()

()

..

2002

: ..

.. : . : - , 2002. 205 .

.., 2002 , 2002

3

.................................................... 6 .............................................................................................................. 7 1 ........................................................ 8

1.1 ............................................................................... 8 1.2 ...................................................................... 8

1.2.1 ......... 8 1.2.2 ............................... 10 1.2.3 .............................. 11 1.2.4 . ................................ 11 1.2.5 .................................. 13

1.3 ........ 14 1.3.1 ........................................... 14 1.3.2 () .......................................... 17

2 ..................... 20 2.1 ....................................... 20 2.2 .............................................. 23 2.3 ............................................ 25 2.4 .................................. 33 2.5 ........................ 34 2.6 ............................................................ 37

2.6.1 ................... 39 2.6.2 ............................. 41

3 ................................................................... 44 4 .................................. 47 5 .................................. 60

5.1 ........................................................................ 60 5.2

............................................................................... 62 5.3 L- () ............................ 65

5.3.1 L- .. 66 5.4 ....................................................... 67 5.5 ............ 70

5.5.1 ...................................................................... 72

5.5.2 ................................................... 73

6 ......................................... 76 6.1 .................................................................... 76 6.2 ................................................... 79 6.3 ........................................................................ 81

4

6.4 ......................................................................... 82

6.4.1 ...................................................... 83 6.4.2 ................................................ 85 6.4.3 ................................ 86

6.5 ................................................ 87 6.5.1 - .................... 88 6.5.2 .................................................. 91 6.5.3 ....................................................... 97 6.5.4 ..................................... 105

7 ........................................... 109 7.1

....................................................................... 110 7.1.1 .................. 111 7.1.2 ......................... 116 7.1.3

.............................. 119 7.2

............................................. 121 7.2.1

............................................................................ 126 7.2.2 .............. 128

7.3 ........................................ 130 7.4

.............................................................................. 136 7.5 ................................. 138

7.5.1 ............... 140 8 ............... 142

8.1 .................................................................................. 142 8.2 ....................................................... 147 8.3 ............................ 149 8.4 ............................. 152

9 ......................................................... 158 9.1 -................................................................ 158 9.2 ....................................................................... 163 9.3 ...................... 164

9.3.1 ................................. 164 9.3.2 ........................................................................... 169 9.3.3 ....................................... 170 9.3.4 ............................................ 171 9.3.5 ...................... 172

9.4 - ................................... 173

5

10 ............................. 175 10.1 ....................................................................... 175 10.2 ..................................................... 177 10.3 ........................................... 184 10.4 ...................... 190

10.4.1 .................................................... 191 10.4.2 ............ 193 10.4.3 ................ 197

10.5 ..................... 202

6

. () , , ().

. : 1) , , -

; 2) ,

, ; 3) -

, -, - .

1. .., .. - . .: , 1983.

2. / . .. . .: , 1986.

3. .., .., .. - . .: , 1987.

4. .., .. - : / . .. . .: , 2000. 479 .

5. .. . - / . .. . .: , 1987.

6. ., . . .: , 1988.

7. .. . .: , 1987. 8. . ., . -

. .: , 1985. 9. ..

. .: - , 1993. 10. .. PCAD Pspice -

. . 1-4, .: , 1992.

7

-

, . - - -, . - .

- - (). - . . - . - , , - . - , , .

8

1 1.1 -

. - , . -, - , .

, - () () , .

, , - , - . .

, . , - (- ) ( ) , .

, , - . , - , -. .

1.2 -

: 1) ; 2) ; 3) -

. 1.2.1

, , - .

9

, - ( ) .

( ) , , - - , - .

, - - () .

- . S () Si. - Si , , , Sij .. , Sij - . , , . S .

, ( ) - , -

S 1

2 S2 S1 S3 Sn

S1m S11 S2 Sn1 S2m Snm 3

. . .

1.1 S

10

() - .

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

()

, , ,

..

, , , .

, , , .

, ,

.

-

1.2.2

- , ( ), . - - . :

; ; .

- , -, . - , , , - . .

- , .. , , - , . , - ..

11

, .. . - , - . : , ..

. , - : , -, , .

- .

1.2.3 -

. - , - . - - ( , ).

- , .. . , - .

, - ( , - ). - - . , , .

1.2.4 .

. - / .

, - , - ( ). , , - ( ).

12

- : - , ; - , - . - . - -

- Sk

Sk

- Ski

(k1)-

Ski;

Ski

k-

Sk

, -

Sk

(k+1)-

Sk

1.2

13

. - .

, - ( - ).

- , . . , , .. .

.

, , , .

1.2.5 :

, . , -

, () . - - ().

: : - ; - ; - ; - -

; : - ; - ; - - ; - () . ,

-. , ..

14

-, . : , .

, , .

, - :

1. , ( -). , - , . .

2. , - . . , - , ( -).

3. , - , -. - . , - , - . , , .

1.3

1.3.1 ()

, , . , , - .

, , ,

15

. - - .

, .

.

, , , - ..

- .

, - . - . , , , , - , - . - , , . () , - .

- , , (. . 1.3).

- , - (, , ..). - , , -, , , , .., - , .. - , , .

: , , , . 3, 2, 1- . - , .. - , , (, , , , , ..). - -

16

- -, , , , , , .., . - . - , ( , - .).

- . . , - (, ..) , . .

- .

1

2

21

22

2M

221

222

22N

1.3 ( )

17

1.3.2 () , -

. - ( ) . , .

- . 1.4.

3.

2.

4.

1. ()

1.1

1.2

1.4 (-)

: 1. . -

. , , , - . - . :

, , - .

-

18

. - .

2. . - - . , - . -- .

, : ; , ;

;

, .

( ).

3. . , - - , - , - (, , , - ..), , , .. - -, .

, : - ( -

, ); ; ,

; .

( , - , ..), .

4. . - - , .

, :

19

; ; ; .

- . , - -. , .

-, . , - :

, ; , ; ; -

; .

20

2

- .

2.1 -

, - - , - . - . ().

- (, , , , .) , .

, , .

- , - ( ). - , , - .

, , , - ..

- ( ), -, ( ), - ( ).

, - . , , . - , .

- ,

21

. - . , .

( ); .. , , .. (, , .) - .

: , ,

, ..; , ,

() .. - (.. );

, - , ..

m , l n . - ) , ... , ,(X 21 nxxx=

G ;

) , ... , ,(Q 21 lqqq=G

; ) , ... ,,(Y 21 myyy=G

.

)Q,X(YGGG

F= . (2.1)

XG

QG

. - (2.1) , Y

G.

, (2.1) . , - , .. (-) Y

G X

G

QG

.

( ) , - . - :

+ iii y TTTT ( mi ,1= ), (2.2)

22

iTT , +iTT yi.

iTT , +iTT ( mi ,1= ), -

YG

,

)TT , ... ,TT ,(TTTT 21 =

m )TT , ... ,TT ,(TTTT 21 ++++ =

m . -

YG

TT +TT .

(2.2), - :

+ TTYTTG

. (2.3) -

, , - (, -, , , - ..). .

) , ... , ,(V 21 svvv=

G, -

, . , - V

G

. - , .

YG

- X

G, QG

, VG

. - :

(V, X, Q) 0; =GG G (2.4)

Y (V),= G G (2.5) , .

23

(2.4) V

G. (2.5)

YG

VG

.

VG

, (2.4) (2.5) . , - (2.4). (2.4) Y

G

(2.5) . 2.2

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

; ; ; .

. ( ). , , -, , , - , - .

, , , , , ..

, , - . - . (, -).

, - , , .

24

- , - . - ) , ... ,,(Y 21 myyy=

G. , -

j- yj yj , - yj || jjjj yyy = , mj ,1= .

m ),...,,( 21 m= .

- . ()

jmj

=== ,1

max|||| 11 (2.6)

=

==m

jj

1

222 |||| . (2.7)

, .. Q

G -

XG

. XG

QG

- Y

G. , -

, X

G QG

, .

- X

G Q

G. -

1 ( 2) 0Q

G , -

XG

QG

.

Q

G -

(): }|Q{ 1 =K

, > 0 , .

: (. . 2.1).

25

I 0,1 [40o; +50o].

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

2.3

, - , 2.1.

U

I

T1 T2

U1

I2

I1

- I

T

10 %

40o +50o

(T | 0,1) = = [40o; +50o]

2.1

26

2.1

, -

, ,

, (-) , , - () ,

, , , -, , ..

, . . , , . - . , - ( - , , ..) (, , ..). , , , ..

