Sbornik zadach (1)

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Mathcad 6.0 Plus
2001 2
621.391.2(07) .. : - Mathcad 6.0 Plus. , - , 2001. 189.
: , , - - . Mathcad 6.0 Plus. . " - " , .
. 2. . 155. .: 14 .
.. , . . , . 3
1. 1.1. 1.1.1. -- x(t) = x(t+mT), T -- , m - - , m= 1, 2, .... x(t) - x(t ) = a 0 + (a k cos k1 t + b k sin k1 t ) =a 0 + A k cos(k1t + k ) (1.1) k =1 k =1 1 = 2 -- 1- ; a 0 , a k b k -- T , : t +T t +T t +T 1 2 2 a0 = T t x(t )dt ; a k = T t x(t ) cos k 1tdt ; b k = T t x(t) sin k tdt ; 1
bk A k = a 2k + b 2k ; k = -arctg , k = 12 , ,3... , ak A k -- k- ; k -- k- ; a 0 -- ( ); k 1 = k -- - k- ; t -- , - . A k k k -- . 1 x(t) = 2 A& k e jk t . 1 (1.2) k = - & (1.2) A k t +T & = 2 x ( t )e - jk1t A k dt . (1.3) T t 4 (1.2) (1.3) -- . A & = A e jk -- - k k x(t). A = A & k k -- . k -- . (1.2) x(t) = C& ke jk t , 1 (1.4) k = - t +T & A 1 x ( t )e - jk1t C& k = k = dt . (1.5) 2 T t
1.1.2. 1.1.1. x ( t ) , - 1 V m 4 . volt . sec , T 2 . sec t 0 1 . sec x( t ) V m. t t 0 if T 0 otherwise T t 1.5 . T , 1.5 . T .. 2 . T .1.1.1. 500
5 volt
x( t ) 4 2 0 2 4 6
5
t sec .1.1.1 5 . - , - . - . t 0 .. T 2. () 1 T rad ( 1 = 3.142 ) k 1 .. 5 . sec 1) T 1. a0 V m. t t 0 d t, a 0 = 0 volt . T 0 2) ( k 1 ) T 2. ak V m. t t 0 . cos k . 1 . t d t. T 0 V m, T 1 2 . sec 2 . volt . 1 . ak 4. ( t 1 . sec ) . cos k . . t dt. . 2 sec sec sec 0 . sec ( cos ( 2 . k . ) k . . sin ( 2 . k . ) 1) ak 4 . volt . . 2. 2 (k ) , a 1 = 0 volt ; a 2 = 0 volt ; a 3 = 0 volt ; a 4 = 0 volt . . 2. T 1 t0 , T 2 t 0 1 T , T 2. T . 2. . ak V m. t cos k . t dt; T 2 T 0 6
1. . ( cos ( 2 . k . ) k . . sin ( 2 . k . ) 1 ) T V m. ak . 2 2 2 ( k . ) , k>0 ak . 3) C ( k 1 ) T 2. bk V m. t t 0 . sin k . 1 . t d t. T 0 t 0 1 T , T 2. T . 2. . bk V m. t sin k . t dt; T 2 T 0 1. . ( sin ( 2 . k . ) k . . cos ( 2 . k . ) k. ) bk T V m. . 2 2 2 ( k . ) T. V m bk k>0. k. k- 2 2 Sk ak bk k 1 T. V m . Sk k. , A( k ) C 0 if k 0 .
S( k ) if k 0 bk k atan . ak ak=0 bk 0 2 2 ( ) F u( ) . 2 2 2 ( ) - (1.14) R 1 . assume , > 0 , R 2 2 1 . 1 . E( ) d .R 2 2 2 ( 4. R ) 3 ( ) 0 2 1 . E u = 0.625 sec watt . Eu ( 4. R ) 3 (1.22), , , 57
c 2 1 . E c d .R 2 2 2 ( ) 0 c atan . 2 c 2 . c 2 . E c . c 2 ( 2. . R ) 3 . 2
, (1.22) , c , c atan . 2 c 2 . c 2 2 . . . 4. R . 3 c 2( 2. . R ) 3 . 2
. , Mathcad Find (x), x - , - ( c ). - : 1 c 1 . sec - ; GIVEN - , .E u E c ; Find c - , - c Find(x). 1 = 18.374 sec , c. , 0.95 fc f c = 2.924 Hz . 2.
1.4.4. U m 1 . volt 1 0.1 . sec 0.95 . 58
(.1.4.5 T 50 . sec T t 0.4 . T , 0.4 . T .. 0.4 . T ) 500 .t E(t) U m. e if t 0 .
0 otherwise
1 volt
E( t )
40 20 0 20 40 t sec .1.4.5 . t=tm - 0.95 tm 1 2 .t 2 .E . U m. e dt , R 0 - , , t m . assume R , U m , > 0 2 tm 2 U m 2. . t U m 1 . 1 . e dt . exp 2 . . t m . R R . (2 ) ( 2. ) 0 R 1 . (1.11) assume U m , R , > 0 2 1. 2. .t 2 1 .U m U m e dt . R ( 2. R ) 0 , 1 . 2 1 E U m . E = 5 sec watt . . (2 R) 59 2 2 1. . U m U m . 1 . 1 exp 2 . . t m ; 2 R R ( 2. ) ( 2. ) 1 . ln ( 1) - , Mathcad. 2 , tm - . 1 . tm ln ( 1 ) , 2. t m = 14.979 sec . E ( 1 ) . E - E = 0.25 sec watt . E . . = 22.361 %. E
1.4.3. 1.4.1. 0.95 - 1.4.4 1 . volt 1 U m 0.1 . sec . . .R. c . tan . E . , 2 U m 1 . 2 1 E U m . - . . (2 R) 1 c = 1.271 sec .
1.4.2. 0.95 - U(t) : 1 V m 4 . volt . sec , 1 . sec. (.1.4.6) 60
U(t) V m. t if 0 t . 2 0 otherwise
2
volt U( t) 2 1 0 1 2 3
2
t sec .1.4.6 . 2 2 A c 4 . cos . c cos . c . . c ; 2 2 A c cos . c . . c 4 ; 3 3 B c Si . c . . c 4 . sin . c . . c; 2 2 C c 4 3. . c , 2 3 2 V m . Vm A c B c C c . . 12 . R ( 6. ( . R ) ) c 3
. Find c . c, -
fc , .. f c = 6.067 Hz . 2.
1.4.3. 0.95 - 1 1.4.3, 2 . volt . sec 1 0.5 . sec . .
61
2 2 2. . t m 2 1 . 2. 2. . t m 1 2. . t m . e 1 . 3 . 3 4. R . 4 R t m . t Find t m t = 6.296 sec .
1.4.4. 0.95 - u(t) 1.4.7, 1 . sec U m . 1.5 volt .
u(t) 2 Um
u( t )
volt 3 2 1 0 1 2 3 4 Um
2
t sec .1.4.7 . n c 7 . 1 c n c. c = 21.991 sec c f c f c = 3.5 Hz . 2.
1.4.5. 0.95 - 3. ( ms 10 sec) u ( t ) , t 0 1 1 (.1.4.8) 2 . volt . ms , 0.5 . ms 0.2 . ms . .t u( t ) . ( t ). e if t 0 . 0 otherwise 62
4 volt u( t) 2
50 0 50
3 t . 10 ms
.1.4.8 2 2 . B 2. . 2 . . 1 , - c atan . 2 2 . B . . ( 2. . 1 ) . 2 2 c c .B . . 4. R 3 2. ( . R ) 3 . 2 c 2
1 Find c = 381.144 sec . c, f c , .. f c = 60.661 Hz . 2.
2. 2.1. 2.1.1. - - X (t ) . (t) x(t). x(t) - . - .[ m 2 (t ) = M X 2 (t ) = ] x 2 p(x, t )dx. (2.2) - 5) o o2 D = m 2 = M X (t ) = (x - m 1 ) 2 p(x)dx , - (2.3)
t. 6) () o o R ( ) = M X( t ) X( t - ) = [1 x 1 - m1 ] [x 2 - m1 ] p( x 1 , x 2 , )dx 1dx 2 , (2.4) 424 3 1424 3 - - o o x1 x2 R() D = R (0) . 7) k
k = | ( ) | d , (2.5) 0 R ( ) R ( ) ( ) = = - . (2.6) 2 D : ) - - m k = M X k (t ) ; [ ] ) - tm 2 1 t t k x (t ) = lim x k (t ) dt, t - m , m . t m t 2 2 m t m - 2 , - , . 64
, m k = x k (t ) . . (t) R=1 . : 1) m1 = x(t ) - ; 2) m 2 = x 2 (t ) = P - ;
o o2 3) m 2 = D = x (t ) =P - , .. ; 4) = D - , .. - . 2.1.2. 2.1.1. X(t) - 2 0.5 . volt , 1 . volt 0.2 . sec - 2 . () ( ) e - X(t1) 2 1 x1 . p x1 . exp . . 2 . 2 2 ( - m1, D) ( - R() k). . , (2.1), X(t1) assume , 2 1 x1 m1 . x 1 . exp dx 1 . 2 . 2. 2. , m 1 , .. m 1 = 1 volt . (2.3), , X(t2), assume , 65
2 1 x2 . 2. 2 D x2 m1 exp dx 2 . 2 . 2. 2. 2 2 , D D = 0.25 volt . (2.6) 2 2 2 R( ) D . exp ( . ) R ( ) . exp ( . ) . , (2.5), assume , > 0 2 1 . k( ) exp ( . ) d . 0 . 2 , 1. k ; k = 1.982 sec . 2
2.1.3. 2.1.1. U ( t ) 1 U m 2 . volt 0 0.2 . sec u( t , ) U m. cos 0 . t . , [-,], .. 1 p( ) if . 2. 0 otherwise , - . . m 1 = 0 volt . 1. 2 D U m . 2 1. 2 R( ) U m . cos 0 . . 2 66
2.1.2. X (t ) - 1 2 . volt . x p( x ) .e if x 0 . 0 if x 2.1.3. (- 1 2 1 . volt 1 . volt ) f( x ) 2 1 .x - X(t) t. . . 1.
