电磁讲义_长庚电子 Ch9.pdf

1 電磁學 2 Chapter 9 Waveguides and Cavity Resonators 國生 副教授 長庚大學電子系 國生 副教授 長庚大學電子系 2 ztjo yxEtzyxE e Re jt z Assume that the waves propagate in the z direction with j For harmonic time dependence with an angular frequency 2 D Vector phasor General Wave Behaviors Along Uniform Guiding Structures z tjzztjtjz eeeee yxE o 3 The electric and magnetic field intensities satisfy the homogeneous vector Helmholtz s equations k HkH EkE 0 0 22 22 wavenumber 4 Laplacian 0 0 0 222 222 22 22 2 2 2222 22 2 HkH EkE EkE EE E z E yx yx yx x yzx y zyx Cross sectional Longitudinal The exact solution depends on the cross sectional geometry and the boundary conditions 5 Interrelationships among the 6 components of E H FromH j E FromEj H 9c 9 9b 9 9a 9 o z Hj y o x E x o y E o y Hj x o z E o x E o x Hj o y E y o z E 10c 9 10b 9 10a 9 o z Ej y o x H x o y H o y Ej x o z H o x H o x Ej o y H y o z H 6 mode TE 0 wavesTE mode TMor 0 wavesTM mode TEM 0 and wavesTEM z z zz E H HE 先找 可得其餘分 藉由 表示 o y o x o y o x HHEE及 o z o z HE y H j y E h E y H j x E h E y E j y H h H y E j x H h H o z o zo y o z o zo x o z o zo y o z o zo x 2 2 2 2 1 1 1 1 o z o z HE 222 kh 波導中之 傳播常 自由空間 中之波 波導中之 傳播常 自由空間 中之波 可能解 Eq 9 11 7 TEM waves can t exist in a single conductor hollow waveguide of any shape 0 and zz HE m A Ea Z H j j H E Z k u j jk k h H E z TEM TEM TEM o y x TEM p TEM TEM zz TEM 1 1 00 0119 0 222 代由 uniform plane wave in an unbounded medium TEM波之相速與頻 無關 TEM waves 8 如果導波管內有TEM波 則B H的field line會在 transverse面形成closed loop 根據安培 loop上之磁場線積分 軸向上之傳導電 位移電 無磁場線之closed loop 存在 TEM波無法存在於single conductor hollow waveguide 內無導體 0無Ez 分 0 Why can t TEM waves exist in a single conductor hollow waveguide 9 TM waves Hz 0 2 2 2 2 14 9 11 9eq and 0 22 From y o z E h o y E x o z E h o x E x o z E h j o y H y o z E h j o x H o z Eh o z E x y 10 2 1 22 2 22 0 If 2222 m A 1 wavesTMfor c f f hh h c f h c hkh E z a TM Z H j j o y H o x E TM Z Cutoff frequency 截止頻 11 uu f f u u f u fk f f f f k f f jkj ff f f g c p c g c c c c 2 2 2 2 2 1 12 where 1 2 1 1 or 1 if a 波長 在導波管中 在自由空間中 虛 可傳播 為相位常 frequency dependent dispersive transmission 12 always 1 2 f f Z c TM 2 2 1 or 1 if b c c c f f h ff f f 實 波會衰減 e z 操作頻 低於截止頻 時 波無法傳播 13 A waveguide exhibits the property of a high pass filter For a given mode only waves with a frequency higher than the cutoff frequency of the mode can propagate in the guide 14 TE waves Ez 0 m V H z a TE ZE j j o x H o y E o y H o x E TE Z x o z H h j o y E y o z H h j o x E y o z H h o y H x o z H h o x H o z Eh o z E x y 2 2 2 2 14 9 11 9 eq and0 22 rom F 15 2 2 2 2 2 2 1 1 or 1 If b always 1 wavesTM 1 or 1 If a c TE c c c c TE pg c c c f f h jZ f f h f f f f f f Z u f f jkj f f f f 似 Reactive No power flow for evanescent waves 16 Example 9 1 graph thefrom determined becan guide in the wavegpropagatin a of velocities group and phase thehow discuss and modes TE and TMfor graph Plot the waveguideain modes gpropagatin of and betweenrelation thegives 1 29 9Equation 2 f ffk c 17 velocity group theis at curve theof slope