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= = =