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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 components
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()
()
..
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. - (