電磁場與電磁波第八課

1、8 8 Guided Wave Guided Wave圖 各種載波體低、中頻區(qū)(雙導(dǎo)體）中高頻區(qū)(微帶線）高頻區(qū)(金屬波導(dǎo)）Waveguide Forms下 頁上 頁返 回(導(dǎo)波類型 ) Guided Wave 導(dǎo)波的種類導(dǎo)波的種類TM波波(E波波)TEM波波TE波波(M波波) 導(dǎo)波的種類導(dǎo)波的種類的導(dǎo)波的導(dǎo)波0zE0zH 的導(dǎo)波的導(dǎo)波0zE0zH 的導(dǎo)波的導(dǎo)波0zE0zH Guided Wave導(dǎo)波場的求解方法：縱向場法導(dǎo)波場的求解方法：縱向場法q由麥克斯韋方程組導(dǎo)出由麥克斯韋方程組導(dǎo)出橫、縱向場關(guān)系式橫、縱向場關(guān)系式；q由麥克斯韋方程組導(dǎo)出由麥克斯韋方程組導(dǎo)出電場或磁場電場或磁場縱向分量縱

2、向分量滿足滿足各坐標(biāo)系中的亥姆霍茲方程各坐標(biāo)系中的亥姆霍茲方程;q由各種情況下的由各種情況下的邊界條件邊界條件（波導(dǎo)內(nèi)壁：波導(dǎo)內(nèi)壁：Et=0=0）求求解各種情況下的亥姆霍茲方程的解各種情況下的亥姆霍茲方程的電場或磁場電場或磁場縱向縱向分量分量特解；特解；q由橫縱向場關(guān)系式由橫縱向場關(guān)系式求各橫向場分量。求各橫向場分量。在規(guī)則導(dǎo)行系統(tǒng)中：在規(guī)則導(dǎo)行系統(tǒng)中： Guided Wave8.1 Rectangular Waveguides這種空芯金屬導(dǎo)波裝置通常稱為這種空芯金屬導(dǎo)波裝置通常稱為波導(dǎo)波導(dǎo)，電磁能量在波導(dǎo)管內(nèi)部，電磁能量在波導(dǎo)管內(nèi)部空間被引導(dǎo)前進(jìn)?？臻g被引導(dǎo)前進(jìn)。在微波波段，為了減小傳輸損耗

3、并防止電磁波向外泄漏，采用在微波波段，為了減小傳輸損耗并防止電磁波向外泄漏，采用空芯的金屬管作為傳輸電磁能量的導(dǎo)波裝置。空芯的金屬管作為傳輸電磁能量的導(dǎo)波裝置。 Guided Wave規(guī)則金屬波導(dǎo)管壁材料：銅、鋁，有時其壁上鍍金或銀。規(guī)則金屬波導(dǎo)管壁材料：銅、鋁，有時其壁上鍍金或銀。使用范圍：使用范圍：3000MHz（3GHz）300GHz導(dǎo)波模式：（非導(dǎo)波模式：（非TEM波）波）TE波，波，TM波，混合波。波，混合波。形狀：橫截面有矩形、圓形、脊形、橢圓形、三角形等。形狀：橫截面有矩形、圓形、脊形、橢圓形、三角形等。金屬波導(dǎo)優(yōu)點(diǎn)：導(dǎo)體損耗和介質(zhì)損耗小、功率容量大、金屬波導(dǎo)優(yōu)點(diǎn)：導(dǎo)體損耗和介質(zhì)

