Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hbar.phys.msu.ru/hbar/pages/vyat/conswork.pdf Äàòà èçìåíåíèÿ: Mon Mar 20 00:00:00 2006 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:15:58 2012 Êîäèðîâêà:

. M.B.

25 2005 .





""

(2 , 4- )

2005


1


, . . . 2- . . . . , . . . , , . . . , (. 3-30 ) (vyat@hbar.phys.msu.ru). : http://hbar.phys.msu.ru




2


1 2 . 2.1 . . . . . 2.2 . . . . . . . . . . . . . . . . 2.3 2.4 . . . . . . . . . . . . 2.5 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 6 7 7 9 9 11 11 12 13 13 15 16 17 19 19 20 20 22 22 22 23 23 23 23 24 27 27 28 28 30 31 32 35 36 37 38 39 40 40 41 42 42 43 45 45 46

3.1 . . . . . . . . . . . . . . . . 3.2 . . . . . . . . . . . . . . 3.2.1 K() . . . . . . . . . . . 3.2.2 h(t) . . . . . . . . . . 3.2.3 . . . . . . . . . . . . . 3.3 K(), h(t), g(t) . . . . . . . . . . . . 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 . . . . . . . . . . . . . . . . . . . . . . . . 3.6 . . . . . . . . . . . . . . . . 3.6.1 3.6.2 3.7 . . . . . . . . . . . . . . . . . . 3.7.1 . . . 3.7.2 . .

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4 4.1 . . . . . . . . . . . . . . . . . . . . . 4.1.1 . . 4.1.2 . 4.1.3 . . . . . . . . . . . . . . . . . . 4.1.4 : - . . . . . . . . . . . . . . . . . . 4.1.5 : . . . . . . . . . . . . . . . . 4.2 . . . . . . . . . . . . . . 4.3 . . . . . . . . . . . . . . . . . . . . . . . 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 . . . . . . . . . . . . . . . . . . . . . . . 5.3 . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 - . . . . . . . . . . . . . . 5.3.2 ("") . . 5.3.3 - 5.4 . . . . . . . . . . . . . . . . . . . . . . 5.5 - . . . . . . . . . . . . . . . 5.6 - . . . . . . . . . . . . . . . . . . 5.6.1 - . . . . . . . . . . . . 5.7 . . . . . . . . . . . . . . . . . 5.8 ( ) . . . . . . . . . . . .


6 6.1 . . . . . . . . . . . . . . . 6.1.1 . . . . . . . . . . . . . . 6.2 . . . . . 6.2.1 6.3 . . . . . . . . . . . . . . . . . 6.4 . . . 6.5 . . . . . . . . . . . . . . . . . . . . . . . 6.6 N . . . . . . . . . . . . . . . . 7 7.1 . . . . . 7.2 p-n 7.3 . . . . 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 46 47 47 48 49 50 50 52 53 54 54 55 57 58 59 59 59 60 61 62 64 64

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8 8.1 . . . . . . . . . . . . . . . . . . . . . 8.2 8.3 . . . . . . . . . . . 8.4 . . . . . . . . . . 8.5 . . . . . . . . . . . . . 8.6 . . . . . . . . . . . . 8.7 . . . . . . . . . . .

....... ....... ....... ....... ....... .......

9 65 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9.2 (FET) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10 10.1 . 10.2 10.3 . . . . . . . . . . . . . . . . . . . . 10.4 . . . . 10.5 . . . . . . . . . . . . . . . . . . . . . . 10.6 . . . . . . . . . . . . . 10.7 . . . . . . . . . . . . . . . . . . 10.7.1 . . . 10.7.2 (, ) . . . 10.8 . . . . . . . . . . . . . . . . . . . . . 10.8.1 . . . . . . . . . . . . . . . 10.8.2 . . . . . . . . . . . . . . 10.8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 . . . . . . . . . . . . . . . . . . . 10.9.1 . . . . . . . . . 10.9.2 . . . . . 11 11.1 . . 11.1.1 . . 11.1.2 . . . . . . 11.1.3 - . . . . . 11.1.4 . . . . . . . . . . . . . . 11.2 11.3 . . . . . . . . . . . 11.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 69 70 70 71 72 73 74 75 76 77 78 78 79 79 79 81 82 82 83 83 84 87 87 88 88


11.3.2 . . . . . . . . . . . . . . 11.3.3 - . . . . . . . . . . . . . . 11.4 . . . . . 11.4.1 11.4.2 11.4.3 . . . . . . ...... ...... ...... ...... ...... . . . . . ........ ........ ........ ........ - ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 90 92 92 92 93 94 95 95 96 97 98 98 101 102 104 104 106 108 110 110 111

12 12.1 . . . . . . . . . . . 12.2 LC- . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 . . . . . . . . . . . . 12.2.2 . . . . . . . . . . . . 12.3 RC- . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 . . . . 12.5 . . . . . . . . . . . . . . . . . . . . . . . . 12.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 . . . . . . . . . . . 12.8 12.9 . . . . . . . . . . . . . . . . . . . . . 12.10 . . . . . . . . . 12.10.1 . . . . . . . . . . . . . . . . . . . . . . . . 12.10.2 . . . . . . . . . . . . . . . . . . . . . . .


1

5

1



"" radiare ­ , . , . , , . , , . ( ) ( "" ). . . . : " , , , , ; . . . , " 1. : 1. . 2. . 3. . 4. . 5. . , , "" . , , : 1. ( ). 2. .. . 3. , , -. 4. , , . . . . , , : . . . . . . . . . . . . . . . .

, : 1. , , . 2. . 3. . , : 1. . 2. .


2 .

6

1831 1873 1887 1895 1897 1905 1906 1907 1914 1930 1935 II 1948 1954 1958 1960

. . . . . . . . , , .

1: - .. , -" " - - .. - . . ( ) - . () - - - - - - - . - - - ( 2000 .) -

3. . 4. . , . (, ): 1. , : ¯ h T ¯ T , ¯ ­ , , T h h . : ¯ h T . 2. , , 10 -2 , L , : L L . : L 10-2 .

2
2.1

.


L ( ), c , T (T = 1/, ). : L c T L 1, = c (1)

. : = 50 6 000 . 30 . (600 ) (1) . : = 100 (FM ) 3 . (1) ( ).

, , . L (2)


2 .

7

R replacements

C

L

. 1: : C, R L , : C, R L . . 1). , , , , , . , . 2.

C replacements r
p

p

L

p

r

p

R

p

. 2: , .

2.2



: dU = R = const, (3) dI d = L = const, (4) dI dQ = C = const. (5) dU . , I U: I = U/R, I: = LI. , (3, 4, 5) I, U, Q, . . , . , . 3 () . , I < 1 U < 0.01 ( I 5 ). . 3: . e2 /¯ (e , ¯ h h ) 20 .

2.3



. .


2 .

8

I 10 mA

replacements 0.1

U

1/R replacements U I 10 mA 0.1 N
2e2 ¯ h

B>0 U 50

20 . 3: : . : (N ). R : UR = IR R, [R] = ; G = 1/R, [G] =
t t

W P

R

=
0

I2 R dt = R
0

U2 R dt = R

t

IR UR dt,
0

R

= J2 R = R

U2 R = U R IR . R , WR

IR , , UR , G PR , . C : QC = UC C, [C] = (), q IC = ddtC = C dUC dt

U W

C

C

IC (t) d + UC (0), C 0 CU2 (t) CU2 (0) C C = WC (t) - WC (0) = - 2 2 =

t


2 . QC , IC UC , , W . L : L = LIL , [L] = (),

9

C

I W

L

L

UL (t) d + IL (0), L 0 LI2 (t) LI2 (0) = WL (t) - WL (0) = L -L 2 2 =
L

t

L , IL UL , , W .

-

2.4



. , , . 4 ( ) RH . : U = I(Ri + RH ), U
R
H

U

R

H

=U
i

U

R

i

, R i RH . , , , U . 4 ( ), RH . : U
R
H

R

RH , Ri + R H RH

=

1 Ri

I +

1 RH

,
i

I

R

H

=
i

IRi , RH + R i R
H



I

R

H

I

R

, R i RH . , , , I . , " " ( 4 ) Ri , . A 4 . " " ( 4) Ri . . ( A 4 ). . ( ).

R

2.5



- (" " ). c I R I c U R U :


2 .

10

A
U
0

A
R
i

R

i

R replacements

H

RH
I
0

PSfrag replacements U0 R
H

I

0

. 4: : ( ) R H . : ( ) RH

replacements

R I
0

R



I



R



r U
0

U



. 5: ag replacements U0 I0 R r

R



I



R



U



=U =I =R

xx

,

(6) (7) = U xx I (8)

I U




,

R

U

I

U xx ( ), I ( ). , : R U = R I , . 5. , , , : r rR + I0 , r+R r+R Rr U xx = . = I R+r
0

U R

xx

=U

I



= I0 +

U0 , R




3



11

3



( -) . 1. . o, . , . 6 :

replacements

a I

b I
1

c
3

R

1

I

2 2

R

C U
g1

L f d e U
g2

. 6: dI1 , dt

(acdf) : (bcde) : I
1

U U

g1

= I 1 R1 + I 2 R2 + L
t

g2

=
3 -

I3 () d - I2 R2 , C

= I2 + I

( ) : , . . ( " " " " , ) 2 . , . - , . , . - .

3.1



( ) e n = ~ ¯ En cos(t + n ), in = ~n cos(t + n ). I , . ( ).
1 , . (""). . 2 , .


3



12

U U replacements

. 7: , . U . U U . , , : Z = a + ib = a2 + b2 ei , = arg(Z), b a sin = , cos = , 2 + b2 2 + b2 a a ( , Z = 0), = cos x + i sin x, ( ).

Z = 0

e

ix

, , : U(t) = |A| cos(t + ) = = |A|e
i i

|A|e Ae

i(t+)

=

e

it

=

it

,

A = |A|e ,



. ( ) :
df(t) dt

f(t) = A cos(t + ) = -A sin(t + ) A f(t) dt = sin(t + )

, : U (t) U (t) = =


- - -

df(t) dt

f(t) = Aeit = iAeit A f(t) dt = i eit

-

~ U () e

K() =

~ U () ~ U ()

-

d , ( ) 2 d ~ U () eit , 2
it

.

(9)

3.2



- , , "" "". U . . 7. U U . , U . , . .


3

K()


13

3.2.1

. U (t) = K():
-

¯ U () e

it

d 2

K() =

¯ U () ¯ U ()

K() (9) . , |K()| (- ), argK() (- ).

1
|K| replacements

C R
U U /2 arg(K) 1/RC

1/RC . 8: .