2.1.4. X (t ) t 1 1 1 . volt 2 . volt . x p( x ) .e , - p x p . , . 2. . 1. 2 D 4. D = 0.5 volt . 3 2.1.5. , d 2 . volt ( - 2 a 2 . volt ) 67
p( x ) a . x if 0 x d . 0 otherwise "a" - . 2 2 . a , a = 0.5 volt . 2 d 1 . . 4. 2 2 4 2 D a d ( 8. a . d 2. a . d 9 ) , D = 0.222 volt . 36
2.1.6. - 2 Z(t) 0.5 . volt , 0.2. sec - 2 ( ) exp ( . ) Z(t1) Z(t2) - 2 2 1 z 1 2. ( ) . z 1. z 2 z 2 p z 1,z 2, . exp . 2 2 2 . 2. ( 1 ( ) ) 2 2. . . 1 ( ) R z ( ) . 2 2 . R z ( ) . exp ( . )
2.2. 2.2.1. x(t ), t 0, t m [ ] - . - (. - ) | F t ( j) |2 S() = lim m . t m tm 2 | F t ( j) | = F t ( j) F t* ( j) m m m
x(t ) F t ( j) x( z) F t ( j) z = t - , , - m m
S() - R() ,.. 68
R ()e - j S() = d . (2.7) - , -
1 R ( ) = S()e j d . (2.8) 2 - , R() S(), - . , - , - . R() - , - - :
S() = 2 R ( ) cos( )d ; 0 (2.9) 1 R ( ) = S() cos( )d . 0 S() . . . - . S() D , .. P , 1 D = P = S()d . (2.10) 0 (2.10) (1.22) - : c 1 P = S()d . 0 (2.11)
(f>0) - 2S(2f ) f 0; W (f ) = (2.12) 0 f p 0. 69
2.2.2. 2.2.1. - U(t), () 1 3 . volt 0.2 . sec , 2 2 R( ) . exp ( . ) ; R ( 0 . sec ) = 9 volt . U(t) .2.2.1 T 8 . sec 2. T 2. T , 2. T .. 2 . T . 400 10
volt * volt R ( ) 5
20 10 0 10 20
sec .2.2.1 . - (2.7) 0 2 . . j . . 2 . . j . . S u( ) . e e d . e e d . 0 assume , , > 0 2 2 . . j . . S1 u ( ) . e e d j . . ( j . ) 0 0 2 . . j . . S2 u ( ) .e e d S1 u ( ) , S() -
. , S2 u ( ) S1 u ( ) , ..
2 S2 u ( ) ; ( j . ) 70
2 ( j . ) . S2 u ( ) - . 2 2 ( ) S u( ) S1 u ( ) S2 u ( ) . 2 2 ( j . ) S u( ) . ; ( j . ) ( 2 2 ) 2 S u( ) 2. . - . 2 2 ( ) , R 1 . - 2 2. . S u( ) , R ( 2 2) 1 S u ( 0 . sec ) = 90 sec watt . ( ) 1 .2.2.2 W 3 . sec 2. W W, W .. W . 50 100 2 2. R . watt * sec
S u( ) 50
4 2 0 2 4
rad / sec .2.2.2
2.2.2. 0.95 - U(t), 3 . volt 1 0.2 . sec () 2 2 R( ) . exp ( . ) ; R ( 0 . sec ) = 9 volt . .2.2.1. . - (2.9) 71
2 . . S u( ) 2. . e cos ( . ) d . 0 Mathcad : . 2. . . 2 e cos ( . ) d ; 0 2 ( exp ( . ) . ( . cos ( . ) . sin ( . ) ) ) 2 . . limit , , left . 2 2 ( ) , (limit) , 2 S u( ) 2. . . 2 2 , R 1 . - 2 2. . S u( ) , R ( 2 2 ) 1 S u ( 0 . sec ) = 90 sec watt . 2.2.1. () , : 1) , (2.10), assume R , , > 0 2 2 1. 2. . D u( ) d ; R ( 2 2) R 0 2) , D=R(0), assume R , , > 0 2 2 . . lim e . 0 R R 2 , D u P u D u, R P u = 9 watt . (2.11) 72
c 2 2 . 1. 2. . d ; R R ( 2 2 ) 0 1. . . tan - . 2 , 1 c . tan . . 2 1 c = 2.541 sec rad . . ( ) 1 .2.2.3 W 3 . sec 2. W W, W .. W . 50 100 2 2. c c R . watt * sec
S u( ) 50
4 2 0 2 4
rad / sec .2.2.3
2.2.3. X ( t ) : P 0 0.5 . watt . sec 1 ( ) c 5 . sec . S x( ) P 0 if c c .
0 otherwise - . 73
.2.2.4 1 W W 10 . sec W, W .. W . 200
1 c c
watt * sec S x( ) P0
10 5 0 5 10 15
rad / sec .2.2.4 . (2.8) , assume P 0 , c , complex c 1 . j . . 1 . R x( ) P 0. e d sin c. . P 0 . 2. ( . ) c , sin c. R x( ) P 0. . ( . ) , - c
sin ( z ) Sa ( z ) . z P 0. c R x( ) . Sa . . c T T 5 . sec 1.0 . T , 1.0 . T .. 1.0 . T - 400 .2.2.5. . 74
1 P 0. c watt * sec
R x( )
6 4 2 0 2 4 6
sec .2.2.5 D=R(0), assume P 0 , c P 0. c lim . Sa . P . c , .. c 0 0 c Dx P 0. D x = 0.796 watt , , (2.10) c 1. Dx P 0 d ; 0 c Dx P 0. - .
2.2.3. 2.2.1. - U(t) , 2 3 . volt 0.2 . sec 2 2 2 R( ) . exp ( . ) ; R ( 0 . sec ) = 9 volt . . 1. k ; k = 1.982 sec . 2 R 1 . 2 . . 1 . 2 S u( ) exp , . (4 ) R . 75 1 S u ( 0 . sec ) = 35.67 sec watt .
2.2.2. Y( t ) 1 - 1.5 . sec , 1 2.5 . volt 0 10 . sec . 2 1 1 S y( ) .. . 2 2 2 2 0 0 .2.2.6. . 6
0 0 2
volt * volt * sec 4 S y( ) 2
40 20 0 20 40
rad / sec .2.2.6 . 2 R y( ) . ( exp ( . ) . ( ) exp ( . ) . ( ) ) . cos 0 . , (t) - . exp ( . ) . ( ) > 0 , - exp ( . ) . ( ) - 2.2.3. U ( t ) 2.1.1 . .
Dirac ( z ) if z 0 . 0 if z 0 76 1. 2 S( ) U m . . Dirac 0 Dirac 0 . 2 , - 1 ± 0 U 2m . 2
2.2.4. Y( t ) 1 10 . sec 1 1.5 . sec , 2.5 . volt 0 2 . . R y( ) .e cos 0 . . . .
2 1 1 S y( ) .. . 2 2 2 2 0 0
2.2.5. V(t), 3 . volt 1 0 20 . . sec () sin 0 . 2 R v( ) . . 0. . - signum ( x ) 1 if x 0 1 if x 1 . . . signum 0 signum 0 2 S v( ) . 2 R 0 77
V(t) 1
0 0 2
watt * sec S v( ) . R . 0
200 0 200
rad / sec
.2.2.7 2 2 D v D v = 9 volt .
2.2.6. 0.95 - U(t), 3 . volt 2 a 0.2 . sec () 2 R u( ) . 2 1 a. . 1 c ln ( 1 ) . a c = 1.34 sec rad .
2.2.7. - Z(t), 3 . volt 1 2 . sec () 2 2 ( . ) .e . . R z( ) . 1 . 3 . 2 16 . 5 S z( ) . . 3 2 2 3 ( )
2.2.8. - G(t), 3 . volt 1 2 . sec 3 2 S g( ) 4. . . 2 2 2 ( ) 78 2 2 . D g D g = 9 volt . 2 . R g ( ) .( 1 . ). e .
2.2.9. S(t) (.2.2.8) W( f) 1 . watt . sec if 200 . Hz f 400 . Hz . 0 otherwise .
watt * sec
W( f) 1
0 200 400 600
f Hz
.2.2.8 . W 0 1 . watt . sec , f 1 200 . Hz
400 . Hz (.2.2.9 ms 3 f2 10 . sec)
1. sin 2 . . f 2 . sin 2 . . f 1 . R s( ) W 0. , 2 ( . ) S(t) W 0. f 2 f1 200
R s( ) watt
15 10 5 0 5 10 15
200
3 . 10 ms .2.2.9 79
=0 R s0 W 0. f 2 f 1 R s0 = 200 watt .
2.2.10. (t) " ", - S 0 0.002 . watt . sec 3. 3.1. 3.1.1. () . - - . () . 1. . 1.1. (.3.1.1)
x (t) y(t) x (t) y(t) Fx(j) Fy(j) X(p) Y(p)
.3.1.1 .3.1.2 F y ( j) K ( j) = , (3.1) F x ( j) Fx(j) = [x(t)] Fy(j) = [y(t)] ­ () - x(t) y(t) . 80
K ( j) K ( j) = P() + jQ() = K () e j( ) , (3.2)
K () = | K ( j) | = P 2 () + Q 2 () - - - (); Q ( ) () = arctg - ( ). P ( ) 1.2. (.3.1.2) Y ( p) K ( p) = , (3.3) X ( p) =c+j - ; X(p) = L[x(t)] - ; Y(p) = L[y(t)] - - . () - j , .. K ( j) K ( p) j p . K ( p ) - K ( ) j p . - K ( p) = M ( p) N ( p) . - ( p - z1 ) ( p - z 2 ) ( p - z m ) K ( p) = K 0 , (3.3 ) ( p - p1 ) ( p - p 2 ) ( p - p n ) K 0 - ; z 1 .. z m p 1 .. p n - , - n m. M ( p) = 0 , - - N ( p) = 0 . p 1 , p 2 .. p n - p , - . 2. . 2.1. g(t), t>0 ( - ) - (t) Dirac(t) Mathcad (.3.1.3). g(t)=0 t0 - 1(t) (t) Mathcad (.3.1.4). - h(t)=0 t0 ( =2f ) A1( f ) A( 2 . . f ) . (3.2), () 86
L R. . 2 2 2 2 ( . L. R. C R) .L ( ) atan ; 2 ( . L. R. C R) R. 2 2 2 2 ( . L. R . C R) .L L ( ) atan . - . 2 ( R . ( . L. C 1 ) ) f>0 (=2f) L ( f) atan ( 2 . . f ) . . 2 ( R . ( ( 2. . f ) . L . C 1 ) ) .3.1.6 3.1.7 F0 2000 . Hz F0 f 0, .. F0 . 100
10
1 o . ( 2. ) R . C L A1( f ) 5
0 500 1000 1500 2000 f Hz .3.1.6 2
( f) rad
0 500 1000 1500 2000
2
f Hz .3.1.7 : assume R , L , C 87
R R . lim C. 1 4 2 2 2 2 2 2 2 2 .L .R .C 2. . L . R . C R .L L
L. C , C . K 0 R. L (R10KoM) .
3.1.2. .3.1.8, - c K1 100 0, . . C2 Z2 Z1 C1 R2 - . K1 . R1 +
.3.1.8 Z1 Z2 - , - - : R1 1 . K C1 0.001 . F ; R2 8 . K C2 0.005 . F . Z2 , . . RC- - T 1 R1 . C1 T 2 R2 . C2 . Z1(p) Z 2(p) R1 R2 . Z 1( p ) Z 2( p ) p. T 1 1 p. T 2 1 , - 88
-Z 2 K= . Z2 1 + Z1 K1 K1 + 1 - Z 2( p ) K(p) . Z 2( p ) 1 Z 1 ( p ). K1 K1 1 K1>>1, Z 2( p ) . R2 . p T 1 1 K(p) K ( p ) . Z 1( p ) R1 p . T 2 1 .
