ЛИНЕЙНЫЕ ЭЛЕКТРИЧЕСКИЕ ЦЕПИ ПЕРЕМЕННОГО ТОКА Часть II. Линейные электрические цепи переменного тока 978-5-7262-1004-9

Citation preview

(

)

. .

И

,

Ы Э

. .

И

И

И

II

«

»

2008 1

621.3.11.7(075) 31.211 7 18 . .,

Э. . Л II: ч

Ч

.:

И И,

2008. – 88 .

« »( И И 2004 .).

-

Э. .

. .

, : ;

; . .

щ

-

«Э

-

, «

». ,

щ

». И

И И. -

.

В. .

,

ISBN 978-5-7262-1004-9 ©М (

а

ы

2

-

ч ), 2008

И

C

............................................................................................................ 4 1. Э ............................................................ 5 1.1. .......................................... 5 1.2. ...... 7 1.3. К ....................................................................... 10 1.4. ..................... 11 .......................................................................................................... 14 2. ............................... 18 2.1. .................................................................... 18 2.1.1. ........................................ 18 2.1.2. Э ....................................................... 20 2.1.3. Ч ..................................................... 22 2.1.4. ..................................................... 25 .......................................................................................................... 27 2.2. ............................................................................... 30 2.2.1. ................................................... 30 2.2.2. Э ......... 37 2.2.3. Ч ..................................................... 39 .......................................................................................................... 42 3. ( ) ....................................................... 47 3.1. ........................................................................ 47 3.2. К ................. 50 3.3. ............. 56 .......................................................................................................... 57 3.4. Х ( ) ................................ 65 3.5. ( ) ................................................................ 74 .......................................................................................................... 77 ......................................................................................... 87 3

; ; . -

,

, -

, . ,

щ

,

-

,

-

,

. , , , -

, . , -

. , ,

,

. ,

ё

, .

4

А А 1.

Э

Т

Т

к

1.1.

А

А

я Э

.Э

,

-

, .

.1

.1

(

),

-

. i1 ,

.

i2 . .1

,

щ 11

i1

–

-

,

щ

,

-

,

щ

,

-

i1 ; 22

-

. :

–

–

i2

i2 ; 5

12 –

21

,

щ – щ

i1 ,

; ,

i2 ,

. i1 :

, =

1

11

,

+

12 .

(1.1)

,

-

i2 : =

2

22

+

21 .

(1.2)

, (

(1.1)

) =

1

И,

1

,

±

21 .

(1.3)

±

12 .

(1.4)

: =

2

«+»

)

2

(1.3) (1.4) «–» – (

,

. -

: 1

, L1 ,

(1.2)

:

= L1i1 ,

2

= L2 i 2 ,

(1.5)

L2 –

.

-

: 12

=

⋅ i1 ,

12

21

=

21 ⋅ i 2 .

,

12 12

Э

. , e1

=

21

=

(1.6) 21

:

.

(1.7)

(1.3)–(1.7) : =−

d

1

dt

= − L1 6

di di1 ∓M 2 ; dt dt

(1.8)

=−

e2

d

1

= − L2

dt

(1.8)

di 2 di ∓M 1 . dt dt

(1.9) Э

(1.9)

-

: e1 L = − L1

di1 , dt

e 2 L = − L2

(1.8)

di 2 . dt

(1.10) Э

(1.9)

: e1 M = ∓ M

di 2 , dt

di1 . dt

e2 M = ∓ M

(1.11)

, (1.9)

(1.8)

, щ

-

: u1

= L1

di di1 ±M 2 , dt dt

(1.12)

u2

= L2

di di 2 ±M 1 . dt dt

(1.13)

(1.8), (1.9), (1.12)

(1.13)

, (

).

L1 ,

L2 , M

. ч

1.2.

к щ . (

) . (1.12)

«+» ( . 2, ).

, y(

(1.13)

,

. 2),

,

y

, ,

. 7

.2

«–»

(1.12) (

(1.13)

, i1 –

, , i2 –

.2, ).

,

. . 3.