. RC-, . 8: K() = |K()| = IR iRC , = 1 1 + iRC I R + iC RC , arg K() = - arctg RC 2 2 1 + RC

3.2.2

h(t)

, . H(t), (. 9 ) 1 t > 0, 1/2 t = 0, H(t) = 0 t < 0
t

, H(t). U (t) =
-

~ U ()H(t - ) d

(10)

~ ~ U H(t) : U(t) = t U(t). , . 9: U(t0 ) = U (t) dt H(t - t0 ),


3



14

H 1
PSfrag replacements U

U(t0 )

t

t

t
0

. 9: : . : .

h(t)

1

C
replacements U t

R

U

RC . 10: RC-. ~ U (t) = t U (t), , (10):
t

dt U (t) = = =

t -

~ U ()H(t - ) d
t -

=

~ U (t)H(0) + ~ U (t) 11 + 22

~ U ()(t - ) d =

~ = U (t)

h(t) H(t). h(t)
t

U (t) =

. RC-, . 10, , :
t

-

~ U ()h(t - ) d,

(11)

U (t) = RI(t) +

-

Q(t) I(t) dt = R t Q(t) + C C


3


t

15 U () e R 1-e

Q(t) = Q(t) = CU
- 0

-(t-)/RC

d,

U () = U0 H(),

(12)

-t/RC

H(t),
-t/RC

U = RI(t) = Rt Q(t) = U0 e

H(t),

(13)

H(t) , : h(t < 0) = 0 ( ). (13) (12):
t

U = Rt Q = U () - : = U () - U () +
t

- t

U ()



e

-(t-)/RC

d,

e
-

-(t-)/RC

U () d,

=
-

dU () e d

-(t-)/RC

d, dH() = () : d

U () = H(), h(t) = H(t) e
-t/RC

.

, ( RC RL) . (.. ) . : U


= H(t)e

-t/

,

U L R



= H(t) 1 - e

-t/

,

= RC,

=

( ) . , . 10 , , , . , . : UR (t) = H(t)e
-t/

,
-t/

UC (t) = H(t) 1 - e

, - , , , . , , . 3.2.3

-.
t

U (t) = U (t) =

- t -

U ()(t - ) d, U ()g(t - ) d. (14)

g(t), -


3



16

U (t)

U (ti )(t - ti )

replacements t
i

t

. 11: U -: U (t) = U (t) . , - (t) 3 . , U () U () . 11
t

U
i

, g(t), (14), U U . . (. .10) (12) U (t) = (t):
t

U (ti )(t - ti )



U () (t - ) d

Q(t) =
-

U () e R

-(t-)/RC t

d (12), U () e
-(t-)/RC

U (t) Rt Q(t) = U (t) - H(t) e g(t) = (t) - RC
-t/RC

.

-

d , RC

3.3

K(),

h(t),

g(t)

, K(), h(t), g(t). . : K() =
3

- (t) , , (x) 2 2 lim0 1 2 e-x /2 . (, ):
2

-



g(t)e

-it

dt,

(15)
=

- . 5.3.

-



d f() (x - ) =

1 [f(x - 0) + f(x + 0)] 2


3



17

g(t)

C
replacements U 1

RC

t

R

U

. 12: RC-. d , 2 K() it d e , 2 - i + d h(t) . dt K() e
it -

g(t) = h(t) = g(t) =

(16) ( 0) (17) (18)

, , (16). , -. g(t): U (t) = (t) = U (t) =


e

it

K() e

-

it

, K() g(t) c . - . 5.3. (15). (17). -: U = H(t) = lim e
0 -t 0

-

d , U () = 1, 2 d = () g(t - ) d = g(). 2 -

,

t > 0, 1 , i +

H() = U =

e

-(i+ )

d =

K() e i +

it

d = h(t) 2

. 5.3.

3.4



(15, 16, 17, 18) RC , . 13. : U () = RI() = R â K() = R 1 R + iC U () , 1 R + iC iRC = 1 + iRC


3



18

t>0

replacements

C R
U U

i/RC t

RC t<0

. 13:

, (17, 18), : h(t) = =


-

iRC d eit = i + (1 + iRC) 2 RC eit d = (1 + iRC) 2
-

= 2i ( = i/RC) = H(t) e

-

d eit = i( - i/RC) 2
-t/RC

.

. t > 0 , t<0 . (t < 0) 4 , .. . , (16) , .. K() ±. : K() = 1 iRC =1- , 1 + iRC 1 + iRC 1 d = g(t) = 1- eit 1 + iRC 2 - H(t) -t/RC = (t) - e . RC

(19)

(20) (21)

- - (. 5.3). , (18): g(t) = dh(t) H(t) = (t) - e dt RC
-t/RC

.

(22)

, h(t) g(t) , (16, 17). - , , (. ). , , . . , , .
4

H(t)).


3



19

3.5



, . 14.

replacements

R U C U U

L R

U

. 14: RC RL , RC = L/R. , RC = L/R: K
RC

() =

K() =


1 1 , KLR () = , 1 + iRC 1 + iL/R 1 L , = RC = , 1 + i R

. : h(t) = =


= H(t) 1 - e g(t) =


-

-

d eit = (i + )(1 + i ) 2 1 1 eit - i( - i ) i( - i/ )
-t/


d = 2

,
-t/


g(t) =

eit d H(t) = e 1 + i 2 - : H(t) dh(t) = e-t/ . dt

,

3.6



" " , . dU (t) , (23) dt a . : " " ? (23) : U


(t) = a

U



(t) =

=a

-



U



() e

d dt

: K
.

= i a

-

d = 2 d U () eit = ia 2
it

(24)


-

U () e

it

d 2 (25)

, " " R, C, L , . , . , , . 15, .


3



20

replacements C R U t>0 t<0 U U R L U

. 15: . ( ): K() = i L , = RC = , 1 + i R h(t) = H(t) e-t/ , g(t) = (t) - H(t) e (26)
-t/


.

(27)

, , , . , RC = R/L. 3.6.1

, (25) (26). , . 15 :


1

, , . , , , (28). , , . , (28) . (28) , , . 3.6.2



K()

i



(28)

, . t 0 , . 16 . () ( . 16). , - (U. . 16 ). , ( ) - (. 5.3). , (27) , . 16 . , , , .. U U , t0 .


t

0

(29)

, . , t0 . , (28) 29) , , ( ) .


3



21

U PSfrag replacements U U t0 U. t - U. R C - L t t t0

U

t

0

t

U.

replacements

R C L



t R C L U

U

t






t

. 16: : . : .

replacements R U C t>0 t<0 U U L R

U

. 17: .


3



22

3.7



" " , .
t

U



(t) = b
-

U () d ,

(30)

. : " b " ? (30) , : K
.

=

b i

(31)

, " " R, C, L , . , , . 17, . ( ): K() = 1 , 1 + i = RC =
-t/


L , R g(t) = H(t) e
-t/


(32) . (33)

h(t) = H(t) 1 - e

,

, , , , RC = R/L . 3.7.1

, (31) (32). , . 17 :


1

, , . , . ( ) (34) , , . 3.7.2



K()

1 .

(34)

, . t 0 , . 16 . () ( . 16). , (U. . 16 ). , (33) , . 16 . , , , .. U t0 U. ,


t

0

(35)

, . , t0 . , (34) (35) , , ( ) .


4

23

4
4.1





4.1.1

, . 18. q ( I = dq/dt) , : L q dI + RI + dt C
U
R L

= 0,

I=

dq dt (36)

U

U

C

d2 q dq + 2 + 2 q = 0, 0 2 dt dt = R , 2 = 0 L :
1 LC

. q = Ae -2 Ae
it

it

, (36) = 0,

replacements U U

PSfrag replacements , , : U U Cq R 0 > q 0 < L g(t) t C R L C R L t

+ 2 i Aeit + 2 Aeit 0 2 - 2 i - 2 = 0 0

g(t) . 18: . U 0 . , 2 - 2 , 0



1,2

q(t) q(0) q(0) q(t) q(t)

, , . 4.1.2 .

¯ ¯ = A1 e-t+i0 t + A2 e-t-i0 t , = CU0 A1 + A2 = CU0 , = 0 A 1 - A2 = CU0 , ¯ i 0 ¯ ¯ = CU0 e-t cos 0 t + sin 0 t , 0 > , ¯ 0 = CU0 e-t ch 2 - 2 t + sh 2 - 2 t , 0 0 2 - 2 0

¯ = i ± 0 ,

¯ 0 =

0 < .

. ... E () t = 0 . 19.


4 : dq E d2 q + 2 + 2 q = , 0 2 dt dt L dq q(0) = 0, = 0. dt t=0
0

24

, E

. : q(t) = A1 e
¯ -t+i0 t

+ A2 e

¯ -t-i0 t

+ CE , CE , ¯ i 0

A1 , A2 : q(0) = 0 A1 + A2 = -CE , q(t) = CE - CE e E U Cq R L g(t) CE C R
-t

¯ cos 0 t +

¯ sin 0 t , ¯ 0

q(0) = 0 A2 - A1 =

2 ¯0 2 = 0 - 2 .

PSfrag replacements

E replacements U C R g(t) L

E

q t t

. 19: , . ( ), . 19 . . 19 ... 4.1.3

, . 20. , .. t 1/, . () Ug (t) = U0 eit . L dI dt
L

t

+

RI +
U
R

U

I U

L

C R L

U

U

= IR = IC = I eit , 1 1 = Ie IC dt = C iC = R I eit , dIL = iL I eit . =L dt

-

I() d = Ug (t) = U0 e C
U
C

it

,

it

,

U(t) = UC + UR + UC ,

U0 = I

1 + R + iL , iC


E 4 Ug C R UL t replacements q E UR C Ug L R R UC t t q q

25

. 20: : , . : . I () = U0 R + i L - = U0 , Z()

1 C

Z() = R + i L - |I ()| = 1 0 = , LC U0

1 C =

= R + i, L , C = R Q= U
0

1 = R R , +
2

L , C

=

0 - 0

R + L -

2

12 C

1 Q2

= arg(I ()) = arctg -

0 - 0

.

Z() , Q , , . : I () = |I ()| = |UR ()| = |UL ()| = |UC ()| =
1 Q

U

0

,



+ i
0

UR = RI ,

UL = iL I ,

UC =

I iC



1 Q2

U

, +
2

I = arg(I ()) = arctg (-Q ) ,
U
R



1 Q2

RU

0

,
2 0 2

= I , = I + , 2 , 2

+

LU
1 Q2

,



U

+ U
0 1 Q2

L

, +
2



U

C

C

= I -

() : L = 1 , C =
0

: U U
L



I ()

max

=

U0 . R

= i0 L =

C

Ug it e = iQ U0 eit , R 1 Ug it e = -iQ U0 eit i0 C R

UL UC , .


4

26

u
U
c

U

L

U replacements

R



0



|U|
2

L




C

0

. 21: () () , . 19 . UL UC Q . Q ( . 4.4) R: Q= . R (37)

Q 1. , , : 2 W W = 2 = LI 2
2

RI 2 2 · 2 0

=

0 L = = Q. R R

, . . 21 . . , ( Q 1). , . ( . 21 1 2 ), 2. , (38) I
2

()

=

I

max



() , 2

(38), Q 1
1,2

0 ±

R2 =

L -

1 C

2

.

