modelling the system
DESCRIPTION
This methodical have step by step description how mathematicals models of electronics components founded. Describe methods which can be used to calculate electrical circuits with electronics componentsTRANSCRIPT
-
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..
2002
-
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.. : . : - , 2002. 205 .
.., 2002 , 2002
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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
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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
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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
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6
. () , , ().
. : 1) , , -
; 2) ,
, ; 3) -
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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.
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1.2 -
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. 1.2.1
, , - .
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, - - () .
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, ( ) - , -
S 1
2 S2 S1 S3 Sn
S1m S11 S2 Sn1 S2m Snm 3
. . .
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Sk
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(k1)-
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(k+1)-
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14
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1. , ( -). , - , . .
2. , - . . , - , ( -).
3. , - , -. - . , - , - . , , .
1.3
1.3.1 ()
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, .
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: , , , . 3, 2, 1- . - , .. - , , (, , , , , ..). - -
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1
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3.
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1. ()
1.1
1.2
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18
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21
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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
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; ; ; .
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, , , , , ..
, , - . - . (, -).
, - , , .
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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
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QG
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G QG
, .
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G Q
G. -
1 ( 2) 0Q
G , -
XG
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.
Q
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(): }|Q{ 1 =K
, > 0 , .
: (. . 2.1).
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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 -, - . ; - , , : , - , .. - , , -, , - -. .
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27
, : +b+z+d = 0, - (/)2 +(/b)2+(z/c)2+ d=0, , , z -, , b, , d . , - .
- , . - .
- -.
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28
, ..
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2
2
=+
+
= tQy
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tTtyxT yx ,
Q
x
y
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29
x, y , Q(t) . - .
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: RC-. :
;)()(;)()(
RtU
tidt
tdUCti RRCC == .
);()()();()( tEtUtUtiti RCRC =+= .
.0)()()( ;
=
+
=
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dttdUC
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dtdUC
CC
CC
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R
C
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UC
i(t)=iR=iC
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31
- .
MM , , (.. ). .
, , - . - , - . - - .
. , - .
.
, .. )Q,X(Y
GGG = . , - , , - , - .
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G X
G QG
.
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.
. , -, , -
-
32
, .. - , - -, , . , , , - , - .
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-, ( ). . - .
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- .
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-
33
2.4 -
: 1. , .
- .
2. . - , - , - , , , , - ..
3. . MM - - - . , . - - . , .
4. . - , ..
)X(min G
Xx
, (2.8)
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G; , -
(2.6) (2.7), yj XG
, j - , - ( , - ).
5. . j, - (2.8).
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-
34
, . -
, - , . , . Q
G=(q1, q2) -
. 2.2, j=1, j=2 j=3, |)Q(| Gj =, j = 1,2,3. . - - Q
G,
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+ 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
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(-, - )
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1 8 11
5 7
4
6
9
2
12
3
10
2.3
-
36
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(- ) , - . 4 -, 2 3.
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, - , - . . 2.3 , - , . - - .
-
37
2.6 ,
. .
() . -
, ..
TT QG
,
XG
( - ).
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G QG
YG
.
- - .
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, . 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) . -
. () . -
, - -. , - , - - , - -.
-, - .
- - - . , .
- , - , - , - . . , - .
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-
43
, , - .
(, - - ) -, - . - , , .
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44
3 () -
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:
(); (); (); (); (); - -
.
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45
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46
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3.1
, : -
; , ; , -
; -
; .
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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. - (