3.1.3. .3.1.9, K1(p) K2(p), - K 0 1.
.3.1.9 1- - T 1 0.001 . sec T 2 0.004 . sec . 1 1 K 1( p ) K 2( p ) . 1 p. T 1 1 p. T 2 . - , .. K(p)=K1(p)K0K2(p). , K 0 K(p) . 1 p. T 1 . 1 p. T 2 p - . c 1 2 , p c j . . 89 (3.5), g(t) - . Mathcad Inverce Laplace Transform 1-1(*), * - - . K 0 1 g( t ) 1 . . 1 p. T 1 . 1 p. T 2 1. 1. exp t exp t T1 T2 g( t ) K 0. T 1. T 2. K 0. , t>0. . 2 2 T2 T1 T1 T2 T 2. T 1 (3.8) h(t) , K 0 1 h( t ) 1 . . p. 1 p. T 1 . 1 p. T 2 1. 1. exp t exp t T1 T2 2 2 h( t ) K 0 K 0. T 1 . K 0. T 2 . ; . 2 2 T2 T1 T1 T2 T 2. T 1
t t T1 T2 T2 T1 T 1. e T 2. e h( t ) K 0. , t>0 - . T2 T1
.3.1.10 3.1.11 3 , ms 10 . sec ,
M M 20 . ms t 0, .. M . 100 90
200
1 / sec g( t ) 100
0 5 10 15 20 3 t. 10 ms .3.1.10 1
h( t)
( t)
0 5 10 15 20 3 t. 10 ms .3.1.11 , (t), (3.16). t 1. 1. T1 T2 K 0. T 1. e T 2. K 0. e h( t ) (t ). d . 2 2 T 2. T 1 T1 T2 T 2. T 1 0 1. 1. T2 T1 T 1 . exp t T 2 . exp t T1 T2 h( t ) K 0. , t>0. T2 T1
3.1.4. g(t) - 3 1 3.1.1 (.3.1.5), o 7 . 10 . sec - 91 1 500 . sec . 2- 2 o K(p) . 2 2 p 2. . p o 1 p - . c 1 . sec 1 2 . sec , p c j .. . (3.5), c j . 2 1 . o g( t ) . ep . t d p . . 2 j. 2 2 p 2. . p o c j . - , .. g(t)=L-1[K(p)]. , (3.14). 2 2 N(p) p 2. . p o . C Mathcad 2 2 p 2. . p o 0;
2 2 o - . 2 2 o , K(p) () 2 2 2 2 p1 o p2 o , 3 1 3 1 p 1 = 500 + 6.982 10 j sec ; p 2 = 500 6.982 10 j sec . -. (3.12) (3.14), Res p1 ( t ) K(p)ept p1
2 o Res p1 ( t ) lim . ep . t .. p p 1 d N(p) dp 92 2 o lim . ep . t ; d 2 2 p 2 o 2 p 2. . p o dp 2 2 1. . exp j . o .t 2 j o . - . 2 2 2 o , 2 2 1. . exp j . o .t 2 Res p1 ( t ) j o . . 2 2 2 o Res p2 ( t ) K(p)ept p2 2 o lim . ep . t ; d 2 2 p 2 o 2 p 2. . p o dp 2 2 1. . exp j . o .t 2 j o . - . 2 2 2 o , 2 2 1. . exp j . o .t 2 Res p2 ( t ) j o . . 2 2 2 o (3.14) t>0 g( t ) Res p1 ( t ) Res p2 ( t ) , ..
2 2 2 2 j . o .t j . o .t j . 2 e j . 2 e g( t ) o . o . , t>0. 2 2 2 2 2 2 o o 93
2 .t o .e A( t ) 2 2 o 2 2 2 2 j . o .t j . o .t e e g( t ) A( t ) . ,t>0. 2. j 2 2 sin o .t 2. g( t ) o exp ( . t ) . , t>0. 2 2 o .3.1.12 , - 3 M ms 10 . sec, M 10 . ms t 0, .. M . 100
10
3 5 g( t ) . 10
1 / ms 3 A( t ) . 10 0 5 10
5
3 t. 10 ms .3.1.12 , (.3.1.5), , - - () 2 2 o .
3.1.3. 3.1.1. - RC- (.3.1.13, R 10 . K C 200 . F ) T . R C, T = 2 sec . 94
R
. C .
.3.1.13 . - () 1 A( ) . 2. 2 1 T () ( ) atan ( . T ) . 1. t g( t ) exp , t>0. T T t T h ( t ) 1 e , t>0. . - (t>0), (t), 1 t>0 0 t0. 2 2 o 96 3.1.5. RC- (.3.1.16, R 20 . K C . 0.01 F ), t 1 . R .C g( t ) Dirac ( t ) e . R.C C . - . R . j . . R.C K( ) . 1 j . . R.C
.3.1.16
3.1.6. RL- (.3.1.17, R 2. K L 0.1 . henry ). - . L . R. R. . . R . g( t ) exp t ( t ). L L
.3.1.17
3.1.6. RL- (.3.1.18, R 2. K L 0.1 . henry ). - . R . R. R. . . L . g( t ) Dirac ( t ) exp t ( t ). L L
.3.1.18
3.1.7. - .3.1.19, R 0.5 . K , - L . 0.1 henry C 0.2 . F . 97
C
. L R .
.3.1.19 .
2 ( 2. . f ) A( f ) . 2 2 4 2. ( 2. . f ) 1 ( 2. . f ) ( 2. . f ) L. C 2 2 L .C 2 2 R .C
.3.1.20 . - 1
A( f ) 0.5
0 0 500 1000 1500 2000 2500 3000 f Hz
.3.1.20
3.1.8. , - - RC- - , .3.1.21, - R 1 . K C 1. F . 98 C
R R - . K1 . +
.3.1.21 . 1 . 1 . . g( t ) exp t ( t ). . (R C) . (R C) 1 . . t ( t ). h( t ) 1 exp . (R C) - 1 A( ) . 2 2 2 1 .R .C
3.2. 3.2.1. . 3.2.1.1. " " - (t) 1(t). g(t) h(t), (3.6) (3.9) (j) (). 3.2.2.2. y(t) (3.1). F ( j) = K ( j) F x ( j) = K ( j) [x(t )] . (3.15) , (3.6), F ( j) = [g(t )] [x(t )] . (3.16) - . 99 - . - (3.2), (3.15) (1.8)
F y ( j) = A y ()e j y ( ) = K ()e j() A x ()e j x () = A x ()K ()e [ j ( ) + x ( ) ]. A y ( ) = A x ( ) K ( ) ; (3.17) y ( ) = x ( ) + ( ) . (3.18) , K (t ) , , , , F y ( j) = [y(t )] = [K (t ) x(t )] . (3.19) 3.2.1.3. (t) - x(t) (j) () . 1. .< - .3.2.1. . . - , . , , .
x (t) Ù Fx(j) y(t) ÙFy(j) K(j) Fy(j)= Fx(j)·K(j)< y(t ) = L-1[Y ( p)] = L-1[ X ( p) K ( p)] = L-1 K ( p) L [x(t )] . (3.21) - . 3. . F y ( j) = F x ( j) K ( j) = F x ( j) F g ( j) , F g ( j) - g(t), 100 t t F x ( j ) F g ( j ) x( )g(t - )d = x(t - )g()d = x(t ) g(t ) , 0 0 - . t , y(t ) = x( )g(t - )d = x(t ) g(t ). (3.22) 0 3.2.1.4. - , . - , .. (t ) = y (t ) - y (t ) . (3.23) (t ) ,
[ ] [ ] F ( j ) = K ( j ) - K ( j ) F x ( j ) ( p ) = K ( p ) - K ( p ) X ( p ) , (3.24) K(j) K(j) - - . (t ) = -1 F ( j) . [ ] , , . - (, , ..) , - . - . () - . , . - , , . , - . , x(t) , y(t ) = K x(t - t 0 ) , K - , - K(0) , t 0 - - . - jt 0 F y ( j) = K ( j)F x ( j) = K (0)F x ( j)e . 101 , () - j ( ) - jt 0 K ( j) = K () e = K (0)e . (3.25) , K(0), .. K () = K (0) . , , .. () = - t 0 . , () = 0 . 3.2.1.5. - . : ) 2 2 E y () = F y ( j) = K ( j)F x ( j) = K 2 ()E x () = K p ()E x () , (3.26) E x () - , 2 2 = A 2x () , K p () = K ( j) E x () = F x ( j) = K 2 () - - ; ) 1 2 1 1 Ey = F y ( j) d = K 2 ()E x ()d = K 2 ()A 2x ()d ; (3.27) 0 0 0 ) 2 2 Fy ( j) K ( j)F y ( j) tm tm S y () = lim = lim = K 2 ()S x () , (3.28) t m tm t m tm S x () - ; ) 1 1 Py = S y ()d = K 2 ()S x ()d . (3.29) 0 0 (3.29) (1.10)
1 Py = K 2 ( k )Px ( k ) = a 20x K 2 (0) + 2 A 2kx K 2 ( k ) . (3.30) k =0 k =1 102 3.2.1.6. - - . - . (.3.2.2).
x(t) "-" y(t) x(t) y(t) K(j) K(j) K1(j)
Koc(j)
.3.2.2 K ( j) = K = const , K ( j) = K ( j)K 1 ( j) . , K K 1 ( j) = . (3.31) K ( j) , K ( j) K ( j) = , 1 + K ( j)K oc ( j) K(j)Koc(j)>>1. 1 K ( j) , (3.32) K oc ( j) .. . , Koc(j)= - K ( j) 1 .