(38)

, 2



0 , Q


4 0 Q

27

Q

1

1 2 . : 1 Q = 2 - 1 2 = 0 0 1, 1. Q 1



1

Q0 ,



2

Q 4.1.4 : -

, , . 22: Z U () = , U () Ri + Z Ri Z(), K() 0, Ri Z PSfrag replacements(), K() 1 K() =



R

i

r . 22 E -, .. ( CU CU K C Ri 1 Z q Ri
1 2

replacements E R
i

CU Z Z

r L
C

r2 r+Ri r r+R



r

CU q

i



0



. 22: : -. : - . , ) , . : K() = 0 4.1.5 Z r + i 1 = , Z = r + iL + = r + i Z + Ri Ri + r + i iC 1 Ri , .. r Ri , = , Q = 1 Ri 0 Q

:

(.. ) . 23 . , . 22 , ( ). Ri , . . , r, R i . : K() = 1/iC 1 0 U () = = = = U () Z + Ri iC(Ri + r + i) i(Ri + r + i)


4

28

replacements

E

p

U

C

r C L
1

L

U

0

. 23: : . : . Q 0 , i 1 + iQ Q = (Ri + r) 1

=

1

( = 0 ) |K(0 )| = Q ( 0 , 0 ) |K()| : 0 Ri + r 1 = , Q Q 1 0 || 1

4.2



: : I U, L C,

, . 24: 1: 2: IR = U0 iL ; Z1 = , R + Z1 1 - 2 LC I0 1 iC UG = ; = G + 1/Z2 Z2 1 - 2 LC



R G, . .,

4.3



, (. . 25). . : 1 Z() = 1 1 + i C - R L = 1 1 + i , Q


4

29

replacements Ep

1
R

2
C L U
0

G C I
0

L

. 24: E replacements Ep PSfrag replacements L U U I C R G L C
p

R

G

I

. 25: : . : . R 0 , = - 0 I0 Q it = I0 Z() = , I = I0 e , 1 + iQ -i0 Q I0 Q I0 , IL = , = R 1 + iQ 1 + iQ L , C Q=

U I
R

=

IC =

i Q I0 . 0 1 + iQ

Z replacements

Zmax = R = 1/2Q Zmax / 2



0



. 26: . . 26 . " 1/ 2", : Q = 1


4 replacements I
C 2

30

U 1 2

Q 1 >Q

2

Ir
R

0



, I0 , U I
L - 2

. 27: arg(U ) : Q Q = 0 =
1 LC

1: 1:



1,2

= 0 ± , Q0 ,
2

0 Q

0 , Q

1



U (0 ) = I0 R, I0 R = -iR i0 L IC (0 ) = I0 R â i0 C = R Q = R 0 C = 0 L IL (0 ) = C I0 = -iQI0 , L iQI0 , R R =. = L
C

(. arg(U ) . 27). Q . , Q .

4.4



. . Q: Q= Q= Q= 2 â ( ) , 0 2L/r ( ), , = 2RC ( ), 2 0 1 , = 2

A A e-t/ . (, ) ( ) . , -


4

31

2: LC T < 4 K Q Q Q Q Q Q Q 50 . . . 300 50 . . . 10
5 10

f = 105 . . . 108 f = 109 . . . 10 f = 109 . . . 10 (T = 1.3 K) (, T = 4 K) (. , f 10 (CaF2 , f 10
15 15 12 12



106 . . . 10 5 â 10 3 â 10 10
10 11 9

1.5 â 10
9

)

)

. 28: . . , (c. . 28) . Q 1/T 5 , , . 2 .

4.5



. . , . 29. d ( ) d , , ­ . U C UC , . , UC d (. 3 ) UC UC = Q d , 2d

Q . , d : UC Q = 200, = 1 â 10-5 , d = 1 â 10-2 . UC d = 1 â 10
-9

.


4

32

replacements L

d

U

C

U d C R

C



. 29: . R replacements I
1

0

1

R

2

M
U L U C
1 1

L

2

C

2

I

2

. 30: d = 6 â 10-17 Q = 5 â 104 , d = 3 â 10-4 = 1 ()5 . . () , d B T d , Q W
B

, T

, W

, .

4.6



, . 30. U (t) = U0 eit , U (t) = V0 eit (U0 , V0 ). . I1 I2 : L dI1 Q1 dI2 + R 1 I1 + +M dt C1 dt Q2 dI1 dI2 + R 2 I2 + +M L2 dt C2 dt
1

= U (t), = 0,

dQ1 =I dt

1

(39) (40)

dQ2 = I2 . dt

5 , d = 6 â 10-17 ( ). , 1 , 1016 . , 10 12


4 M . I1 (t) = i1 e . (39, 40) : d i, dt dt 1/i. + iM I + iM I
2 2 1 1 it

33 , I2 (t) = i2 e
it

, i1 , i

2

I I

1

2

1 iC 1 iL2 + R2 + iC iL1 + R1 +

= U0 , = 0, (41) (42)

: + iM I + iM I
2

L1

01 I1

L2

02 I2

R1 01 +i - L1 01 01 02 R2 +i - L02 02

= U0 , =0

1

. , : L1 = L2 = L, C1 = C2 = C, R1 = R2 = R. 0 = 1/ LC, = 0 - , 0 = R , 0 L = M . L 0

(41, 42) 0 L, : : ( + i)I1 + i I2 = U0 , 0 L i I1 + ( + i)I2 = 0, (43) (44)

, I1 . I2 , V0 K: (43) â i - (44) â ( + i) I1 â 0 - 2 + ( + i)2 I2 = U () = U 0 0 I2 () =- 2 , iC + ( + i)2 - 0 K() = 2 + 2 - 2 + 2i i U0 , 0 L

, .. Q 1/ 1,

1. , , k const. N : N = ( 2 + 2 - 2 )2 + 42 2 , N = 2 - 2 2 - 22 + 22 + 4 (45) : 1 = 0,
2 2, 3

= 2 - 2 ,

= 0, 1 = . Q

2

(45) (46)

, < K() , > . , . 31 . .


4

34

|K()| k> k<

1 Q

|K()|

replacements

1 Q


0



1

2

. 31: () ()

R replacements C1 C2 I
1

1

M
R L U I
2 1 2

L

2

U . 32:


4 replacements A L n2 R
2 1

35

R R I
2 2 1

R2 /n I U
0 1

2

nU

0

. 33: .

4.7



( . 32) . . . U1 (t) = U0 eit , I1 (t) = i1 eit , I2 (t) = i2 eit . , : (R1 + iL1 )i1 - iMi2 -iMi1 + (R2 + iL2 )i2 = U0 , = 0. (47) (48)

(-) M. : 1. M , .. M
2

L 1 L2 .
1

2. , .. R 3. : R
2

L1 .

L2 . L2 /L1 , (48): M 1 =, L2 n

, n = i i U U
2 1

= =

L2 L1

M L2 - iR/ i 2 L2 n. i 1 L1

n ( ). (47,48), : D = (R1 + iL1 )(R2 + iL2 ) + M2 2 = = 2 (M2 - L1 L2 ) + i(L1 R2 + L2 R1 ) + R1 R i(L1 R2 + L2 R1 ), = iM U0 , 1 = U0 R2 + iL = = 2 D MU0 = L1 R 2 + L 2 R 1 L2 U 0 = L1 R 2 + L 2 R 1
MU0 L1 R1 L 2 L1 + R2 n2 2

2

I I

2

iL2 U =
2

0

(49) (50) n= L L
2 1

2

R
1

nU0 , R2 + n 2 R1

(51) (52)

1

1 D

U0 +R

(51) (52) , . 47 , . 33. , " " ,


5



36

R2 . " " , R 2 U 0 . (51) U2 = I2 R2 K: K I 2 R2 nR2 = U0 R2 + n 2 R (53)
1

, ( K n) , .. R2 n 2 R1 . , : P P
2

= =

I2 R 2 2

2

(nU0 )2 R2 2(n2 R1 + R2 )
R2 n2 R2 2 n2

2

A,

2

U2 â 0 2 R1 +

.

5



. , . (). . 1. 2. () : "0" . . "1" .

3. 1957 . . , ,

Sfrag replacements 1 p Hg 2 t C

4. "" . , ( - ), (, - - ). ( m 1), . -.

Sfrag replacements


5

PSfrag replacements

37

replacements n f
0

n 0 30 0 /tau1



0

n fa t 0 1 0 30 0 /tau1 t

n

n



n

0

1

. 34: () ().

5.1



f(t) ­ "" 6 0 = 2/0 , ( ): f(t) = f(t) = a0 + 2 a0 + 2 cn = a b
n n= n=1 n= n=1

(an cos(n0 t) + bn sin(n0 t)) , cn sin(n0 t + n ), tgn = an , bn

a2 + b2 , n n

= =

2 0 2 0

0 /2

f(t) cos(n0 t) dt,
-0 /2 0 /2

n

f(t) sin(n0 t) dt, ~ Cn e
in0 t

f(t) =

, . 34 . , - a n , an = 2f
1 0

n=-

-0 /2 n=

,

1 ~ Cn = 0

0 /2

f(t)e
-0 /2

-in0 t

dt.

sin

n 0 n 0
1

1

0

. 34 (, 0 1, ). n n 1 . , n 90% n < n :
n=n


a
n=1

2 n

n= n=1

a

2 n

0.9

: 90% 1/1 . ( n n 0 = 2/1 .) , 0 0 ( ).
6

- 0 /2 0 /2


5



38

a

n

replacements

n

0

0 0 /2

n

0





0

3

0



0

. 35: 0 , . 34 ( ) ( ). n . . n 0
1

=





= n 0 =



n



0 , 1 2 0

2 , 1

0 ( ) ( ). n . , .

5.2



, "" 7 F(t) :
it

F(t) = F() =

F()e F(t)e

-

d , 2 dt.

(54) (55)

-it

F(t) - F() . , .. (t ) , 8 . U0 :
/2

-

U() = U

0 -/2

e

-it

dt =

, â sinc 2 2

sinc (x)

sin x x

: F1 (t) + F2 (t) + F3 (t) F1 () + F2 () + F3 (), F(t) F(), - const dF(t) i â F(), dt F() F(t) dt , i


(56) (57) (58) (59)

7 8

- . , 1/2 , : F(t) = F() e
it -

d , 2

F() =

. (54, 55).

-



F(t) e

-it

dt . 2


5

1 F


39

F(t)


, . (58) - : t F(t) (60): F () =


- const d W= , F2 (t) dt = F2 () 2 - - f( - 0 ) f( + 0 ) F(t) = f(t) cos(0 t) F() = + . 2 2 F(t - ) F()e
i



,

,

- const

(60) (61) (62) (63)

=

t

-



F() e

it

d =

-



iF() e

it

d.

F(t) e

-it

dt =

F(t) e

-i(/) t

(61) : F () =


-

-

d(t) 1 =F



,

F(t - ) e

-it

dt =

F(y) e

-iy

e

-i

dy = F() e

-i

(62) . F(t) = F(t) (62): W = = = :


-

-

, F(t)


F() e

it

dt = F (t) =

F () e

-it

dt

(64)

- it

-

-

-

-

d d dt = (2)2 - - d d F() F ( ) 2 ( - ) = (2)2 - 2 d . F() 2 F() e F ( ) e
-i t

e

ix

dx = 2 ().

(63) : F() = = =


-

f(t)

1 f(t) e-i(-0 )t + e 2 - f( - 0 ) f( + 0 ) + 2 2

-

1 e 2

i0 t

+e

-i0 t

e

-it

dt = dt =

-i(+0 )t

5.3



- , .


