正弦量的向量表示法課件講解

VectorRepresentationofSinusoidalQuantities正弦量的向量表示法運用復數(shù)分析電路的方法稱為相量法(phasormethod)。Themethodofusingcomplexnumberstoanalyzecircuitsisreferredtoasthephasormethod.1.復數(shù)ComplexNumbers設一個復數(shù)A=a+jb，其中a、b都是實數(shù)，a為復數(shù)的實部，b為復數(shù)的虛部，為虛數(shù)單位(虛數(shù)單位在數(shù)學中用i表示，因電工學中用i表示電流，故改用j以示區(qū)別)。ConsideracomplexnumberA=a+jb,wherebothaandbarerealnumbers.Here,aistherealpart,bistheimaginarypart,andistheimaginaryunit(oftenrepresentedasjinelectricalengineeringtoavoidconfusionwiththesymboliusedforcurrentinmathematics).例如，3+j4對應于圖中復平面上的P1點。Forexample,3+j4correspondstothepointP1inthecomplexplane,asillustratedinfollowingfigure.復平面complexplaneRealAxis復數(shù)還可以用復平面上的一個矢量來表示。Complexnumberscanalsoberepresentedbyavectorinthecomplexplane.如A=3+j4可以用一個從原點O到P點的矢量來表示。Forinstance,A=3+j4canberepresentedbyavectorOPfromtheoriginOtothepointP.這種矢量稱為復數(shù)矢量。Thistypeofvectoriscalledacomplexvector.復平面中的矢量vectorinthecomplexplane矢量與實軸正方向的夾角θ稱為復數(shù)A的輻角。TheangleθbetweenthevectorandthepositiverealaxisistermedtheargumentofthecomplexnumberA.RealAxis任意一個復數(shù)A可對應一個復數(shù)矢量OP，如圖所示。AnycomplexnumberAcorrespondstoacomplexvectorOP,asshowninthediagram.矢量的長度定義為復數(shù)A的絕對值，稱為復數(shù)的模。即ThelengthofthevectorisdefinedastheabsolutevalueofthecomplexnumberA,knownasthemodulusofthecomplexnumber,denotedas因為Because根據(jù)歐拉公式BasedonEuler'sformula(1)式(1)又可以寫為Equation(1)canberewrittenas復平面中的矢量vectorinthecomplexplaneRealAxis復數(shù)的四則運算規(guī)則如下Therulesforthefourfundamentaloperationswithcomplexnumbersareasfollows.設Set則Then設有時間t的復值函數(shù)，利用數(shù)學中的歐拉公式Consideracomplex-valuedfunctionoftimet,utilizingEuler'sformulainmathematics：ej

t=cos

t+jsin

t

顯然Obviouslycos

t=Re(ej

t)sin

t=Im(ej

t)這里，Re(·)和Im(·)分別為取實部和取虛部的符號。進一步，有Here,Re(·)andIm(·)representthesymbolsfortakingtherealandimaginaryparts,respectively.Furthermore,wehave:Imej(

t+

)=Imcos(

t+

)

+jImsin(

t+

)2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors顯然正弦電流i(t)＝Imsin(

t+

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

)canbeexpressedasi(t)＝Im[Imej(

t+

)]=Im[Imej

ej

t]Imej(

t+

)稱為復指數(shù)電流。記復數(shù)電流Imej(t+)isreferredtoasthecomplexexponentialcurrent.Denotethecomplexcurrentas:

式中，Im為的模，則WhereImisthemagnitudeof,then

i(t)＝Im

[]2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors在電路中，通常把電流的振幅（或有效值）與其初相角構成的一個復數(shù)稱為電流相量。例如Incircuits,theamplitude(oreffectivevalue)ofacurrent,alongwithitsinitialphaseangle,isoftenreferredtoasacurrentphasor.Forinstance,i(t)＝5sin(

t

30

)A對應電流相量為Thecorrespondingcurrentphasoris（最大值相量Themaximumvaluephasor）（有效值相量TheEffectivevaluephasor）2.用相量表示正弦量RepresentationofSinusoidalQuantitiesUsingPhasors類似地，若有電壓Similarly,ifthereisavoltageas

u(t)＝10sin(

t+50

)V電壓相量為thenthevoltagephasoris

（最大值相量Themaximumvaluephasor）若用有效值為相量的模，則有效值相量為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，還可以用相量圖來表示。Afterrepresentingsinusoidalquantitiesusingphasors,theycanalsobedepictedusingphasordiagrams.設電流相量Letthecurrentphasequantity它們的相量圖如下圖所示Theirphasordiagramsareshownbelow二者之和按平行四邊形