MM -, - . ; - , , : , - , .. - , , -, , - -. .

- , - (-, ). MM

27

, : +b+z+d = 0, - (/)2 +(/b)2+(z/c)2+ d=0, , , z -, , b, , d . , - .

- , . - .

- -.

, , . , , , - . - - , , , . - .

- R(u, v), R =(x, , z), v . R(u), , .

, -, , . - .

, - , . , , , .

. , , . - , , -

28

, ..

. - - .

, - . - , , -, .

- . , , , , .., -. , , - .

- - - . - - . - , -, - . , -, - .

, . - - (). - . -

-, , , ..

: :

0)()()( ),,( 22

2

2

=+

+

= tQy

tTx

tTtyxT yx ,

Q

x

y

29

x, y , Q(t) . - .

- , - , , - .

- ( , , - .), .

- , (). t, V

G , -

-. , , - .. - , , . - . - .

- . - , -, , . - . - t. - , - .

. - , .

30

: RC-. :

;)()(;)()(

RtU

tidt

tdUCti RRCC == .

);()()();()( tEtUtUtiti RCRC =+= .

.0)()()( ;

=

+

=

RtUtE

dttdUC

RUE

dtdUC

CC

CC

, , - UC(t).

- ( , , -, .). -. , , .

- . - - . , - , , - - , .

. , - , . , . - , , - . - , - , -, .

- . , .

R

C

E UR

UC

i(t)=iR=iC

31

- .

MM , , (.. ). .

, , - . - , - . - - .

. , - .

.

, .. )Q,X(Y

GGG = . , - , , - , - .

- , .. - - Y

G X

G QG

.

MM , - . - -, .

.

. , -, , -

32

, .. - , - -, , . , , , - , - .

- ( -, ). - -, , MM . - .

-, ( ). . - .

- , , - , , .. , , . -, , , ..

- .

, , - , . . -.

33

2.4 -

: 1. , .

- .

2. . - , - , - , , , , - ..

3. . MM - - - . , . - - . , .

4. . - , ..

)X(min G

Xx

, (2.8)

XG

- ; - ( ) X

G; , -

(2.6) (2.7), yj XG

, j - , - ( , - ).

5. . j, - (2.8).

- , - , (). . -

34

, . -

, - , . , . Q

G=(q1, q2) -

. 2.2, j=1, j=2 j=3, |)Q(| Gj =, j = 1,2,3. . - - Q

G,

- ( - ): + 111 qqq ;

+ 222 qqq .

, - . , - .

2.5

. - ( ) :

1) ; 2) -

. -

(- ), . - , - . - , .

q1

q2 j=1

j=2

j=3

j=1 +1q 1q

+2q

2q

2.2 QG

=(q1, q2)

35

, , . 2.3.

- , , - . 1 . 2.3 , , . . - -; -.

- . ; , . (.. ),

(-, - )

-?

-?

1 8 11

5 7

4

6

9

2

12

3

10

2.3

36

- , ( 2 . 2.3). (.. - ), : 1) ( 3), - ; 2) ( 4).

(- ) , - . 4 -, 2 3.

- ( 5), , ( 6), . - () ( 7) , .

8 . 2.3 - , , . , - 4, 6, 7 4, 5; , - ( 9).

- ( 10), , - , - - ( 11). - ( 12) .

, - , - . . 2.3 , - , . - - .

37

2.6 ,

. .

() . -

, ..

TT QG

,

XG

( - ).

- , , . - , .. X

G QG

YG

.

- - .

- , . ( ) - .

, . n- , n xi (i = 1,n ) . - X

G

YG

()

2.4

38

QG

(. . 2.5). - (2.4)-(2.5), .

X

G.

, , .

- )X(

GS , ..

-.

x1

x2

y

x10

x20

y0

y0 = ( , )x10 x20f

2.5 ),(),( 0201021 xxFyxxFy ==

. QK , XG

YG

()

.

QK , TT , +TT )X(

GS

(-)

. QK , )X( GS ,

TT , +TT

XG

2.6

39

- (.. X

G)

)X(G

S ( QK , TT +TT ), .. - , ( ). , - - Y

G -

XG

.

2.6.1 -

(. . 2.7).

k- , - (k+1)- - .

(k+1) - , .. . : - , - . - , . -, . - , -, (k+1)- () . , - .

-, X

G.

XG

, . X

G

- , - . , -. , - .

40

, . , ( ), - k- . k- , - .

)Q,X(YGGG =

XG

- ?

X

G

k

k+1

2.7

2.7 - . - (. . 2.8).

41

(), - , .. - , - . , - - .

, , . . -, .

- . - . - , .

2.6.2 :

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

2.8

42

, - . :

1) , - , ;

2) ; 3) . -

. () . -

, - -. , - , - - , - -.

-, - .

- - - . , .

- , - , - , - . . , - .

- . , .. , -, . - ( ) . - - -. ,

43

, , - .

(, - - ) -, - . - , , .

44

3 () -

- , , - , -.

:

(); (); (); (); (); - -

.

- , , .

- , - , , -.

-, . , .

- . . . , - . - - . - , - .

-, . , - -

45

. - .

, - . () - , . , -, ..

, - . - . - .

- , , -, - .

, , , - . - , , , .

-:

- . , - . , - , .

. : - , , .

-. , , -, .

. - . , .

46

-. - . , - , .

-

-

-

-

-

-

3.1

, : -

; , ; , -

; -

; .

47

4

, - , , . . - , - . - , , - .

- G(V, S), vij - , - (- ).

Si . , - . - . -, .

. Si Sj , , , - Si Sj. - .

. G(V, S) , Si Sj.

. , .

. , .

. , - , . , - m+1 , . , , . , - , .

. , - . m+1 m , - m .

48

. 4.1, , -, . 4.1, . - . - v10, v24, v30, v40. v12, v23, v34 -. . 4.1, .

)

E1 C1

R1

R2

L1 L2

C2

0

2

1

4

3

0

1

2 3

4

v10=E1

v12=R1

v30=C1 v40=R2

v24=C2

v23=L1 v34=L2

4.1 : ; ; ;

) )

0

C1

C3 C2

4 1

3 2

C4

)

v12

v30 v40

v24 v23 v34

K1

K2

K3

, - . - (). , - .

G(V, S) (m+1) n = [aij], ( 1,1 += mi , nj ,1= ), - aij=1, , aij = 1, , aij=0, - . , . 4.1, :

49

() v10 v30 v24 v40 v12 v23 v34

1 1 0 0 0 1 0 0 2 0 0 1 0 1 1 0 3 0 1 0 0 0 1 1 4 0 0 1 1 0 0 1

=

0 1 1 0 1 0 0 0

- ( , ), [ij] 1 1, . - , . - , . -, , -. - , .. .

- ( ). - , - -.

- : +1 , .

, , , ( ). , - , - , , , I

G,

, . :

0 = IG

A , (4.1) T

, ],...,,...,,[ 12010 nnij iiiiI =G

, - , . (.4.1, ) -

50

.0

1001100110001001101000010001

34

23

12

40

24

30

10

=

iiiiiii

:

=+

=+

=+

=+

.0;0;0

;0

344024

342330

231224

1210

iiiiiiiii

ii

- . , - vij ui uj uij = ui uj. - :

0]0...0...1...0...1...0...0[ UuijG

= ,

T],...,,[ 210 nuuuU =G

. -

, , - . , -

0T UUGG

A= . (4.2) (4.2) :

.

11000110001110001010

01000001

4

3

2

1

34

23

12

40

24

30

10

=

u

u

u

u

u

u

u

u

u

u

u

, : -

51

, , -. . , . D = [dij], ( mi ,1= , nj ,1= ) dij =1, j i () (-) , . dij = 1, j - i dij =0, j i. , . 4.1, .4.1, ( ), - :

v10 v30 v24 v40 v12 v23 v34

1 1 0 0 0 1 0 0 2 0 1 0 0 0 1 1 3 0 0 1 0 1 1 0 D= 4 0 0 0 1 1 1 1

- D :

0 = IG

D , (4.3) T

, ],...,,...,,[ 12010 nnij iiiiI =G

, - D. , , . 4.1, , - :

;0

1111000011010011000100010001

34

23

12

40

24

30

10

=

iiiiiii

=++

=+

=+

=

.0;0;0

;0

34231240

231224

342330

1210

iiiiiiiiii

ii

: -, .

52

- :

[ ] 0|

=

IIGG

DD , (4.4)

, . D, -, D : D= [1]. , (4.4) :

[ ] 0|1

=

IIGG

D .