5

-

40

5.3.1

- (t) :


f() (x - ) d =

f(x) - . - . -. . : (x) = lim D(x, ), D(x, ) =
0

-

1 f(x - 0) + f(x + 0) , 2

(65) -x2 2 2 ,


1
2

2

exp

D(x, ) dx = 1

(66)

, - "" , "". D(x, ) "" -9. - . D(x, ) D(, ) = D(, ) = :


-

D(x, ) e 1

-ix

dx. D(x, ) = -x2 - ix 22

-x2 - ix = - 2 2

-

-

2

2

exp

dx = exp
2

-



D(, ) e

ix 2

d 2 , (67)

-2 2

x i + 2 2

-

2 2

2

- -, : () = lim D(, ) = lim exp (x) = lim D(x, ) = lim
0 0 0

-2 2

2

= 1,
ix

(68)


, (69) -. .36. , "" D(x, ) "" . 5.3.2 ("") H(t) t

0 -

D(, ) e

d = 2

e

i x

-

d . 2

(69)

: 1 t > 0, 1/2 t = 0, PSfrag replacements H(t) = 0 t < 0
1

. : H() =
9

, - "" , sin(x/) ~ , D(x, ) = x
-

-



t

H(t) e

-it

0

dt = lim

t0 0

e

-it

dt = lim

t0 0

1 - e-it0 =? i

D(x, )



~ D(x, ) dx = 1


5

replacements D(x, )

1.5

1

D(, )

x D(x, ) 0 1 -1 2 1 2 -2 = 1/5
0.5

41

= 1/5

= 1/2 0 1 2
0.5 0

= 1/2 0 1 -1 22 -2 =1
0 -5 0 5

=1 -2 -1
0 1

x



H(t, ). -t e 1/2 H(t) = lim H(t, ) = 0 0 H(, ) =
0

. 36: : "" - D(x, ) . : "" - D(, ) . , D(x, ) , D(, ) ( 1). " " t > 0, t = 0, t < 0

(70)

"" H(t, ) e
- t-it

dt =

1 . i +

: > 0 , , H(, ) . : H(t, ) =


t > 0 : H(t > 0, ) = e- t , t < 0 : H(t < 0, ) = 0, t = 0 : H(t = 0, ) =
-

-

eit i +

d , 2

: = i



1 - i d = 2 + 2 2 2

(, "- "), . , - ( ). , , . 5.3.3 -
t

H(t) = () d,
-

"" - (66) (71):
t

1 2

dH(t) = (t) dt

(71)

H(t, ) =
-

D(, ) d =

1 + erf



t 2

,

(72)


5


1

42

replacements =1 =2 =5

H(t, , )

0.8 0.6 0.4 0.2 0

­0.8

­0.4

t

0

0.2 0.4 0.6 0.8

1

1/3 1/30

0.1

. 37: H(t, ) = 1/3 ( ), = 0.1 ( ) = 1/30 ( ). 2 erf(x)
x -u
2

e
0

du

(73)

5.4

erf(x) . 37 "" H(t, ) : , "" 0 .



. - , c : 1 dA(t) U (t) = A(t) cos t, A(t) dt - , c : U


(t) = A cos t + (t) ,

d(t) dt



- -.
t

U ,



(t) = A0 cos 0 t +
-

() d ,

|()|

0 .

(74)

d dt
0

= (t). () .

(75)

, (t)

5.5

-
a(t) = A0 (1 + m cos t) cos 0 t, m m a(t) = A0 cos 0 t + cos(0 + )t + cos(0 - )t . 2 2 (76)

- () a(t):

0 , , m . , . . 38 , . ():


5



43

A

mA/2



mA/2 replacements

mA/2 A mA/2

0 -
0

0 +

. 38: (). (). A0 0 , 0 ± . ± , A0 . : UAM (t) = Aslow (t) cos 0 t, Aslow (t) . , a() A() ( ): UAM () = = =


1 dt Aslow (t) e-i(+0 )t + e-i(- 2 - 1 Aslow ( + 0 ) + Aslow ( - 0 ) 2

-

dt UAM (t) e

-it

=
0

)t

=

Aslow () replacements

UAM ()

-
0


0

. 39: ( ) . 39.

5.6

-
d dt

, - () U


(t) = A0 cos(0 t + (t)),

0 ,

(77)


5



44

A mA/2 replacements

0 -
0

mA/2 0 + A mA/2

-mA/2

. 40: m ().

1().

, . : U


(t)

= A0 cos(0 t + m sin t),
(t)

(78)

m . , (78) (76) ± , ± 2, ± 3, ± 4 . . . (78) : U e


(t)

= A0 e =


i0 t+im sin t ikt

, ,
ikt

im sin t

Jk (m)e
i0 t

U



(t)

= A0 e

k=-

Jk (m)e

, 0 ± k (k ). : m 1. (78) m m m2 : U


k=-

(t) =

A A

0

cos 0 t - m sin t sin 0 t - 1- +

m2 cos t sin2 t 2

= (79)

0

m2 m cos 0 t + [cos(0 + )t - cos(0 - )t] + 4 2

m2 [cos(0 + 2)t + cos(0 - 2)t] 8

, m . . 40 (79), . (): A0 0 , 0 ± . ± , A0 . (79) , , m2 , 0 ± 2. , m3 0 ± 3 .. , , 0 ± 2, 0 ± 3 . . .


5



45

Jk (m) 0.4

m=1 m=2 m=3

replacements

0.2

-2

-1

1

2

k

. 41: - m = 1, 2, 3. . 41 - m = 1, 2, 3. , - - . 5.6.1 -

- : UM d dt UM = = = >1 <1 A cos (t), (t) = 0 t + sin t + 0 , (80)
0

0 + cos t, A cos 0 t + sin t + ,

(81)

, .

(80) (78) , - () . . (75).

5.7



( "" "") - ^ K: ^ U (t) = KU (t - t0 ). , U () = K()e
it
0

U (),

K() ( ). K , , , ^ K.


6

46

(- ) U (t), . , ? 10.

5.8

( )
hn = h(n),

h(t) :



, 1/

(c. . 42).

h(t) h
n

t replacements

. 42: h(t) hn c . ( ): 1 fc = 2 , hn : h(t) =


h

n

. , fc . , . , , T . , 5.2. "" f c , "" ( fc ) . . f
c

n=-

sin[2fc (t - n)] 2fc (t - n)

6



. . , ( , , , ) ( ). .
10

(sampling theorem).


6 replacements I(x) I(x + x)

47

L

R

L

R L U(x + x) d

R G C x + x

Z

G I(x) C

U(x) C x

G

x

. 43: .

6.1



, d = c/, . d ( " ") : L [/], C [/], R [/] , G [/]. (. . 43). , I(x) I(x + x) (- , "" x ): U(x, t) - Gx U(x, t), t I(x, t) U(x, t) - · x = Cx + Gx U(x, t), x t U(x, t) I(x, t) - =C + G U(x, t). x t I(x + x) = I(x) - Cx

(82)

, ( x L R): 0 - = U(x + x) - U(x) + Lx I(x, t) + Rx I, t

U(x, t) I(x, t) · x = Lx + Rx I(x, t), x t I(x, t) U(x, t) =L + R I(x, t). - x t

(83)

(82 83) . 6.1.1

(82, 83) (). , , .. R = 0, G = 0. : L â (82) t 2 U(x, t) x2 v
0

-

â (83) x 2 U(x, t) = LC , t2 1 =, LC

(84) (85)


6 v

48

0

. , U(x, t) = F1 (t - x/v0 ) + F2 (t + x/v0 ), (86)

F1 F2 . U+ (x, t) = F1 (t - x/v0 ) ( ) , x. U - (x, t) = F2 (t + x/v0 ) , , x. . 44. replacements t=0 F(t - x/v0 ) t=t t=t
0 0

F(t + x/v0 ) t=0

0 I(x) U(x + x) I(x + x)

x x0 = v 0 t
0

0 x x = -v0 t
0

. 44: , . (86) (82 83) a I + , I+ , : I(x, t) = I+ (x, t) + I- (x, t) = I+ (x, t) = U+ (x, t) , U+ (x, t) U- (x, t) - , -U- (x, t) I- (x, t) = = L , C (87) (88)

L ( (87) 11 ). = U = . I C (87): , , . W , x:

W

= =

[U+ (x, t) + U- (x, t)][I+ (x, t) + I- (x, t)] = 1 [U+ (x, t) + U- (x, t)][U+ (x, t) - U- (x, t)] = U2 (x, t) U2 (x, t) + --

, , , . "­" , x.

6.2



, , .. U, I eit . ,
11 (87) (87). , (, , ). (, , ). .


6

49



replacements

Z

0

x

. :

. 45: , Z

H

U(x, t) = U+ eit-ikx + U- eit+ikx , 1 U+ eit-ikx - U- eit+ikx . I(x, t) =

(89) (90)

U+ U- . , x0 t0 U
phys

(x0 , t0 ) =

1 (U(x0 ) + U(x0 ) ) 2

.. , , |U + |, |U- | . (89) Wcp , x: W cp W= |U+ |2 |U- | - 2 2
2

, . eit . , , ZH (c. . 45). (89, 90), : U(0) U(0) I(0) . = U + + U- , = U+ + U U+ - U
- -

U+ - U - , U(0) = I(0) ZH , U- ZH - = ZH , = . U+ ZH + I(0) =

1. ZH = , (U- = 0). . . Z H , . 2. ZH , .. . U- = U+ . , 2U+ .

3. ZH = 0, .. . U - = -U+ , I- = I+ . , 2U+ /. 6.2.1

G, R = 0 - U = U(x)eit , I = I(x)eit


6 (82 83), : 0 2 =- 2 U(x) + 2 U(x), x2 = (R + iL)(G + iC), = + i, R L, G C, R 1 i G + . + , ± v0 2 U- e x
it

50

U+ e replacements

it

e

(-ik-)x

e

(ik)x

x

. 46: , . 46: U(x) = U+ e
(-i/v0 -)x

+ U- e

(i/v0 +)x

.

.

6.3



( , ) , , , , . 47 . , ( ). , . . , , 12 . ( ) . , , ( . 47). , , , . , .

6.4

.

, .. . , . ( cos2 , ). . , . . 48. W. , , ( = /D1 1) : W


=W



â

D L

2 2 2

=W



D1

â

D1 D L

2 2

,

12 , . , , . , ( ) () v() .


6

51

'$' $ D 5 D D D 5 D 5D 5 D D m mmD D DD D D D &%& % () & T &D D & E& D mE B D
( )

D D DT & T DT T T && DT T . . ()

L0 @ @ @@¨ ¨ ¨




@@@
@@ @ & & & $ %

&

& &

&

& &

'

TT E TT

C0



. 47:

&B

replacements E

D
1



/D

1

D L
2

. 48: .


6

52

D1 D2 . , . W = 10 , D1 = 1 , D2 = 30 , = 3 , L 1012 . W


10 â

W T ( , T ). W T . T = 30 K , 500 â 1500 , , : 500 · 1500 · 3 ·
7

30 · 1 1012 · 0.03

2

1 â 10

-17



4 · 10

-5



5


107 ,



, : W


4 · 10
-17

-5

â 10

400 .



6.5



U2 0 2 = 100 U 10 =

0

4.5 · 10

-8

.