4、損耗小、功率容量大、沒有輻射損耗、結(jié)構(gòu)簡單、易于制造。沒有輻射損耗、結(jié)構(gòu)簡單、易于制造。 Guided Wave Guided Wave波導(dǎo)方程及其解的條件：波導(dǎo)方程及其解的條件：(1)波導(dǎo)內(nèi)壁是理想導(dǎo)體，電導(dǎo)率為無限大；波導(dǎo)內(nèi)壁是理想導(dǎo)體，電導(dǎo)率為無限大；(2)波導(dǎo)內(nèi)的介質(zhì)是均勻無耗、線性及各向同性的理波導(dǎo)內(nèi)的介質(zhì)是均勻無耗、線性及各向同性的理想媒質(zhì)，電導(dǎo)率為想媒質(zhì)，電導(dǎo)率為0；(3)波導(dǎo)內(nèi)無激勵源，即無自由電荷和傳導(dǎo)電流波導(dǎo)內(nèi)無激勵源，即無自由電荷和傳導(dǎo)電流(=0, J=0) ；(4)波導(dǎo)的橫截面是均勻的波導(dǎo)的橫截面是均勻的;(5)電磁波沿電磁波沿 z 軸傳播，且隨時間作正弦變化。軸傳播，

5、且隨時間作正弦變化。 Guided WaveRectangular waveguide：rectangular cross sectionAssume that the four sides of the waveguide are bounded by perfectly conducting walls and the medium enclosed by the walls is perfect dielectric. Consider that the waveguide has its long axis parallel to the z direction and the rec

6、tangular cross-section is invariant along its direction. ， Guided WaveFor time-harmonic electromagnetic wave, the Helmholtz equations are220kEE220kHH22.k (1)(2)whereAssume that the wave propagation is along the longitudinal direction (+z direction). we can write the E and H as( , , )( , )ezx y zx yE

7、E( , , )( , )ezx y zx yHHjwhere the propagation constant( , , )( , )( , )( , )ezxxyyzzx y zE x yEx yE x yEeee( , , )( , )( , )( , )ezxxyyzzx y zHx yHx yHx yHeee Guided Wave222( , )()( , )0 xyx ykx yEE222( , )()( , )0 xyx ykx yHHThus, the Helmholtz equations can be written as(6)(7)The first term of (

8、1) can be expressed as22222xyxywhere(5)2222222222222222222()()()( , )e( , )e( , )( , )ezzzxyxyzxyzx yx yxyx yx y EEEEEEEE Guided WavejHEj EHWithin the waveguide, E and H are determined by Maxwells equations(8)(9)222222()0zzzHHkHxy222222()0zzzEEkExyWe can obtain the solutions of and by solving scalar

9、 equations( , )zE x y( , )zHx y縱向場分量滿足亥姆霍茲方程：縱向場分量滿足亥姆霍茲方程： Guided WavejzyxEEHy jzxyEEHxjyxzEEHxy jzyxHHEyjzxyHHEx jyxzHHExyWe can express these equations in scalar forms as(8.1.10a)(8.1.10b)(8.1.10c)(8.1.10d) (8.1.10e)(8.1.10f)橫橫,縱向場關(guān)系式縱向場關(guān)系式: Guided WaveWe can express the x and y components of the

10、 electric and magnetic fields in term of their z components as(11)橫縱向場關(guān)系式橫縱向場關(guān)系式xyxy Guided Wave( ) ( )zEX x Y y2222221( )1( )()( )( )X xY xkX xxY yy zETM Mode 模式The magnetic field has no component in the longitudinal direction. component satisfies(12)(12)Apply the method of separation of variables.

11、 Let(13)Substituting (13) into (12) and divided by (13), we get(14) Guided WaveLetLet2221( )( )xX xkX xx 2221( )( )yY ykY yy xkyk2222.xykkk11( )cossinxxX xCk xDk x22( )cossinyyY xCk yDk y(15)(16)whereandare two arbitrary constants and satisfy The solutions to (15) and (16) areand(17)(18) Guided Wave

12、Thus, we have1122( , )(cossin)(cossin)zxxyyE x yCk xDk x Ck yDk y121,C CD2D1122( , , )(cossin)(cossin)ezzxxyyE x y zCk x Dk x Ck yDk ywhere0 x (0, , )0zEy z xa( ,0, )0zE xz 0y ( , , )0zE a y z yb( , , )0zE x b z and are arbitrary constants.Thus,The four boundary conditions are AtAtAtAt(19)(20)， Guid