3.2.2. 3.2.1. () , - () K 0 1.5 f c 350 . Hz , - () (f)=-2ft0 t0. 103 3 ms 10 . sec. x(t) : U m 1 . volt , - 2.5 . ms T 4. . t 0 y(t) - 2 , . . K ( f) K 0 if 0 f f c .
0 otherwise ( f) 2 . . f. t 0 if 0 f f c .
0 otherwise
x( t ) U m if 0 t
U m if T t U m if 2 . T t 0 otherwise
t T , T 0.01 . ms .. 3 . T , . f 0 .. 0.5 KHz .3.2.3, 3.2.4 3.2.5.
1
volt x( t )
10 0 10 20 30 3 t. 10 ms .3.2.3 104
2 0 500
( f) 2
rad K( f ) 1
4 0 500 f f Hz Hz .3.2.4 .3.2.5 - . , - , . (f>0), - - (0 0 , , . 2 2 .3.2.9 , 1 W W 0.8 . sec W, W .. W . 100
1 2
A k( ) ( ) rad
1 0 1
2 1 0 1
rad / sec rad / sec .3.2.9 x ( t ) U m. ( ( t ) ( t ) ) - (.3.2.10 t 0 , 0.01 . ms .. 2 . ms).
2 - (1.7). - volt
x( t ) 1 - . 1) 0 2 4 t. 10 3 j . . t ms F x( ) U m. e dt; .3.2.10 0 109
U m. exp ( j . . ) U m F x( ) j . ; sin ( . ) U m. cos ( . ) U m U m. F x( ) j . . 2) A x ( ) F x( ) . sin 1 cos ( . ) 2 A x( ) 2 . U m. A x ( ) 2 . U m. . , A x = 0.955 ms. volt . 3) x ( ) arg F x ( ) ( cos ( . ) 1 ) . x( ) atan , .. x ( ) , sin ( . ) 2 tan(-/2). (3.15) U m. exp ( j . . ) U m F y( ) j . . ( j . . T ); F y( ) U m. T . ( 1 exp ( j . . ) ) - ; F y( ) U m. T . ( 1 cos ( . ) ) j . U m. T . sin ( . ) - , a+jb.
2 2 A y( ) U m. T . ( 1 cos ( . ) ) U m. T . sin ( . ) ;
1 cos ( . ) A y( ) 2 . U m. T . 2
. 2 . U m. T . sin A y( ) . 2 sin ( . ) . y( ) atan y ( ) atan cot . ( 1 cos ( ) ) 2 . y( ) atan tan . 2 2 , 110
. 1 y( ) if 0 . 2 2 . if 5 10 2. U m volt * sec
A y( ) 1 y( ) 20 0 20 rad 20 0 20
5 10
. 10 3 . 10 3 rad / ms rad / ms .3.2.11, .3.2.11,
(3.17) (3.18). y(t) - Fy(). - Mathcad Inverse Fourier Transform 1-1(*), * - - - Dirac ( t ) if( t 0 , , 0 ) . Mathcad - 10 307 . 1 1 . U m. T . ( 1 exp ( j . . ) ) . y( t ) 1 . y( t ) U m. T . ( 2 . . Dirac ( t ) 2 . . Dirac ( t ) ) ( 2. ) y ( t ) U m. T . ( Dirac ( t ) Dirac ( t ) ) . .3.2.12 t 0, .. 2 . 25 . 111
307 2 10
3 . 10
volt g( t ) 0 0.5 1 1.5 2 2.5
307 2 10
3 t. 10 ms .3.2.12 , - , UmT. - - . -- .
3.2.3. 3 ms 10 . sec. RC- (.3.2.13) - R 20 . K , C 0.01 . F T R. C ( T = 0.2 ms) p. T K(p) . 1 p. T t=0 - c 1 . ms U m . 1.5 volt . (.3.2.14) , - Mathcad, u( t ) U m. ( ( t ) ( t ) ) . , (3.21). 2
volt u( t) 1
5 0 5 3 t. 10 ms .3.2.13 .3.2.14 112
. p .t U(p) U m. e dt ; 0 . ( exp ( p ) 1 ) U(p) U m. - . p , ,
(1 exp ( p . ) ) . p. T U R(p) U m. . p 1 p. T
, ( ) UR(p) exp ( p . ) , .. , . - ,
T T . exp ( p . ) U R(p) U m. U m. . (1 p. T ) ( 1 p. T )
- UR(p), .. uR(t)=L-1[UR(p)]. - , (3.14). U m. T U1 R ( p ) . ( 1 p. T ) U1R(p) N ( p ) 1 p . T , 1 p 1 . (3.12) (3.14), T U1R(p) Res1 p1 ( t ) U1R(p)ept p1. Mathcad U m. T Res1 p1 ( t ) lim . ep . t . p p 1 d (N(p)) dp 113
assume U m , T U m. T t lim . ep . t , t>0. U m. exp 1 d ( 1 p. T ) T p T dp , t>0 . , (t). t . Res1 p1 ( t ) U m. exp ( t ). T p . U m. T . e U2 R ( p ) . 1 p. T assume U m , T ,
p .( t ) U m. T . e ( t ) lim U m. exp , t>. 1 d T p (1 p. T ) T dp
, t> . , (t-), - (t ) . Res2 p1 ( t ) U m. exp (t ) . T - ,
t . (t ) . u R(t) U m. exp (t) U m. exp (t ). T T
RC- .3.2.15. 114
2 volt
u R( t) 3 2 1 0 1 2 3
2
3 t. 10 ms .3.2.15 , RC- . (.3.2.16) u C( t ) u ( t ) u R ( t ).
2
Um u C( t ) volt
1 u( t)
3 2 1 0 1 2 3 3 t. 10 ms .3.2.16
. - , Mathcad - Laplace Transform Inverse Laplace Transform . , 1 p ( 2 j . 5 ) . sec , 1 ( 1 exp ( p . ) ) . p . T u R(t) 1 . U m. , p 1 p. T Inverse Laplace Transform t>0 t (t ) exp exp T T u R(t) U m. T . ( t ). , t>0. T T , . 115
3.2.4. RC- (.3.2.17) T 1 . sec 1. t g( t ) exp . T T t=0 - U 0 1.5 . volt T0 2.0 . sec . (.3.2.18 M 8 . sec M t M, M .. M ) 250 t . x( t ) U 0 . exp ( t ). T0 , .
x(t) 2
x( t )
volt 1
10 0 10 .3.2.17 t ms .3.2.18 . t>0 (3.22) t . 1. t y( t ) U 0 . exp ( ). exp d ; T0 T T 0 t 1. exp exp t T0 T y( t ) U 0. T 0. , t>0 - - T0 T . .3.2.19 N 10 . sec N t 0, .. N . 250 116
1
1.36. sec 0.75 . volt
volt y( t )
0 5 10 t sec .3.2.19 , RC- . (1.7) t t T0 T e e F y( ) U 0. T 0. . exp ( j . . t ) d t. T0 T 0 , t t U 0 T0. . T0 . j . . t T . j . . t F y( ) e e dt e e dt . T0 T 0 0 assume T 0 > 0 t T0 .e j . . t j .T . F1 y ( ) e dt 0 0 j .T 0 - assume T > 0 t T. j . . t T F2 y ( ) . e e dt j . 0 (j T. ) , 117
U 0. T 0 j j F y( ) . .T .T ; 0 T0 T j T 0. (j T. ) T0 F y( ) U 0. - . j T 0. . ( T . j ) 1 , F y ( 1 . sec ) = 0.3 0.9j sec volt . assume U 0 , T 0 , T , U 0 > 0 , T 0 > 0 , T > 0 , T > T 0 T0 A y( ) F y( ) U 0. . 2. 2 2. 2 T0 1. T 1 (3.26), - assume U 0 , T 0 , T , U 0 > 0 , T 0 > 0 , T > 0 , T > T 0 2 T0 2 2 . F y( ) U 0 . 2 2 2 2 T 0 . 1 .( T . 1) , R 1 . 2 2 U 0 T0 E y( ) . . R 2 2 2 2 .T 0 1 .( T . 1) 1 2 , E y ( 1 . sec ) = 0.9 sec watt . 1 W .3.2.20 W 2 . sec W, W .. W . 100 10 2 2 1 U 0 .T 0 .R
watt * sec * sec E y( ) 5
2 1 0 1 2 3
rad / sec .3.2.20 118
(3.27) signum ( z ) if( z > 0 , 1 , 1 ) 2 2 1. U 0 T0 E y T,T 0 . d ; R 2 .T 0 2 1 .( T . 2 2 1) 0
1. T 0 . signum T 0 T . signum ( T ) 2. 2. E y T,T 0 U 0 T0 . 2 2 2 R. T T0 signum T 0 = 1 signum ( T ) = 1 , 2 2 1. U 0 . T0 Ey 2 R T T0 E y = 1.5 sec watt .
3.2.5. RC- (.3.2.17) - T 1 . sec 1 K( ) 1 j ..T 1 c 2 . sec , 2 E 0 0.5 . sec . watt 0 c. . . -
assume T complex 2 1 2 T K p( ) K ( ). K ( ) . 2 2 2 2 2 2 (1 .T ) (1 .T )
1 K p( ) . 2 2 ( 1 .T ) (3.27), assume c , E 0 , T 119
c E0 1 E0 Ey . d . atan . T . c (1 2 2 .T ) ( .T ) 0 assume c , E 0 c 1. 1. Ex E 0 d E 0 . c. 0 assume c , E 0 , T . Ey 1 . atan c T h ; h = 0.554 . Ex T c , h - T , .