, , , . d

/2 - replacements d sin . 49: . , d , 49. , , a ­ . , , A ( ): = A
2

2 d sin + , d sin + 2 = 2

= a2 + a2 + 2a2 cos = 4a2 cos2 = 4a2 cos
2

= 0 . (A2 ) sin = ±m d , m = 0, 1, 2 . . . . . 50 ( ). , = /2 .


6

53

d = /2, = 0
1 0.4

d = , = 0
1

d = 2, = 0

replacements d = , = 0 = 2, = 0 /4, = 0 /4, = /4, = /2 4, = 3/4

0.2 0 0.2 0.4 1

PSfrag replacements d = /2, = 0 d = 2, = 0 d = /4, = 0 d = /4, = d = /4, = /2 0.5 d = /4, 0.5 = 3/4 0 1

0.5 0 0.5 1

PSfrag replacements d = /2, = 0 d = , = 0 d = /4, = 0 d = /4, = d = /4, = /2 0.5 d = /4, 0.5 = 3/4 0 1

0.5 0 0.5 1

1

1

0.5

0

0.5

1

d = /4, = /2
1 1

d = /4, = 3/4
1

d = /4, =

replacements /2, = 0 d = , = 0 = 2, = 0 /4, = 0 /4, = 4, = 3/4

0.5 0 0.5 1

1

0.5

PSfrag replacements d = /2, = 0 d = , = 0 d = 2, = 0 d = /4, = 0 d = /4, = d = /4, = /2
0 0.5 1

0.5 0 0.5 1

PSfrag replacements d = /2, = 0 d = , = 0 d = 2, = 0 d = /4, = 0
1 0.5

0.5 0 0.5 1

d = /4, = /2 0 1 d = /4, 0.5 = 3/4

1

0.5

0

0.5

1

. 50: .

6.6

N

N , . 51 (N 1). A , (k = 2/):
n=N

A=a
n=0

e sin

inkd sin

=a .

1 - eiNkd sin sin(Nkd/2 â sin ) , =a ikd sin 1-e sin(kd/2 â sin ) (91)

2 2

A

2

= a2 â

N

sin

d sin d sin

, M , m sin m = . d = m (91) . , /(dN). . , d = /2 . , , "" . , : N 1 , , "" . (.. " "), , () . . . , L. -


7

54

/L. , "", ( L 1013 ) = 3 , 3 · 10
-13

(!)

, , , () f/f < 10-14 . d = replacements
Nd

A

2

NA

2

. 51: N , . d = /2 .

7



: , . 1. . 2. , . () . , T 4 · 10 -21 = 0, 03 ( ). , E 5, 4 T . 3. , , , ESi 1, 1 , EGa 0, 67 . - T , . 4. : , , "" . n- (, As Si) p (, In Si). n 1014 . . . 1017 -3 . ( ) , , EAs 0, 01 . . . 0, 04 T () . , .

7.1



n p . n p . , ( ). : p


7

55

U = 0

U > 0 () p n

U < 0 ()

replacements

p

n

n

U x

U x

U x

. 52: . (U > 0) . (U > 0) . . . p n , (. . 52a, ). , (. 52a, ). , . , , p , n , . 52, . , , . . , . , , , . . . . . , , , . 52. , , . . . , , . . . . , p - n . . . 53 () . U 0.5 , I 0.3 . , . . 53 . C S , C/S 103 /2 . , , . p-n- f = (RC)-1 1012 , f = (RC)-1 109 p-n-.

7.2

p-n

, . 54. - (Wv ) (Wc ) (Wc ) , (Wf ).


7

56

I replacements 100 R


C/S

U

U

0, 5 0, 2

-10

-5

. 53: () () (c) , n-. n- p- , . 54 0 . W

c

p
W W replacements W
f v



0

E
fv

W W W
fc

c f

n



0

W

v

. 54: p-n U, ( . 55, ), ( . 55, ). , . , - n-. ( ) .


7 PSfrag replacements Wc Wv Wf 0 - qU qU W- 0 c qU Wfv Wfc qU W Wfv Wfc

57

W

c

W

c

p E
replacements W
f

p

0 + qU
E n

W W

f

W

c

v

qU

n
v

0 + qU W W W
fv fc v

0 - qU

W

v

. 55: p-n . .

,

7.3



(. 56): R , C Rs . replacements R I R
s



I



U = 0 + Rs I

C



0

V -

. 56: : R p-n , C , Rs p- n- .

- ( ) ( ): I = I 0 =
0

exp

kB T ln e

e0 -1 , kB T I +1 I0

0 , I0 , . , C . U = 0 + IRs . - : : U : kB T e kB T dU = R + Rs = +R dI e(I + I0 ) 0 + Rs I,
0

(92)
s

(93)

R kB T eI0 25 . I0 /1 (94)


7

58

7.4



D replacements m, v H m, v

. 57: mgH, mv2 /2 (. 57 ). mgH > mv 2 /2, "" . , (. 57 ), , .. eU > mv2 /2. , ( eU - mv2 /2) ´ h ( D - = 2 ¯ ). mv p n . - - , n- . , . , . (- p- + n-) . - . . , , , , . , . 58. N- ( N). . , (.. -1 1011 ), replacements f (RC) .

I mA Wfv 10 Wfc

W W

c

p



0

v

W W 0, 2 0, 1 UB 0.3 . 58:
f

f c

nW
0

W

v


8

59

8



.

8.1



, , (), () (). 5.5, 5.6. . replacements R U(t) I t t
0

L

C(t)

. 59: () () . 59. (). . ( ) . , , ( ).

8.2



R



U0 (t)

replacements U (t)

R



. 60: - R, C L. , . 60 ( V0 = U0 sin 0 t V = U sin t, 0 ), R R.


8 I = S1 U + S2 U
2

60

S1 U0 sin 0 t + U sin t + S2 U0 sin 0 t + U sin t

2

( ). , R R. U


IR +S
2



R



S1 [U0 sin 0 t + U sin t] + = (95)

U2 sin2 0 t + U2 sin2 t + S2 U0 U [cos(0 - )t - cos(0 + )t] 0 S1 [U0 sin 0 t + S2 U0 U [cos(0 - )t - cos(0 + )t] + S1 U sin t + S
2

=R +R





U2 sin2 0 t + U2 sin2 t 0

, 0 ± . , ( (95)), . , "" (, 2, 2 0 ). , , 0 , 0 ± . S3 U3 + S4 U4 + . . . , . : S3 (V0 + V )3 3V0 V
4 2 2 0 U 3

4 S4 (V0 + V ) 4V0 V U1 U 2
3 2



3U

= 3U0 U2 sin 0 t sin2 t = 4U0 U3 sin t sin3 t

sin( + 2)t + sin( - 2)t ,

cos( + 3)t + cos( - 3)t

, (0 ± 2, 0 ± 3). , , S3 , S4 . replacements8.3



(. . 61) . S R a 0 - Sab 2 0 -
0

0 +

R U (t)

b

20 - 20 + 0 + 2
0

. 61: () () - : U(t) = U0 (1 + m sin t) sin 0 t =


8 m [cos(0 - )t - cos(0 + )t] . 2

61

=U

0

sin 0 t +

. I = S1 U + S2 U2 ( ). , R R, m 1. : U
ab

(t) =

R I = R S1 U(t) + S2 U(t)2 + . . . R S1 U0 (1 + m sin t) sin 0 t +
2

=

+R S2 U2 (1 + m sin t) sin2 0 t + · · · = 0
1+2m sin t 1/2

=

S1 · · · + S2 R S2 U

2 0

11 + 2m sin t + . . . 22

, , . : U
ab

(t)

R S2 U2 â m sin t 0

, . 61 . , (0 , 0 ± ) , "": (0, , 2), (0 , 0 ± ), (20 , 20 ± ). , S3 U3 + S4 U4 + . . . , "" : (30 , 30 ± , 30 ± 2, 30 , 30 ± 3). , .

8.4



I = S1 U + S2 U2 , . , . 62: I = U /Ri (Ri ), . ) ) U

U

U replacements U ) I )
D

R



C

t

U

I

D

U

D

2

t

. 62:


8

62

. 62. , U(t) = U0 cos 0 t. RC : R C 1/0 , .. 2/0 . , .. U U . R . , . , . 62.. t0 0 t 0 = 2 (. . 62). , , : tan - = , R
i

Ri . R , , 1,

R





3

3 Ri . R

. , -: U(t) = U0 (1 + m sin t) sin 0 t, , 0 R




0 .

C

1,

R



C

1

(96)

, .. U (t) U0 m sin t. , (96) , 2/0 . (96) . , C .

8.5



. U (t) = U0 cos(0 t + (t)), (t) 1 : U (t) = U0 cos(t + (t)) = U0 cos cos t - U0 sin sin t. , (t) 1. , , . . . 63 . . : U(t) = U (t) + U (t) = U0 cos cos t - U0 sin sin t + - U0 cos(t) - U1 sin t = (t) -U0 sin (t) sin t + U1 sin t - U0 (1 - cos ) cos t
U (t) 2 /2 1 U



(t)

-U

1

1+

U0 (t) U1

sin t




8

63

R ! U (t) U + U PSfrag replacements U (t) U U + U U ! U (t)



replacements

R U (t)



. 63: : , , ( ). : PSfrag replacements . A B 0 U E U A U Es + E U D1 U Es 2 U replacements 0 B D
2

Es - E 2



I1 (Es + E )

2

I2 (Es - E )

2

. 64: : . : , . , , , . 63 . U0 U1 . , . 64 . , . U
A0

= U - U ,

U

B0

= U + U ,

U = U1 cos(t + )

­ . , . . U2 - U2 : B0 A0 U (U + U )2 - (U - U )2 = 2U U = -U0 U1 cos[ - (t)] + . . . : - U0 U1 sin (t), = 2


8

64

, .. -, - , . , , . 64 .

8.6


U(t) = U0 sin {(1 + m sin t) 0 t}

-

, . , , 0 (. . 65). U t
filter


replacements

U



. 65: - .

8.7


g(t)

replacements U U R C

. 66: . . (), . . 66. U (t) = Um (t) cos(t + (t)), g(t) = g0 + g1 cos(t + ), R, 1 iC 1 g

, g(t) , . I(t) g(t)U (t) = g0 Um (t) cos(t + (t)) + g1 Um (t) cos(2t + (t) + ) + 2


9 g1 Um (t) cos((t) - ), 2 .. g1 Um (t) sin (t), = 2 2

65

+

I

=

, , . ( - RC-.).

9



, . . ( , ), . (, ) . ( , , , ). , ( ""), . .

9.1


E p n p E

replacements

R I I I

E

E

. 67: . p, n, p ( n, p, n) . . , , ( emit ), ( col lect ), . 67 , . (. 7.1),


9

66

, . 67 . . ( ) , . . , . "", . , . ( ) , , . I (mA) I = 0.3 mA I = 0.15 mA I = 0 -10 U (B)

replacements

I (mA)

2 0, 1 U (B)

. 68: , I I , I , I I . I I I . 68 replacements .

9.2

(FET)
n p U = 1





U = 0

IC mA 0, 4



-15 -10 -5

0, 2 UC

. 69: . n- p- ( 69 n-). , , . ( 69 -). p-n , ,


9

67

n- . . . (--) (--) , . . . 69 . , , I U . I S = - , U , , S 0, 2 mA/B. R R , : R


=

U I

108 . . . 10

12

,

R



=

U I

104 .

