電磁場(chǎng)與電磁波第17講時(shí)諧場(chǎng)課件

FieldandWaveElectromagnetic電磁場(chǎng)與電磁波第17講11.Faraday’sLawofElectromagneticInductionReview22.Maxwell’sEquations3.ElectromagneticBoundaryConditionsTheintegralformThedifferentialform

SignificanceFaraday’slaw(電磁感應(yīng)定律)Ampere’scircuitallaw(全電流定律)Gauss’slaw(高斯定理)Noisolatedmagneticcharge(磁通連續(xù)性原理)34.PotentialFunctions5.WaveEquationsandTheirSolutions4Maxwell’sequationsandalltheequationsderivedfromthemsofarinthischapterholdforelectromagneticquantitieswithanarbitrarytime-dependence(時(shí)間任意相關(guān)).Theactualtypeoftimefunctionsthatthefieldquantitiesassumedependson(取決于)thesource(源)functions

andJ.Inengineering,oneofthe

mostimportant

casesoftime-varyingelectromagneticfieldsisthe

time-harmonic(sinusoidal)field(時(shí)諧場(chǎng)、正弦場(chǎng)).Inthistypeoffield,the

excitation

sourcevaries

sinusoidally

intimewith

a

singlefrequency(單一頻率).In

alinearsystem(線性系統(tǒng)),asinusoidallyvarying

source

generates

fields

thatalsovarysinusoidallyintimeatallpointsinthesystem(正弦變化的源產(chǎn)生正弦變化的場(chǎng)).1)whatisTime-HarmonicFields3.Time-HarmonicFields52)討論時(shí)諧場(chǎng)（正弦信號(hào)）的原因Whenfieldsareexaminedinthismanner,thereisnolossingeneralityas(a)Theyareeasytogenerate(b)anytime-varyingperiodicfunctioncanberepresentedbyaFourierseriesintermsofsinusoidalfunctions(c)theprincipleofsuperpositioncanbeappliedunderlinearconditions.Inotherwords,wecanobtainthecompleteresponseoftimevaryingperiodicfieldsbyusinglinearcombinationsofmonochromaticresponses（a）正弦信號(hào)容易產(chǎn)生，50Hz交流電，通信的載波都是正弦信號(hào)（b）從信號(hào)分析的角度來(lái)看，正弦信號(hào)是一種簡(jiǎn)單基本的信號(hào)。正弦信號(hào)進(jìn)行各種運(yùn)算（加減微分積分后仍為同頻率正弦信號(hào)）（c）傅立葉分析：任意周期信號(hào)分解為不同頻率的正弦之和（d）線性系統(tǒng)的疊加原理63.1

電路中的相量表達(dá)式Incircuittheory,youhavealreadyusedthephasornotation(相量)torepresentvoltagesandcurrentsvaryingsinusoidally

intime(1)Instantaneous(time-dependent)expressionofasinusoidalscalarquantity(瞬時(shí)值）三角函數(shù)表達(dá)式3Parameters:

angularfrequency:

amplitude:Im

phase：(2)

復(fù)數(shù)的表示xjyP（x,y）復(fù)平面上一點(diǎn)P7(3)正弦表達(dá)式和相量表達(dá)式的對(duì)應(yīng)關(guān)系相量的模正弦量的幅值初位相復(fù)角頻率是已知？頻率相量乘以ejt，再取實(shí)部8EXAMPLE7-6P337-33893.2

Time-harmonicElectromagneticsFieldvectorsthatvarywithspacecoordinatesandaresinusoidalfunctionsoftimecansimilarlyberepresentedbyvectorphasors(矢量相量)thatdependonspacecoordinatesbutnotontime.Asanexample,wecanwriteatime-harmonicE

fieldreferringtocostaswhereE(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)istoberepresentedbythevectorphasor

E(x,y,z),thenE(x,y,z,t)/tandE(x,y,z,t)dtwouldberepresentedby,respectively,vectorphasors

jE(x,y,z)

andE(x,y,z)/j.Higher-orderdifferentiationsandintegrationswithrespecttowouldberepresented,respectively,bymultiplicationsanddivisionsofthephasor