13、ed Wave120CC0,1,2,xmkmaL0,1,2,ynknbL0( , , )sinsinezzmnE x y zExyabApplying these boundary conditions, we obtainThus, the z component of the electric field is(22)(23)(24)(25)where E0=D1D2 is determined by exciting source.縱向場分量縱向場分量 Guided Wave(11)Thus, we can determine the other field components for

14、 the TM mode applying (11) asxyxy Guided Wave0zH (26)0sinsinezzmnEExyab橫向場分量橫向場分量TM Mode 場分量場分量沿沿x x，y y方向均為駐波，電磁波沿方向均為駐波，電磁波沿 z z 軸方向傳播。軸方向傳播。From (26), TM00 ,TMm0,TM0n can not exist in a waveguide. Guided WaveTM .mnFrom these equation, we note that a rectangular waveguide can support an infinite n

15、umber of TM modes for each m and n,and they are denoted by However, if either m or n is zero, no field will exist in the waveguide. Thus, the lowest possible TM mode is the TM11mode.由場解可知，矩形波導(dǎo)中可能存在的電磁場有無限多由場解可知，矩形波導(dǎo)中可能存在的電磁場有無限多個解，即個解，即TEmn和和TMmn模式，或?qū)⒋朔Q為模式，或?qū)⒋朔Q為“波型波型”。 Guided Wave)zHx,y（TE Mode 模式模式

16、The electric field is entirely in the transverse direction.(0),zE The magnetic field has a component in its direction of propagation. component satisfiesSimilarly, applying the method of separation of variables and boundary conditions, Guided Wave0zHyyb0zHy0y 0zHxxa0zHx0 x AtAtAtAtwe obtain(28a)(28b

17、) (28c)(28d)where H0 is determined by exciting source.，22zxHjEky 22zyHjEkx Guided Wave(11)Thus, we can determine the other field components for the TM mode applying (11) asxyxyxy Guided Wave0zE From (29), TE00 can not exist in a waveguide. (29)TE Mode 場分量場分量沿沿x x，y y方向均為駐波，電磁波沿方向均為駐波，電磁波沿 z z 軸方向傳播。

18、軸方向傳播。 Guided Wave,mnmnIfis real,The wave experiences attenuation and disappears after travelling a very short distance.If is imaginaryj.mnmnThe wave will travel along the waveguide with no attenuation.2222xykkk，the propagation constant is0mnIfis zero,臨界情況。傳播模式非傳播模式矩形波導(dǎo)的傳輸特性矩形波導(dǎo)的傳輸特性Since Guided Wav

19、eThe cutoff frequency22cmnkab221cmnabThe cutoff wave numberThe cutoff angular frequencyThe cutoff wavelength within the waveguide is2222cckmnab Guided WavecmncmnffIf the operating frequency , we havewill be imaginary; that isj.mnmn 當(dāng)工作頻率（信號源發(fā)出頻率）當(dāng)工作頻率（信號源發(fā)出頻率） 或或 時，信號可以時，信號可以通過波導(dǎo)，否則截止。通過波導(dǎo)，否則截止。Cff

20、C m m，n n 的任何整數(shù)的組合構(gòu)成的任何整數(shù)的組合構(gòu)成TMTMmnmn模，最低模式模，最低模式TMTM1111；0截止頻率截止頻率 fc 最小的模式稱為最低模式最小的模式稱為最低模式: : m m，n n不同時為不同時為0 0的任何整數(shù)的組合成的任何整數(shù)的組合成TETEmnmn模，最低模式為模，最低模式為TETE1010；波導(dǎo)中波導(dǎo)中截止波長最長截止波長最長的模式稱為的模式稱為主模主模，或稱基?；蚍Q基模、最低型模；、最低型模；其它的模稱為高次模。矩形波導(dǎo)中主模為其它的模稱為高次模。矩形波導(dǎo)中主模為TE10模模。 Guided Wave0zH (26)0sinsinezzmnEExyab橫