3.2.6. K 0 1.5 , t 0 2 . sec 1 c 5 . sec . - K( ) K 0 . exp j . . t 0 if c.
0 otherwise (t) - Dirac(t) Q0 1 . volt . sec , .. (t)=Q0Dirac(t). y(t) . . . - assume Q 0 F ( ) Q 0 . Dirac ( t ) . exp ( j . . t ) d t Q 0. (3.19) 120
c 1 . j . . t 0 . ej t d ; . . y( t ) Q 0. K 0. e 2. c sin c. t t 0 y( t ) Q 0. K 0. - . . t t 0 y(t) .3.2.21 T T 10 . sec t T, T .. T 500 4
t0 U 0. K 0 2 . c volt
y( t )
10 0 10
2
t sec .3.2.21
3.2.7. RC- (.3.2.17) - T 1 . sec - : 1) 1. t g( t ) exp ; T T 2) 1 K( ) . 1 j ..T t=0 - 1 v 1.5 . volt . sec x ( t ) v. t. (t). . , . assume v 121
j . . t F x( ) v. t. e dt undefined . 0 Mathcad . a>0 a.t v. t. e dt; 0 ( a . t. exp ( a . t ) exp ( a . t ) 1 ) limit v. , t , left - 2 a . ( limit ) (left) : ( a . t. exp ( a . t ) exp ( a . t ) 1 ) v. - ; 2 a v. t v v - a exp ( a . t ) 2 ( a . exp ( a . t ) ) 2 a . , 2 d d ( v. t ) ( v) d t2 dt lim 0 - lim 0. t d2 t d ( a 2. ea . t ) a.t ( a. e ) dt d t2 , v/a2. "a" j v F x( ) . 2 ( j . ) 1 L [x(t)]=1(*), * - . Mathcad s. X( s) 1 . ( v. t ) ; v X( s) . 2 s s j, assume v 122
v v F x( ) . 2 2 ( j . ) (3.20) j . . t 1 . v. e y( t ) d . 2. 2 ( j . ) .( 1 j ..T ) - , . 2 ( j . ) . ( 1 j . . T ) 0; 0 0 - j - . T , : - j 1 0 2 . T (3.10), Res 1 ( t ) 1 assume v , T , t j . . t d . ( )2 v. e lim j . v. ( t T ). 0 d . 2. (j ) (1 j T) . . , Res 1( t ) j . v. ( t T ) . Res 2 ( t ) 2 assume v , T , t j . . t v. e . j t . lim j . v. exp T. 2. j ( j . ) ( 1 j ..T ) T T T t . , Res 2 ( t ) T. j . v. exp T , (3.13), t>0 t . y( t ) j . ( j . v. ( t T ) ) j . v. exp T , t>0; T 123
t . y( t ) v. t T , t>0 - . T exp T , (3.22) x(t) g(t),
t t 1 T y( t ) v. . . e d ; T 0 t . y( t ) v. t T. v T . exp v , t>0. T
(t>0) (t), t y( t ) v. t T . 1 exp . ( t ) ; y ( 1 . sec ) = 0.552 volt . T , , - 1 K ( 0 . sec ) = 1 assume v 1 y ( t ) K ( 0 . sec ) . x ( t ) . ( t ) v. t. ( t ) .
. y(t) , " assume v" Mathcad - "v" . , - y(t), - (. .3.2.22) .
(3.23) t (t) v. t T . 1 exp . ( t ) v. t. ( t ) ; T
t (t) v. T . 1 exp . ( t ) - . T
- .3.2.22 124
M M M M 3 . sec t , .. M . 10 10 200
6
y( t ) 4 y ( t ) . volt volt
2
1 0 1 2 3 4 t sec .3.2.22
t ( ) (t)=-vT=const y(t) x(t) T. t 0 T , assume v , T 1 y1 ( t ) K ( 0 . sec ) . x t t 0 . ( t ) v. ( t T ). ( t ).
t 1( t ) v. t T. 1 exp . ( t ) v. ( t T ). ( t ) ; T
t . 1( t ) v. T . exp ( t ) - . T
- .3.2.23 M 4 . sec t 0 , 0.001 . M .. M . 125
2 v. T
( t )
volt 1( t ) 0 1 2 3 4 5
v. T
2
t sec .3.2.23 , x(t) y(t), (t) . - - .
3.2.8. , - . T 2 . sec , - , 1 K( ) . 1 j ..T - RC-. 10 . T 0 0.2 . sec ( ) - . . (3.31) - K 0 1 K 1( ) K 0. ( 1 j . . T ). . (.3.2.24) - . 126
- .3.2.24 : R 1 1. K ; R 2 1. K ; C1 1. F . .3.2.24
: 1) R 2 K 0 ; R 1 R 2 2) T 1 R 1. C 1 T 0 K 0. T 1. - 1 j ..T 1 K 1( ) K 0. . 1 j ..T 0 , 2>1, . t=0 : 1 Vm 4 . volt . sec , 2 . sec. - x( t ) V m. t. ( ( t ) ( t ) ) . . . (.3.2.31)
132
1. (t) T . V m. 1 exp t if t T t t t T T T. V m. e . e .T e T if t >
0 otherwise 10
y( t ) 5 x( t ) volt
( t ) . volt 0 1 2 3 4
5
t sec .3.2.31
3.2.6. 3.2.5 , .3.2.29 . . (.3.2.32)
2 1. . 2 2 2. T 2. T . t 2 . exp t T t 1. T ( t ) V m. if t 2 T t t Vm . T T. 2 2 2. e . T. ( T) 2. e T if t > 2. T 0 otherwise 133
10
y( t )
y ( t )
volt ( t ) . volt 0 1 2 3 4
10
t sec .3.2.32
3.2.7. () - , H 2.0 1 c1 4 . sec c2 3 . c1 ( 2 . c1 ). K( ) H. c1 c2 . A 2.0 volt . sec , 0 . t0 . 2 sec sin 0 . t t 0 x( t ) A. , 10
t0 0 c1 H .A . 5 y( t ) volt
5 0 5 10
5
t sec .3.2.33
3.2.8. - 2 . A 10 . volt . sec 1 2 . sec A. x( t ) (1 cos ( . t ) ) . ( t ) . 2 t=0 1- - (.3.2.34, R 50 . K C 200 . F ). q , q 20 T . 1 K(p) . . q 1 p . C
R R - . . K1 +
.3.2.34 . (.3.2.35) 135
. t 2. q 2 q e q. sin ( . t ) cos ( . t ) 1 q y( t ) A. . 2 2 ( .( 1 q )) x(t) y(t) 5
y( t )
x( t )
volt y( t ) 0 10 20 30 40
5
t sec .3.2.35
3.2.9. () 3.1.1, 3 1 0 2 . . 10 . sec ( f 0 1000 . Hz ) - 1 400 . sec , t=0 : U m 8 . volt ; 0.8 . 0 1 400 . sec ; 1 2 0.4 . 1 . .t x( t ) U m. e . sin 1 . t sin 2 . t if t 0 .
0 otherwise , , y(t). . (.3.2.36)
2 2 2 2 2 2 . U m. o cos o .t cos o .t y( t ) . . ( t ). exp ( . t ) o 2 2 1 2 o 2 2 2 2 136
50
x( t ) . volt volt
y( t ) 5 0 5 10
50
3 t. 10 ms .3.2.36
3.2.10. (.3.2.37, R 50 . K , C 100 . F K1>>1) , - 1 1 v 1 . volt . sec 0.05 . sec t . x( t ) v. t. e , t>0. y(t) - .
C R R - . K1 . +
.3.2.37 . T R . C (.3.2.38) t v. T . . t. 2 T y( t ) e ( t. . T .t 1) e , t>0. 2 ( T. 1) (.3.2.38) t T. v . T . e t . . ( t . 2. T . T ( t ) .t T. 2) e , t>0. 2 ( T. 1) 137
5
y( t )
y ( t )
volt ( t ) 0 50 100 150
5
t sec .3.2.38
3.2.11. - (.3.2.39, R1 0.5 . K , R2 5. K C . 1 F ) R1 . - 1 v 10 . volt . sec . - x( t ) v. t. ( t ) .
C R1 R2 - . K1 . +
.3.2.39 . 1 . . ( t ) v. R2 . C. exp t ( t ). ( R1 . C )
3.2.12. - (.3.2.40, R 1. K C 10 . F ) ( K1 100 ). 138
U m 1 . volt . x ( t ) U m. ( t ) .
x(t) R y(t) - . K1 . +
3.2.40 . (.3.2.41) 1 U m ( t ) R . C. K1 K1 . exp . t . R.C t . . ( R . ( C. ( 1 K1 ) ) ) ( R. C ) 100
y( t )
y ( t ) 0 0.5 1 1.5 volt
( t ) 100
200
t sec .3.2.41
3.2.13. X - 1 K 0 1 . volt . m , - M 0.008 . kg , 2 315900 . kg . sec P 1 G 0 2822 . kg . sec . - 1 K(p) K 0. . 2 M.p P. p G 0 139 - G 0 P 0 - M . 2 M 2 0 . K(p) K 0. 2 2 0 p 2. . p - .3.2.42 R 2. K , L 0.253 . henry C 0.2 . F . (.3.2.42), Z1 Z2. A(f) A(f) - , A1(f) .
Z1 C Z2 - K1 . L R + .
.3.2.42
. 1 1 Z1 ( p ) Z2 ( p ) R2 p . L2 . . p C1 . p C2 R2 0.5 . K C1 C2 2. . 1 C2 ( C2 = 0.253 F ) 2 0 R2
1 . L2 R2 ( L2 = 0.1 henry ). ( 2. ) K0 .3.2.43. 140
4
A( f )
A 1( f ) 2 A ( f)
0 500 1000 1500 2000 f Hz .3.2.43
3.2.14. 3.2.8 - (.3.2.44) K1 500 0 T 0.02 . sec T 2 . sec .
1 R2 - . K1 . R1 +
.3.2.44 . R1 20 . K , T C1 ( C1 = 100 F ) R1 T 0 . ( K1 1 ) R2 . R1 ( R2 = 101.212 K ). T T0 1 R1 . 1 1 a 1 K y ( K y = 5 ). K1 R2 K1 a (.3.2.45) t 1. t h( t ) 1 exp h (t) 1 exp . T a T0 141
1 h ( t ) .a
h( t )
0 0 0.5 1 1.5 2 t sec .3.2.45
3.2.15. l = 20 km - , 1 : R 0 32 . . km , 1 3 1 L 0 0.822 . mH . km , C 0 26.5 . 10 . pF . km 6 1 G 0 2.3 . 10 . siemens . km . - .3.2.46, R 1.28 . K , L 0.016 . henry , C 0.53 . F R y 21.739 . K . - - .3.2.46 T0 q , q 50 .
