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1、3.6 AC Power 3.6.1 Instantaneous Power In AC circuits,if the current of a resistor is m()cosi tIt=,its voltage will bem()()cosv tRi tRIt=.The instantaneous power of the resistor is 222mm(1cos2)()()()cos2RItp tv t i tRIt+=(1)The waveform of p(t)of the resistor is shown in Fig.1.Figure 1:Instantaneous
2、 power waveform of the resistor.If the current of an inductor is m()cosi tIt=,its voltage will be m()()sindi tv tLLItdt=.The instantaneous power of the inductor is 22mm()()()sincossin22LIp tv t i tLIttt=(2)The waveform of p(t)of the resistor is shown in Fig.2.Figure 2:Instantaneous power waveform of
3、 the inductor.In the same way,we can get the instantaneous power of a capacitor 2mmm1()()()sincossin22Ip tv t i tItIttCC=(3)The waveform of p(t)of the capacitor is shown in Fig.3.Figure 3:Instantaneous power waveform of the capacitor.From(1)-(3),the instantaneous power of AC circuits varies with tim
4、e periodically.Therefore,it is difficult to measure the power of AC circuits quantitatively by the instantaneous power.3.6.2 Average Power(Active Power)In order to measure the power of AC circuits quantitatively,the instantaneous power varying with time periodically can be averaged during a period T
5、 of sinusoidal voltage and current.av01()TPp t dtT=(4)The average value Pav of the instantaneous power is called average power or active power.For a resistor,from(1)and(4),the average power is 22mmRav0011(1cos2)()=22TTRItRIPp t dtdtTT+=(5)For an inductor,from(2)and(4),the average power is 2mLav0011(
6、)sin2=0 W2TTLIPp t dttdtTT=(6)For a capacitor,from(3)and(4),the average power is 2mCav0011()sin2=0 W2TTIPp t dttdtTTC=(7)From(6)and(7),the average power of the inductor and capacitor in AC circuits is zero.The expressions(5)-(7)show that in AC circuits the resistor consumes average power,while both
7、the inductor and the capacitor do not consume average power.For any branch in AC circuits,we can calculate its average power as ()mmavmm0011()cos()cos()=cos2TTviviV IPp t dtVtItdtTT=+(8)where vi is the phase difference between the branch voltage and branch current.For a resistor,0vi=.According to(8)
8、,the average power of the resistor is ()ommmmmmRavcoscos0222viV IV IV IP=(9)For an inductor,o90vi=.According to(8),the average power of the inductor is ()ommmmLavcoscos900 W22viV IV IP=(10)For a capacitor,o90vi=.According to(8),the average power of the capacitor is ()ommmmCavcoscos(90)0 W22viV IV IP
9、=(11)The expressions(9)-(11)are in accordance with the expressions(5)-(7).3.6.3 Reactive Power From(10)and(11),the inductor and capacitor do not consume average power.Although the inductor and capacitor do not consume average power,they keep absorbing and delivering power,as shown in Fig.2 and Fig.3
10、.In order to measure the ability of power throughput of AC circuits quantitatively,we need to introduce another important concept:reactive power Q.For any branch in AC circuits,its reactive power is ()mm=sin2viV IQ(12)Q has been given the unit volt-ampere reactive(VAR).For a resistor,according to(12
11、),the reactive power is ()ommmmsinsin00 VAR22RviV IV IQ=(13)For an inductor,according to(12),the reactive power is ()ommmmmmsinsin90222LviV IV IV IQ=(14)For a capacitor,according to(12),the reactive power is ()ommmmmmsinsin(90)222CviV IV IV IQ=(15)The expressions(13)-(15)show that in AC circuits bot
12、h the inductor and the capacitor have the ability of power throughput,while the resistor does not have the ability of power throughput.The resistor only has the ability of consuming average power,as shown in(9).3.6.4 Root-Mean-Square Value The average power expression(8)and the reactive power expres
13、sion(12)seem a little imperfect for there is a coefficient 12.In order to remove the coefficient 12,it is necessary to introduce the concept of root-mean-square(rms)value.For any periodic waveform x(t),its rms value is defined as 2rms01()TXx t dtT=(16)For any sinusoidal voltage m()cos()vv tVt=+,acco
14、rding to(16),its rms value is 22mrmsm0011()cos()2TTvVVv t dtVtdtTT=+=(17)For any sinusoidal current m()cos()ii tIt=+,according to(16),its rms value is 22mrmsm0011()cos()2TTiIIi t dtItdtTT=+=(18)Substituting(17)and(18)into(8),we get ()avrmsrmscosviPVI=(19)Substituting(17)and(18)into(12),we get ()rmsr
15、ms=sinviQ VI(20)From(19)and(20),the coefficient 12 is removed and the expressions seem more perfect than(8)and(12).After introducing the concept of rms value,we can define the phasors of voltage and current with rms value.mrmsrms22vvjjVeVe=VV(21)mrmsrms22iijjIeIe=II(22)3.6.5 Complex Power The comple
16、x power is defined as avPjQ=+S(23)Substituting(19)and(20)into(21),we get ()()()()()()avrmsrmsrmsrmsrmsrmsrmsrmsrmsrms*rms rmscossincossinviivvivivivijjjPjQVIjVIVIjVIeVeIe=+=+=+=SVI(24)The complex power S has been given the unit volt-ampere(VA)to distinguish it from the average power unit watt(W)and
17、the reactive power unit volt-ampere reactive(VAR).From(24),we can obtain Pav and Q through the calculation of complex power according to*rms rms=SV I.Another important fact about the complex powers can reflect the energy conservation.The sum of the complex powers associated with n branches is equal
18、to zero:10nii=S(25)From(23)and(25),we can obtain av10,niiP=10niiQ=(26)3.6.6 Apparent Power and Power Factor From(24),the amplitude of the complex power S is rmsrmsVI=S(27)We call the amplitude of the complex power S as apparent power S rmsrmsSVI=S(28)The apparent power is so called because it seems
19、apparent that the power should be the voltage-current product.In fact,the apparent power represents the ability of power generation of a circuit element.In AC cirtuits,it is possible that not all power generated by the circuit element is converted to the average power(active power).The ratio of the
20、average power Pav to the apparent power S is called the power factor pf,and it is given by avrmsrmsrmsrmscos()cos()viviPVIpfSVI=(29)The expression(29)shows that the power factor pf represents the ability of converting the apparent power to the average power(active power).If pf=1,all apparent power is converted to average power.If pf=0,no apparent power is converted to the average power.We usually prefer to the high power factor.