1、Field and Wave Electromagnetic電磁場與電磁波電磁場與電磁波第第17講講21. Faradays Law of Electromagnetic Induction (V)ddt () (V)CuBdl() (V)CSCE dlB dSuBdlt (V)SddB dSdtdt Review32. Maxwells Equations3. Electromagnetic Boundary Conditions ()CSDH dlJdSt CSBdE dldStdt 0SB dS SD dSQThe integral formDHJtBEt 0BDThe differen

2、tial form SignificanceFaradays law(電磁感應(yīng)定律電磁感應(yīng)定律)Amperes circuital law(全電流定律全電流定律)Gausss law(高斯定理高斯定理)No isolated magnetic charge(磁通連續(xù)磁通連續(xù)性原理性原理)212()0naEE212()0naBB212()nSaDD212()SnaHJH44. Potential Functions5. Wave Equations and Their Solutions2221EJEtt 222HHJt BAAEVt VAt 222AAJt222VVt1( , )d4VtuVt

3、VRRR,RRR,( , )d4VtutVRRJ RA RRR5Maxwells equations and all the equations derived from them so far in this chapter hold for electromagnetic quantities with an arbitrary time-dependence(時間任意相關(guān)時間任意相關(guān)). The actual type of time functions that the field quantities assume depends on(取決于取決于) the source(源源)

4、functions and J. In engineering, one of the most important cases of time-varying electromagnetic fields is the time-harmonic (sinusoidal) field(時諧場、時諧場、正弦場正弦場). In this type of field ,the excitation source varies sinusoidally in time with a single frequency(單一頻率單一頻率). In a linear system(線性系統(tǒng)線性系統(tǒng)), a

5、 sinusoidally varying source generates fields that also vary sinusoidally in time at all points in the system(正弦變化的源產(chǎn)生正弦變化的場正弦變化的源產(chǎn)生正弦變化的場).1) what is Time-Harmonic Fields3. Time-Harmonic Fields 62) 2) 討論時諧場（正弦信號）的原因討論時諧場（正弦信號）的原因When fields are examined in this manner,there is no loss in generality

6、 as (a) They are easy to generate (b) any time-varying periodic function can be represented by a Fourier series in terms of sinusoidal functions (c) the principle of superposition can be applied under linear conditions. In other words, we can obtain the complete response of time varying periodic fie

7、lds by using linear combinations of monochromatic responses（a a）正弦信號容易產(chǎn)生，）正弦信號容易產(chǎn)生，50Hz50Hz交流電，通信的載波都是正弦信號交流電，通信的載波都是正弦信號（b b）從信號分析的角度來看，正弦信號是一種簡單基本的信號。正弦信號）從信號分析的角度來看，正弦信號是一種簡單基本的信號。正弦信號進(jìn)進(jìn) 行行 各種運(yùn)算（加減微分積分后仍為同頻率正弦信號）各種運(yùn)算（加減微分積分后仍為同頻率正弦信號）（c c）傅立葉分析：任意周期信號分解為不同頻率的正弦之和）傅立葉分析：任意周期信號分解為不同頻率的正弦之和（d d）線性系統(tǒng)

8、的疊加原理）線性系統(tǒng)的疊加原理73.13.1 電路中的相量表達(dá)式電路中的相量表達(dá)式In circuit theory, you have already used the phasor notation(相量相量) to represent voltages and currents varying sinusoidally in time( )cos()( )cos()( )cos()mmumittu tUti tIt交變電動勢：交變電壓：交變電流：(1) Instantaneous (time-dependent) expression of a sinusoidal scalar qua

9、ntity(瞬時值）三角函數(shù)表達(dá)式瞬時值）三角函數(shù)表達(dá)式3 Parameters: angular frequency: amplitude: Im phase： (2) 復(fù)數(shù)的表示復(fù)數(shù)的表示22;:(cossin );:;cos:sinjAxjyAjAeyxytgxxy代數(shù)表達(dá)式：三角表達(dá)式指數(shù)表達(dá)式復(fù)數(shù)的模：；復(fù)數(shù)的幅角：復(fù)數(shù)的實部：；復(fù)數(shù)的虛部xjyP（x,y）復(fù)平面上一點復(fù)平面上一點P8(3) (3) 正弦表達(dá)式和相量表達(dá)式的對應(yīng)關(guān)系正弦表達(dá)式和相量表達(dá)式的對應(yīng)關(guān)系()()( )Re()Re()Re()Resin()cos()( )Recos()Re()Resin()cocos()s(

