电磁场与电磁波第17讲时谐场概要

FieldandWaveElectromagnetic电磁场与电磁波第17讲21.Faraday’sLawofElectromagneticInduction(V)ddt()(V)CuBdl'()(V)CSCEdlBdSuBdlt'(V)SddBdSdtdtReview32.Maxwell’sEquations3.ElectromagneticBoundaryConditions()CSDHdlJdStCSBdEdldStdt0SBdSSDdSQTheintegralformDHJtBEt0BDThedifferentialformSignificanceFaraday’slaw(电磁感应定律)Ampere’scircuitallaw(全电流定律)Gauss’slaw(高斯定理)Noisolatedmagneticcharge(磁通连续性原理)212ˆ()0naEE212ˆ()0naBB212ˆ()nSaDD212ˆ()SnaHJH44.PotentialFunctions5.WaveEquationsandTheirSolutions2221EJEtt222HHJtBAAEVtVAt222AAJt222VVt1(,)d4πVtuVtVRR'R',RRR',(,)d4πVtutVRR'JR'ARRR'5Maxwell’sequationsandalltheequationsderivedfromthemsofarinthischapterholdforelectromagneticquantitieswithanarbitrarytime-dependence(时间任意相关).Theactualtypeoftimefunctionsthatthefieldquantitiesassumedependson(取决于)thesource(源)functionsandJ.Inengineering,oneofthemostimportantcasesoftime-varyingelectromagneticfieldsisthetime-harmonic(sinusoidal)field(时谐场、正弦场).Inthistypeoffield,theexcitationsourcevariessinusoidallyintimewithasinglefrequency(单一频率).Inalinearsystem(线性系统),asinusoidallyvaryingsourcegeneratesfieldsthatalsovarysinusoidallyintimeatallpointsinthesystem(正弦变化的源产生正弦变化的场).1)whatisTime-HarmonicFields3.Time-HarmonicFields62)讨论时谐场（正弦信号）的原因Whenfieldsareexaminedinthismanner,thereisnolossingeneralityas(a)Theyareeasytogenerate(b)anytime-varyingperiodicfunctioncanberepresentedbyaFourierseriesintermsofsinusoidalfunctions(c)theprincipleofsuperpositioncanbeappliedunderlinearconditions.Inotherwords,wecanobtainthecompleteresponseoftimevaryingperiodicfieldsbyusinglinearcombinationsofmonochromaticresponses（a）正弦信号容易产生，50Hz交流电，通信的载波都是正弦信号（b）从信号分析的角度来看，正弦信号是一种简单基本的信号。正弦信号进行各种运算（加减微分积分后仍为同频率正弦信号）（c）傅立叶分析：任意周期信号分解为不同频率的正弦之和（d）线性系统的叠加原理73.1电路中的相量表达式Incircuittheory,youhavealreadyusedthephasornotation(相量)torepresentvoltagesandcurrentsvaryingsinusoidallyintime()cos()()cos()()cos()mmumittutUtitIt交变电动势：交变电压：交变电流：(1)Instantaneous(time-dependent)expressionofasinusoidalscalarquantity(瞬时值）三角函数表达式3Parameters:angularfrequency:amplitude:Imphase：(2)复数的表示22;:(cossin);:;cos:sinjAxjyAjAeyxytgxxy代数表达式：三角表达式指数表达式复数的模：；复数的幅角：复数的实部：；复数的虚部xjyP（x,y）复平面上一点P8(3)正弦表达式和相量表达式的对应关系()()()Re()Re()Re()Resin()cos()()Recos()Re()Resin(())cocos()s()uuijjtjtjtmmmmumujtjtmmmimummiiutUeUeeUejUtUtitIeIejUtItItIt相量的模正弦量的幅值初位相复角频率是已知()cos()()cos()uijmummjmimmutUtUUeitItIIe三角表达式相量表达式（复数表示正弦量）？频率()cos()()cos()mumiutUtitIt相量乘以ejt，再取实部9EXAMPLE7-6P337-338103.2Time-harmonicElectromagneticsFieldvectorsthatvarywithspacecoordinatesandaresinusoidalfunctionsoftimecansimilarlyberepresentedbyvectorphasors(矢量相量)thatdependonspacecoordinatesbutnotontime.Asanexample,wecanwriteatime-harmonicEfieldreferringtocostas(,,,)Re[(,,)]jtExyztExyzewhereE(x,y,z)isavectorphasor(矢量相量)thatcontainsinformationondirection(方向),magnitude(振幅),andphase(相位).Phasorsare,ingeneral,complexquantities.weseethat,ifE(x,y,z,t)istoberepresentedbythevectorphasorE(x,y,z),thenE(x,y,z,t)/tandE(x,y,z,t)dtwouldberepresentedby,respectively,vectorphasorsjE(x,y,z)andE(x,y,z)/j.Higher-orderdifferentiationsandintegrationswithrespecttowouldberepresented,respectively,bymultiplicationsanddivisionsofthephasorE(x,y,z)byhigherpowersofj.1112已知正弦电磁场的场与源的频率相同，因此可用复矢量形式表示麦克斯韦方程。jm(,)Re(j()e)tttErErjRe(j2()e)tEr考虑到正弦时间函数的时间导数为jjRe(2e)Re2j2ettHJD或jjjRe(2e)Re2eRej2etttHJD因此，麦克斯韦第一方程可表示为tEHE上式对于任何时刻均成立，实部符号可以消去，即22j2HJDjHJD130BEtDHJtDBJjDEBH'JEJ瞬时值由相量值代替时间求导由jω代替Wenowwritetime-harmonicMaxwell’sequations(时谐麦克斯韦方程组)intermsofvectorfieldphasors(E,H)andsourcephasors(,J)inasimple(linear,isotropic,andhomogenous)mediumasfollows./0EjHHJjEEH14Thetime-harmonicwaveequations(时谐波动方程)forEandHbecome,respectively,2221EJEtt222HHJt221EEjJ22HHJ222AAJt222VVt22AAJ22VVThetime-harmonicwaveequationsforscalarpotentialVandvectorpotentialAbecome,respectively,Letkuiscalledthewavenumber.15kuRR'RR'ThenConsiderthetimedelayfactor,forasinusoidalfunctionitleadstoaphasedelayof.uRR'uRR'Weobtain1(,)d4πVtuVtVRR'R',RRR',(,)d4πVtutVRR'JR'ARRR'j()e()d4πkVVRR'JR'ArRR'j1()e()d4πkVVVRR'R'rRR'16ThecomplexLorentzconditionisThecomplexelectricandmagneticfieldscanbeexpressedintermsofthecomplexpotentialsasVAt()()ArjVrAEVtBA()()BrArAEjAVjAj173.3source-free(无源)fieldsinsimplemediaInasimple,nonconducting(非导电)source-freemediumcharacterizedby=0,J=0,=0,thetime-harmonicMaxwell’sequationsbecome/0EjHHJjEEH00EjHHjEEH18220AkA220VkVwhicharehomogeneousvectorHelmholtz’sequations(齐次矢量亥姆霍兹方程).andwaveequationsforAandVbecome22