21、向場分量橫向場分量TM Mode 模式場分量場分量沿沿x x，y y方向均為駐波，電磁波沿方向均為駐波，電磁波沿 z z 軸方向傳播。軸方向傳播。 Guided Wave0zE From (29), TE00 can not exist in a waveguide. (29)TE Mode 模式場分量場分量沿沿x x，y y方向均為駐波，電磁波沿方向均為駐波，電磁波沿 z z 軸方向傳播。軸方向傳播。 Guided WaveThe cutoff frequency or cutoff wave length is different for different mode. The distr

22、ibution of the cutoff wavelength for the waveguide.(2 ).abcutoff region截止波長只與模式和波導(dǎo)尺寸有關(guān)截止波長只與模式和波導(dǎo)尺寸有關(guān) Guided Waveccff l l導(dǎo)模的傳輸條件：導(dǎo)模的傳輸條件：ccff l l導(dǎo)模的截止：導(dǎo)模的截止：高通濾波器高通濾波器 Guided Wavel“簡并”模式：不同的模式不同的模式具有具有相同的截止頻率（波長相同的截止頻率（波長）等特性參量的現(xiàn)象）等特性參量的現(xiàn)象稱為稱為“簡并簡并”。具有相同波型指數(shù)具有相同波型指數(shù)m和和n的的TEmn和和TMmn模式為簡并模（雙重簡模式為簡并模（雙

23、重簡并），但由于并），但由于TM模無模無TM0n和和TMm0模，故模，故TEm0和和TE0n模無簡模無簡并模。并模。 Guided WavePropagation constant 傳播常數(shù) The phase constant is The waveguide wavelength is (35)(36)(8.1.37)22221cmnmnfjjkabf 22222gmnab j22221cmnmnfkabf When the operating frequency ,cmnffwe have波導(dǎo)波長波導(dǎo)波長 Guided Wave22=1 ()1 ()pmncmncmnffvvvv1v(3

24、8)whereis the phase velocity in the unbounded medium. The phase velocity of the wave traveling along the waveguide is221 ()1 ()gcmncmnff2 whereis the wavelength in the unbounded medium. Guided Wave波阻抗：導(dǎo)模的橫向電場和橫向磁場之比稱為該導(dǎo)模的波阻抗矩形波導(dǎo)矩形波導(dǎo)TE導(dǎo)模的波阻抗導(dǎo)模的波阻抗:221 ()1 ()xTEyccEZHff矩形波導(dǎo)矩形波導(dǎo)TM導(dǎo)模的波阻抗導(dǎo)模的波阻抗2211xcTMyc

25、EfZHf其中ZTE和ZTM為實(shí)數(shù),為電阻 Guided Wave0 xzyEEHTE10 modeThe electromagnetic fields are(39a)主模主模2ca2210a 12cfa Guided Wave( , , , )( , , , )( , , , )0 xzyE x y z tE x y z tHx y z t010( , , , )sin()sin()yaxEx y z tHtza10010( , , , )sin()sin()xaxHx y z tHtza 010( , , , )cos()cos()zxHx y z tHtzaThe electromag

26、netic fields in time domain are Guided WaveElectromagnetic Fields Distribution電磁場的分布 Guided Wave10j0000()ezSxxxzzyzyxxxHHHH Jeeeee10j0()ezSxxxzzyzyx ax ax aHHHH Jeeeee000()SyxxzzzxxzyyyHHHH JeeeeeSJ,SnJeHneThe surface currenton each wall can be determinedby the boundary conditionwhereis theunit normal to the surface pointing inside the waveguide. Guided WaveFigure shows the distribution of the surface current density on the wall. Guided Wave電磁場分布特性電磁場分布特性在矩形波導(dǎo)中大多采用在矩形波導(dǎo)中大多采用TETE1010模式，因?yàn)檫@種模式具有如下優(yōu)點(diǎn)：模式，因?yàn)檫@種模式具