,
LdI dt
. , :
dQ Q dI
I Ldt C dt = = (1)
, :
2
2
d Q QL
dt C= (2)
, ,
( )0 0cosQ Q t= (3),
( )0 1
2
1
LC
= (4).
, , , .
LRC , :
8/6/2019 LRC
2/18
II
7: LRC
2 18
1 3020 2980
2
2
20
d Q dQ QL R
dt dt C + + = (5)
,
LRC, . LRC R . , R , .
, ,
2L
R = . (6)
21
T Q :
( )1 22
ln 2L
TR
= (7)
QF , :
1
20
1
2
LQF
R C
= =
(8)
, , :
1
22
0 2
1
=
(10)
::
2, LRC, ,
0 cosV V t= (11). 2 Kirchhoff
, , :
2
02cos
d Q dQ Q L R V t
dt dt C + + = (12).
, Taylor
, :
8/6/2019 LRC
3/18
II
7: LRC
3 18
1 3020 2980
0
0 1
2 22 1
VQ
R LC
=
+
(13).
0
Q
max 0
0
0
VQ
R= (14)
( )L C 2 = .
::
::
, 4 :
[ ]mHL 7,22%57,22 =
[ ]120 5% 120C nF=
L R R r r = = +
50r =
[ ]25, 3 0,1r =
, 1 2
T :
[ ]1 2 0,4 0, 2T DIV DIV = , 0, 5 s DIV m=
, T :
[ ]0, 4 0, 2T DIV DIV = , 1 s DIV m=
4
8/6/2019 LRC
4/18
II
7: LRC
4 18
1 3020 2980
, :
, 6, :
[ ]3
2 2 45, 4 100,60 0,03
75 3
L
L L ms
R r r ,
= = = =
+
2 2 22
2
2 20 03
L L R L R , ms
L R R R
= + = + =
( ) ( )2 2
2 2
0,1 L L L
L
R R R r r r r r
r r
= + = + = =
, :
[ ]1 21 20, 2
ln 2 0, 29 0,14ln 2 ln 2
TT ms= = = =
2
1 2
1 2
10,14
ln 2
T ms
T
= = =
, :
0, 60 0, 29% 100% 100% 51,6%
0,6
= = =
0
, , :
( ) ( )[ ]0 1 1
3 92 2
1 1 19,2 0,68
22,7 10 120 10
rad ms
LC
= = =
2 22 2
0 0
03 3
1 10 68
2 2 L C L C , rad ms
L C C L L C
= + = + =
0
, :
8/6/2019 LRC
5/18
II
7: LRC
5 18
1 3020 2980
[ ]
1 2
22 2
0 02 2 2
1 1 2 116,08 2,47 rad ms
T
= = + = + =
( )
22
0 0
0
2 21 1
2 22 22
32 2 3
2 1 4 2 1 12,47
TT
T rad msT TT
= + =
= + + + =
0
0
, :
0 0
0
19,20 16,08% 100% 100% 16,3%
19,20
= = =
QF , 8, :
( ) ( )
11 13 22 2
9
1 1 1 22, 7 105, 8 0, 2
50 25, 3 120 10L
L LQF
R C C r r
= = = =
+ +
( ) ( ) ( )
2 2 2
2 221
2
3
1 1
2 2
0 2
L L L
QF QF QF QF R L C
R L C
L L R L C
Cr r r r LC r r C
,
= + + =
= + + = + + +
=
QF , :
0 16,08 0,29 4, 7 1, 22 2
QF
= = =
2 22 2
0
0 0
0
1, 22 2
QF QF QF
= + = + =
8/6/2019 LRC
6/18
II
7: LRC
6 18
1 3020 2980
QF QF, :
5,8 4,7% 100% 100% 19%
5,8
QF QF
QF
= = =
, , 1:
::
11..
, 4 :
[ ]mHL 7,22%57,22 =
[ ]120 5% 120C nF=
R =
50r =
[ ]25,3 0,1r =
, , . , :
[ ]0,397 0,001R K=
R
,:
1
8/6/2019 LRC
7/18
II
7: LRC
7 18
1 3020 2980
[ ]3
9
22,7 102 2 0,87 0, 03
120 10
LR K
C
= = =
22 2 2
3
10 03
R R L R L C L C , K
L C LC C
= + = + =
, , , :
0i
i
r R R= =
, R
R
, :
0,87 0, 40% 100% 100% 54, 0%
0,87
R R
R
= = =
, , , , :
0i L
i
r R R r r = + +
[ ]3
0, 397 10 50 25, 3 0, 472 0, 001L R R r r K = + + = + + =
( ) ( )
2 22
2 2
0,001 L L
L
R R R R R r r R r K
R r r
= + + = + =
, R R , :
0,87 0,47% 100% 100% 46, 0%
0,87
R R
R
= = =
22..