10、)uuijjtj tj tmmmmumujtj tmmmimummiiu tU eU eeU ejUtUti tI eI ejUtItItIt相量的模相量的模正弦量的幅值正弦量的幅值初位相初位相復(fù)角復(fù)角頻率是已知頻率是已知 ( )cos()( )cos() uijmummjmimmu tUtUU ei tItII e三角表達(dá)式相量表達(dá)式（復(fù)數(shù)表示正弦量）？頻率頻率( )cos()( )cos()mumiu tUti tIt相量乘以相量乘以 e ej j t t，再取實部，再取實部9EXAMPLE 7-6 P337-338103.2 Time-harmonic ElectromagneticsF

11、ield vectors that vary with space coordinates and are sinusoidal functions of time can similarly be represented by vector phasors(矢矢量相量量相量) that depend on space coordinates but not on time. As an example, we can write a time-harmonic E field referring to cos t as( , , , )Re ( , , )j tE x y z tE x y

12、z ewhere E(x, y, z) is a vector phasor (矢量相量矢量相量) that contains information on direction(方向方向), magnitude(振幅振幅), and phase(相位相位). Phasors are, in general, complex quantities. we see that, if E(x, y, z, t) is to be represented by the vector phasor E(x, y, z) , then E(x, y, z, t)/ t and E(x, y, z, t)d

13、t would be represented by, respectively, vector phasors j E(x, y, z) and E(x, y, z)/j . Higher-order differentiations and integrations with respect to would be represented, respectively, by multiplications and divisions of the phasor E(x, y, z) by higher powers of j .1112 已知已知正弦正弦電磁場的電磁場的場場與與源源的的頻率相

14、同頻率相同，因此，因此可用可用復(fù)矢量復(fù)矢量形式表示麥克斯韋方程。形式表示麥克斯韋方程。j m( , )Re(j ( )e)tttE rErj Re(j2( )e)tE r考慮到正弦時間函數(shù)的時間導(dǎo)數(shù)為考慮到正弦時間函數(shù)的時間導(dǎo)數(shù)為 jjRe( 2e)Re2j 2ettHJD或或jjjRe( 2e)Re2 eRe j 2 etttHJD因此，麥克斯韋第一方程因此，麥克斯韋第一方程 可表可表示為示為 tEHE 上式對于上式對于任何時刻任何時刻均成立，均成立，實部實部符號可以符號可以消消去去，即即22j 2HJDj HJD130BEtDHJtDB Jj DEBH JEJ瞬時值瞬時值由由相量值相量值代

15、替代替時間求導(dǎo)時間求導(dǎo)由由jj代替代替We now write time-harmonic Maxwells equations (時諧麥克斯韋方程組時諧麥克斯韋方程組) in terms of vector field phasors (E, H) and source phasors ( , J) in a simple (linear, isotropic, and homogenous) medium as follows./ 0EjHHJjEEH 14The time-harmonic wave equations(時諧波動方程時諧波動方程) for E and H become,

16、respectively,2221EJEtt 222HHJt 221EEjJ 22HHJ 222 AAJt 222VVt 22 AAJ 22VV The time-harmonic wave equations for scalar potential V and vector potential A become, respectively,Let ku is called the wavenumber.15ku RRRRThen Consider the time delay factor , for a sinusoidal function it leads to a phase de

17、lay of . uRRuRRWe obtain1( , )d4VtuVtVRRR,RRR,( , )d4VtutVRRJ RA RRRj ()e( )d4kVVR RJ RA rRRj 1 ()e( ) d4 kVVVR RRrRR16The complex Lorentz condition is The complex electric and magnetic fields can be expressed in terms of the complex potentials as VAt ( )( )A rjV rAEVt BA( )( )B rA r AEjAVj Aj173.3

18、source-free(無源無源) fields in simple mediaIn a simple, nonconducting(非導(dǎo)電非導(dǎo)電) source-free medium characterized by =0, J=0, =0, the time-harmonic Maxwells equations become / 0EjHHJjEEH 00EjHHjEEH 18220A k A220Vk Vwhich are homogeneous vector Helmholtzs equations(齊次矢量亥姆霍齊次矢量亥姆霍茲方程茲方程). and wave equations

19、 for A and V become22 AAJ 22VV The time-harmonic wave equations for E and H become, respectively,220Ek E220Hk H221EEjJ 22HHJ Let ku is called the wavenumber.19If the simple medium is conducting (0)(導(dǎo)電介質(zhì)導(dǎo)電介質(zhì)), a current J= E will flow, and the equation should be changed toHJjE()()cHjEjEjEjwith (F/m)c

20、jjjThe other three equations in Maxwells equation are unchanged. Hence, all the previous equations for nonconducting(非導(dǎo)電非導(dǎo)電) media will apply to conducting media if is replaced by the complex permittivity c . Meanwhile, the real(實數(shù)實數(shù)) wavenumber k in the helmholtzs equations should be changed to a c