.3 8

y , -

-

( i1 = i 2 = i ) .

u12

. 3, : u12 = ( L1 di + M di ) + ir1 + ( L2 di + M di ) + ir2 = dt dt dt dt = i ( r1 + r2 ) + ( L1 + L2 + 2 M ) di dt .3 : u12 = ( L1 di − M di ) + ir1 + ( L2 di − M di ) + ir2 = dt dt dt dt = i ( r1 + r2 ) + ( L1 + L2 − 2 M ) di . dt

(1.14)

, ,

-

–

. (

щ

)

: i ( t ) = I m sin( ω t + ϕ ) .

:

i (t )

i ( t ) = Im [ Ie jωt ] , I = I m e jϕ .

I– u = M di dt

щ

: U = jωM I . jωM

-

, . ,

. 3, щ

1–2 . 9

. 3, : U 12 = ( jωL1 I + jωMI ) + r1 I + ( jωL 2 I + jωMI ) + r2 I =

(1.15)

= ( r1 + r2 ) I + jω ( L1 + L 2 + 2 M ) I .

.3 : U 12 = ( jωL1 I − jωMI ) + r1 I + ( jωL2 I − jωMI ) + r2 I = ( r1 + r2 ) I + jω( L1 + L2 − 2 M ) I .

1.3.

фф

я щ

щ

K ,

(

.

.1): K

12

=

21

⋅

1

.

(1.16)

2

, , . 21 = Mi 2 , (1.16)

1 = L1i1 , :

K

И 12

2

=

(1.16) ≤ 1, 21 ≤

,

12

, = Mi1 ,

= L2 i 2 Mi1 Mi2 ⋅ = L1i1 L2 i2

,

M . L1 L2

(1.17) , -

K

2.

,

K

: 0≤ K

≤1.

,

(1.18) 1,

K

-

. 4, . Э .

,

0,

K 10

. 4, , .

.4

щ

(L1+L2–2M), .

, (1.15),

(1.14)

-

:

(

L1 − L2

L1 + L 2 − 2 L1 L 2 ≥ 0 ,

)2 ≥ 0 L1 + L2 ≥ 2 L1 L2 .

L1 L2 = M . K

И (1.17)

(1.19)

(1.19):

L1 + L2 ≥ 2 M . Kc K

≤1 ,

(1.20)

(1.20)

,

:

L1 + L2 ≥ 2 M .

: L1 + L 2 − 2 M ≥ 0 .

1.4. Т щ

ф

.5

ф

ч к ,

щ

-

. ,

( ). 11

,

,

-

.5

щ

:

U 1 = ( r1 + jωL1 ) I 1 − jωMI 2 , − U 2 = ( r2 + jωL2 ) I 2 − jωMI 1 .

(1.21) jωMI 2

,

U1 =

:

[ r1 + jω( L1 − M ) + jωM ] I 1 − jωMI 2 ,

− U 2 = − jωMI 1 + [ r2 + jω( L2 − M ) + jωM ] I 2 .

(1.22)

(1.22) , ,

щ

I1

I2.

. 6.

.6

щ

, (

) 12

-

. ,

.5

-

. , (

)

-

щ

, , .

« »

-

Z ,

.

. 5,

,

E (1.21):

,

U 1 = ( r1 + jωL1 ) I 1 ± jωMI 2 , 0 = ± jωMI 1 + ( r2 + jωL 2 + Z ) I 2 .

(1.23)

(1.23) (

(

И

(1.23) I2 =

+), «–»).

:

∓ j ω MI 1 . r2 + jωL2 + Z

(1.23)

I2

:

Z1

=

U1 I1

= r1 + jωL1 +

( ωM ) 2 . r2 + jωL2 + Z

(1.24)

(1.24) ,

.Э

-

. 7, 7,

7, . (1.24),

( Z1

)

, : Z

(1.24)

. 13

)

) .7

, ( ).

З

ч 1.1

.8

, :

.8. .

2, 3

щ

: 14

1,

I1 − I 2 − I 3 = 0 , I2 − I6 − I4 = 0, I3 + I6 − I5 − J = 0 .