, (4.4) :

IIGG

D= . (4.5) . , , , (4.5) :

1I

II GGG

=

D

. (4.6)

, , , , - (4.6). - I

G -

. , , . 4.1, , - (4.4), (4.5) (4.6), :

;

1000010000100001

=D

=

111011110001

D

=

40

24

30

10

iiii

IG

;

=

34

23

12

iii

IG

;

=

34

23

12

40

24

30

10

111011110001

iii

iiii

.

53

. 4.1, :

+=

=

=

=

.

;

;

;

34231240

231224

342330

1210

iiiiiiiiii

ii

, - . - G(V, S). - , - , . +1 , -, . , , .

[bij] ( mi ,1= ; mnj = ,1 ), bij =1, i j- -

, , bij = 1, - , bij = 0, .

- :

,

v10 v30 v24 v40 v12 v23 v34

1 1 0 1 1 1 0 0 2 0 1 1 1 0 1 0 = 3 0 1 0 1 0 0 1

, , , 1 . , (4.4), : , - . - (. .4.1, ).

- : ]|[ BBB = , - , , : =1. ,

54

m+1.

, , :

0 =UG

B , (4.7) T

, ],...,,...,,[ 12010 nnij uuuuU =G

. , - U

G

. - , , (4.7)

[ ] [ ] 01||

=

=UU

U GGG

BBB .

, , , . - , :

UUGG

B= . (4.8) (4.6), , , (4.8)

1 UUU GGG

=

B. (4.9)

, - , (4.9), , - . - , . , . (4.6) (4.9), , - , - -. ,

55

D - . - (4.6) (4.9) , - . - -.

. - , - :

.0

,0T

T

=

=

BD

DB (4.10)

(4.10) . 4.1, , B D :

=

000000000

100010001111

011110001

101010011100101101001

.

(4.10), ,

[ ]

[ ] ,01

|1

;011|

T

T

TT

=+=

=+=

DBB

D

DBD

B

, .,

T

T DBDB == (4.11)

(4.11) , :

[ ][ ]. |1

; 1|T

T

BDDB

=

=

(4.12)

56

(4.12) , D . . 4.1, :

=

101011101101

B ; [ ]

==

111011110001

10000100

00100001

|1 TBD ,

D , . , - , - -:

0T = BA . (4.13) (4.13) . , , , D (. ): [ ] [ ] [ ]. |1; 1|; | DDBBAAA === (4.13)

[ ] 01

| TT

=+=

ABAB

AA ,

, ,

0 TT == AAB . (4.14) (4.10), (4.12), (4.13) (4.16) , - : [ ]

[ ] .|;1|

11

1

AAAAAD

AAB

==

=

(4.15)

(4.15) B D . , (4.15) , , , - .

- , ,

57

. , , - .

- . (4.6) , , , . -, ( ) .

, (4.9) , - , . -, , , , .. .

, , -. , , - .

- . , , , , , . - , .

, : m+1 - , -. , . , [ ] | AAA = . , - , , .

- (4.1), = 1 , , , -

58

, .. : , - , 1. (4.1). -, (4.1), (4.1), - (4.1) . 1, 1, 0, . . :

. 1. , -

, , , , , .

2. ik j.

3. kj, j, , - j k1, .

4. i > j, i j. 5. j , aij,

j - .

6. aij=1, j 1. 7. j

59

, - . - , Z ( IU

GG= Z ) k, l m, n (. 4.11, ). -

- Z :

U1= z11 i1+z12 i2 = ukl= uk ul, U2= z21 i1+z22 i2 = umn= um un,

U1, U2, i1 i2 -; ukl, umn , uk, ul, um un , .

- , . 4.11, , . . .

, . , , , - . , , . , - , . , -, . , - - .

60

5 5.1

- - . - . .

f(t) - t (). ,

f(t) = 0 t < 0, (5.1) .. - t. f(t) - , , - - .

(5.1), -, . p=+j , . F(p) f(t)

=

0)()( dtetfpF pt , (5.2)

p, . f(t) -. , , , , , - . f(t), (5.2), . - F(p) () p. - , f(t), L- , , f(t). (5.2) :

t

f(t)

0 5.1

f(t)

61

F(p) =L+[ f(t)] F(p) f(t), L+ (5.2), () F(p) f(t).

: 1) -

-

>

0,

, , L-

pApF =)( -

pAALpF == + ][)( ;

2) L-

>

0. >0 +>0, (5.2) . , . :

1. . 2. .

- . f(t) F(p),

t

f(t)

0 5.2 - f(t)

t

f(t)

0 5.3

f(t)

62

. , F(p) - f(t), (5.2), f(t) . : L- L- , ..

)()(1

tftfn

kk

=

= , )()]([)(1

pFtfLpFn

kk

=

+== .

L- ( F(p) =L+[f(t)]), - , - L-. - f(t) =L[F(p)], -. , -

)()(0

pFdtetf pt =

-

f(t). 5.2

- . i(t) RL-

, - E i(0)=I0. - :

.

)()(;)()(dt

tdiLtURtitU LR ==

(. . 5.4)

: )()()( tUtUtE LR += . , -

EtRidt

tdiL =+ )()( (5.3) i(0)=I0.

R E UR

UL

L

i(t)=iL(t)= iR(t) 5.4

RL-

63

:

=+000

)()( dtEedtetRidtedt

tdiL ptptpt .

pEdtEe pt =

0 - , .

==

00)()()( pRidtetiRdtetRi ptpt , i(p)=L+(i(t)) L-

i(t) .

dttdi )(

-

:

[ ])()0()()()(000

ppiiLdtpetietiLdtedt

tdiL ptptpt +=

+=

.

, (5.3) p

pEpRippLiLi =++ )()()0( ;

- )0()()( LipEpRippLi +=+ .

:

RpL

LIpE

pi+

+

=

0

)( . (5.4) (5.4) (5.3) - p. i(t), (5.4) , .. i(t)=L[i(p)]. (5.4) , .. :

pLRLI

pLRpE

RpL

LIpE

pi+

++

=

+

+

=0

0

)( . (5.5)

+

=

+

LRppL

EpLRpE 11/

,

( )teAppA

+1)( . ,

64

=

=

+

tLR

tLR

eRE

eRL

LE

LRppL

EL 1111 .

(5.5) L- -

tAep

A

+ )( , .. t

LR

eI

LRp

IL

=

+0

0.

, (5.4) t

LR

tLR

eIeRE

ti

+

= 01)( .

(5.3), .5.5 . - I0=0, - (. .5.6):

=

tLR

eRE

ti 1)( .

- - :

1. - - (5.3), , - (5.4) . , -- , - . ( ) . - .

2. ( ) .

3. - . - , () .

i(t)

t

RE

5.6 - RL-

65

5.3 L- () , , , () - . .

5.1

f(t) F(p)

1 A(t) A

2 A pA

3 At 2pA

4 tAe +pA

5 tAte ( )2+pA

6 tA

sin 22 +pA

7 A cost 22 +pAp

8 teAt

sin2 , 4

2= ++ pp

A2

9

ttAet

sin2

cos2 , 4

2= ++ pp

Ap2

10 ( )teA

1 )( +ppA

11 ( )tA

cos12 )( 22 +ppA

12

+

tteAA t

sin2

cos2

4

2=

)( 2 ++ pppA

66

5.3.1 L- f1(t), F1(p). , (, -, ..) f1(t) f2(t), L- F2(p). F1(p) F2(p) f(t) F(p) , , L- . )()( 12 tftf dtd= . F2(p) = pF1(p) f1(0), , :

)0()()( fppFtfdtd

. (5.6) (f(0) = 0) :

)()( ppFtfdtd

.

, - p - t=0. n- - (5.6)

)0(...)0()0()0()()( )( 1321 nnnnnn

n

ffpfpfppFptfdtd

,

f(k)(0) k- f(t) t=0. ( 0)0(...)0()0()0( )( 1 ===== nffff ) -

: )()( pFptfdtd n

n

n

.

dttftft

=0

12 )()( . , ,

.

)()(

)(1)()()(

11

1112

00

0000 0

ppFdttf

pe

dtfdtep

dttfp

edtedttfpF

tpt

pttpt

ptt

+

=

=+

=

=

67

p,

0)(lim0

1 =

tptdttf

pe

t, ,

,

pf

ppFpF )0()()( 112 += .

,

.

)0()()(0 p

fppFdttf

t

+ (5.7) (f(0) = 0)

ppFdttf

t )()(0

. ,

t p - t=0.

-- , - , - . - - p.