( R
), . , 2:
8/6/2019 LRC
8/18
II
7: LRC
8 18
1 3020 2980
,
. , 3:
( R
), , , , . , 4:
2
3
4
8/6/2019 LRC
9/18
II
7: LRC
9 18
1 3020 2980
::
, 5 :
[ ]mHL 7,22%57,22 =
[ ]120 5% 120C nF=
[ ]98,0 0,1R =
50r =
[ ]25,3 0,1r =
L R R r r = + +
0
CV , T t 11
f . ,
1 1, 2 & 3:
( ) ( )[ ]0 1 1
3 92 2
1 119,2 0,68
22,7 10 120 10
rad ms
LC
= = =
2 22 2
0 0
03 3
1 10 68
2 2 L C L C , rad ms
L C C L L C
= + = + =
2
T
= ,
2 2
2 2
1 12 2T T T
T T T
= = =
5
8/6/2019 LRC
10/18
II
7: LRC
10 18
1 3020 2980
0
,
2 2
2 2
0 0
0 02
0 0 0 0
1
= + = +
2t
=
, ( ) ( )2 2 2 2
2
2 2 tt T t T
t T T T
= + = +
0
0 1
2 22 1
VQ
R LC
=
+
2 2 2 2 2
0 0 0 0 0
0 0
0
Q Q Q Q QQ V R L C
V R L C
= + + + +
0
20
2
1
1
Q
VR L
C
= +
2
0
0 322 22
2
1 1
2
1 1
L LQ C C
V
R L R LC C
+ = + + +
0 0
3
2 22 1
Q V R
R
R LC
=
+
0
0 3
2 22
1
1
LQ CVL
R LC
=
+
2 2 30 0
3
2 22
1
1
L
Q V C C
C
R LC
=
+
8/6/2019 LRC
11/18
II
7: LRC
11 18
1 3020 2980
V(Volt) V(Volt) t(s) t(s) T(ms) (ms) (rad/ms) (rad/ms) Qo(C) Qo(C) / (/) (rad) (rad)
2,60 0,20 24,00 24,00 1,20 0,20 5,24 0,873 0,310167 0,009 0,27328 0,04657 0,31 0,0303,20 0,20 30,00 30,00 0,64 0,04 9,82 0,614 0,376239 0,008 0,51239 0,03202 0,13 0,100
3,40 0,20 30,00 30,00 1,10 0,10 5,71 0,519 0,405152 0,004 0,29812 0,0271 0,23 0,059
3,80 0,20 40,00 40,00 1,00 0,10 6,28 0,628 0,452156 0,002 0,32793 0,03279 0,16 0,128
4,40 0,20 40,00 40,00 0,46 0,01 13,66 0,297 0,507907 0,003 0,71290 0,0155 0,07 0,546
4,80 0,20 80,00 80,00 0,40 0,10 15,71 3,927 0,547528 0,026 0,81983 0,20496 0,03 0,3512,40 0,20 24,00 24,00 2,80 0,20 2,24 0,160 0,287687 0,039 0,11712 0,00837 0,73 0,010
1,20 0,20 10,00 10,00 2,20 0,10 2,86 0,130 0,143747 0,049 0,14906 0,00678 1,38 0,029
0,80 0,20 8,00 8,00 0,20 0,02 31,42 3,142 0,080368 0,032 1,63966 0,16397 0,16 0,1280,60 0,20 8,00 8,00 1,70 0,10 3,70 0,217 0,071788 0,029 0,19290 0,01135 1,34 0,015
0,42 0,02 4,00 4,00 0,14 0,02 44,88 6,411 0,036845 0,006 2,34237 0,33462 0,22 0,181
1
8/6/2019 LRC
12/18
II
7: LRC
12 18
1 3020 2980
Vc = f()
0,00
1,00
2,00
3,00
4,00
5,00
6,00
0,00 10,00 20,00 30,00 40,00 50,00
(rad/ms)
Vc(Volts)
Qo = f(/)
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,0 0,5 1,0 1,5 2,0 2,5 3,0
/
Qo(C)
1
2
8/6/2019 LRC
13/18
II
7: LRC
13 18
1 3020 2980
=f(/)
0,00
0,20
0,40
0,60
0,80
1,00
1,20
1,40
1,60
0,00 0,50 1,00 1,50 2,00 2,50 3,00
/
(rad)
, , , , , . , .