1, 2, 3 ( ) щ : r1 I 1 + jωL1 I 2 − jωMI 4 + jωL 2 I 4 − jωMI 2 = E , 1 I − r I − ( jωL I − jωMI ) = 0 , 1 2 4 jωC1 3 2 6 r6 I 6 + 1 I 5 − ( jωL 2 I 4 − jωMI 2 ) = 0 . jωC 2 L1 L2 . З

-

-

ч 1.2

.9

,

.9,

-

: E , M, x1 , x 2 , r1 , r2 , ω, r . I1

I2.

: .9

.

x1 r1

-

x 2 r2 15

; (

x2 –

x1

. 10).

. 10

(

1

2

):

r1 I 1 + [ jx1 I1 − jωM ( I1 + I 2 )] + r2 I 2 + [ jx 2 ( I 1 + I 2 ) − jωMI 1 ] = E ; − r I 2 − [ r2 ( I 1 + I 2 ) + jx 2 ( I1 + I 2 ) − jωMI 1 ] = 0 .

З

ч 1.3

. 11

. 11 U , x1 , x 2 , x c , K .

.И ,

x1 = x 2 = x c = x .

:

r .

I2

: щ x1 = ωL1 ,

x1 x 2 = ωL 2 ,

x2 – xc –

xc = 1 . ωC

– ,

x1 , x 2 16

K :

K

=

M = L1 L2

ωM ωM , = ωL1 ⋅ ωL2 K x1 x 2

x1 = x 2 = x , ωM = K x .

(1.25) ,

щ

:

U = I 1 jx1 − I 2 jωM − ( I1 − I 2) ⋅

xc . j

, ωM :

x1 = x 2 = x c = x ,

,

I1 − I 2 . K x (1.25)

-

U = − I 2 jK x + I 2 jx .

: I2 =

U . jx (1 − K )

, ,

-

r .

17

А А 2.

А

Ы

Ы Э

Т

я

2.1. 2.1.1.

Э

-

щ

, .

: Im Z = 0 ,

(2.1)

Z И (

. r, L, C,

. 12).

. 12

1 ⎞ Z = r + j ⎛⎜ ωL − ⎟. ωC ⎠ ⎝ (2.1) ωL −

И

1 =0. ωC

:

, (2.2) -

, . .

, : 18

ω0 =

1 . LC

(2.3) -

щ

I =| I |=

:

| E | E . = | Z | r + j ⎛ ωL − 1 ⎞ ⎜ ⎟ ωC ⎠ ⎝

Im Z = 0 , |Z|

,

-

r, I 0: I0 =

E . r r →0 . Э

-

, . : U L0 = jω 0 LI0 = ω 0 LI0 e

U C 0 =

j

π 2,

π

1  1  − j2 I0 = I 0e . jω 0 C ω 0C

,

(2.4)

(2.4)

U L0 = −U C 0 , , U L

«

. 13. ,  UC

-

»

.

. 13 19

(2.2) :

U L0 = −U C 0

,

-

L – C :

U L −C 0 = U L0 − U C 0 = 0 .

,

L

щ

C)

,

( щ

,

,

-

.

, , . 12, ( Э

щ

-

. 14). -

(2.2),

щ : . 14

щ

.

2.1.2. Э

(

. 12)

.

i(t ) = I m cos ω 0 t . , : π U C (t ) = U C m cos ⎛⎜ ω 0t + ⎞⎟ = U C m sin ω 0t . 2⎠ ⎝

: LI m2 cos 2 ω 0t = W Lmax cos 2 ω 0 t , 2 2 CU Cm sin 2 ω 0t = WC max sin 2 ω 0t . WC (t ) = 2 W L (t ) =

20

(2.5)

, W Lmax = WC max =

2 CU Cm 2

.