5.4 L- (-

), , , .. - . - f(t) =L[F(p)], -, -

)()(0

pFdtetf pt =

.

:

+

=

j

jptdtepFjtf )(2

1)( . (5.8)

(5.8) - .

68

, , - . - L- - . , .. - f(t) =L[F(p)].

1. . , - f(t) F(p), (. 5.1), L-. 2. . L- F(p) f(t) M(p) N(p) , .. F(p) -:

n

mm

ppbpbbpapapaa

pNpMpF

++++

++++==

...

...

)()()(

2210

2210

. (5.9)

(5.9) Fk(p), fk(t). - :

=

=

n

kk tftf

1)()( .

Fk(p) , (5.9). , , - (5.9) -. , N(p) = b0+ b1 p+ b2 p2++ bn pn n - n , ..

N(p) = (p p1)( p p2)( p p3)( p pn). (5.10) p1, p2, , pn, -, , N(p).

b0+ b1 p+ b2 p2++ bn pn

= 0. (5.11) , , .. - N(p), - . pk =k + jk (5.11), pk+1 =k+1 jk+1. (5.10), F(p) (5.9)

69

))...()((...

)()()(

21

2210

n

mm

pppppppapapaa

pNpMpF

++++== . (5.12)

p1, p2, , pn, , . (5.9). , -, F(p) -. , -, (5.9) , .. m

70

tpn

k k

k kepNpM

tf =

=

1 )()()( . (5.14)

(5.12) (5.14) -

))...()((...

)()(

21

2210

n

mm

pppppppapapaa

pNpM

++++=

tpn

k k

k kepNpM

=

1 )()(

.

. k ( nk ,1= ) --, f(t) . -, - , - . -. 5.1. , - (5.9). , , - .

5.5

- , - ( ) (- ). - - - ( ) t. - - . - - .

- - , L- ( -) . .

71

5.2

t

I =

=

N

kk ti

10)(

=

=

N

kk pI

10)(

II =

=

N

kk tu

10)(

=

=

N

kk pu

10)(

ik(t)Ik(p), uk(t)uk(p) L- .

, L- RLC- - .

dt

tdiLtu LL)()( = , , -

, UL(p)=pLIL(p). (5.15, a)

UR(p)=RIR(p); (5.15, )

pCpIpU CC

)()( = . (5.15, ) (5.15) , L- - .

pL, R pC1

(5.15) - . . Z(p), Y(p). , - . , L- , L- - - , . - , , , - L- ,

72

L- , - . , - - , , - L- .

5.5.1

- , RLC- . , L- - . , ,

, - - - . - - - RLC- (. . 5.7)

pCpLRpZ 1)( ++= ;

1)(1)( 2 ++== pRCLCp

pCpZ

pY .

- - .

. RLC- - L- - :

PCpLR

pEpZpEpI 1

)()()()(

++== .

(t)=,

R E UR

UL

i(t)=iL(t)=iR(t)=i(t) 5.7 -

RLC-

U

L

73

pEpE =)( ,

++==

LCp

LRpL

EppZ

EpI1)()( 2

.

, RLC-:

LCp

LRp

pEppLIpU L 1)()( 2 ++== ;

++==

LCp

LRpL

ERpRIpU R 1)()(

2;

++==

LCp

LRppLC

EpC

pIpUC 1)()(

2.

- , - . , , - .

5.5.2

- . - L- . - :

))(()( 0pIpIpLpU LL = ;

74

))(()( 0p

UpUpCpI CC = .

. - I0:

.)()(;)()(

0

0

pIpI

pLpU

LIpUppLI

LL

LL

=

+=

(5.16)

U0:

.)()(;)()(

0

0

pUpU

pCpI

CUpIppCU

CC

CC

=

+=

(5.17)

(5.16) (5.17), :

1. I0 L - , I0.

2. U0 - , U0. L- , -

( pI0

pU 0 ), L- . 5.3 - .

75

5.3

dttdiLtu )()( =

+=t

dttuL

Iti0

)(1)( 0

.)()( 0pIpI

pLpU

LL

=

0)()( LIpUppLI LL +=

dttduCti )()( =

+=t

dttuC

Utu0

)(1)( 0

0)()( CUpIppCU CC +=

pUpU

pCpI

CC 0)()( =

, , , - .

I0 L

pL

LI0

+

U0 C

pC CU0

pL1

pI0

pC1

pU0

76

6 , - . - , , , . - , , . -, . , .

6.1 - , , - , .

, i(t) u(t) :

u(t)=R i(t) = i(t)/G, (6.1) R ; G , . , -.

q = f(u), q . f(u) - u(t) , -

q = C u(t), . , - ,

i(t) = dtdq

.

dttduCti )()( = , (6.2)

+=t

dttiC

Utu0

0 )(1)( , (6.3) U0 t=0.

77

-, L . -

= f(i), . i(t) L,

= L i(t) -. -

dttd

tu)()( = .

dttdiLtu )()( = , (6.4)

+=t

dttuL

Iti0

0 )(1)( , (6.5) I0 , t=0. (6.1)(6.5) -, , - . , - : . , ( -) . - . - . - () 6.1,. , -, - . , , . - , e=0 , - .

78

i

u

e

e

+

i

u

J

J

) ) 6.1 () ()

. 6.1, . , . - J , - . , - - . - , . 6.1, . - J . J , - . , - J = 0 - . 6.1

- --

- -

1

u(t) = R i(t) i(t) = G u(t)

U(p) = R I(p) I(p) = G U(p) R G= R

1

2 -

dttduCti )()( =

+=t

dttiC

Utu0

0 )(1)(

0)()( CUppCUpI =

pU

pCpIpU 0)()( += pC

1 pC

3 -

dttdiLtu )()( =

+=t

dttuL

Iti0

0 )(1)(

0)()( LIppLIpU =

pI

pLpUpI 0)()( += pL pL

1

79

6.2

1

u(t) = e(t) U(p) = E(p)

2

i(t) = J(t) I(p) = J(p)

3

u(t) = e(t) R i(t)

U(p) = E(p) R I(P)

4

i(t) = J(t) G u(t)

I(p) = J(p) G U(P)

6.2 , 6.1, . - . - 6.2 (. 3). R - . u(t) e(t) , (.. R). 6.2 (. 4), G J(t).

e(t) u(t)

i(t)

J(t)

R

e(t) u(t)

i(t)

G J(t)

u(t)

i(t)

80

: - e J, , (. . 6.2) u(t) R i(t) ?

.6.2, u(t) = e(t)R/(R+R),

. 6.2, J(t) = (1/R+1/R) u(t).

u(t) , J(t) = e(t)/R e(t) = J(t) R. (6.6)

, - R - R, - (6.6). - , , - , . , e(t) -' - , J(t) , - -'. -, -', . -' , . 6.2, , - , -. , . 6.2, , - , . , , - : 1. E -' , J, . , J .

R

e(t) u(t)

i(t) A'

A

R R J(t) u(t)

i(t) A'

A

R

) ) 6.2 , R:

a ;

81

2. , - - . . ( ) -' - I, ( U ). R=1/I ( R=U).

6.3 . , , . , - . . () . 6.3, . ,

,

;0

12

1UU

I=

=

(6.7)

. (6.7) - :

=

+

0

00001

100

2

1

2

1

II

UU

.

+

gU1

+

U1 U2

I1=0 I2

, ()

)

, ()

)

, ()

)

, ()

) 6.3

+

+

U1

+

U1 U2

I1=0 I2

+

I1

+

U1=0 U2

I1 I2 +

+

rI1

+

U1=0 U2

I1 I2

82

, (), . 6.3, . , , :

,

;0

12

1gUI

I=

=

(6.8) g . - (6.8) :

=

+

00

1001

000

2

1

2

1

II

UU

g.

, (), . 6.3, . , -,

,

;0

12

1IrU

U=

=

(6.9) r . (6.9)

=

+

0

0000

1001

2

1

2

1

II

rUU

.

, , (), . 6.3, . , , :

,

;0

12

1II

U=

=

(6.10)

. (6.10) :

=

+

00

100

0001

2

1

2

1

II

UU

.

- : , , ..

6.4

- ( ) , . -.

83

. , , . - , , - . - , . - - , , - . , , , . - .

6.4.1 R, . (. 6.4, ), R

: L (. 6.4, ). - , , - -, . 6.5, .

R = R0 l/w, R0 ; l, w . , , , . =

w d / l = w l / d, , d . , -

)

)

L R

C

6.4 () ()

84

, , . 6.5, . - , - - p-n-. - - . - , - , . 6.6,.