::
, 3( ), :
, ,
0 = , ,
1 0 = +
2 0 =
.
, :
( ) ( ) [ ]0 1 1
3 92 2
1 1
19,2 0,6822,7 10 120 10
rad msLC
= = =
3
8/6/2019 LRC
14/18
II
7: LRC
14 18
1 3020 2980
2 22 2
0 0
03 3
1 10 68
2 2 L C L C , rad ms
L C C L L C
= + = + =
3 3
3
173,33,82 10 0,35 10
2 2 45, 4 10
L R r r R rad
L L s
+ + = = = =
( )2 2 2 2
3
2
10,35 10
2 2
R rad R L R L
R L L L s
= + = + =
( ) ( )2 22
2 2
0,14 L L
L
R R R R R r r R r
R r r
= + + = + =
,
3 3
1 015,34 10 0,22 10
rad
s = + =
3 3
2 022,98 10 0,40 10 rad
s = =
( ) ( ) ( )( )2 2
22
0 0
0
i i
i
= + = +
, 1, 2i =
, :
30
3
19,16 105,02 0,45
2 3,82 10QF
= = =
( )( )
( )
22 2 2
0
0 0 2
0
10,45
2 2
QF QF QF
= + = + =
, QF
, ,:
8/6/2019 LRC
15/18
II
7: LRC
15 18
1 3020 2980
5,80 5,02% 100% 100% 13, 0%
5,80
QF QF
QF
= = =
::
1. LC( ), , .
, . , . .
2. LRC , R ;
.
3. (7.26) ,
0Q ,
0
, R ,
R , . , R , , . R
QF . ( , 2 =
20
I Q ).
:
0
02
2 1
VQ
R LC
=
+
2 0
0 2
2 2 1
VQ
R LC
=
+
8/6/2019 LRC
16/18
II
7: LRC
16 18
1 3020 2980
2 0
0
2 2 2 2
2 2
12
VQ
LR L
C C
=
+ +
2 2 2 2
2 2
2
0 0
12
1
LR L
C C
Q V
+ +
=
2 2
2 2
0 0
1 1 12
L A R L
Q V C C
= = + +
2 2 2
2 2 20
2 2 2
2 2
0
2 2
0
1 1 12
1 1 12
12 2
dA L R L L
d V C C C
L R L L
V C C C
LL R
V C
= + + + =
= + + + =
= +
2 2
0
2 2
2 2
2
2
2
2 2
0 2
10 2 2 0
2 2 0
2 2
1
2
2
dA LL R
d V C
LL RC
LL R
C
R
LC L
R
L
= + =
+ =
=
=
=
R > , 2 2
0
12 2
dA LL R
d V C
= +
, 2L
C
, .
8/6/2019 LRC
17/18
II
7: LRC
17 18
1 3020 2980
, 2
0
1
Q 2
0Q ,
0Q
.
R , :
2 2 2 2 2 22 2 0 2 2
L L L R R L
C C +
, R : 2 22 2L
R LC
=
QF:
2 2
22 2
2
2
0
1 1
2 2
1 1
1 22
1
1 2
L LQF QF
R C C LL
C
QF QF C L LC
LL C
QF
= =
= =
=
4. C L ;
.
5. L C , , L C .
, :
( )01
coscQ
V Q tC C
= = +
:
8/6/2019 LRC
18/18
II
7: LRC
18 18
1 3020
( )( )
( )( )
( )
( )( )
( )
220
2 2
0
2
0
2
0
0
cos
sin
cos
cos
cos
L
d Q td QV L L
dt dt
d Q tL
dt
LQ t
L Q t
Qt
C
+= = =
+= =
= + =
= + =
= + +
C =
L L C c = + = +
:
( ) ( )0 01 1
cos cos 0C L
V V Q t Q t C C
+ = + + =
7. C
V
0
V
, ( ) 0CV QF V = .
:
0
0
VR
LVc
=
0L R ,
C
V 0
V .
, :
LC
10
=
:
C
L
RQF
1=
, , :
( )0 0 01
C C C
L LV V V V V QF V
R C R LC= = =