R L R1 L1 R2 - K1 . C Ry + . C1
.3.2.46 . R1 10 . K , - .3.2.47 R2 R y R , R2 = 23.019 K ; R1 L1 R . C. R y L . , L1 = 0.136 henry ; R y. q 142 2 q. R y . L . C C1 , C1 = 0.061 F . R . C. R y L . R1 . R y R
3
A( f )
2 A ( f)
1
0 1000 2000 3000 f Hz .3.2.47 3.3. 3.3.1. (.3.3.1). - K(j) () g(t). X(t) K Y(t) K(j), g(t) t=0 .3.3.1 K t0 x x x z 2 . T T .e . z R( ) e dx. e dz . 2 T 0 () 0 x x z z T . T. .z T. .z f( , x ) dz . e e e dz e e 0 , (limit) z ( .T 1) ( x) ( x ). T T . T. 2. . T e .( 1 .T ) f( , x ) e . (1 . T ). ( . T 1) () ( .T 1) ( x) ( x ). x 2 T . T . 2. . T e .( 1 .T ). T R( ) e e dx . T (1 . T ). ( . T 1) 0 , (limit) x
1. . . exp ( . ) exp T 2 T R( ) . , >0. 2 2 ( 1 .T ) 147
, 1. exp ( . ) exp ..T 2. T R( ) , 0. 2 2 ( 1 .T ) 2 , R ( 1 . sec ) = 0.152 volt . B .3.3.5 B 10 . sec B, B .. B. 100
2
volt * volt R ( ) ( 1 .T )
10 5 0 5 10
sec .3.3.5 2 Dy (1 .T ) 2 D y = 0.238 volt .
3.3.2. (.3.3.6, R 1 40 . K , R 2 80 . K , C 25 . F K1>>1), T R 2 . C -
R 2 1 K( ) . . R 1 ( 1 j ..T ) 148
C
R1 R2 - . K1 . +
.3.3.6 X(t) - m 1x 0.5 . volt 0.5 . volt 2 0.2 . sec () 2 2 2 R x( ) . exp ( . ) , R x ( 0 . sec ) = 0.25 volt . Y(t) m1y, Sy(), Ry() - Dy. . - - (2.7).
assume , , > 0 1 . 2 2 exp 2. . . j . . 2. . (4 ) S x( ) e e d . . assume R 1 > 0 , R 2 > 0 , T 2 R 2 2 K p( ) .( K( ) ) 2 2 2 R 1 .( 1 .T ) (3.36), 2 2 exp R 2 2. 4. S y( ) . . . .2 2 2 R 1 (1 T ) (3.37) assume R 1 , R 2 , m 1x 149
R 2 1 m 1y K ( 0 . sec ) . m 1x .m 1x . R 1 (3.38) 2 2 exp 1 . R 2 . . . 2 4. . R y( ) exp ( j . . ) d . 2. R 2 2. 2 (1 T ) 1 Mathcad , - Inverse Fourier Transform . , , - - (Im,Re) . 2 2 1 . T 0. 1 . 2 - T , 2 T assume T complex 1. 2 1 . 2 j . T T 2 2 T T T , - j j 1 2 . T T 1 C (.3.3.7,) . (3.12) (3.10) 1 >0. Im Im C 1 Re 1 Re 2 2 C
.3.3.7 150
2 C (.3.3.7,) . , , (3.12) (3.10) 2 0 2 R 2 2. 4. . T . ) R1 y ( ) . . j . 1 . j . exp 1. ( 1 2 2 T 4 2 R 1 ( T . ) 2 1 . R 2 . 2. . exp 1. ( 1 4. . T . ) R1 y ( ) , >0 . 4 2 R 2 2 ( T . ) .T 1 , (3.10), Res 2 2: 2 exp 4. . exp ( j . . ) . 1. 2 lim T ; 2. 2 2 1. 2 (1 T ) T T 2 T 1. j . 1 ( 1 4. . T . ) exp . - . 2 T 4 2 ( T . ) , 2 151
1. j . 1 ( 1 4. . T . ) exp . Res 2 ( ) . 2 T 4 2 ( T . ) (3.13) - 1. - RC- T 1 0.4 . sec T2 0.2 . sec 1 1 K 1( ) K 2( ) . 1 j ..T 1 1 j ..T 2
R R + . K1 . C -
.3.3.9 X(t) 2 S 0 0.1 . volt . sec. Y(t) Ry() Dy. . Kp() - assume T 1 > 0 , T 2 > 0 , K 0 2 K 0 2 K p( ) K 0. K 1 ( ) . K 2 ( ) . 2 2 2 2 .T 1 . 1 .T 2 1 (3.36), 2 S 0. K 0 S y( ) . 2 2 2 2 1 .T 1 . 1 .T 2 1 2 , S y ( 1 . sec ) = 0.083 sec volt . (3.38) 2 S 0. K 0 exp ( j . . ) R y( ) . d . 2. 2 2 1 .T 1 . 1 .T 2 2 2
153
, - . 2 2 2 2 1 .T 1 . 1 .T 2 0.
1 . 2 T1 2 T1 - 1 . 2 , T1 2 assume T complex T1 1 . 2 j T . 1 . 2 T 2 T T2 2 T2
1 . 2 T2 2 T2 , : j j j j 1 ; 2 ; 3 4 . T1 T1 T2 T2 >0, C (3.12) , 1 3. - (3.13), (3.10) 1 3. , (3.10), Res 1 1: exp ( j . . ) . j lim ; 2 2 2 2 j 1 .T 1 . 1 .T 2 T1 T1
1. . . T1 j exp - . 2 T1 2 2 T2 T1 , 1 1. . . T1 Res 1 ( ) j exp . 2 T1 2 2 T2 T1 154
, (3.10), Res 3 3: exp ( j . . ) . j lim ; 2. 2 . 2. 2 T2 j 1 T1 1 T2 T2
1. . . T2 j exp - . 2 T2 2 2 T2 T1 , 3 1. . . T2 Res 3 ( ) j exp . 2 T2 2 2 T2 T1 (3.13) >0
. . exp T 1 exp T2 1. T1 T2 2 R1 y ( ) S 0. K 0 . , >0. 2 2 2 T2 T1 >1) , 2 S0 0.2 . volt . sec .3.3.11 . S0 . . . e R .C . 1 R .C . R y( ) ( ) e 1 ( ) 4. R . C R.C R. C S0 . R y( ) . exp 1 . . . 4 R C . R C R. C S0 y 4. R . C y = 5 volt . 156 3.3.2. .3.3.11, R 2. K C 10 . F , K1>>1. - T R . C. X(t) - 0.5 . volt 1 50 . sec c 2 . .e 2 R x( ) , R ( 0 . sec ) = 0.25 volt . Ry() . . 2 T . . ( 2. . T 2 2 3 . . e .T 3. T ) 2. e R y( ) . 2 2 2 ( T. 1 ) . ( T. 1)
3.3.3. 3.3.2 , 2. R . C R x( ) e . . 2 2 . . 3. R y( ) exp 3 . 8 . (R C) R.C R. C
3.3.4. (.3.3.12, R 2. K C 10 . F K1>>1) , 2 S0 0.2 . volt . sec R R R R - K1 . . C +
.3.3.12 157 . 3. S 0 . 1. 2 R y( ) . exp 1 . . 16 R C. . R C 3 R.C R. C 3. S 0 y y = 1.369 volt . 16 . R . C
3.3.5. RL- (.3.3.13, R R 0.1 . K L 20 . henry ), L g ( t ) . . exp ( t ) . t=0 - X(t) m 1x 0.5 . volt - 1 0.5 . volt 0.5 . sec 2 . .e 2 R x( ) , R ( 0 . sec ) = 0.25 volt . L - m1y Ry(t1,t2) Y(t), . R . Ry() - y. .3.3.13 . m 1y ( t ) m 1x . ( 1 exp ( . t ) ) . . t 2 . t 1 . t 2 t 1 2. 2. t 2. ( ) e e R y t 1,t 2 e 1 . . 2 2 ( ) 2 ( exp ( . ) . . exp ( . ) ) R y( ) .. . 2 2 ( ) . y y = 0.477 volt . 158 3.3.6. RL- (.3.3.13, R 0.1 . K L 20 . henry ). X(t) 2 3 . volt 2 . sec 2 . 2 R x( ) .e . Ry() . . 1. 2 . R . exp 1 . ( R R R y( ) . 4. L . . ) . . 2 4 2 .L ( L . )
3.3.7. : 1 K 0 1.5 , c 2 . . 10 . sec . K( ) K 0. 1 c c . , - 2 S 0 0.2 . volt . sec 4. 4.1. . - x(t ) x * (t ) , () t (t ) = x(t ) - x (t ), t t . - x(t ) x * (t ) . 159 ( t ) - : 1) x(t ) ; 2) () , . . x * (t ) ; 3) ; 4) . , x(t ) () x * (t ) : 1) max (t ) = m , t t ; 2) tm 1 2 (t )dt = 2 2 , t t ; tm 0 * (t ) = x(t ) - x (t ) ­ ; ­ ; ­ . x(t ) n x * (t ) = a k k (t ) , t t , (4.1) k =0< k =0 ­ ; n k (t ) a k ­ , x(t ) ; n ­ (4.1). - t k (t )< k =0 . n a k 160 - a k x(t k ) - t k . - . - x(t k ) . - - (.4.1.1). x(t) x(tk ) x(t)
t t
0 tk tm .4.1.1 -< n n t k (t - t 0 ) k . k =0 k =0 . . ­ - , ­ . t - . t = const , () t m . t = var , (). - t . . , .. , . ­ . ­ . n . ­ n = 0 n = 1 . 161
4.2. 4.2.1. : 1) ; 2) . 4.2.1.1. . . . , - . , 1 t = = , (4.2) C 2f C f c ­ x(t ) . - K (t ) , .. () sin C (t - t k ) [ ] x(t ) = K (t ) = x(t k ) C (t - t k ) = x(t k ) Sa C (t - t k ) , (4.3) k =- K =- t k = k t ; Sa (x) ­ . ( t 0, t m ) - [ ] . , , - . . , - (4.2), - c , . () - , . - - , - . A() P P 2 (2 ÷ 3) ; P P (4.4) 162
E E , 2 (2 ÷ 3) E E P E ­ ; P E ­ (- ) ; ­ , .. tm 1 2 2 (t ) dt (t ) dt tm 2 2 t m 2 = 0 tm = 2 = - = , (4.5) P E x 1 x 2 2 (t ) dt (t ) dt tm - 0 ­ - (). , (4.4) (4.5) E 2 E = 2 t m . (4.6) (1.14) (1.22) c 1 F ( j) 2 E - E = d . (4.7) 0 (4.6) (4.7) - (4.2).
