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
预览:

本文由漠然轻寒贡献

ppt文档可能在WAP端浏览体验不佳。建议您优先选择TXT,或下载源文件到本机查看。

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



第1页/共3页 下一页>尾页