- , p-n- . (. 6.6, ),

)

6.6 () (), ()

p

n

p

u1 u2

)

r1 r1 r1

R1 R1 C1 C1

u1 u2

R

C C

)

w

l )

)

R

C

6.5 () ()

C

85

+= )()( 1112

2ul

dtdu

uCrz

u,

u(z1, t) = u1(t), u(z2, t) = u2(t), r1, l1, C1 - , ; z1, z2 ; u1, u2 . , . 4.6, .

6.4.2 - (6.2)-(6.3), - .

- , - (. . 6.7, ). - (0,13,0 /2), 10 . - R, . - , , , - r R.

. 6.7, - - -. R L , r = S / d; r = 2 R0 lw, S - , d - , R0 - -

, l w . - - , - ,

6.7 () ()

)

R

C

)

r L

86

. p-

n-, . - , , - p-, n- (. . 6.8, ). - , , p-n-, - : (u) = C0 / (1u / )m, C0 ; m=0,330,5 , . - - , . 6.8, .

R p-n-, r - n+-, n+--.

6.4.3

(6.4)-(6.5). - r (. . 6.9, ).

- ( /8) - ( ).

10 () , - - (. . 6.9, ). Q=120-150 - 300 . -

p n

p

u 0

a)

n+

r

R

C C

0

)

u

6.8 - () ()

87

, , 10-20 , .

. RLC- . , .

6.5 . -: , - , , - - .

. - - , - ( - ). ( ) - .

, - ( - ) , .

- ,

6.9 - () ()

) )

r

L

88

. , , ( -).

6.5.1 - p-n- -. - , . 6.10, . , ; R , - ( 50200 , 150 ); r ( 110 ); I .

-:

=

1

0m

u

eII , (6.11)

u ; ,

I r C

R

u

I

U

2

1

IB

) ) ) 6.10 :

(), - (), ( 1) ( 2)

89

q

kT= , k=1,381023 / ,

; q=1,60221019 . 27 (300 ) =0,0256 . I0 - , - ( I0=105109 ); m - (- - ); .

(6.11) - () , - . - (6.11) u (u>BV):

)(

+

=NBV

BVu

eIBVI , (6.12) BV (BV=(1,52)Umax); IBV ; NBV .

- :

=+, .

[ ]

>++

=

+, , )1(1)1(

; ,)1(

1

0

0

ku

nk

knu

MunnM

Mu

C

k

k

(6.13)

0 ; k - , (~ 0,7-0,8 , ~ 0,4-0,5 ); n - ( - ), 0,33 ( ) 0,5 ( - ); - ( M=0,51).

>

=

,0 ,0

;0 ,

0

u

ueIm

Cm

u

(6.14)

90

- (2

1f= , f -

).

. , - . . , [6, 8, 10].

U0, I0. (. . 6.11, ). U0 -, -

G u r C(U0)

R

u

I

U

I0

) ) 6.11 :

(), ()

U0

0

0

)(Uudu

dIUG

=

=

6.12

- -

L2 L1

91

00

0

0

)( =

==

m

U

Uue

m

IdudI

UG ,

. G(U0) - U0.

, - (. . 6.12, ) . I , - . L1 L2, - . . 6.12.

6.5.2 - . - -, -, ., - - -. -, - .

. 6.13, .

U

U

U

I

I I

) ) )

6.13 p-n-p- () n-p-n- (); ()

(. . 6.13, ) :

I = f (U, U); I = f (U, U).

- -, -

92

:

,

;

RRF

RFFIII

III+=

= (6.15)

=

=

1 ;1

0

0m

U

Rm

U

F eIIeII , (6.16)

I0, I0 - -; U, U - -; F, R - - ; m, m ; . F, R I0, I0 . - ,

FI0 = R I0 I0. I0, (6.15) , , :

,

;

III

III

F

R+=

= (6.17)

=

=

1 ;1

0

0 m

Um

U

eIIeII .

(6.17) --.

, - (6.15)-(6.16), . 6.14. , - , - .

93

R R

R

IR IF

IFF IRR

R R

I I

IU U

6.14 -

.

R ( 5500 );

R ( 150 , );

R , - - ( 1...50 );

R - - ( 1 );

R - - ( 1 ).

, , . , : = + .

[ ]

>++

=

+, , )1(1

)1(

; ,)1(

1

0

0

MUnnM

MU

CU

n

nU

(6.18)

: = 0,751,0 ; n = 0,330,5; M = 0,50,9.

94

- . (6.18) . : = 0,50,75 ; n = 0,330,5; M = 0,30,6.

>+

=

,0 ,0

;0 ),(

0

U

UIImC R

R

R . - :

>+

=

,0 ,0

;0 ),(

0

U

UIImC F

F

F . F -

f.

F

FF

=

1 - -

. f,

21

fF . - -

, - .

: , - .

- -: , . 6.16.

6.15 f

f

F

1

f

0

95

- ( ) . 6.17. :

r, r, r , , ;

r - -, ;

R = (1+0)/S -,

0

00 1

==

dIdI

-

; 0 - ;

R , - ;

0

=

m

IdUdIS - ;

-;

, - .

6.16

-

L L

L

96

r

R I=SU

r

r

r

RU

6.17 - ( )

, - S, :

+= j

SS1

0,

r ; = 2 f - ; S0 . / - ( ) , . 6.18.

r

I=SU

r

r U

6.18 - ()

- -: r , -

][6,25

II

r

= ;

97

, - , ..

2

1frm

IC F

= ;

r , 12000 ;

S ,

rdU

dIS = , - -

. -

+

+

=

11

)( 0

0

ffjf

fjef

ffjm

,

0 ; f = (1+m)f . r ( 5500 ). C - C = C C =(1) C, C - - (6.18). = S / S, S S -. - - 0,50,8.

. 6.17-6.18 - - . / , . 6.16 -.

6.5.3 (FET) - . (, , ) . , -

98

, . - - (Junction FET) , - ( -) (MOSFET) . - - -. . (Level=1) ( ) . - (, ..). - . [6, 9, 10 .].

p-n- - , - . 6.19, n-. - , . 6.19, .

-n

-p

Rd

Igd

Rs

Rg

Igs

gd

gs

ds

Idrain

) )

6.19 p-n-: () - ()

99

6.3 -.

6.3 c p-n-

U0 U 2

- 104

0 /2 IS p-n- 10101014 m 1 ISR p-n- 0 A

mR 0

0 1/ Rg 020 Rd 050 Rs 050

Cgd0 - - 150

Cgs0 - 150

n 0,330,5

M 0,5

z p-n- 0,51

Uk , 0

:

Ig = Igs + Igd, Igs -, Igs = In + Ir Kg + Ii;

In ,

=

1mU

n

gs

S eII ;

Ir ,

=

1Rgs

SRm

U

r eII ;

100

Kg , 22

005,01

n

z

gsg

UK

+

= ;

101

- . 6.20. - (. 6.19, ) . ,

gs

drainm dU

dIG = -

. -

gs

gs

fjm

GCfj

eSG+

=

21

20 ,

- ; S0

,

=

000 1

1UU

RS gs

k,

000

=

=

=

gdgs

UUds

dsk I

UR (6.19)

- (. . 6.21).

Rd

Ggd

Rs

Rg

Ggs

gd

gs

ds

UgsGm

6.20 p-n-

Ids

Uds Uds

IdUgs

= 0

6.21 Rk0

102

(- , MOSFET) -, (- LEVEL). (LEVEL=1) -, - , - . - . . - . 6.22, . . 6.22, - . - .

Rd

Rs

Rg

Cgc

gd

gs

Idrain

-n -p

6.22 - : - () , ()

(Uds0) - :

103

(Uds < 0)

104

6.4 - c

U0 U - 0

- 104

0 /2 Rg 020 Rd 050 Rs 050 Cgd - 150 Cgs - 150 Cbs - 5100

1

0,6

- (), . 6.23. , - . - , - . - 6.5.

6.5 -

- -

Gm 0,110 / 0,130 /

Ri 0,11 0,14

Cgs 215 215

Cgd 0,3 10 0,110

Cds 315 < 1

105

gd

gs

Gm

ds

Ri

Ugs

Ugs

+

_

6.23 -

f )1(2 000

kgs RSC

Sf+

= .

, - . - , 6.5.2.

6.5.4 , - (), - , - . , -. , , - , - . - . , . - - , , . , , , , , - .

106

, . ( ). - . , - , . - .

, , , - , .. . , . - : -, . - , , .

(). , (30200 ), , - ( ) ( ) -, . , - - . 6.24.