,

:

ΔW LC max = W Lmax − WC max =

LI m2 2

−

CU 2

ω 0C ω0 L =

1 , ω 0C

,

m

=

1 2 ⎡ω 0 LI m2 − ω 0CU Cm ⎤. ⎣ ⎦ 2ω 0

-

:

U L U C =| U L | , U Cm

U Lm

2

,

C⎡ ω 0 LI m2 ⎢ ⎣ 2 , щ =| U | ,

(

ΔW LC max =

U Lmax

LI m2 2

)

2

2 ⎤ − U Cm . ⎦⎥

(2.6) , : U L = −U C . ,

C

U Lm = U Cm . = ω 0 LI m ,

,

-

,

ΔW LC max = 0 ,

. . 15

WL

W

-

)

.

t. . 15 ( : W LC = W L + WC =

2 LI m2 LI 2 CU Cm ⎡cos 2 ω 0t + sin 2 ω 0t ⎤ = m = . ⎦ 2 ⎣ 2 2

21

. 15

, , «

»

, C,

L

, .

щ

, ,

.

(r = 0) .

-

2.1.3. Ч

(

.

щ

.12)

:

1 ⎞ 1 ⎤ ⎡ω Z = r + j ⎛⎜ ωL − ⎟ = r + jω 0 L ⎢ − ⎥= ωC ⎠ ⎝ ⎣ω 0 ωω 0 LC ⎦ ⎡ω ω ⎤ = r + jω 0 L ⎢ − 0 ⎥ . ⎣ω 0 ω ⎦ 22

(2.7)

, E = Z ⋅ I ,

Z =| Z | e jϕ ,

φ – .

(2.7): ϕ = arg tg

( .

-

ω0 L ⎡ ω ω0 ⎤ . − r ⎢⎣ω 0 ω ⎥⎦

(2.8)

)

-

Э . 16.

φ (ω)

. 16

φ (ω) ( ω < ω0

. 14.

щ

ω = ω 0,

-

),

φ ,

ω > ω0

,

φ -

, . (

)

щ

: 23

.

I=

UL =

E

1 ⎞ r 2 + ⎛⎜ ωL − ⎟ ωC ⎠ ⎝ E ωL

2

;

; 1 ⎞2 ⎛ r + ⎜ ωL − ⎟ ωC ⎠ ⎝ 1 E ωC . UC = 2 1 ⎞ r 2 + ⎛⎜ ωL − ⎟ ωC ⎠ ⎝ 2

. 17

.

. 17

,

E , r

,

(ω = ω 0 ) . U L =UC . ,

ω

щ

ω

щ

.Э , . 24

, -

И

(2.1)

, ,

-

. . 16, 17. . 18 .

. 18

L=L

=

1

-

ω 02C =

.

=

1 ω 02 L

.

2.1.4.

щ

, Q = ω0

Wmax . P

25

–

-

: (2.9)

Wmax –

, ,P–

-

щ

. Wmax = W Lmax = WC max =

2 LI 02 CU Cm . = 2 2

P=

I 02 r . 2

, :

щ

L ω0 L ρ 1 Q= = = C= , r r r ω 0Cr L C . (2.10)

ρ=

И

(2.10) -

, ω0 L

1 ω 0C

r. U L0 = −U C 0 = jω 0 LI0 ,

, (2.10),

I0 =

:

E U L0 = −U C0 = jω 0 L = jEQ . r

И (2.11)

(2.11)

, Q=

U L0 E

=

U C0 E

.

(2.12)

,

-

L . . 19 , Q1 > Q 2 .

E r

,

. 19, 26

«

-

» .

. 19

З

ч 2.1

=1[

. 12 r = 10 [ ], L = 1 [ ω0 ,

].

],

Q, UC,

10

.

: (2.3) 1 = 10 3 / . LC ω L Q = 0 = 100 . r UC = Q⋅ E =1 . ω0 =

З

(2.10) (2.12)

ч 2.2 ,

. 20, f 0 = 50

. : U = 220

щ , U rL = 204 , U C = 180 – r, L, 27

, I=4

. -

. r1 .

. 20

: UC =U L

1. L=

UC = 0,143 2 πf 0 I

. UC = I

2. I = 70,8 2π f 0U C 3.

C=

2 ⎡U ⎤ r = ⎢⎛⎜ rL ⎞⎟ − ( 2πf 0 L) 2 ⎥ ⎣⎝ I ⎠ ⎦ (

4.

U r1 = − r = 31 I

U = I (r1 + r )

1 , ω 0C

.

U rL = I(r + jω 0 L) .

З

ω 0 LI = U L ,

1/ 2

= 24

. .2

3)

41

,

.