E(x,y,z)byhigherpowersofj.1011已知正弦電磁場(chǎng)的場(chǎng)與源的頻率相同，因此可用復(fù)矢量形式表示麥克斯韋方程?？紤]到正弦時(shí)間函數(shù)的時(shí)間導(dǎo)數(shù)為或因此，麥克斯韋第一方程可表示為上式對(duì)于任何時(shí)刻均成立，實(shí)部符號(hào)可以消去，即12瞬時(shí)值由相量值代替時(shí)間求導(dǎo)由jω代替Wenowwritetime-harmonicMaxwell’sequations(時(shí)諧麥克斯韋方程組)intermsofvectorfieldphasors(E,H)andsourcephasors(,J)inasimple(linear,isotropic,andhomogenous)mediumasfollows.13Thetime-harmonicwaveequations(時(shí)諧波動(dòng)方程)forEandHbecome,respectively,Thetime-harmonicwaveequationsforscalarpotentialVandvectorpotentialAbecome,respectively,Letiscalledthewavenumber.14Then

Considerthetimedelayfactor,forasinusoidalfunctionitleadstoaphasedelayof.

Weobtain15ThecomplexLorentzconditionis

Thecomplexelectricandmagneticfieldscanbeexpressedintermsofthecomplexpotentialsas

163.3

source-free(無(wú)源)fieldsinsimplemediaInasimple,nonconducting(非導(dǎo)電)source-freemediumcharacterizedby=0,J=0,=0,thetime-harmonicMaxwell’sequationsbecome

17whicharehomogeneousvectorHelmholtz’sequations(齊次矢量亥姆霍茲方程).andwaveequationsforAandV

becomeThetime-harmonicwaveequationsforEandHbecome,respectively,Letiscalledthewavenumber.18Ifthesimplemediumisconducting(0)(導(dǎo)電介質(zhì)),acurrentJ=Ewillflow,andtheequationshouldbechangedtowithTheotherthreeequationsinMaxwell’sequationareunchanged.Hence,allthepreviousequationsfornonconducting(非導(dǎo)電)mediawillapplytoconductingmediaifisreplacedbythecomplexpermittivity

c.Meanwhile,thereal(實(shí)數(shù))

wavenumber

kinthehelmholtz’sequationsshouldbechangedtoacomplex(復(fù)數(shù))

wavenumber:19Theratio’’/’

iscalledalosstangent(損耗正切)becauseitisameasureofthepowerlossinthemedium:Thequantityc

maybecalledthelossangle(損耗角).Amediumissaidtobeagoodconductor(良導(dǎo)體)if>>,andagoodinsulator(良絕緣體)if<<.Thus,amaterialmaybeagoodconductoratlowfrequencies(低頻)butmayhavethepropertiesofalossydielectricatveryhighfrequencies(高頻).201.Faraday’sLawofElectromagneticInductionReview212.Maxwell’sEquations3.ElectromagneticBoundaryConditionsTheintegralformThedifferentialform

SignificanceFaraday’slaw(電磁感應(yīng)定律)Ampere’scircuitallaw(全電流定律)Gauss’slaw(高斯定理)Noisolatedmagneticcharge(磁通連續(xù)性原理)224.PotentialFunctions5.WaveEquationsandTheirSolutions236.Time-HarmonicFields相量的模正弦量的幅值初位相復(fù)角頻率是已知？頻率相量乘以ejt，再取實(shí)部24dx25P.7-7P34926P.7-13P35127梯度運(yùn)算符合以下規(guī)則：C為常數(shù)散度運(yùn)算規(guī)則旋度運(yùn)算規(guī)則28P.7-25P3522930P.7-30P35331Theelectricfieldintensityinasource-freedielectric()regionisgivenas(V/m),whereangularfrequency，allareconstants.Find:Example.(1)Thephasorrepresentationofelectricfieldintens