电工学武汉理工大学课件chapter5-2

电工学武汉理工大学课件chapter5-2
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5.4 R, L and C in series connection alternating circuit
1. Relation of voltage and current
By Kirchhoff’s Voltage Law: i uR uL uC R L C
di 1 u = uR + uL + uC = iR + L + ∫ idt dt C u t Supposed that i = I sinω
uR = RIm sinωt =URm sinωt
uL
m
d(Im sinω ) t =L = ImωL cosω t dt
= ULM sin(ω + t
π
uc = Xc Im sin(ωt ) = Ucm sin(ωt ) 2 2
π
2
)
π
uL lead i 90° °
uL = ImωLsin(ωt + 90°) = ULm sin(ωt + 90°)
uc lag i 90° ° Im uC = sin(ωt 90°) = UCm sin(ωt 90°) ωC Then: URm = UR = R
.
Im
I
UL
.
U
.
ULm UL = = ωL = XL Im I
UCm UC 1 = = = XC Im I ωC
I
.
UR
.
UC
u = uR + uL + uC =Um sin(ωt +)
According to phasor diagram, voltage triangle:
i i i i
U ,U R ,U L + U C
U = U + (UL UC )
2 R 2 2 2
.
UL
.
= (IR) + (IX L IXC )
2
U
.
= I R + (XL XC)
or
2
.
I
.
UR UC
U 2 2 = R + (XL XC) = z I
z = R + (XL XC) = U I
2 2
| Z | is defined as Impedance amplitude(
).
Z = R 2 + X 2 impedance amplitude
Impedance rectangular form: is composed by | Z | 、R、(XL–XC) . 、 angel : phase difference of u and i. Phase difference is :
|Z|
R
.
XL–XC
U
UL + UC
.
.
.
UR
UL UC XL XC = arctg = arctg UR R
f=constant, phase difference is determined by parameters of circuit. If XL>XC, >0, u lead i , inductive circuit; If XL<XC, <0,u lag i , capacitive circuit; If XL=XC, = 0, u and i in phase, resistive circuit. Because u=uR + uL + uC then
U = UR + UL + UC = I R + jI XL jI XC
Relation of value, phase and phasor
Um U Z= = , = ψu ψi , U = ZI Im I
If define the ratio of the phasor voltage to the phasor current as complex impedance.
Z=
V
Z = R + j( XL XC ) = Z e
where
j
I
XL XC = arctg R
Z = R + ( XL XC )
2
2
Real part is resistance, imaginary part is impedance. Complex impedance: value relation of voltage and current in the circuit; Angel : phase relation of voltage and current in the circuit. Complex impedance: complex calculation value, not phasor.
R :
V = R I or
V
= R → Z = R
I
L : V = jω L I or
V
= jω L → Z = jω L = jX L
I 1 C: V = I or jωC
V
1 1 1 = →Z = =j = jX C jωC jωC ωC I
1 (XC = ) ωC
2. Power of RLC series connection circuit
Instantaneous power
p = ui = 2Usin( ωt + ψu ) 2I sin( ωt + ψi ) = UI cos(ψu ψi ) UI cos(2ωt + ψu ψi ) = UI cos UI cos(2ωt + )
average power(or active power) ( )
1T 1t P = ∫ pdt = ∫ [UI cos UI cos(2ωt + )]dt T0 T0 U2 = UI cos = IUR = I2R = R R

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