正弦量的向量表示法课件讲解

VectorRepresentationofSinusoidalQuantities正弦量的向量表示法运用复数分析电路的方法称为相量法(phasormethod)。Themethodofusingcomplexnumberstoanalyzecircuitsisreferredtoasthephasormethod.1.复数ComplexNumbers设一个复数A=a+jb，其中a、b都是实数，a为复数的实部，b为复数的虚部，为虚数单位(虚数单位在数学中用i表示，因电工学中用i表示电流，故改用j以示区别)。ConsideracomplexnumberA=a+jb,wherebothaandbarerealnumbers.Here,aistherealpart,bistheimaginarypart,andistheimaginaryunit(oftenrepresentedasjinelectricalengineeringtoavoidconfusionwiththesymboliusedforcurrentinmathematics).例如，3+j4对应于图中复平面上的P1点。Forexample,3+j4correspondstothepointP1inthecomplexplane,asillustratedinfollowingfigure.复平面complexplaneRealAxis复数还可以用复平面上的一个矢量来表示。Complexnumberscanalsoberepresentedbyavectorinthecomplexplane.如A=3+j4可以用一个从原点O到P点的矢量来表示。Forinstance,A=3+j4canberepresentedbyavectorOPfromtheoriginOtothepointP.这种矢量称为复数矢量。Thistypeofvectoriscalledacomplexvector.复平面中的矢量vectorinthecomplexplane矢量与实轴正方向的夹角θ称为复数A的辐角。TheangleθbetweenthevectorandthepositiverealaxisistermedtheargumentofthecomplexnumberA.RealAxis任意一个复数A可对应一个复数矢量OP，如图所示。AnycomplexnumberAcorrespondstoacomplexvectorOP,asshowninthediagram.矢量的长度定义为复数A的绝对值，称为复数的模。即ThelengthofthevectorisdefinedastheabsolutevalueofthecomplexnumberA,knownasthemodulusofthecomplexnumber,denotedas因为Because根据欧拉公式BasedonEuler'sformula(1)式(1)又可以写为Equation(1)canberewrittenas复平面中的矢量vectorinthecomplexplaneRealAxis复数的四则运算规则如下Therulesforthefourfundamentaloperationswithcomplexnumbersareasfollows.设Set则Then设有时间t的复值函数，利用数学中的欧拉公式Consideracomplex-valuedfunctionoftimet,utilizingEuler'sformulainmathematics：ej

t=cos

t+jsin

t

显然Obviouslycos

t=Re(ej

t)sin

t=Im(ej

t)这里，Re(·)和Im(·)分别为取实部和取虚部的符号。进一步，有Here,Re(·)andIm(·)representthesymbolsfortakingtherealandimaginaryparts,respectively.Furthermore,wehave:Imej(

t+

)=Imcos(

t+

)

+jImsin(

t+

)2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors显然正弦电流i(t)＝Imsin(

t+

)可以表示为Clearly,thesinusoidalcurrenti(t)=Imsin(t+

)canbeexpressedasi(t)＝Im[Imej(

t+

)]=Im[Imej

ej

t]Imej(

t+

)称为复指数电流。记复数电流Imej(t+)isreferredtoasthecomplexexponentialcurrent.Denotethecomplexcurrentas:

式中，Im为的模，则WhereImisthemagnitudeof,then

i(t)＝Im

[]2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors在电路中，通常把电流的振幅（或有效值）与其初相角构成的一个复数称为电流相量。例如Incircuits,theamplitude(oreffectivevalue)ofacurrent,alongwithitsinitialphaseangle,isoftenreferredtoasacurrentphasor.Forinstance,i(t)＝5sin(

t

30

)A对应电流相量为Thecorrespondingcurrentphasoris（最大值相量Themaximumvaluephasor）（有效值相量TheEffectivevaluephasor）2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors类似地，若有电压Similarly,ifthereisavoltageas

u(t)＝10sin(

t+50

)V电压相量为thenthevoltagephasoris

（最大值相量Themaximumvaluephasor）若用有效值为相量的模，则有效值相量为Ifweusetheeffectivevalueasthemagnitudeofthephasor,theeffectivevaluephasoris:

（有效值相量TheEffectivevaluephasor）2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors正弦量用相量表示后，还可以用相量图来表示。Afterrepresentingsinusoidalquantitiesusingphasors,theycanalsobedepictedusingphasordiagrams.设电流相量Letthecurrentphasequantity它们的相量图如下图所示Theirphasordiagramsareshownbelow二者之和按平行四边形