, : I I = I U â = SR U I


10

-4

â 10

12

= 108 (!)

: f = 0 . . . 2 · 10 11 . , S , R R U U -. , . U U (.., , ) S , R R . . .

9.3
replacements U1 , I



1

U2 , I

2

I



R U




U R




SR



I



. 70: (). , - (97, 98) () , 4 . . , H- ( . . 70): U I
1 2

= H11 I1 + H12 U2 , = H21 I1 + H22 U2 ,


10

68

, . : U
1

= =

U1 I1 I I
2

I1 +
U2 =0

U U

1 2 I1 =0

U2 , U2 ,
I1 =0

I

2

I1 +

1 U2 =0

I2 U2

. , H - U


=

U I
R


I
U




+

=0

U U I U

U
I 0




,

(97)

=0

108 ...10

12

I



=

I I
SR

I
U
9



+

U
I
4



.

(98)

=0

=0



10

1/R

C

1/10

, (97, 98), , .70 ). , , .. - (97, 98) .

10



. , . : 1. (). 2. (). 3. . 4. . 5. . 6. . 7. . ( ) ~ K() = U () , U ()

~ , . |K()| ~ ()) , arg(K - . K(U ) ( ). , , , : N = 10 lg U W = 20 lg , W U


10 U U N = 40 .

69

K= W


,W



. : K = 100,

( , ), : K N


= K1 â K2 â K3 ,

= N1 + N2 + N3 ,

.

10.1



, . 71, R. I(U) .

replacements R U
0

I

I0 = U0 /R I (U)

a R b U
ab

U
U
R

U

0

. 71: I(U) - , . 71, (U1 , I1 ) , . , . , . 72 , . - , . 72 I (U ) . - R . U = -2 , 0, 5 . : K
U

=

U -4 B = = -8, U 0, 5 B

-, KU () . -, . , U = -1, 5 , , 0, 5 . KI KW ., , 0,5 B R = 107 . I = 107 5 · 10-8 A. , I = 5 K
I

=

I = 105 , I

K

W

=

I2 R R = K2 I I2 R R

10

10

â

10 10

3 7

106 .


10 R = 20 O R U= = 20 B U C 20 10 -4 10 I mA

70

Uab UR I0 = U0 /R

U = -0, 5 B U = -1 B U = -1, 5 B U = -2 B UB 20

. 72: () ()

10.2



. 72 . -, , U= = const, R I


+ U



= 0,

(99)

-, : I


=

I U U
S



+

I U U
1/R




.

(100)

(99) (100): -U R SU


= SU



+

= -U



U , R 1 1 + R R

(101)

KU R RC R U = -S · = -µ · U R + R R + R


,

µ = SR



.

(102)

(102) , , . 73. µ "" . R R , KU -SR . R R KU -SR = µ. , S RC , " ".

10.3



, Z , . 74. : 0 = R I + U U U I = SU + + Ri Z (103) (104) : SU = -U 1 1 1 + + R Ri Z , (105) (103) (104)


10

71

U R

-SU

U R

R replacements U R

i

R

i

U

R -µU PSfrag replacements U= R . 73: () Z (U Ri R . ). U C R -SU U U R replacements R Ri U= R Z C U U R U Ri U R -µU Z R Z . 74: . SRi U = µU = -U â R= + R i , R= 1 1 1 = + R= R Z (106)

(106) , , . 75 . , . , Z R .

10.4



. 75, 76 . Rg Cg , . , I0 , Rg -I0 Rg ( U= ). I0 Rg . Cg , R g . Rg , Cg . C , . . , , . 76 . , .


10

72

R replacements C C

R R U a b

C Cg Rg U

R

C

. 75: () R


, , R .

. C = C R (R = R R /(R + R )), R : iR C . Uab = U 1 + iR C C ( R ), . . . C C , . , C 1/R C , . C 1/R C (1/R = 1/R + 1/R ). . , , C . , . 76 . , , 76 , : C


1 , C R C


R



0

,

1


1 , C0 R 1 1 = + , R R

C R



C0 R



.

10.5



. 75 , Z = , R , . 77. (102, 106), ,


replacements10 replacements U C R C R C R Ri R C0 R µUab U U a K b

73

a

Ri U

C µUab

R

C0

b

µUab U 1/(C R ) 1/(C0 R )

. 76: : , . 75 . : . replacements R K U
=

U



C U Z
0



. 77: R : K() = -µ ·

Z () Z () + R



-µ ·

Z () , R

, R : R R . , . . , . , , . .

10.6



. , l = c. .


10

74

R C0 replacements R U= R C R2 C2 Ri C0 R µU C U
=

U



C

C U C
2

Ri R µU U C
2

C U R2

R2

U t

t

U
1

U
2

t

t . 78: (). () . , . 1 C2 . . 78 C U
1 2

= C 2 R0 = C = R C , = H(t),
2

2

Ri R Ri + R 2 1 , U


R C2 ,
-t/



R

R



R


i

=U



SR

0

1-e

1

10.7



, . . 79. , . 79 . K, . U U
ab

=U



+ U
ab



,



= KU

.

K , : K


=

U

ab

U - U



=

K . 1 - K

, K = K() = () : K = K0 e
i
k

,

= 0 e

i



,


10

75

U a replacements U b K PSfrag replacements U U a b U U

U U

K



. 79: : 1 replacements R R KU 2
12

U

R



. 80: K0 , 1 - 0 K0 K0 |K | = 1 + 0 K0 (107) (108)

k + k +



=0 =

|K | =



(107) . , . 0 K0 = 1 , . (108) . K0 0 1, 1 K: K 0 . 10.7.1



. , . 80. Z = U /I . ( c d ): I U U


=

ab

Uab , Z = U + U
ab

(109)


,


(110) KU
ab



= KU

Z Z




+Z

,

(Z



Z ).

(111)


10

76

a replacements R R

U

=

U U b R

. 81: (110, 111) (109), : I


=

U , Z (1 - K)

, , Z Z (1 - K). K < 0 , . I . : I




Z



= Z (1 - K)

(112)

=

Z

KU + Z



(113)

( ): U
ab

=U



+ U



=U



+ KU

Z
ab



Z



+Z



,

U

ab

= 1 - K U


U



Z Z




,


+Z

I



=

KUab = Z + Z Z



KU K == + Z (1 - K) 1 - K
K


Z +Z (1 - K)

.


(114)

(113) (114), , Z Z : Z






=

Z (1 - K)

(115)

, K < 0 , . 10.7.2 (, )

, . 81. , ( ) ( ). : R R U
ab

U ,

R

R ,


- +10 1 2 K replacements E+ E- a) ) U Z1 Z2 K U 1 2



a) 1 - K +

E

+

2

E ) Z 1
1

-

Z1 1 Z2 2 K E+ U E- 1 a) ) 2 ) U Uout Z1 Z2 K U U 1 2 ) Z
2

77

Z

1
1

Z K

2

2

U

2

K U U

Z

2 2

Z

1
1

Z K

0

U

U

1

2

U

. 82: (). (), () (), . : U R U U U , U
ab

= = = =

Uab - R Uab - Uab 1 + R S

R I = U Uab U , R I = U ,

ab

ab

- R SU , R R , - R SU ,

R IC = R SU = U

ab

R S 1 + R S

U

ab

,
U

= U , K K
U

KI .

= =

K

I

R S U 1, U 1 + R S R I = KU 1 I R

: R R .

10.8



( , ), : · K 106 . · R · R




. 0.

· (, ) ().


10

78

, . 82, "-"( 13 ) "+"( ) . (E + -E- ). , . 10.8.1

, . 82. , . I I= U - U Z1 + Z 2
12

(116)

U U (116 117), : K U Z2 Z1 + Z 2 - U


(12):
12

= -KU12 ,

U

=U



+ IZ

1

(117)

=U =-



+ K

U



U



- U Z1 , Z1 + Z 2 KZ1 1+ Z1 + Z 2

K : U =- U KZ2 Z1 + Z 2 , KZ1 1+ Z1 + Z 2 Z Z

K



=

K

lim K = -

2 1

, K , (to invert ", "). , (K ), K Z2 : K - Z1 . Z1 () Z2 () . 10.8.2

, . 82, : - U K U


=U =

12

=+

U



K

U Z1 - U , Z1 + Z 2 KZ1 1+ Z1 + Z 2

K : K


=

U = U

K . KZ1 1+ Z1 + Z 2 K

K



Z1 + Z 2 , Z1

K ("" ).

13 "", .


10 10.8.3

79

, . 82, I1 I2 , U1 U2 : U
12

= U 1 + I 1 Z1 , = U 2 + I 2 Z2 ,

U

12



I1 =

I1 I2 : U U



12

- U1 , Z1 U12 - U2 . I2 = Z2 U
12

=U =U

12

+ (I1 + I2 )Z0 = U 1+ Z0 Z + Z1 Z
0 2

+
0



12

-Z

- U1 U12 - U + Z1 Z2 U1 U2 + . Z1 Z2 U
12

2

Z0 ,

, U K : U


= -KU12 .

1+

1 K

1+

U



, n , :
n

Z0 Z0 U1 U2 + + = -Z0 , Z1 Z2 Z1 Z2 U2 U1 , K + -Z0 Z1 Z2

U



-Z

0 i=1

Ui , Zi

, Z 1 = Z2 = · · · = Zn Ui ( ). Zi , U ki Ui , ki .

K

10.9



. . 10.9.1

, , , . , . 83. d d. , (1) . 83. T/4 , , d. , E . UC = Ed = d UC . d . . , . 2T/4 , .. . 3T/4 U C = d UC d , T/4. UC = d . UC d


10

80

C(t)

C C(t) 2 0 t L

U PSfrag replacements

C

R 1

q = CUC =const

2

0 t

UC /U

C

-C/C

psfragRR

U UC â C/C . 83:

, (2) . 83. T/4 , .. . 2T/4 , .. d . . . . 3T/4 , .. . 4T/4 , 2T/4. . , -: ( (1)) , ( (2)) . , . - UC UC =- 1-e
rel -T/2


-

T 2/0 =- =- . 2 4Q/0 2Q

T = 2/0 , = 2Q/0 , Q . (1) (2) . 83 : UC UC =± d = ±m d

param

: UC UC =-
full

±m= 2Q 2Q

, : Q


=

Q 1
2mQ

,


10

81

"-" ( (1) . 83), "+ ( (2)). 2mQ > 1 : . 14 . (, ). 10.9.2

. 84: 0 . C C = C0 (1 - m sin(20 t + )). (118) : U+ (t) = U+ sin(0 t - 0 z/c). 0 , U- (t) .

replacements R U U
+ -

C(t)

L z

. 84: , q , I , : U+ + U U+ - U
-

= =

q = LI , C q + I.

(119) (120)

-

(119) , . (119) . I q, (119), (119, 120), U+ + U U+ - U
-

= =

q , C ¨ q+ q . LC

(121) (122)

-

(121) (122). , : 2U
+

=
-

¨ q+

q 0 q q ¨ + =q+ + 2 (1 + m sin(20 t + ))q, 0 C LC Q

(123) (124)

U

=

q - U+ . C

14 , .


