电磁场与电磁波第八课

1、8 8 Guided Wave Guided Wave图 各种载波体低、中频区(双导体）中高频区(微带线）高频区(金属波导）Waveguide Forms下 页上 页返 回(导波类型 ) Guided Wave 导波的种类导波的种类TM波波(E波波)TEM波波TE波波(M波波) 导波的种类导波的种类的导波的导波0zE0zH 的导波的导波0zE0zH 的导波的导波0zE0zH Guided Wave导波场的求解方法：纵向场法导波场的求解方法：纵向场法q由麦克斯韦方程组导出由麦克斯韦方程组导出横、纵向场关系式横、纵向场关系式；q由麦克斯韦方程组导出由麦克斯韦方程组导出电场或磁场电场或磁场纵向分量纵

2、向分量满足满足各坐标系中的亥姆霍兹方程各坐标系中的亥姆霍兹方程;q由各种情况下的由各种情况下的边界条件边界条件（波导内壁：波导内壁：Et=0=0）求求解各种情况下的亥姆霍兹方程的解各种情况下的亥姆霍兹方程的电场或磁场电场或磁场纵向纵向分量分量特解；特解；q由横纵向场关系式由横纵向场关系式求各横向场分量。求各横向场分量。在规则导行系统中：在规则导行系统中： Guided Wave8.1 Rectangular Waveguides这种空芯金属导波装置通常称为这种空芯金属导波装置通常称为波导波导，电磁能量在波导管内部，电磁能量在波导管内部空间被引导前进。空间被引导前进。在微波波段，为了减小传输损耗

3、并防止电磁波向外泄漏，采用在微波波段，为了减小传输损耗并防止电磁波向外泄漏，采用空芯的金属管作为传输电磁能量的导波装置。空芯的金属管作为传输电磁能量的导波装置。 Guided Wave规则金属波导管壁材料：铜、铝，有时其壁上镀金或银。规则金属波导管壁材料：铜、铝，有时其壁上镀金或银。使用范围：使用范围：3000MHz（3GHz）300GHz导波模式：（非导波模式：（非TEM波）波）TE波，波，TM波，混合波。波，混合波。形状：横截面有矩形、圆形、脊形、椭圆形、三角形等。形状：横截面有矩形、圆形、脊形、椭圆形、三角形等。金属波导优点：导体损耗和介质损耗小、功率容量大、金属波导优点：导体损耗和介质

4、损耗小、功率容量大、没有辐射损耗、结构简单、易于制造。没有辐射损耗、结构简单、易于制造。 Guided Wave Guided Wave波导方程及其解的条件：波导方程及其解的条件：(1)波导内壁是理想导体，电导率为无限大；波导内壁是理想导体，电导率为无限大；(2)波导内的介质是均匀无耗、线性及各向同性的理波导内的介质是均匀无耗、线性及各向同性的理想媒质，电导率为想媒质，电导率为0；(3)波导内无激励源，即无自由电荷和传导电流波导内无激励源，即无自由电荷和传导电流(=0, J=0) ；(4)波导的横截面是均匀的波导的横截面是均匀的;(5)电磁波沿电磁波沿 z 轴传播，且随时间作正弦变化。轴传播，

5、且随时间作正弦变化。 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纵向场分量满足亥姆霍兹方程：纵向场分量满足亥姆霍兹方程： 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)横横,纵向场关系式纵向场关系式: Guided WaveWe can express the x and y components of the

10、 electric and magnetic fields in term of their z components as(11)横纵向场关系式横纵向场关系式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.纵向场分量纵向场分量 Guided Wave(11)Thus, we can determine the other field components for

14、 the TM mode applying (11) asxyxy Guided Wave0zH (26)0sinsinezzmnEExyab横向场分量横向场分量TM Mode 场分量场分量沿沿x x，y y方向均为驻波，电磁波沿方向均为驻波，电磁波沿 z z 轴方向传播。轴方向传播。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.由场解可知，矩形波导中可能存在的电磁场有无限多由场解可知，矩形波导中可能存在的电磁场有无限多个解，即个解，即TEmn和和TMmn模式，或将此称为模式，或将此称为“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 场分量场分量沿沿x x，y y方向均为驻波，电磁波沿方向均为驻波，电磁波沿 z z 轴方向传播。

18、轴方向传播。 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,临界情况。传播模式非传播模式矩形波导的传输特性矩形波导的传输特性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 当工作频率（信号源发出频率）当工作频率（信号源发出频率） 或或 时，信号可以时，信号可以通过波导，否则截止。通过波导，否则截止。Cff

20、C m m，n n 的任何整数的组合构成的任何整数的组合构成TMTMmnmn模，最低模式模，最低模式TMTM1111；0截止频率截止频率 fc 最小的模式称为最低模式最小的模式称为最低模式: : m m，n n不同时为不同时为0 0的任何整数的组合成的任何整数的组合成TETEmnmn模，最低模式为模，最低模式为TETE1010；波导中波导中截止波长最长截止波长最长的模式称为的模式称为主模主模，或称基模或称基模、最低型模；、最低型模；其它的模称为高次模。矩形波导中主模为其它的模称为高次模。矩形波导中主模为TE10模模。 Guided Wave0zH (26)0sinsinezzmnEExyab横

21、向场分量横向场分量TM Mode 模式场分量场分量沿沿x x，y y方向均为驻波，电磁波沿方向均为驻波，电磁波沿 z z 轴方向传播。轴方向传播。 Guided Wave0zE From (29), TE00 can not exist in a waveguide. (29)TE Mode 模式场分量场分量沿沿x x，y y方向均为驻波，电磁波沿方向均为驻波，电磁波沿 z z 轴方向传播。轴方向传播。 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截止波长只与模式和波导尺寸有关截止波长只与模式和波导尺寸有关 Guided Waveccff l l导模的传输条件：导模的传输条件：ccff l l导模的截止：导模的截止：高通滤波器高通滤波器 Guided Wavel“简并”模式：不同的模式不同的模式具有具有相同的截止频率（波长相同的截止频率（波长）等特性参量的现象）等特性参量的现象称为称为“简并简并”。具有相同波型指数具有相同波型指数m和和n的的TEmn和和TMmn模式为简并模（双重简模式为简并模（双

23、重简并），但由于并），但由于TM模无模无TM0n和和TMm0模，故模，故TEm0和和TE0n模无简模无简并模。并模。 Guided WavePropagation constant 传播常数 The phase constant is The waveguide wavelength is (35)(36)(8.1.37)22221cmnmnfjjkabf 22222gmnab j22221cmnmnfkabf When the operating frequency ,cmnffwe have波导波长波导波长 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：导模的横向电场和横向磁场之比称为该导模的波阻抗矩形波导矩形波导TE导模的波阻抗导模的波阻抗:221 ()1 ()xTEyccEZHff矩形波导矩形波导TM导模的波阻抗导模的波阻抗2211xcTMyc

25、EfZHf其中ZTE和ZTM为实数,为电阻 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电磁场的分布 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电磁场分布特性电磁场分布特性在矩形波导中大多采用在矩形波导中大多采用TETE1010模式，因为这种模式具有如下优点：模式，因为这种模式具