4.2.1.2. . - ­ x(t), tm . . n = 0 ( ) n = 1 ( ) n = 0 ( - ) n = 1 ( ). ­ . , , , .4.1, 163
M 1 M 2 ­ - 1- 2- , M 1 = max x (t ) , M 2 = max x (t ) , t t m ; ­ - .
4.1 n t 0 / M1 1 2 / M 2 0 2 / M 1 1 8 / M 2 0 3 / M 1 - 1 2 5 / M 2 0 2 3 / M 1 1 2 30 / M 2
4.2.2. 4.2.1. (.4.2.1 T 2 T 10 . sec t 0, .. T ) a 0 1.5 . volt . sec 100 1 1 . sec , 2 a 0 . t . exp ( . t ) , t>0. x( t ) ­ (.. ), - 0.1 . 164
volt 1
x( t ) 0.5
0 0 5 10 t sec .4.2.1 . t>0, (1.7) - . - Laplace Transform 1(*). 2 X( s) 1 . a 0 . t . exp ( . t ) ; a0 X( s) 2. - , s - - 3 (s ) . , s j, 2. a 0 F x( ) . 3 ( j . ) 1 , F x ( 1 . sec ) = 0.75 0.75j sec volt . .
assume , a 0 > 0 a0 F x( ) 2. . 3 2 2 2 ( ) (1.14) assume a 0 , > 0 165
2 2 1. a 0 3. a 0 Ex 2. d . 3 4 5 2 2 2 ( ) 0 , R 1 . 2 3 .a 0 Ex E x = 1.688 sec watt . 4. R 5 , (4.4) (4.5), 2 E E x . E = 0.017 sec watt . , c, c 2 1. a0 Ex c 2. d . 3 2 2 2 ( ) 0 c 3 3 . 2 2 2 2 5. . c 3. . c 3 . atan c a0 Ex c . . 2. 5. 2 2 2 c (4.7), c - 2 E x E E x c , 1 . E x E x c , = 0.99 . - : 1 c 1 . sec - ; Given - c 2 3 3 . 2 2 2 2 5. . 3. . c 3 . atan c c 3. a 0 . a0 . ; 4 5 . 2 5. 2 2 2 c 166
c1 Find c - , Mathcad - 1 Find(z). c1 = 1.803 sec . , (4.2) t t = 1.742 sec . c1 t m 10 . sec . tm N p N p = 5 , floor t floor(z) - . - (4.3) N p sin c1 . ( t i. t ) K(t) x ( i. t ) . . c1 . ( t i. t ) i= 0 .4.2.2 . 1
K( t ) volt
x( t )
0 2 4 6 8 10 t sec .4.2.2
4.2.2. (.4.2.3 T 10 . sec T 2 1.5 . volt . sec 1 t 0, .. T ) a 0 1 . sec , - 100 2 x( t ) a 0 . t . exp ( . t ) , t>0. , - 0 0.2 . volt , (). 167
1
volt x( t ) 0.5
0 0 5 10 t sec .4.2.3 . .4.1, - x(t) t>0. 2 d 2 x2 ( t ) a 0 . t . exp ( . t ) ; d t2 2 2 x2 ( t ) a 0 . exp ( . t ) . ( 2 4 . . t . t ) - . 2- : 2 1 x2 ( 0 . sec ) = 3 sec volt x2 ( ) 0 . sec . volt. 2- . 3 d 2 . t x3 ( t ) a 0. t . e ; d t3 2 2 a 0 . . exp ( . t ) . ( 6 x3 ( t ) 6. . t . t ) - - . 2 2 6 6. . t . t 0 ; 1 . 6. 2. 3. 2 ( 2. ) - 1 - . . 6. 2. 3. 2 ( 2. ) , 1. t1 3 3 , t 1 = 4.732 sec . 168
1. 3 3 , t 2 = 1.268 sec . t2 exp(-t)=0 t3= . 2- 2 2 x2 t 1 = 0.072 sec volt x2 t 2 = 0.618 sec volt . 2- - 2- M 2 x2 ( 0 . sec )
8. 0 , t = 0.73 sec . t M 2 - .4.2.4 j 0 .. 12 tj j. t
1
x t t 6. t j volt
x t j
0 2 4 6 8 10 t j sec .4.2.4
4.2.3. 4.2.1. 5 . % 1 1 50 . volt . sec 10 . sec , - .t u( t ) . t. e . ( t ) . 1 . c = 54.646 sec . t = 0.057 sec . 169 4.2.2. 10 . % 2 1 a 10 . volt . sec 0.5 . sec , - . t a. e . . x( t ) ( t sin ( . t ) ) , t>0. 2 1 . c = 0.847 sec . t = 3.707 sec .
4.2.3. - 0 5 . volt - 2 1 a 10 . volt . sec , 0.5 . sec q 10 , - . t q a. e .( .t x( t ) sin ( . t ) ) , t>0. 2 2 . - 2- M 2 7.42 . volt . sec . t = 2.322 sec .
4.2.4. . = 2 % U m . 1 volt . 1 0.1 sec , x( t ) U m. exp ( . t ) . ( t ) . . 2 . c . cot . 2 . . 2 t tan , t = 0.02 sec . 2
4.2.5. - 0 0.2 . volt 1 10 . volt . sec, 2 . sec , 170
. z(t) sin ( . t ) , t>0. t 1. . 3 . - 2- M 2 . 3 1 0 4 t 2. 3. 5 . , t = 0.183 sec . 3 2 .
4.2.6. - 0 0.2 . volt - 2 1 A 10 . volt . sec , 2 . rad . sec q 10 , . t q A. e . ( sin ( . t ) x( t ) . t. cos ( . t ) ) , t>0. 2 2. 1 . - 1- M 1 8.657 . volt . sec . t = 0.046 sec .
4.2.7. 4.2.6 - 0 0.2 . volt . . t = 0.04 sec .
4.2.8. 4.2.2 , - . . - 1- a 0 M 1 . 2 2 . exp 2 2 . 2. 0. 2 t . . exp 2 2 , t = 0.289 sec . 2 a 0. 2 2 171
4.3. 4.3.1. pp p p, pp p [ mx, p Dx - pp Rx()] . p p pp pp p. p p pp pp p. p p p p. p p P p pp pp p p .4.2. 4.2 n 2 0 R x ( t ) = R x (0) - 2 1 2 2 = 2R x (0) - 2R x ( t ) + t R x (0) - 2tR xx ( t )
t 2 0 R x = R x (0) - 2 2 1 2 = 15 . R x (0) + 0.5R x ( t ) - 2R x ( t 2) .4.2 : R x ( t ) - = t ; R x (0) - 1- = 0; R xx ( t ) - 1- - =t. p pp p pp - p
k d 2 k R x ( ) R ( ) = ( -1) , (4.8) x( k ) d 2k pp 1- p 172
dR x ( ) R xx ( ) = . (4.9) d - . - , .. . c , ( ). c 1 S()d 2 P = . (4.10) c
4.3.2. 4.3.1. X(t) 1 0.5 . volt 0.5 . sec 2 R( ) . exp ( . ) . 0 0.2 . volt - . 2 . D x R ( 0 . sec ) , .. D x . (. .4.2) t 2 2 2 2 1.5 . 2 . . exp . 0.5 . . exp . t 0 . 2 .5 . 2 2 2 ln 4. . 2. . . 2. . 0 2 t 1 2. . ; .5 . 2 2 2 ln 4. . 2. . . 2. . 0 2 t 2 2. . . 173
t 1 = 4.588 sec ; t 2 = 0.645 sec. , .5 . 2 2 2 ln 4. . 2. . . 2. . 0 2 t t 2 t 2. . .
4.3.2. X(t) 2 2 . volt 0.5 . sec
2 2 R( ) . exp ( . ) . 0 0.1 . volt (). . .4.1, - 2- - . X(t). - (4.8) 4 2 d 2 2 R x2 ( ) ( 1) . . exp ( . ) ; d 4 2. 2. 2 2 2 4 R x2 ( ) 4 exp ( . ) . ( 3 12 . . . 4. . ) . 2- assume , , > 0 2 2 D x2 R x2 ( 0 . sec ) 12 . . . 2 2 4 2 , D x2 12 . . , D x2 = 12 sec volt . - 2- - " ", x2 D x2 . 2 M 2 3 . x2 M 2 3 . . . 12 , M 2 = 10.392 sec volt . - 2- assume x2 , M 2 174
M 2 1 x2 2 1. M 2 P 1 2. . exp d x2 1 erf 2. . 2 . x2 2 2 x2 x2 . 2 . 0 1. M 2 , P 1 , P = 0.003 . erf 2. x2 2 .4.1 assume 0 , , 1 2. 0 1. 0 .3 4 . t . M 2 3 , - P = 0.003 1 1. 0 .3 4 t t = 0.139 sec . 3 .
4.3.3. X(t) 2 2 . volt 10 . sec 2 2 R( ) . exp ( . ) . 0 0.1 . volt . . - , - . - (2.7), assume , , > 0 2 2 2 j . . 1 . 2 . . S x( ) . exp ( . ) . e d exp . ( 4. ) (2.10), assume , , > 0 175
2 1. 1 . 2 . . 1. 2 P x( ) exp d . 4. . (4 ) 2 0 2 2 , P x P x = 4 volt . . 2 P 0 - . Pc P x P . (2.11)
c 2 1. 1 . 2 . . Px P exp d . . (4 ) 0 assume , , c c 2 1. 1 . 2 . . 1 . . 2 exp d erf c . . (4 ) 2. 0 P x ( ) P - 2 2 1 . 2 0 erf c . . 2. : 1 c 1 . sec - ; Given - 2 2 1 . 2 0 erf c . ; 2. 176
c Find c - , Mathcad - 1 Find(z), c = 13.521 sec . , (4.2) -
t t = 0.232 sec . c
4.3.3. . 4.3.1. 4.3.1 - . . . 2 2 1. 2 0 ln 2 2 t , t = 0.167 sec .