11

82

C(t) . , C(t) q/(C) . , (m = 0). q = q 0 sin 0 t (123), (124), 20 U+ 0 U- q0 q0 = 2U+ C 0 C = U+ sin(0 t + 0 z/c). 0 =

: . , (118). q q = q0 sin 0 t (123), , 3 0 : 20 U
+

cos 0 t

= q0 sin 0 t(-2 + 2 (1 + m sin(20 t + )) + 0 0 = q 0 2 m 0 sin 0 t sin(20 t + )
1/2(cos(0 t+)-cos(30 t+))

2 q 0 0 cos 0 t = Q

+

2 q0 0 cos 0 t = Q

2 q0 2 m q 0 0 cos(0 t + ) + 0 cos 0 t 2 Q 2 : = 0 = . = 0 : q
0

=

2U+ C0 0 , 1 + mQ/2

U- = U

+ 0

1 - mQ/2 sin(0 t + 0 z/c) 1 + mQ/2

, . , (2) .83). = : q
0

=

+ 2U0 C0 , 1 - mQ/2

U- = U

+ 0

1 + mQ/2 sin(0 t + 0 z/c) 1 - mQ/2

, . ( (1) .83).

11



() . , , , (, ) . : , ( ) , , (). .

11.1



, ( ) . . ( ), , ( ). , .. , . , . , , 15 .
15 , , () . ( ). .


11 ¯ x : ¯ x= lim 1 T
T/2 -T/2

83

¯ x(t) = x(t) - x,

T

x(t) dt x ,

¯ = x2 = (x(t) - x)2 .

¯ , x . ( ) u(t). u(t)2 ( 1 ). u1 (t) + u2 (t) ( 1 ). ( 1 ) (u1 (t) + u2 (t))2 = u1 (t)2 + u2 (t)2 + 2u1 (t)u2 (t). . . u1 (t) u2 (t) , . , . 11.1.1

. , (, u(t) = 0): B() = u(t)u(t - ) = =
T

lim

1 T

T/2

u(t)u(t - ) dt.
-T/2

(125)

. = 0 . , (125), || 16 . : B(0) = = lim 1 T
T/2

B() = B(-).

T

u(t)2 dt,
-T/2

, = 0 . , "" |B( > 0)| < B(0). = 0 , ´ ( |B(0 )| = B(0)/e, e ), " " . 11.1.2

. . . , . , . 85: . . u2 . : u2 lim 1 T
T/2

16 , , , ||.

T

u
-T/2

ab

(t)2 dt

~ Su () â 2 â 2

(126)


11

84

1 replacements U


a

b

. 85: . ~ Su () . : , . , : =


, ~ Su () . "" 2 (126). Su () ( Su () ). , : ~ Su () = 2Su (). u(t): u() =


-

d ~ S() 2

(127)

u(t) e

-it

dt

(. 85), u
ab

-

=


u() e

-it

d 2

(128)

, u
2 ab

=


u()u ( ) e

-i(- )t

d d (2)2

(129)

(126) , : u() u( )


~ = 2 â ( - ) â 2S()

. 11.1.3 -


. u(t): u() = u(t) e
-it

dt,

u() = 0.

(.. u(t) = 0). u() u( )


-

=

-



u(t)u(t ) e

-i(t- t )

dt dt =


11


85

=

B(t - t ) e

-i(t- t )

dt dt .

, . : = t-t , t - t = t+t , 2

-

= ( + ) + ( - ) 2

, -: (x) = : u() u( )


1 2

-



e

ipx

dp

(130)

~ = 2 ( - ) S


u

+ 2

~ = 2 ( - ) Su (),

(131) (132)

~ Su () =

B() e

i

d .

~ , Su () u. (131) , u() u( ) , ~ Su (). . -, . v(t) T . WT = 1 T
T/2

-

v(t)
-T/2

2

dt

(133)

, v() v(t): v(t) = W
T

v() e

it

=

1 T

- T/2 -T/2

d , 2 v()v( ) e
i(- )t

(134) d d dt (2)2 (135)

, v(t) v(t) = v (t). T :
T

-

lim W

T

=

T

lim W

T

= = =

T -

lim

1 T

T/2

v(t)

2

dt

-T/2

= B(0),

(136)

lim

- T

- T

lim

- T



lim

v()v( ) d d ei(- )t dt = T (2)2 d d v()v( ) 2 ( - ) = T (2)2
2

lim

|v()| T

d = B(0) (2)2

(137)

T

1 T

T/2

v(t)
-T/2

2

dt

=

- T



lim

|v()| T

2

d (2)2

(138)


11

86

S() ~ S() replacements

labelspdens3

. 86: .

Sv () = lim (132) (139), (137):
T

|v()| T

2

(139)

~ S() = Sv () = lim

, (132) (139) . -. (132) , . , . -: ~ Su () = B() =
-

T

|v()| T

2

(140)

B() e

i

d, d 2

(141) (142)

~ Su () e

-i

~ , Su () ~u () ( (141) . , S B()). ~ 17 S() = 2Su () . . 87. - (141, 142) : Su () = 4 B() =


-

B() cos() d d 2

(143) (144)

0

0

Su () cos()

(144) , u : u = B(0) =


Su ()

0

d . 2

(145)

u = u2 () u2 (), /2, : u2 () Su () = 2 lim 0 (145) .
17 .. S() . B() .


11 11.1.4

87

, , .. Su () = S0 . (144), (130), B() = S0 ()/2, Su () = S0 . (146)

. , " " ( ). x, . , ( ) . , x 2 . N yi = {xi }, 1 < i < N, i . yi ( , u(t)). , ( ): 1 y = lim N N y2 = lim
N N

xi = 0,
i=1

1 N

N

(xi )2 = x2 ,



u = lim

T

1 T 1 T

T/2

u(t) dt,
-T/2 T/2

i=1

Bj : Bj = lim 1 N
N



u2 = lim

T

u(t)2 dt.
-T/2

0j 18 . . , |j| 1 Bj = 0. (147). , 0. , x2 = B0 (B0 ). - ij / () (. (146): Bj = (x)2 0j , , ( ). , , - , , . "" ( ). "" , 1/ ´ . "", - . B() = B0 () (148)

N

xi x
i=1

i-j

= (x)2 0j , B() = lim

T

1 T

T/2

u(t)u(t - ) dt.
-T/2

(147)

11.2



. , , . K() (. 87 ) , . . : u () u ( )
18



~ = 2 ( - ) â S (),



ij

= 1 i = j

ij

= 0 i = j.


11 PSfrag replacements u u K() E u replacements u t q a b U


88

C R L

K()

. 87: : , . : . u () u ( )


~ = 2 ( - ) â S (),

, , : ~ ~ u ( = K() u , S () = |K()|2 S (), 2 S () = |K()| S () (149) (150)

, . 87 . S() U . , SL () UL L? . UL = U â K(), SL = S() â K() K() =
2

=

iL , R + iL + 1/iC S() 2 L2
2

R2 + L - 1/C

11.3



: 1. ( ). 2. ( ). 3. ( " 1/f"-). ( ) ( ). . 11.3.1

, , . ´ . o , , . 88 ( , , ). , . Su () = S0 19 . 88. (145)
19 S () S () = S ()/R2 ( u I I ).


11

89

a) R U PSfrag replacements U I R a)

R replacements

) R U
C

R

I



U



C

. 88: , (). (). , , , : CU 2
2 C

=

C 2

0



T d SC () = 2 2
0

SC () S : UC () SC () U CU 2
2 C

= = = =

U ()/(iC) U () = , R + 1/(iC) 1 + iRC S0 , 1 + (RC)2 S0 d S0 = , 2 1 + (RC)2 4RC 0 T , 2 U2 = C T , C SI () = 4T R (151)

2 C

SI () (. 88a) ). (151) 20. , f ( ) ( /) : U
2

Su () = S0 = 4TR,

4TR f =

2 TR ,

. (.. T 0) : Su () = 4TR 4R


¯ h . f , . f f , f f
20

T = 300 K f

¯ h ¯ h +¯ h/T - 1 2 e T = = 6 · 1012 . 2 h

,

(152)

.


11 PSfrag replacements I replacements I
0

90 I
0

I t

0

B(

t
1 0

-

0



0





1

. 89: (). B(). , (151) T ¯ /2, f f h (152). , , . ( R) , , . . , , . - (). , , . 11.3.2

, . , , , , "" e. , , . N , N2 =N, .. ( ). , , . . , 0 , . 89. , , e, I0 = e/0 . , I = I0 â 0 /1 . 0 1 . I
0

=

e , 0



0



1

I=

e . 1

(153)

, , .. . : I(t)I(t - ) = B() = lim =
T/2

1 T â T 1

T
0

1 T

T/2

I(t)I(t - )dt =
-T/2

(154) (155)

I(t)I(t - )dt.
/2

-

0

, -T/2 (154) T/1 -1 1 /2 . , || 0 , (155). I2 (0 -||) 0 , || < 0 , 1 (0 - ||) = e I 2 (156) B() = 0 0, || > 0 .


11

91

B() . 89 . , (143) -, . 1/ 0 ( , 1/ ) -: B() = e I () (143) f: S () I
2

=4

0



B() cos d = 2eI, = 2eI f. 2

(157) (158)

= S () ·

. S



1/f



2eI

replacements f

103

108

. 90: : 1. : 0 1. 0 1, S () . 2. . ( , () .) 3. , .. . ( S () .) ( ), : I
2

(f) (f) 0.05

= 2eI f (f) , 1 = , 1 f < 103 , f 0.5 f 1/0 , < 1.

(159)

(159) , (f)). , . . 90.


11

92

, , , . 89. , n- p- , . 11.3.3 -

, SF () : A SF () , 0.8 < < 1, 4 A , . 1/f. , : , , , .. . , . -, - , , , . - , , , .

11.4
11.4.1




, R ( Us Un ), . 91. : U2 = 4T R f, f , T n . , , .

R replacements Uc

R R


R Uc



U

. 91: : Un = IR + U+ , U+ = I+ , I=I
+

W+ , : W+ = , W
+

U (R + )2

2 n

R = W
m +

=

4T R f U2 n = = T f 4R 4R


11

93

, R , m , R , = R W+ . , , . m W > W+ = T f 11.4.2 -

, K. Ws Wn = T0 f. W


= K2 (Ws + T0 f) + W = K2 (Ws + T0 f + T f)

" " T W = K2 T f . , T , . N . : N N


= =



Ws Wn Ws Wn

=


Ws , T0 f Ws T0 f + T f

=


- ( ) : F = N T0 + T T = =1+ , N T0 T0 T = (F - 1)T0 . (160)

, , T 0 , . T0 = 290 K ( ). (160) . , . , . , . K 1 T1 , K2 T2 . W1 ( ) W2 W W
1 2

=K

2 1

Ws + (T0 + T1 ) f ,
2 2

= K2 K 1

Ws + (T0 + T1 ) f + K

2 2

T

2

f .

-: N N


= =

Ws , T0 f W
s



(T0 + T1 ) f + F= N T1 =1+ N T0

, T2 f K1 1 T2 + â . K1 T0

, - K 1 . 4TR f 4 · 10-9 .