21、omplex(復(fù)數(shù)復(fù)數(shù)) wavenumber: ku cck 20The ratio / is called a loss tangent(損耗正切損耗正切) because it is a measure of the power loss in the medium:tan.cThe quantity c may be called the loss angle(損耗角損耗角). A medium is said to be a good conductor(良導(dǎo)體良導(dǎo)體) if , and a good insulator(良絕緣良絕緣體體) if . Thus, a material

22、 may be a good conductor at low frequencies(低頻低頻) but may have the properties of a lossy dielectric at very high frequencies(高頻高頻). (F/m)cjjj211. Faradays Law of Electromagnetic Induction (V)ddt () (V)CuBdl() (V)CSCE dlB dSuBdlt (V)SddB dSdtdt Review222. Maxwells Equations3. Electromagnetic Boundary

23、 Conditions ()CSDH dlJdSt CSBdE dldStdt 0SB dS SD dSQThe integral formDHJtBEt 0BDThe differential form SignificanceFaradays law(電磁感應(yīng)定律電磁感應(yīng)定律)Amperes circuital law(全電流定律全電流定律)Gausss law(高斯定理高斯定理)No isolated magnetic charge(磁通連續(xù)磁通連續(xù)性原理性原理)212()0naEE212()0naBB212()nSaDD212()SnaHJH234. Potential Functio

24、ns5. Wave Equations and Their Solutions2221EJEtt 222HHJt BAAEVt VAt 222AAJt222VVt1( , )d4VtuVtVRRR,RRR,( , )d4VtutVRRJ RA RRR246. Time-Harmonic Fields ( , , , )Re ( , , )j tE x y z tE x y z e/ 0EjHHJjEEH 22 AAJ 22VV 相量的模相量的模正弦量的幅值正弦量的幅值初位相初位相復(fù)角復(fù)角頻率是已知頻率是已知？頻率頻率( )cos()( )cos()mumiu tUti tIt相量乘以相量乘以

25、e ej j t t，再取實部，再取實部 ( )cos()( )cos() uijmummjmimmu tUtUU ei tItII e三角表達(dá)式相量表達(dá)式（復(fù)數(shù)表示正弦量）()()cHjEjEjEj221EEjJ 22HHJ 25 (V)SddB dSdtdt 670.667023 10cos(5 10) ()0.2320.6 10cos(5 10)3zzSSB dSatxadxtx dx dx26 (V)SddB dSdtdt 3335 10cos()5 10cos5 10cos0.70.35(1 cos)0.2zzSSSB dSata dstdstt P.7-7P34927P.7-13

26、P3510()BBAA()() or ()0()()AEAEttAAEVEVtttVVVVttVVt 222222()()0VVtAAttVAttt BAAEVt VAt 28梯度運(yùn)算符合以下規(guī)則：梯度運(yùn)算符合以下規(guī)則：20()()(/)()/()()CCCFF C C為常數(shù)為常數(shù)散度運(yùn)算規(guī)則散度運(yùn)算規(guī)則()()()ABABCACAAAA 旋度運(yùn)算規(guī)則旋度運(yùn)算規(guī)則()()()ABABCACAAAA 290BEtDHJtDB / 0EjHHJjEEH ( , , )0.1sin(10)j zyE x y zax e()0.1sin(10)cos(10)0.1 sin(10)j zxyzyj zj

27、 zzxEaaaax exyzax ea jx e(/2)cos(10)0.1 sin(10)0.1cos(10)sin(10)0.1cos(10)sin(10)j zj zzxj zj zzxjzj zzxEEjHHjax ea jx ejjax eax eax eax e P.7-25 P3523022220.1()cos(10)sin(10)100.1100.1sin(10)sin(10)()sin(10) 0j zj zxyzzxj zj zj zyyyyHJjEHjEjHaaaax eax exyzjjjjax eax eax ejEa 22222222100.1.1sin(10)0

28、.110010054.4 (rad/m)j zjjjx ej (/2)(/2)0.1( , ,)cos(10)sin(10)0.1( , , )Recos(10)Resin(10)0.1cos(10)cos(/2)sin(10)cos(jzj zzxjzj tj zj tzxzxH x zax eax eH x z tax eeax eeaxtzaxt) z31tan.c998712 100 1010369.75 105.7 10()()cHjEjEjEj220cHk Hcck P.7-30 P35332The electric field intensity in a source-free dielectric( ) region is given as (V/m),where angular frequency ，all are constants. Find:, ,0 0sincosxEa Extkz0, ,EkExample. E