. 4.3.2. 4.3.1 - . . 2 2 . 0 .. c tan 2 2 2 2 . . 0 t cot t = 1.613 sec . 2 2
. 4.3.3. 4.3.1 - . . 2 d d d signum ( ) 2 . Dirac ( ) , signum ( ) d d 2 d 1- 2 . . . 2 R x1 ( ) ..e ( 2 Dirac ( ) . signum ( ) ) . , 1- . , . . 177 . 4.3.4. 4.3.2 - . 1. 2 . t 0. , t = 0.033 sec . 3 .
. 4.3.5. 4.3.3 - . . 2 2 2 2 2 2 4 2 2 2 . t 2 . t . . 2 0 2. 2. t . . 4. t . . 2. . e 4. . e t t = 0.335 sec .
. 4.3.6. 4.3.3 - . . 2 2 2 2 2 2 0 1.5 . .5 . . exp ( 1. . . t ) 2. . . exp ( .25 . . t ) t = 0.109 sec.
4.4. 4.4.1. . - p P - p Np K = , (4.11) N ax + N at
Np - p P; Nax - x(t ) p ; Nat - p p . (4.11) p p p p - p. , . - (4.11) , . 178 , - K c , - K . p p. - (4.11) p Np ta K = = , (4.12) 2N ax 2t p
Np=tm/tp; Nax=tm/ t a ; tp - p p P; t a - p p p . pp pp p p ( p p) p p p ta = - ; (4.13 ) x
2 ta = - , (4.13 ) x
x - p 1- p , tm 1 x = tm x (t ) dt - ; 0 x [ = M X = ] x p(x )dx - , - - ; p(x ) - p 1- p . p p (4.13) p .4.1 p - p , (4.12) - p p M1 Kc = . (4.14) 2 x
p (4.14) p - p pp pp p. 179 p pp p p p p
2 ta = - ; (4.15 ) x
8 ta = - , (4.15 ) x
x - p 2- p , tm 1 x = tm x (t) dt - ; 0 [ ] x p(x )dx - , M X = - p(x ) - p 2- p . p (4.15) p .4.1 p - p , p
1 M2 Kc = , (4.16) 2 x
p (4.16) p pp pp p.
4.4.2. 4.4.1. (.4.4.1 T 12 . sec T 2 1.5 . volt . sec 1 t 0, .. T ) a 0 1 . sec , - 100 t m T 2 a 0 . t . exp ( . t ) , t>0. x( t ) , (). 180
1
volt x( t ) 0.5
0 0 5 10 15 t sec .4.4.1 . (4.14) x(t) tm. d 2 x1 ( t ) a 0 . t . exp ( . t ) ; dt x1 ( t ) a 0 . t. exp ( . t ) . ( . t 2 ) - - . 1- .4.4.2 1 2 volt / sec
x1( t )
0 5 10 15
t sec .4.4.2 , - . - . - . 1- x1(t)=0 . , a 0 . t. exp ( . t ) . ( . t 2 ) 0 ; 181
0 2 - . 2 , t 11 0 . sec t 21 , .. t 21 = 2 sec . 1- t 21 tm a0 .t . t x1 . t. e . ( . t 2 ) d t t. e . ( . t 2 ) dt . tm 0 t 21 2 . t 21 2 . t m 2 . t 21 . e t m .e x1 a 0. . tm 1 , x1 = 0.135 sec volt . t21 tm
2 2 ( 8 . exp ( 2 ) T . exp ( . T ) . ) x1 a 0. . 2 ( .T ) 1- 2- - (. 4.2.2) , 1- 1. t 2 2 . - 1- 2 1 M 1 a 0. 2 2 . exp 2 . , M 1 = 0.692 sec 2 volt . (4.14) 2 2. .T K ( ) 1 2 .e . 2 2 . T. 2 8. e T .e K ( ) = 2.558 . , , 0.02 . volt - - 2. t , t = 0.296 sec . x1 182 4.4.2. X(t) 2 2 . volt 0.5 . sec
2 2 R( ) . exp ( . ) . (). . - (4.16) X2(t) - X(t). - . ( ). (4.8) - X2(t) X(t). 2- 4 2 d 2 2 ( 1) . R x2 ( ) . exp ( . ) . d 4 2 2 2 2 2 4 R x2 ( ) 4 . . . exp ( . ) . ( 3 12 . . 4. . ) . =0 2- assume , x2 R x2 ( 0 . sec ) 2. 3. . . - 2- - " ". 2 M 2 3 . x2 M 2 3 . 2 . 3 . . , M 2 = 10.392 sec volt . 2-
assume , , > 0, > 0 2 x2 x2 2 X2 x2 2. . exp d x2 2. 3. . . . 2 x2 . 2 . 2 . x2 0 , 2- 183
2 2 X2 2. 3. . . , X2 = 2.764 sec volt . (4.16) - assume , 3 1 1. M 2 1. 4 4 K 3. 2 K = 0.97. . 2 X2 4 0.1 0 . .4.1 M2 X2 - -
2. 5. 0 t . X2 X2 0
. 30 . t , t = 0.569 sec. 6.
4.4.3. . - 1 0.5 . sec . volt 1- . p 1( x ) exp ( . x ) . 2 - [-M1,M1], M1 - - 1- X1(t), - " ", .. M1=3x, x - - 1- . . 1- assume , > 0 . 2 2 D x( ) 2 x . p 1 ( x ) dx . 2 0 184
2 2 2 , D x ( D x = 8 sec volt ) - 2 assume ,
2 . x Dx - 1- 1- 2 1 M 1 3. , M 1 = 8.485 sec volt . 1- assume , > 0 1 X 1 x 2. x. p 1 ( x ) dx . 0 , 1- 1- 1 1 X 1 , X 1 = 2 sec volt . p(x)=1/2M1 - 1- assume , > 0 M 1 x 3. 2 X 2 x2 2. dx . . 2 M 2 1 0 , 1- 2- 3. 2 1 X 2 , X 2 = 4.243 sec volt . 2 (4.14) - 1- assume 1. M 1 3. K 1 2 K 1 = 2.121 . 2 X 1 2 2- assume 185
1. M 1 K 2 1 K 2 = 1 . 2 X 2
4.4.3. 4.4.1. 4.4.1 - (). . - 2- x2(t) t=0, .. 2 M 2 x2 ( 0 . sec ) M 2 = 3 sec volt .
2 2 2 2 S 22 2. 2. 2 2 .e S 12 2. 2. 2 2 .e 1. x2 ( 0 . sec ) K ( ) , K ( ) = 2.2. 2 S 12 S 22 . T. a 0. e (2 .T ) .T
4.4.2. 4.4.2 - (). . - 1- M 1 3. 2. . . C 1- 1 X1 2. . , X1 = 1.596 sec volt . - 1. M 1 3. . K 2 K = 1.88. 2 X1 4
4.4.3. , 1 5 . volt . sec - 1- 1 p 1( x ) . 2 2 . x 186
. - 1- M 1 . C 1- X 1 2 . . K K = 0.785 . 4
4.4.4. , 2 5 . volt . sec 2-
2 1 . exp x . p 2( x ) 2 . 2. 2. . - 2- M 2 3 . . C - 2- 2 X 2cp . . 3 1 1. 4 4 K c 3. 2 . K c = 0.97. 4
4.4.5. - 1- - 1 M 1 10 . volt . sec . [-M1, M1] 1. . 1 p 1( x ) , M 1 x M 1. 2 atan . M 1 1 2. x 2 . . atan . M 1 K c M 1. . , K c = 3.188 . 2 2 ln 1 . M 1 187
1. . ., . ., . ., - . .: , 1977. 2. .. - . .: , 1983. 3. .. , .. - . .: , 1977. 4. .. . .: , 1971 . 5. .. . .: , 1973. 6. .. - . .: , 1967. 7. .. . .: , 1973. 8. . . - . , , 1978. 9. . ., . ., . . - . .: , 1970. 10. . . . .: , 1966. 11. . . . .: , 1975. 12. .. . .: - , 1988. 13. ., . - . .: , 1973. 14. MATHCAD 6.0 PLUS. , Windows 95. /. . - - , "", 1996. 188
1. ......................................................................................3 1.1. ......3 1.1.1. ...................................................... 3 1.1.2. .................................................................................. 4 1.1.3. .................................................................................... 10 1.2. .................................................................................................14 1.2.1. .................................................... 14 1.2.2. ................................................................................ 14 1.2.3. .................................................................................... 30 1.3. ..............................38 1.3.1. .................................................... 38 1.3.2. ................................................................................ 40 1.3.3. .................................................................................... 46 1.4. ..................................................50 1.4.1. .................................................... 50 1.4.2. ................................................................................ 51 1.4.3. .................................................................................... 59 2. ...........62 2.1. ..................................62 2.1.1. .................................................... 62 2.1.2. ................................................................................ 64 2.1.3. .................................................................................... 65 2.2. .................................................67 2.2.1. .................................................... 67 2.2.2. ................................................................................ 69 2.2.3. .................................................................................... 74 3. ...............................................................................79 3.1. ............79 3.1.1. .................................................... 79 3.1.2. ................................................................................ 84 3.1.3. .................................................................................... 93 189
3.2. ............................................................................................. 98 3.2.1. .....................................................98 3.2.2. ..............................................................................102 3.2.3. ..................................................................................128 3.3. ........................................................................................... 142 3.3.1. ...................................................142 3.3.2. ..............................................................................144 3.3.3. ..................................................................................155 4. .............158 4.1. ........................................................................... 158 4.2. ..................... 161 4.2.1. ...................................................161 4.2.2. ..............................................................................163 4.2.3. ..................................................................................168 4.3. ...................................... 171 4.3.1. ...................................................171 4.3.2. ..............................................................................172 4.3.3. ..................................................................................176 4.4. ................................................ 177 4.4.1. ...................................................177 4.4.2. ..............................................................................179 4.4.3. ..................................................................................185 ..............................................................................187 190
Mathcad 6.0 Plus
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; 020565 23.06.97. 60x84 1/16 . . ..- 12. .-. .- 11. 100 . "C"
17, , 28, , 44 17, , 28, , 1