ч 2.3 f = 50 – 9

.

-

,

C = 51

? 28

: (

r

L)

-

: Z = r + jω L .

f = 50

41 2

41 = r + (ωL) ,

|Z| = 41 , 1 = | Z | 2 − r 2 = 0,127 2 πf = 62,5 .

З

, . .

2

L= f0 =

.

1 = 2π LC

ч 2.4 Э R = 16

-

E = 1,6 r

. -

щ ? :

-

r. -

Э (

. 21

. 21).

щ

, *

Z =Z ,

,

Z

Z

. Z = R,

щ

Z =r,

r = R = 16

: 29

щ-

I Ig =

E

2 ⎛ E ⎞ P = I g2 ⋅ r = ⎜ ⎟ ⋅r , ⎝R + r⎠ щ

–

I E х , Eg = х. 2 2

P = 20

Э

:

. к

2.2. 2.2.1.

Э

,

-

, (

),

(

). : Im Y = 0 ,

(2.13)

Y–

(

щ

-

). (

. 22)

.

-

r( 1 g = ). r

,

L,

–

C.

. 22 30

,

:

1 1 + jωC = g − j ⎛⎜ − ωC ⎞⎟ . j ωL ⎝ ωL ⎠ 1 − ωC = 0 . Э , : ωL 1 щ ω0 = . , LC

Y =g+

(

.

(2.3)).

,

Y (ω 0 ) = g !

IC ,

щ

-

U I , I , g L

, , I = I g + I L + IC (

щ

(2.13)

). ,

,

ω. U 1 − ωC ⎞⎤ : I = U ⋅ Y = U ⎡⎢g − j ⎛⎜ ⎟ . ⎣ ⎝ ωL ⎠⎦⎥ Y (ω 0 ) = g :

Y

I = U ⋅ g .

Э

,

I

,

U

щ

. ( . 31

-

(2.14)

, щ

-

)

-

щ

:

1 I =| I |=| U | ⋅ | Y |= U g − j ⎜⎛ − ωC ⎟⎞ . L ω ⎝ ⎠ Im Y = 0 , , I0 =U ⋅ g

|Y| = g. Э ( ω ≠ ω0 ) . (2.14)

,

,

-

I = I g , . .

,

щ .

1 = yL = jωL =−j

1 = − j ⋅ bL , ωL

bL =

y C = jωC = j ⋅ bC ,

1 – ωL

.

bC = ωC –

,

.

, π −j ⋅e 2 ,

IC = U ⋅ y C = U ⋅ jωC = ωCU ⋅ e

1

IC

j

(2.15)

π 2.

I L

, ,

I L

-

:

1 1  I L = U ⋅ y L = U ⋅ = ⋅U jω L ω L

Э

-

IC

-

1 = ω 0C , ω0 L

-

.

-

щ

1

:

ρ=

« , . 32

L », C

-

I L = − IC = −ω 0CU ⋅ e

j

π 2

=−

. 23. Э

щ

,

U L C

⋅e

j

π U j 2 =− e . ρ

π 2

:

I0 = I g + ( I L + IC ) = I g , (I L + IC ) = 0 .

(2.15), (2.16) , IL

. 23

-

-

IC

Ig . Э

I0

I L = IC =

, | U | L C

(2.16)

>>

| U | = I g = I0 . r

L ρ , ⎨ ⎩r2 > ρ

>0

⎧ r1 < ρ . ⎨ ⎩r2 < ρ

r 1 = r2 ≠ ρ , r1 = r2 = ρ ,

, ω p = ω0 . Im Y = 0 (

щ

ωp

,

b(ω) = 0 )

,

-

Z =ρ.

(2.19) r2 = 0

,

. 26:  p = ω0 ω

( (

.

27

. 25 28),

ρ 2 − r12 ρ2

= ω0 1 −

r12 ρ2

.

. 26) , . Y p = g (ω p )

щ . 25,

,

(

-

,    . 26): I p = U ⋅ Y = U ⋅ g (ω p ) .