U +

U1

U2

U1U2

U

U

,

lg( f ) f1 f2

6 /

12 /

0

) ) ) 6.24 :

(), () - ()

107

- , , - . - . 6.25, . R - U1 U2; Ui = K (U1 U2) ( - ), R - ( ). , - , . 6.25, . VD1, VD2 , V1 V2, . R1, 1 R2, 2 - .

3

0

1)(

+=

pKpK ,

0 = S3 R3 S4 R ,

1333 2

1fCR == , f1 -

. - RC-, , .

U

+

_

U1

U2

R1

U = K U

R2

0

I

+

I2_

U1

U2

R1

R2

1

2

I1U

=S U3

R 3 C3U3

U1'= U

U

VD1 VD2

E2E1R

3 I =S U4 34

) ) 6.25 :

, (); ()

R . - , -

108

. - U'1=U . , - , I1 I2 - . , - . . - . - - : , Model, Micro-CAP, Parts - DesignCenter (PSpice).

109

7

, , - - - () (2, 3, ) . - . - , - . , - , - . - , - , - , ., - - . . , K(j) , K(j) g(t) - :

+

= dtetgjK tj)()( ,

+

=

j

j

tj dejKtg )(21)( . (7.1)

, (7.1) -; - ( ) . K(j)=K()e j (), , K() - -, () . g(t) h(t)

)()( thdtd

tg = , =t

dgth0

)()( . (7.2)

(t) - (t) -

110

(): = dtxtgtx

t

0 )()()( . (7.3)

(j) = K(j)(j), (7.4)

(j) (j) . : - , - . (7.1) - (7.4) . , (7.1) . - (7.4), - . - , , - ( ) . , . - .

7.1

, - , z(j) = r() + jx() y(j) = 1/z(j) g() + jb(), - . . :

; .

() Z, Y, , . S .

111

- N (. 7.1). - -.

7.1.1

, (. 7.1), - . , - , ().

, , . - -. . ( ) Z - ,

+=

+=

2221212

2121111 ,

IzIzUIzIzU

(7.5)

IUGG

= Z ;

=

=

=

2221

1211

2

1

2

1 ;;zz

zz

II

IUU

U ZGG

.

Z z- :

01

111

2 =

=

IIU

z ; 02

112

1 =

=

IIU

z ; 01

221

2 =

=

IIU

z ; 02

222

1 =

=

IIU

z .

z11 z22 , z12 z21 . Z z12 = z21. ( ) Y

R1

e1() U1

I1

N

R2

e2() U2

I2

7.1

112

U1 U2 -,

+=

+=

2221212

2121111 ,

UyUyIUyUyI

(7.6)

UIGG

= Y ;

=

2221

1211

yyyy

Y .

Y y- :

01

111

2 =

=

UUIy ;

02

112

1=

=

UUIy ;

01

221

2 =

=

UUIy ;

02

222

1=

=

UUIy .

y11 y22 , y12 y21 -. Y - . , y11 1/z11 .., , : Z = Y1. ( ) - , , - U2 I2,

+=

+=

2222211

2122111 ,

IaUaIIaUaU

(7.6)

=

2

2

1

1

IU

IU

A ;

=

2221

1211

aa

aaA .

-. :

02

111

2 =

=

IUU

a ; 02

112

2 =

=

UIU

a ; 02

121

2 =

=

IUI

a ; 02

122

2 =

=

UII

a .

11 22 - . 12 21 . , detA=1.

, I1 U2 -,

113

+=

+=

2221212

2121111 ,

UhIhIUhIhU

(7.6)

=

2

1

2

1

UI

IU

H ;

=

2221

1211

hhhh

H .

h-. :

01

111

2 =

=

UIUh ;

02

112

1=

=

IUUh ;

01

221

2 =

=

UIIh ;

02

222

1=

=

IUIh .

h11 h21 - , h22 h12 - . - h12 + h21 = 0. . , : Z

Zi =

=

m

ii

1ZZ .

- . Y Yi

=

=

m

ii

1YY .

-, - . -

i : =

=

m

ii

1AA .

H - , . : , . i -

114

: =

=

m

ii

1HH .

- 7.1. - :

22

11

21

12det

1dethh

a

a===

YZ ;

21

12

21

12

21

12dethh

yy

z

z===A ;

22

11

11

22

22

11deta

a

yy

z

z===H .

7.1 -

Z Y A H

Z

2221

1211

zz

zz

1121

1222

det1

yyyy

Y

22

11

21 1det1a

a

a

A

1det1

21

12

22 hh

hH

Y

1121

1222

det1

zz

zz

Z

2221

1211

yyyy

11

22

12 1det1a

a

a

A

Hdet

1121

12

11 hh

h

A

22

11

21 1det1z

z

z

Z

11

22

21 det11

yy

y Y

2221

1211

aa

aa

1

det122

11

21 hh

hH

H

1det1

21

12

22 z

z

z

Z

Ydet1121

12

11 yy

y

21

12

22 1det1a

a

a

A

2221

1211

hhhh

- 7.2.

7.2

Z Y A

yz1

=

yyyy

10

1 z

y z1

=

zz

zz

101

y

115

7.2

Z Y A

l

W -

ljWl

jWl

jWljW

ctgsin

sinctg

Wlctgcsc

cscctg

jW

ljW

ljW

lj

lW

ljljWl

cossin

sincos

-

u1 u2

u2/u1= n

n

n

100

u2u1

u2= u1

0001

i 2

u1

g

i2=g u1

000

g

0010 g

u2

i1

r

u2=r i1

000

r

0100

r

i 2i1

i2= i1

10

00

116

7.1.2

- (. 7.2). - -:

=

+=

),(2

1

),(2

1

1111

1

1111

1

IRUR

b

IRUR

a

=

+=

).(2

1

),(2

1

2222

2

2222

2

IRUR

b

IRUR

a

, - , S .

+=

+=

2221212

2121111 ;

asasbasasb

(7.7)

ab = S ;

=

2

1

a

aa ;

=

2

1

bb

b ;

=

2221

1211

ss

ssS .

S s- - ( 1 2, . 7.2):

01

111

2 =

=

aa

bs ,

02

112

1 =

=

aa

bs ,

01

221

2 =

=

aa

bs ,

02

222

1=

=

aa

bs . (7.8)

(7.8) - , :

11

1111 RZ

RZs

+

= ,

22

2222 RZ

RZs

+

= ,

1

2

2

112

2RR

EU

s = , 2

1

1

221

2RR

EU

s = , (7.9)

Z1 Z2 , . , s11 s22 , s12 s21 . - : s12 = s21. - : SS+=1, S+ - (-) .

117

S - - .

. - , , - . - b2 2, b1 1 :

+=

+=

;

;

2222211

2122111

atbtbatbta

(7.10)

=

2

2

1

1

a

bba

T ;

=

2221

1211

tt

ttT .

t- :

02

111

2 =

=

aba

t , 02

112

2 =

=

ba

at ,

02

121

2 =

=

abb

t , 02

122

2 =

=

ba

bt . (7.11)

, , .

: =

=

m

ii

1TT .

S -

=

=

12

21

112221

1211

1det1

t

t

tss

ss TS ;

=

=

ST

det1111

22

212221

1211

s

s

stt

tt.

S 7.3.

R1

1 U1

I1

N

R2

2 U2

I2

7.2

1 b1 2

b2

118

U1

U2

U3

Un

Uk

Y

I1

I2

I3

Ik

In

7.3 Y-

7.3 -

Z Y A H

detZ+z11+z22+1 detY+y11+y22+1 a11+ a12+a21+a22 detH+h11+h22+1 1 detZ+z11z221 detYy11+y22+1 a11+ a12a21a22 detH+h11h221

2 detZz11+z221 detY+y11y22+1 a11+a12a21+a22 detH+h11h22+1

S

221

121

221

z

z

221121

221

yy

2

1

2det21 A

221

121

221

hh

. S Y . Y- (. 7.3). Y- , ..

UI = Y ; 21 ],...,,[ nIIII = ; 21 ],...,,[ nuuuU = . , , -

: 01

==

n

iiI , 0

1 1=

= =

n

i

n

jiji uy ,

.. Y- . - ui ,

11

==

n

ijiy ,

.. . (, k-) , , - . Y k- k- .

, Y-, . - ( ) , - - . -. Y- - ,

119

-: , .. Y- . - . , Y- , Y-. .

7.1.3

- - , . - 7.4.