11

94

: T = 300 K, R = 1 , f = 1 . T = 330 K ( < 1 ), T = 17 K ( < 2 ). (): T = 6 . . . 10 K. ( 4 K). , : T = 1 K. 11.4.3

. . , Us , U2 , ´ : n N1 = U2 s U2 n 1

, "" . , T n T/. U i = Us s ( , ), . : V= 1 n
n

(Ui + Ui ) s n
i=1

, V = Us . : 1 (V - V ) = 2 n
2 n 2

U
i=1

i n

=

1 n2

n

U
i=1

i2 n

=

( Un ) . n

2

: N2 = n U2 s = nN1 . U2 n

, n , N2 . (n = T/). . , . , , , , . , . c Tn Ws , f. , , N= Ws Tn f

, , . f, . , .


12

95

S

K2 ()

S

n

S replacements

s



. 92: Sn () Ss (). K2 () , , , . 92. K (), , , - . , , , . , ( ) .

12
12.1





, K K K = , 1 - K



K , . , K 1 K . , . , , . , K() = |K| eK () = || e . : |K| || = 1, K +


(161) n = 0, 1, 2 . . . (162)

= 2 n,

(161) , (162). K , (161, 162) , . ( ""). ( (161, 162) ): , . , . : . , .


12

96

L

M L replacements C

R . 93: LC-

12.2

LC-

LC-, . 93 ( ). q C U , , I L . ... . M , ... . . , , . . (- ) . LC-. q : ¨ Lq + R q + q C =M dI . dt

, . . I I = SU = Sq/C, ( S ), ¨ Lq + R - MS C q+ q C =0 (163)

(163) , S , : q(t) = q0 exp - R - MS t CL cos(0 t + ), 1 0 = . LC

, MS , , . C R > MS , R < MS C C . . , . S(U ) ( ). .


12

97

I



¯ S

¯ R = MS/C

replacements U


q/C

q(t)/C

q0 /C

. 94: S(U ), (). I (U ) , (). 12.2.1

, . 94 . ( ) . 94 . , R < MS(0)/C q 0 , R= MS C
q0 C

,

(164)

. . , q q3 I = S0 - S1 (165) C C . , : q(t) = q 0 cos t. I : I


S

q0 3 q0 cos3 t = cos t - S1 C C q0 q0 3 3 1 = S0 cos t - S1 cos t + cos 3t = C C 4 4 2 q0 3 cos t + [. . . ] cos 3t = S cos t + [. . . ] cos 3t, = S0 - S1 4 C 3 q0 2 = S 0 - S1 . 4 C =S
0 0

3. , S , q (164): C q0 2 4 S0 - R . = C 3S1 M

, (165) , ( U0 ) , I (U ) . . 94 . , .


12 12.2.2

98

, (. . 95 ) , . , . 95 , , . , (q1 ) . , , , I , , ( ). .. , (q2 ) . , ( ) . , , , , q1 , . q1 , q2 . . ¯ S ¯ R = MS/C replacements U


I



q/C

q(t)/C

q1 /C

q2 /C

. 95: S(U ), (). I (U ) , (). , q/C q/0 C. , , ( ). . , q0 . , , . . , , , . . 96.

12.3

RC-

, , . 97. . , . (161 162),


12

99

q/L

0

q

q replacements

t

q(t)/C

p

q/L replacements q q/L0 q(t)/C t

0

q

t q

q(t)/C

. 96: . : = Z Z
1

2

Z2 , Z1 + Z 2 1 = R1 + , iC1 R2 , = 1 + iC2 R2 R2 = (1 + iC2 R2 )[R1 + 1/(iC1 )] + R2 1 . 2R 12 1 + R1 + C2 + 1-iCRRC1 C2 R2 C1 12

= =

(162) , , ..
2

= 2 = 0

1 R1 R2 C 1 C

2

. R1 = R2 , C1 = C2 (0 ) = 1/3. (161) K 3 . , K(U ) . . 97 RC- . K 1 + R3 /R4 , . " , K = 30, , , ,

(0 ) = 0.


12

100

0 . , () (. . 98). R C

1

1

R

3

R C replacements PSfrag replacements
2

4

R

R
2

2

K K C
2

R

1

C

1

. 97: RC- () ().

1/3

U

replacements 1/30

t

. 98: RC- " () RC- . (.. ), RC , . 3 RC-, .. /2. RC-, . 99. R0 , rg cg . . , . : = U2 = U1 1 i 5 + 1- (RC)2 RC 1 -6 (RC)2

,


12

101

R

0

replacements A C U C C c
1 g

R

R

R

r

g

. 99: RC- .

= 0 = (0 ) = -

1 6 RC

1 . 29

(166)

" -" , ( ). (166) , 0 . 1 K(0 ) > 29

12.4



, . . 100. replacements A I R U
=

U U U C U U U U
2 =

1

t
1 2

t

0

t



. 100: (), - ( ) . . ,


12

102

I

1

U R R C A C I

=





2

replacements A

1

2

B

C



F

D

C



R

3

R

3

. 101: . U2 , , . U1 , U2 (. . 100 ). . U= C. . U2 , U1 . . U2 .. , . 100 . RC.

12.5



, . 101. . K = K1 K2 1 (K1 K2 ), .. K 1 "". , R C C1, 2 R3 , C . , , , . , , .. . , . - (. . 101) I 1 +I10 . A . C1 , D , , , B: U U
A

= -I10 R , U = -I10 R


C

1

0,
D K

D

- , C2 , B F .

UB = -R I2 = - R S2 U
2


12

103

U

D



front

R C



t U


R3 C replacements A

1

U

B

t . 102: I1 , I 10 : U
C
2



. , , , , . , . , front C : front R C . , , R 10 C 10-11 front 103 â 10-11 = 10-8 . , . UC1 . UA ( I1 ), C1 . C1 ( ) . C2 UC2 U= . ( ). C 1 , . , : . . . . . 102 . . . , . 103. , " " . 103. , , . 103. , . , (. 103). . 2. . . 30 . . .

I

1

=

0 UF = UB ,
K
1

+S1 UB = S1 R K2 I

10

I10 .


12

104

U t replacements A

. 103: .

12.6



( ). : , . (1- , 2- , ). . . 104

12.7



LC- . , , , .. . . (.. "") . , . : 1. ( ): (a) . , ( ) : 0 = ... T, 0 ... (...). ... T , .. ... T . : T 2 · 10-5 1/K , T = 300 K . -3 T 5 · 10 K , 0 2 · 10-5 â 5 · 10-3 10-7 . 0 . , SiO2 LC- , . - , - . . . ...


12

105

I

1

+U R


=

R



I

2

A replacements A

B

C



F
-U R
3 =

D

C



R

4

U



U



replacements A +U= -U= R U C A B R3 R4 I1 I2

t

U

B





t

dUD /dt t

. 104: .


12 ... 10-8 1/K : T 0 0

106

(b) ( , ). , - 0 1 · 10-7 . 0

. 1 · 10-10 1 .

(c) . . , . ( !) .

2. ( ): (a) . () , , , . (b) ( ). (c) . ¯ 0 > T , h , ( .).

12.8



, . . 105, . 105. ) ) PSfrag replacements A ) L C


q/

0

q q
0

replacements A

R L



U C R

R




R q U


. 105: LC- (a). (). q : ¨ Lq + (R + R )
=0

q+

q C

=U



. : q(t) = (q0 + q) cos(0 t + ),


12

107

q0 , q . , "": , , , , . q0 . : , , - . . , . ( ). , , , . t0 , , . , ( , ) t 0 q (. . 105) . C , , U0 t0 q L¨ + q C = U0 cos(0 t), 0 = 1/ LC, q (t0 ) = U 0 t0 . 20 L

q (t) =

U0 t sin(0 t), 20 L

U0 - , : U0 = 4TR/t0 . q T T t0 : 4TR t0 (q t0 )2 2 q (t0) (t0 )2 , . (167) 20 L q0 t0 t0 . . ± t0 . : N , x, . : y . :
N

y=
i=1

xi ,

xi i- . , y = 0, .. xi = 0 ( , ). y2 = 0. :
N 2 N

y

2

=
i=1

x

i

=
i=j

xi xj +
=0 i=1

(xi )

2

= N(x)

2

t0 ( ), , t t0 ()2 : t ()2 = (t0 )2 â N = (t0 )2 , t0 4TR t0 TRt t ()2 â = , 2q0 0 L t0 Lq0 0 2W Q 2 q2 R 2W I2 R , W = 0 = 0 0 q2 = 2 = 2 â 0 2 2 0 L 0 R 0


12 T2 0 . 2Q2 W , c
T

108

(T )2 = Dt,

D=

(168) ,

W , , Q . (168) , D (168) ). , (T )
2

. , (.. "" : L


= 2 = D



,

= c



.

(169)

L , , , , . , :


=D=

T2 0 2Q2 W

. . :
T

= t

D , t T 0 T 0 ¯0 h 2Q2 Wt (170)

T 0

T , 2Q2 Wt

. , (170) T ¯ 0 /2 ( ), h . , (). (170). . ( ) . . (170) , . . Q = 10 2 , W = 1 mW, T = 300 K , t = 1 , 0 = 107 -1 . , T 0 1.4 · 10
-11

,





5 · 108 ,

L



1.5 · 10

17

.

: Q 10 7 . 1 -16 : T ( 1 ). 0 4 · 10 . 106 .

12.9



LC (). , "" . (170). , , . 106. "" , , (, , ). . . , .


12

109

d replacements 10 A
-9

T



10

-10

10
-11

t
-1 1 2 3

10

1

10

10

10

. 106: . , . 107 . f00 13- ( ): f
00

= 1 420 405 751, 786 ± 0, 004 ,

f00 f00

3 · 10

-13

. 107 . , , . , , . , "" . , (, ) : Microwave Amplification by Stimulated Emission of Radiation = MASER. , , , ´ . W , : W 1012 â 2¯ 00 10-12 . hf , . , , B 10-3 3 · 10-12 . T 1 K 1.5 · 10-13 . , : 3 · 10-16 3 , , . ( 4-5 ) 10-17 . . . 10-18 (!) 1976 . (R. Vessot) . . "" "" . .


12

110

replacements replacements A A 10
16

/

10

12

/

H

2

( )

( )



. 107: H2 , (). (). =
Rel

grav c2

7 â 10

-10 H=10
4

km

2 â 10 =
Rel

-15

1 .

grav 1 ± 2 · 10 c2

-4

12.10



. .. . 12.10.1

(. . 108) , . . , : . , . L (. . 108 ). , . , - . . , . . . . , L/2 "".. , . ,


12

111

x L L replacements e A V U t V0 - V V0 + V t eU/md

. 108: . . , 10 . , . 1 20 . ... ( 70% ). 12.10.2

. ( q m), , , . 109. , xy. : md2 x = q dt y B, t md2 y = q E - q dt x B. t , x V 0 , x = x + V0 t. : md2 x t
2 t

= q dt y B,

md y = q E - qV0 B - q dt x B. , V0 = E/B, , B. , , , V 0 . ( ). V 0 . , , . , . 109. , ( ) , , - ( ) . , , , . , . . , "", . ,


12

112

A

y

E x replacements A K z B

. 109: : . : . , "": , ( "") , . , : , . , . : 1 ( ) 1 . . .