И ωp

, rp =

(2.19)

ωp

1 . g (ω p )

-

(2.18)

-

, (

)

: g (ω p ) =

r1 + r2

rp =

r1r2 + ρ 2 36

r1r2 + ρ 2 . r1 + r2

(2.18) Im Y = 0

, ωp,

|Y|

щ

ωm ,

ω p ≠ ωm . ωm ,

I щ

(

-

ωp. Э

. 26) . . 23),

-

( Im Y = 0

. 25 -

min | Y |

ω0 .

2.2.2. Э

Э ,

. 22,

R–L–C . I L = − IC . Э

, (

π).

iC (t ) =

i L (t )

iC (t )

: = I C cos(ωt + ψ ) .

: i L (t ) = I L cos(ωt + ψ − π) = − I C cos(ωt + ψ ) . u (t ) :

щ

p L + p C = u (t ) ⋅ i L (t ) + u (t ) ⋅ iC (t ) = u (t )[i L (t ) + iC (t )] = = u (t )[−iC (t ) + iC (t )] = 0.

:

u (t ) = U sin(ωt + ψ ) . ,

,

:

37

u (t ) = -

π iC (t ) = I C sin ⎛⎜ ωt + ψ + ⎞⎟ = I C cos(ωt + ψ ) , 2⎠ ⎝ π i L (t ) = I L sin ⎜⎛ ωt + ψ − ⎟⎞ = − I L cos(ωt + ψ ) . 2⎠ ⎝

(

,

uC (t ) = u L (t ) = u (t ) ): W L (t ) =

Li L2 LI L2 = cos 2 (ωt + ψ ) = 2 2

(2.20)

= W L cos 2 (ωt + ψ ) = W L [1 + cos 2(ωt + ψ )], WC (t ) =

Cu C2 CU 2 = sin 2 (ωt + ψ ) = 2 2

= WC sin 2 (ωt + ψ ) = W [1 − cos 2(ωt + ψ )].

,

W L (t )

.И p L = − pC ,

,

WC (t ) p L + pC = 0

,

, d d (W L (t )) = − (WC (t )) . dt dt

(2.21)

(2.20) (2.21)

: W L [− sin 2(ωt + ϕ)] ⋅ 2ω = −WC [sin(2ωt + ϕ)] ⋅ 2ω .

щ

: W L = WC . ( W L (t )

ψ = 0) . 38

.15, WC (t ) ( ,

.

2.1.2). -

. ,

( ) ) (

, щ

),

(

щ

-

.

g. 2.2.3. Ч

(

. 22)

:

⎛ 1 ω⎞ 1 − ωC ⎟⎞ = g − jω 0C ⎜ − Y = g − j ⎜⎛ ⎟= ⎝ ωL ⎠ ⎝ ωω 0CL ω 0 ⎠

,

⎡ω ω⎤ = g − j ω 0C ⎢ 0 − ⎥ = Y ⋅ e − jϕ . ⎣ ω ω0 ⎦ φ– ( , ϕ(ω) = arctg ϕ(ω)

щ

ω 0 C ⎡ω 0 ω ⎤ . − g ⎢⎣ ω ω 0 ⎥⎦ . 27.

. 27 39

) :

ϕ(ω 0 ) = 0 . ϕ > 0,

ω < ω0

, ( b L > bC ).

ϕ ω0 ,

. 28.

. 28

Ig =U ⋅ g ,

:

IL =

I C = U ⋅ ωC .

U , ωL -

. 29. ω0

щ

–

.

I,

U( )

IL, IC, Ig

(

): 40

,

-

. 29

U (ω) =

I

2 1 g + ⎜⎛ − ωC ⎟⎞ ⎝ ωL ⎠

;

2

I⋅ I L (ω) =

I C (ω) =

1 ωL

2 1 g 2 + ⎜⎛ − ωC ⎟⎞ ⎝ ωL ⎠

I ⋅ ωC

2 1 g + ⎜⎛ − ωC ⎟⎞ ⎝ ωL ⎠

;

.

2

. 30 (

.

щ

. 30)

-

, . 17.

щ

щ U (ω) , I L (ω)

, ( U C (ω) , I C (ω)

) U L (ω) . 41

: I (ω)

-

. 30

З

ч 2.5 ,

U = 1,4 , : r = 50 ω0 , I g , I L IC И

( , L = 0,25

,

= 2,5

. 22),

.