7.4

Z-

Z=Z1+ Z2

Y-

Y=Y1+ Y2

A-

A=A1 x A2

H-

H=H1+H2

Z1

Z2

Y1

Y2

A1 A2

H1

H2

120

7.2, - , 7.1 7.3 , 7.4 , - , . , - - -. , - - , -, - (7.1)(7.4).

- . - , , 30 -. , D : D , . , -, :

(Z, Y, H, S, T) A, A (Z, Y, H, S, T).

, 10 . -, - . -. , - 7.4, , . , -,

=

2221

1211

aa

aaA ;

=

2221

1211

aa

aaA ,

7.4 - :

121

+

+

+

+

++

+

=

2121

22212221

2121

21212121

222211111212

2121

21112111 ))((

aa

aaaa

aa

aa

aa

aaaaaa

aa

aaaa

A ;

+

+

+

+

+

+

+

=

1212

22122212

1212

222211112121

1212

1212

1212

12111211

))((aa

aaaa

aa

aaaaaa

aa

aa

aa

aaaa

A ;

-

+

+

++

+

+

+

=

2222

2222

2222

212122212222

21122212

2222

212112121111

))((

aa

aa

aa

aaaa

aa

aaaa

aa

aaaaaa

A .

-. - - .

7.2

- Y- . . Y-.

. 7.4 , - (n+1) . , k- yki, ni ,0= . , , . , k- Jk. - .

122

- k-

Jk + Ik0 + Ik1 + Ik2 + + Iks + + Ikn = 0

kn

kii

ki JI ==0

. (7.11)

yki

Iki = (uk ui) yki. (7.12) (7.12) (7.11),

(uk u0) yk0 + (uk u1) yk1 + (uk u2) yk2 + + (uk us) yks + + (uk un) ykn = Jk

kkin

kii

ik Jyuu ==0

)( . (7.13)

(7.13) , - ui ( ni ,0= ). (7.13)

0

yk1

1

2

s

n

p

k yk2

yks

ykn

yk0 k

7.4 Y-

Ik0 Ik1

Ik2 Iks

Ikn

Ikp=Jk

123

kn

kii

kiin

kii

kik Jyuyu = =

= 00

.

=

=

n

ikikk yy

0 k-

,

=

n

kii

kiy0

k-

i- . , :

=+++++

=+++++

=+++++

=+++++

nnnnininn

knknikikk

nnii

nnii

Juyuyuyuy

Juyuyuyuy

JuyuyuyuyJuyuyuyuy

......

...

;.........

;......

;......

1100

1100

111111010

000101000

JU =Y , (7.14)

=

nnnn

n

n

yyy

yyyyyy

...

...

10

11110

00100

###Y ;

=

nu

u

u

U #1

0

;

=

nJ

JJ

J #1

0

.

:

124

1. Y , k- , ..

=

=

n

ikikk yy

0.

2. Y , k- i- ,

, .. =

=

n

kii

kiki yy0

.

3. - , k- . (7.14) Y, - , - . Y- - . 0 - (7.14). Y- - , (n+1) , (n+1)- . , Y- (, , ) nxn. . , - . 7.5

3

1 2

R1 R3

R2

R4 L

1 3 2

0

J1 J2

7.5

125

Y 33.

1 2 3

1 3121

11 pCpCRR

+++ 3pC 12

1 pCR

2 3pC 324

1 pCpCR

++ 2pC Y =

3 12

1 pCR

2pC pLpCpC

RR111

2132

++++

- :

=

02

1

JJ

J .

: p . - p=j2f. . - . - , - . y, j- k- (. 7.6). , I j k. , j- k- , , :

j- : + I = ; k- : I = ,

. I (uj uk) y:

j- : + y (uj uk) = ; k- : y (uj uk) = .

, : j- : + y uj y uk = ; k- : y uj + y uk = .

, , j- k- , Y j k, (j, j) (k, k) (j, k) (k, j). 22

126

yyyy

kj

k j

y (ej ek) (ej ek), (7.15)

ej j- , , j-, , j- . j- . - - . Y, . , yi i- , j- k- .

)ee)(ee( kijikijii

iy =Y . , J j- k- k. j- k-

+

JJ

kj

.

: J (ej ek). (7.16)

, - (7.15)-(7.16) . .

7.2.1

. , () (). :

1. , -.

2. .

k j y

I +

7.6 y Y-

127

3. - . .

, , - , , . , - .

- () i, j k, l 7.5. - , .

7.5

, - k l i j. , (), - (S),

+

+

SSSS

lk

ji

+

+

yyyy

lk

ji

+

U

J=SU

k

l

i

j

+

J=I

k

l

i

j

y

I

128

(+S). S. , (), y, , (). - Y (+ ), ( ).

- . . 7.6 - . - - . Y- 7.6

7.6

- Y-

gd

gs

Gm

ds

Rds

Ugs

Ugs

+

_

g

s

d

+++

+++

+

mgsgsgs

dsgs

mgs

mdsgs

dsgdgs

mgd

gsgdgdgs

GCCpR

pCR

GpC

GpCR

CCpR

GpC

pCpCCCp

s

d

g

sdg

)(11

1)(1)(

r

R I=gU

R

RU

'

''

+++

++

++++

11110

11110

11)(11110011

'

'

pCRR

gR

pCR

g

RgpC

RRpC

Rg

pCR

pCR

CCpRRrr

rr

, Y-, , .. Y- - .

7.2.2

-. . . Y- , , ,

129

, . ( ) , . - :

1. . 2. -

. 3. -

-.

Y- (. 7.7), , , .

T

y1

y2

12

31 T2

4

7.7

, - ( ):

=

333231

232221

131211

)2()3()1(

)2()3( )1(

1

yyyyyyyyy

e

c

bY

ec b

T;

=

333231

232221

131211

)3()4()0(

)3()4( )0(

1

yyyyyyyyy

e

c

bY

ec b

T;

=

11

111 0

20 2

yyyy

Y;

=

22

222 4

14 1

yyyy

Y.

Y- -. -

130

+

+

+

+

=

222232

3233222321

3213331

21213211

0

0

4321

4321

yyyyyyyyy

yyyyyyyyy

Y

.

, , :

1. Y- - , .

2. . 3. Y-

.

. 7.3

- , .. . - , , - -. , - .

YbUb =Ib, (7.17) Yb ; Ub, Ib . , . 7.8, - :

=

6

5

4

3

2

1

6

5

4

3

2

1

6

5

4

3

2

1

11

iiiiii

u

u

u

u

u

u

pCpC

ypL

pLy

.

131

3

) ) 7.8 () ()

6

5

y1 L2

y4

L3

1 2 3

0

J1 J2

J3 1

2

0

y1

J1 J1

L3

y4

J3 6

5

Ad - . 7.8,

[ ].110110100

000011010101101001

3

2

1

321654321

Jd

JJJCCyLLy

AAA =

=

Yb -.

[ ] 0=

=

b

bJbd J

IAAIA ,

bJbd JAIA = ,

Jb . , (7.17),

YbUb = AJ Jb. (7.18) , -,

ntJ

t

J

bn

td UA

AUU

UAU

=

== ,

UJ . Un. -

132

(7.18), Yb

tUn = AJ Jb.

Y=Yb

t, (7.19)

Jn = AJ Jb, (7.20)

: YUn = Jn. (7.21)

, . 7.8, (7.19) (7.20),

Y=Ybt =

=

++

++

++

36556

52

544

64641

1

1

pLpCpCpCpC

pCpL

pCyy

pCypCyy

,

Jn = AJ Jb =

+

=

32

31

3

2

1

0110000101

JJ

JJ

JJJ

.

-, , (. . 7.9).

I = YkU, (7.22) I = Ik Jk; U = Uk Ek; Yk - . (7.22) ,

(Ik Jk) = Yk(Uk Ek). (. .4) -,

Ik = 0; Jk = Jn;

Uk =tUn.

Jn =Yk tUn Yk Ek,

7.9

yk

Ek Jk

Ik

U

I

133

, (7.19), Y

Un = Jn + Yk Ek, Jn , ; Yk Ek , -, - -. - . , . - , , (. 7.10).

ie , ye, - , , - ie, () .

=

110011111001

c

pe

A

ecbe yygy

.

.7.10 -

7.10 -

ie

yc

ie

ye e

gb

c p

134

=

0ec

b

e

b

yy

gy

Y .

: - , -, . , (7.19)

++

==

ccee

ecbee

ee

b

yyyyyygyy

yy)1()1(

0

c

pe

cpe

tAYAY .

, (. 7.10). - , , -. , - , . 7.11.

