2.3.1. ()
, (. 2.9):
(2.21)
: (2.22)
(Im Um) . , .
1 :
f = 1/T (2.23)
f () -1.
ω =2π f (2.24)
/.
. 2.9
, . . (ωt + φ), . ( ) t.
: , .
50 , . 60 .
: , , , () .
. . ( ) , , . . . . ( ) . . . . , . . . , : e(t ) i(t).
(2.21) (2.22) φ = 0, :
u = RImsinωt =Umsinωt (2.25)
:
i = GUm sinωt = Imsinωt (2.26)
R = 1/G, R , G ,
Um Im .
(2.25) (2.26), .
p = ui = Ri2 = Gu2
, t t0 > 0:
u i :
(2.27)
, , , t :
, .. i = Imsin(ωt + φ0) (2.27) :
(2.28)
(2.28) , UmL = ωLIm, XL ωL.
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, π/2
0, .. .
() :
(2.29)
, :
UmC = Im/ωC,
RC 1 /ωC,
(2.29) , π/2
φ
, , . :
p = iu = imumsinωtsin(ωt + φ)
, (, ) . , φ , < 0, . . . , , , . . .
, , . , , , < 0, . :
(2.30)
:
(2.31)
( ≈ 0,707 ) ( ) , . φ = 0, = IU, . .
. ( ), , , .
cosφ , . , cosφ < 1 I, U, cosφ = 1. - . , cosφ < 1 I U .
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( , . . cosφ), UI, , :
P = UI
, ,
Pa = UIcosφ
φ = 0, .
P = I2R
, , cosφ = 0, . , .
, .
. ,
(2.32)
. 2/π ≈ 0,638. .
, ,
.
I1, ω. y (. 2.10).
. 2.10
φ1 , 1 . i1 1 t :
i1 =I1sin(ωt + φ1)
M1 ( , ).
, :
i1 =I1sin(ωt + φ1) i2 =I2sin(ωt + φ2)
, i1 i2 1 2, . ω. 2.10, , I1sin(ωt + φ1) I2sin(ωt + φ2). I1cos(ωt + φ1) I2cos(ωt + φ2).
2.10 .
, , , ω,
. 2.10, t = 0, = I φ , , :
Isin(ωt + φ) = I1sin(ωt + φ1) + I2sin(ωt + φ2). (2.33, )
Icos(ωt + φ) = I1cos(ωt + φ1) + I2cos(ωt + φ2). (2.33, )
, . , (. . ). , (2.23 , ), . . , , , , .
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, (.. , ), . , .
, . , , , , , ( , . .). ( ), .
. 2.11, . . . , , (. 2.11, ).
, ω . , , , (√2).
. 2.11.
.
. , , . ( ), . , , ,
2.3.2. .
, ( ) .
, .
() . , , , .
. 2.12.
, , , . . (.2.12, ). , , ; , , , . .
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() . , , , . , , .
jω , jω. - .
, , , .
, :
(2.34)
:
a = Acosφ; b = Asinφ;
,
i = imsinφt ( , ) (, j) :
(2.35)
i. i t = 0.
,
i = imcosωt :
, ( sin, cos). , .
, :
(2.36)
φ ( ωt) (2.36) . φ ( ωt) , .
, φ, = Umsin(ωt + φ), Umt , (ωt + φ). Umt , :
Umt = ú + jű = Um [cos(ωt + φ) + jsin(ωt + φ)] = Umej(ωt+φ) =
, j =
, ( t = 0):
. .
, , , , , . . .
, , , i İm; R, Ri, Rİm, İm; İmjωL, 90; , 90; . . . Ė. İmjω (2.28), , . L XL= ωL. j , cnb 90.
, (2.29) , , .
90 j.
. 2.13:
uR + uL + uC = e
. 2.13.
(2.37)
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:
İ , .:
(2.38)
, . 2.13
, . , Rİm iR R, jωLİm L, .
(2.38) , Z. :
(2.39)
, Z . z. Z , .. , .
(2.38) : İmZ = Ėm. , İ Ė İ Ė:
(2.40)
(2.40) .
Z R jX, ..:
Z = R = jX
R ; X .
, , 2.13