-

ρ .

: (2.13)

-

: ω0 =

1 LC

1

=

ρ=

0, 25 ⋅ 10

−3

⋅ 2,5 ⋅ 10

−6

= 4 ⋅ 10 4

L 0, 25 ⋅ 10 −3 = = 10 C 2,5 ⋅ 10 −6

/ .

. :

1 1 I g = U ⋅ = 1, 4 ⋅ = 0,028 A = 28 50 r 42

,

U = 0,14 A = 1, 4 ⋅ 10 2 L C , IL ( Ig = I

I L = IC =

.

-

IC )

r

-

ρ З

I L IC r = = = 5. I I ρ

ч 2.6 (

. 25) ωp,

. : r1 = 8

, r2 = 6 ω0 ,

, L = 0,25

, C = 2,5 щ

. .

:

L

,

щ ω 0 ρ.

(2.19) : L − r12 ρ2 − r 2 100 − 64 C ωp = = ω 0 2 12 = ω 0 = L 100 − 36 LC ρ − r2 − r22 C 3 3 / = 3 ⋅ 10 4 / . = ω 0 = ⋅ 4 ⋅ 10 4 4 4 1

3 ω p = ω0 < ω0 . Э 4 , r1 < r2 < ρ

,

,

r2 < r1 < ρ . , ω p > ω0 . 43

C .

-

З

ч 2.7 2.6, Y1 = g 1 − jb1

Y2 = g 2 + jb2 . ,

-

u (t ) = 1, 4sin (ω p t ) [ ].

: (2.18)

-

ωp: Y1 = g 1 − jb1 =

r1 r12

+ (ω p L )

2

ω pL

−j

r12

+ (ω p L)

2

=

32 30 −j , 481 481

1 Y2 = g 2 + jb2 =

r2

⎛ 1 ⎞ r22 + ⎜⎜ ⎟⎟ ⎝ ω pC ⎠

2

+j

ω pC

⎛ 1 ⎞ r22 + ⎜⎜ ⎟⎟ ⎝ ω pC ⎠

2

=

27 30 +j . 962 481

щ

-

: Y = Y1 + Y2 = g 1 + g 2 =

щ ,

, )

91 7 = [ 962 74

–1

щ

U

].

, u. Э

( I

-

,

u (t) 7 : I = U ⋅ Y = 1, 4 ≈ 132 74

. U = 1, 4 ⋅ e j 0 = 1, 4 ,

-

щ

щ :

32 30 ⎞ I1 = U ⋅ Y1 = 1, 4 ⎛⎜ −j ⎟ ≈ 93 − j87,3 481⎠ ⎝ 481 44

.

,

27 30 ⎞ I2 = U ⋅ Y2 = 1,4 ⎛⎜ +j ⎟ ≈ 39 + j87,3 962 ⎠ ⎝ 962

. 31 I2

I1 I1A = 93

щ

, щ

: I1 = I1А + I1 p , I2 = I2 A + I2 p .

. I = − j87,3 1p

I2 p = j87,3

щ

щ

-

.

, ( щ

I1

I2 ) :

-

I1 p = − I2 p ,

. 31

I1A + I2 A = I .

З

щ

-

I2 A = 39

, щ

.

ч 2.8 (

.И-

.26) I2 = 12

щ

I =5

.

I1 .

: u (t ) = U sin (ω p t ) , U = U ⋅ e j 0 . 45

-

: I2 = U ⋅ jω p C =

I2 = U ωpC ⋅ e

jπ

2

. (

щ

(2.18))

:

Y = Y1 + Y2 =

⎛ ⎞ ωp L j C − − ω = ⎜ ⎟ p ⎜ r 2 + (ω L )2 ⎟ r 2 + (ω p L )2 p ⎝ ⎠ r = 2 = g (ω p ). r + (ω p L )2 r

: I = U ⋅ Y = U ⋅ g (ω p ) . , I U , I 2

U

щ . 32,

. ,

U

щ

, ,

I = I

, I2 = I2

. 32

: I = I1 + I2 (

. I1

I1

: I1 = I − I2 . И

) : I1 =| I1 |= | I |2 + | I1 |2 = 52 + 122 = 13

46

-

.