. 7.11 :

=

11100011

dg

ySCC dgdgs

A.

=

d

gd

gs

b

yS

pCpC

0Y .

7.11

SUg yd

g

Cgs

d

s

Cgd

Ug

135

: S .

, (7.19),

+

+==

dgdgd

gdgdgsb ypCpCS

pCCCpdg

dg)(tAYAY

, (. 7.11). - , -. , - . 7.12, , , .

. 7.12

=

100100100100101

321

21

A

ecbyy

,

, , - . - :

=

333231

232221

131211

2

1

yyyyyyyyy

yy

bY .

(. 7.12), - :

.

321

321

3323231

23221211

13121111t

+

++

++

==

yyyyyyyyyyyyyy

b AYAY

T

y1

y2

13

2

7.12

136

7.4

(. 7.13, ). - (. 7.13, ). . . 7.13, (. 7.13, ). , , - RE, JE. . 7.14.

+

R1 R3

R2

R4

C1

C2

C3

) ) 7.13 ()

()

r

SU

r

C

'

U

137

:

1 2 3 4 5 6 7

1 pC1 pC1 0 0 0 0 0

2 pC1 1

21

1

11

pCr

RR

++

++

1r

0 0 0 1

1R

3 0

1r

11

pCpC

Srr

++

++ SpC

r+

1

p 0 0

4 0 0

1 pCr

2

4

11

pCpCrR++

++ 0 0 0

7.14

R1 R3

R2

R4

C1

C2

C3

r

SU

r 2

C

5

3

4

U

R

J = E/RE

1 6

7

138

5 0 0 SpC + S 331 pCpCR

++

pC3 31R

6 0 0 0 0 pC3 pC3 0

7 0 1

1R

0 0

3

1R

0 ERRR111

31++

:

4 3 5 S +S 3 +S S

: JE , J.

7.5

, . - -. , . 1, , , :

1=1=. 1 . , , .. det(A)0. () - :

=+++

=+++

=+++

nnnnnn

nn

nn

bxaxaxa

bxaxaxabxaxaxa

...

...

...

2211

22222121

11212111

#### ,

139

xj ( nj ,1= ) ; ij ( nji ,1, = ) ; bj ( nj ,1= ) .

Ax=b,

nxn ; x=[x1, x2, , xn]

T - ; b=[b1, b2, , bn]

T -

( ). (), .. .

, : x = A1b.

- (. . 7.15). , . , , . , - , , -, . -, , , LU- . -, . -

LU--

-

-

-

7.15

140

, - . . ( 50 ). - , , - .

. (. [7, 8]).

7.5.1 (. .7.4), - (Y-) . . - . (-). . n () - . ( ) k = n/n, n . - 0,010,001, - - 510. . , . k . - (, LU-) - . . , - () - . - . - n! , n -. , , , .

141

, . - . , - , 103105 107 109. () . , aij : i, j . 3n . - - LU-. , - LU-, , () . -, . , - n , - , , (1020)n. - k - , 5060 . -, .

142

8

8.1

. , - . - . , , , - .

, b , R-, L-, -, , . , Q, . -, , - , . , ,

Ib = 0. (8.1)

Ub At Un, (8.2) Ub, Ib ; Un .

-

=

2

1

2

1

2

1

b

bbb W

WZK

KY

IU ,

Y1, Z2 ; 1, 2 ; Wb1, Wb2 -, , - . :

YbUb ZbIb = Wb (8.3) , Yb Zb - : +1, 1, 0, a Wb , 0.

8.1 Yb, Zb, Wb - .

143

8.1

Yb Zb Wb

Ub RbIb = 0 1 Rb 0

GbUb Ib = 0 Gb 1 0

pCb Ub Ib = Cb U0 pCb 1 Cb U0

-

Ub pLb Ib = Lb I0 1 pLb Lb I0

-

Ub = Eb 1 0 Eb

Ib = Jb 0 1 Jb

, 8.1 - pCb, pLb. , , , , , - 1/. , , - - .

Ub At Un, YbUb ZbIb = Wb, (8.4)

Ib = 0

=

0

0

000

01 t

b

n

b

b

bb

n

bb

nbb

WUIU

AZY

A; (8.5)

TX = W. (8.6)

, . - , . - . RC-, - . 8.1.

144

:

=

11000111

21

4321

A

GCRJ

.

8.1 , (8.5) :

=

0000

00000

1100 00111

0

11

110

10111101

01

011

1

2

1

4

3

2

1

4

3

2

1

4

3

2 J

UUiiiiu

u

u

u

GpC

R.

3 u0, 1, 2, - w7 = 3u0. - - , , .

, , , , 8.1 (. . 8.2)

) ) 8.1 RC- () ()

1

0

2

R2 J1 G4

3

3

R2 G4

1 2

0

J1

145

8.2

-

I = 0

-

U = 0

,

- -

=

+

00

1001

000

2

1

2

1

ii

u

u

g

,

-

=

+

00

100

0001

2

1

2

1

ii

u

u

-,

- -

=

+

0

00001

100

2

1

2

1

ii

u

u

-,

-

=

+

0

0000

1001

2

1

2

1

ii

ru

u

-

=

+

00

0100

0001

2

1

2

1

ii

u

u

i j

U

i j

I

gU1 U1 U2

I1=0 I2

I1 U1=0 U2

I1 I2

U1 U1 U2

I1=0 I2

r I1 U1=0 U2

I1 I2

U2 U1=0

I1=0 I2

146

- . 8.2, , .

) ) 8.2 () ()

2

0

3 G3 G2

4 1 C4

C5 U6 U7 E1 G3

G2

U7=U6

1

2

0

1

3

4

C4

C5 U6

- . 8.2:

=

1001000011010000011100000011

A

=

+

000000

01

11

11

1

100

1 1

7

6

5

4

3

2

1

7

6

5

4

3

2

1

5

4

3

2

E

iiiiiii

u

u

u

u

u

u

u

pCpC

GG

.

, - (.. ). (-) :

D= / . 1818 39 . , D=39/182 0,012 = 12%.

147

. , - , .

8.2

, - , , Ub, Ib Un. , -:

Ub = AtUn.

Ub, (8.4). , - (8.4) ,

YbAtUn + ZbIb = Wb, (8.7, ) Ib = 0. (8.7, )

.

00

t

=

bb

nbb

n

bbn

WIU

AZAY (8.8)

(8.7) (8.8) . - , . , (b+n)x(b+n), b , n . - .

. - -. , , , , YbAt. , YbAt. . ya yb, - i, j k, l. Yb At:

.

0000

11000011t

=

=

bb

aa

b

ab yy

yyba

yy

lkji

AY

, YbAt

148

1 yb. -

. i, j , k, l . (22). Yb At

.

11000011t

=

=

ddcc

bbaa

dc

bab yyyy

yyyyba

yyyy

lkji

AY

, a, b i, k , j, l .

. , , -, - ,

YbUb + ZbIb = Wb

Ib = YbUb Jb, (8.9) , 8.1, Zb= 1, Ib = Jb. (8.9) ,

Ib = YbtUn Jb. ,

Ib =(YbtUn Jb) = 0

Ybt = Y, Jb = Jn,

YUn = Jn.

, -, -.

, ( - ) , . , , . - .

149

8.3 ( ) . 7. . .. , - . () :

1) , ( );

2) , , -, ;

3) . Un - I2. Ub = tUn, - I1 = Y1U1. , - , - . ,

[ ] 01

2

1

321 =

JII

AAA . (8.10)

:

n

J

UAAA

UUU

=

t3

t2

t1

2

1

. (8.11)

:

U1 = nUA t1 ; U2 = nUA t2 ; UJ = nUA t3 . (8.12) . , ,

I1 = Y1U1. (8.13) :

150

Y2U2 + Z2I2 = W2, (8.14) W2 . (8.10)

1I1 + A2I2 = A3J. (8.15) (8.13), (8.15)

1Y1U1+ A2I2= A3J. (8.16) (8.16) (8.14) (8.12):

1Y1 nUA t1 + A2I2= A3J, Y2 nUA t2 + Z2I2 = W2.

:

=

2

3

22t22

2t111

WJA

IU

ZAYAAYA n

.

(. . 7) , 1Y1 t1A = Yn1;

3J= Jn, Yn1 ; Jn - . -

=

222

t22

21

WJ

IU

ZAYAY nnn

. (8.17)

, , , - - . , - . - . , (8.17) - (n+n2)x(n+n2), n2 . - , .

151

, - . , - Y2 t2A . , - Y2 t2A .

Y2 t2A 1 y. - (