, , .
, ( ) .
(, , , ) - , u, i, p.
i = I m sin (w t + y), I m , . u = U m sin (w t + y) e = E m sin (w t + y).
− (w t + y) .
, (, ) (, ), f = 1/ T. . 1 , 1 .
, . w = a / t. a = 2p ( ), t = , , , , w = 2p / = 2p f. (w) .
³ . i= I m sin (w t + y) y, , [8]. y = 0, i = I m sinw t.
̳ , , , y u y i , j = y u y i. (. ) , , j ᒺ .
ij , , .
ij . . ij I, U, E.
: ( m, U m, E m) (, U, E) ; (w, f : w = 2p f, f = 1/ T) y.
, .
i = I m sin(w t + y).
, , . y , I m. w = 2p f, , .
|
|
, .
, w. , . (, ), , , .
. , , .
, , . - , . 璺 .
, , . , . , , , . , , , 䳿 .
, (, , , , , ) . ʳ .
[9]:
q Z a jb, Z = a + jb. jb b.
q , jb. :
¨ (a; jb);
¨ , , (a; jb).
q . : . a . , b/a = tg a, a = arctg b/a.
a , arctg b/a 0£ a£ 90. :
|
|
b | a = | ||
+ | + | arctg b/a | |
+ | 180 arctg b/a | ||
180 + arctg b/a | |||
+ | IV | arctg b/a |
ja, Z = |Z|× ja.
q . . . .
a = |Z|× cos a, b =|Z|× sin a.
Z = a + jb = |Z|× cos a + j|Z|× sin a
.
:
Z 1 Z 2 = a 1 + jb 1 a 2 + jb 2 = (a 1 a 2) + j (b 1 b 2).
:
Z 1 ´ Z 2 = | Z 1 | e ja ´ | Z 2| e jb = | Z 1 | ´| Z 2| e j(a + b);
Z 1 / Z 2 = | Z 1 | e ja / | Z 2| e jb = | Z 1 | / | Z 2| e j(a b),
:
Z 1 ´ Z 2 = (a 1 a 2 b 1 b 2) + j (a 1 b 2 + b 1 a 2); .
, ( ), ( ), :
a + jb a jb;
| Z| e ja | Z| e ja .
, , , , . , . , .
, , , , .
, 0. , U , 0 . :
( ) , .
ʳ, , :
.
|
|
:
[10], 0. :
) ,
U a jU L ; U j .
) .
+ , − , -; − − , -.
, :
,
, .
j > 0, U L > U C, ;
j < 0, U L < U C, [11].
Z, X R, : y = 1 / Z, b = X / Z 2 g = R / Z 2.
|
|
(, ) = .
. Z. . I = U / Z.
, , , [12] .
ֳ :
= R + jX = R + j (X L X C),
.
:
: .
:
; .
, :
P = U a ×I = UI × cos j [];
Q = U ×I = UI× sn j [];
S = U×I [ × ].
ʳ , . , ( ʳ, , , , ), . , , , , , .
:
1. , , , .
2. , , .
3. .
1. :
:
R1 = 1 , R2 = 1 , R3 = 2 , R4 = 0,8 , R5 = 4 , C1 = 1 , C2 = 2 , C3 = 0,5 , C4 = 0,4 , L = 4 , 1 = 80∙sin(ω t) , 2 = 120∙sin(ω t + 45) , ω:= 5106.
MathCAD:
ORIGIN:= 1
:
R1:= 1000 R2:= 1000 R3:= 2000 R4:= 800 R5:= 4000
1:= 1109 2:= 2109 3:= 0.5109 4:= 0.4109 L:= 4103
1:= 80 j0 2:= 120 jπ / 4 ω:= 5106
: acdba, cefdc, eghfe.
:
:
:
[13]:
1:= 1 2:= 1 2 3:= 2 4:= 3 2 5:= 3
1 = 0.017 + 7.561j ´ 104 I2 = 0.016 + 2.636j ´ 104
3 = 5.757 ´ 104 + 4.925j ´ 104 4 = 0.016 + 0.025j 5 = 0.016 + 0.025j
:
[14] (i = msin(ωt + ψ):
| I1 | = 0.017 1 = 0.017sin(5106 t + 2,547)
| I2 | = 0.016 2 = 0.016sin(5106 t + 0,944)
| I3 | = 7.576E4 3 = 7.576E 4 sin(5106 t + 40,546)
| I4 | = 0.03 4 = 0.03sin(5106 t + 57,381)
|
|
| I5 | = 0.03 5 = 0.03sin(5106 t + 57,381)
2. . .
:
r 1 = 6 ; L 1= 25,5 ; 1= 1590 ; r 2= 5 ; L 2 = 5 ; 2 = 1000 ; U = 127 ; f = 50 .
:
1. Z 1 璺 r 1, L 1, 1:
X L1 = 2 π f L 1= 23,145025,5103 = 8 ;
C1 = 1/(2 π f 1) = 1/(23,14501590106) = 2 ;
2. y 2 [15] :
X L2 = 2 π f L 2 = 23,14505103 = 1,57 ;
2 = 1/(2 π f 2) = 1/(23,14501000106) = 3,18 ;
3. (Z), (J) (U r1, U L1, U C1, U b):
Z = Z l + Z 2 = Z l + l / y 2 = 8,5 + 2,63 = 11,13 ;
J = U / Z = 127 / 11,13 = 11,4 ;
U r1 = J r 1 = 11,46 = 67,3 ;
U L1 = J X L1 = 11,48 = 91 B;
U C1 = J X C1 = 11,42 = 22,8 ;
U ab = J Z 2 = 11,42,63 = 30 ;
4. :
J r2 = U ab / r 2 = 30 / 5 = 6 ;
J L2 = U ab/ X L2= 30 /1,57 = 19,1 ;
J C2 = U ab/ X C2= 30 / 3,18 = 9,4 ;
5. U ab J:
b L2=1/ L2 =1 /1,57 = 0,64; b 2= 1 / 2 = 1 /3,18 = 0,315;
φ = arcsin(b / y)=arcsin ((b L2 b C2) / y 2) = arcsin(0,335 / 0,38) = arcsin0,885 Þ φ = 62.
6. , : j = 0,4:1 /; U = 4:1 /.
J, , [16].
(b L2 > b C2), φ = 62 .
i . 90, , 90:
. , , , i :