-

А А 3.

Ы

Ы А

(

Т

к

3.1.

щ

Ы Э Т А ТА)

к

,

-

(

-

, , .

(

)

,

). –

( ,

). И

( )

(

-

. 33),

.

-

, ,

:

– (

. 33, ), )

(

, . 33, ).

,

( -

. 33

. 34 .

щ .

47

, . 35

,

-

. 34

. 35

( = C1 .

Ca =

(

),

2

)

-

C1 ⋅ C2 . C1 + C2

,

/

, .

tK = 0 . И Δ tK = 0 . tK = 0 , . . t 3τ = 15 [

( ,

].

. 66) Э

:

-

. IC = 0 ,

: i1

=

E 2, 4 = = 0, 05 [A] . r1 + r3 8 + 40 82

. 66

. 67

4. И

: i1 = A ⋅ e A

−

t τ

+ 0, 05 .

-

.

i1 (0)

, –

uC (0+ ) =

uC (0+ ) . И ).

= uC (0−) = 0 (

t =0+ « i1 (0+ ) : i1 (0+ ) =

»(

. 67).

E 2, 4 = = 0,15 [A] . r1 + r2 || r3 8 + 10 ⋅ 40 10 + 40

А:

Э i1 (t = 0) = A ⋅ e 0 + 0,05 = i1 (0+ ) = 0,15 .

: A + 0, 05 = 0,15 → A = 0,1 [A] . = 0,1 ⋅ e i1 (t ) ,

−

t τ

i1 (t < 0) =

+ 0, 05 [A] .

. 68

E = 0,05 [A] . r1 + r3 83

, i1 (t ) =

. 68

З

ч 3.8 u1 (t )

E = 2, 4 [B]; L = 12 [

. 69, ]; r2 = r3 = 4 [

]; r1 = 2 [ 1.

,

, :

].

( -

щ

): u1 (t ) = B ⋅ e

τ

t τ

. ( . 70) E =0, « »

, λL .

,

= r1 + r3 ,

(3.8): τ=

+ u1

2.

. 69

R

−

L L 12 ⋅ 10 −3 = = =2[ R r1 + r3 2+4

84

].

. 70

t > 3τ = 6 [ I

3. ( . 71) UL = 0 , u1 : u1

= I ⋅ r1 =

E ⋅ r1 = r1 + r3

2, 4 ⋅ 2 = 0,8 [B]. 2+4

.

4.

:

u1 (t ) = B ⋅ e

−

t τ

]. -

+ 0,8 ,

. 71

B.

1 E iL (0+ ) = iL (0−) = ⋅ = 0,3 [A] 2 (r1 + r2 || r3 ) ( . 72) t =0+ ( 3.2). , u1 (0+ ) = J L 0 ⋅ r1 = iL (0+ ) ⋅ r1 = 0,3 ⋅ 2 = 0,6 [B] .

: B:

u1 (t = 0+ ) = A ⋅ e 0 + 0,8 = u1 (0+ ) = 0,6 [B] .

: А + 0,8 = 0,6,

А = 0,2. 85

. 72

u1 (t ) = 0, 2 ⋅ e .73.

, : E ⋅ r1 = 1, 2 [B] u1 (t < 0) = r1 + r2 || r3

−

t τ

+ 0,8 ,

.

. 73

86

-

1.

.И. .

.: Э

2.

. . , 1978. 3. И . ., . .I. , 1976. 4. .И. 5. . ., 1998. 6. .: 7.

И И, 1987. . ., . .:

.

.I.

.

.:

-

, 1978.

.И.

-

. .Э

.

/ И И, 2004.

.:

И И, 1984. . .: Э

. . . Э. .

87

, И. .

, щ

. -

И

Ы Э

И

И

И

II

.В. Ш -

. . 5,5

(

я«

.В.

01.12.2008 .. . 5,5 . № 4/89 № ) 115409, », . , 88

60×84 1/16 150 .

,

., 31 .