local The frequency cutoff a having mode particular afor velocity phase theto equal is curve on the point andorigin thejoining line the of slope The diagram dispersion a called isgraph The 1 1 and withngSubstituti Sol 2 g c p c u Pf u P u uuk 18 Example 9 2 Consider a parallel plate waveguide of two perfectly conducting plates separated by a distance b and filled with a dielectric medium having and The plates are assumed to be infinite in extent in the x direction a Obtain the time harmonic field expressions for TM modes in the guide b Determine the cutoff frequency direction in the propagate wavesLet the a z hyBhyAyE yEh dy yEd EhE eyEy zEH o z o z o z o z o zxy zo zzz cossin 0 0 From phasor 0 2 2 2 22 可能解 Sol 19 0 1 2 sin 0sin at 0 condition boundary sin 0 0at 0 condition boundary cossin n b yn AyE b n hnhbhbA byyE hyAyEB yyE hyBhyAyE n o z o z o z o z o z 代 代 2 2 22 where cos cos 0 and 0 with 14 9 and 119 From b n h b yn A h yE b yn A h j yH x E H n o y n o x o z o z 20 mode dominant thecalled isfrequency cutofflowest thehaving mode The 0TEMTM 0mode TM own its has modeeach 1 2mode TM 2 1 1 mode TM Hz 2 0 makesthat frequency theisfrequency cutoff The b 0 0 22 11 c z TMgpg c c c f E Z u u b fn b fn b n f 21 Two conductors are necessary for the transfer of EM energy L Rayleigh 1897 TE and TM modes propagation in hollow waveguides with rectangular or circular cross sections Experiments in 1936 1 G C Southworth at AT 0 sinsin 0 sinsin 0 0 and 1 formphasor a a hk ztx a HtzyxH tzyxH ztx a H a tzyxH tzyxE ztx a H a tzyxE tzyxE ahnm e z y x z y x ztj 並取實部乘上將推得之場分 之 Sol 37 ly with sinusoidal vary and 0 cos and 1 sin or 2 at plane typicalaIn sin as vary and 1sin when say plane typicalaIn b zHE axax axyz x aHE ztxy xy xy 38 0 sinsin coscos 0 010 0 cos 0 cos 00 0 001 0 0at c 0 2 0 0 0 yJbyJ z x a H a h a zx a Ha HaHa H H a a a HayJ zHaa y z HaaxJ zHa y z Ha H H a a a HaxJ t HaJ ss zx xzzx zx zyx ys yzys yzy zx zyx xs ns 39 40 Example 9 6 GHz 45 12GHz 19 8 95 0 251 101 13 1055 6 22 1 2 1 TE and TE are lowest thehaving modes twoThe Sol 95 0 frequency operating251 befrequency operating that theand modeTE dominant in theonly operates waveguide that thedesired isit If cm 021 cm 292 are nsapplicatio band for suitable waveguidea of dimensionsinner The bands radar for the designedbeen have esr waveguidrectangula filled air Standard 2010 20 10 2010 TETE 9 TE 9 22 TE 2010 TETE 10 ffff a c f a c ab n a m f f f fff b aX cc c c c cc 41 42 zP zP hj L c c d d d d d 2 loss conductor Ohmic Re into ngSubstituti loss Dielectric 22 Attenuation in Rectangular Waveguides cd 43 Attenuation due to wall losses in rectangular copper waveguide for TE10 and TM11 modes 44 Example 9 7 guide within theintensity electric theof valuemaximum the c waveguide theof wallsin the dissipatedpower ofamount total the b waveguide thepower toinput average required the a find dB m 0 05 ofconstant n attenuatioan Assuming GHz 4 5at load matched a kW to 1 2deliver tois long m 0 8 es r waveguidrectangula cm 2 5cm 5 filled airAn mode TE is GHz 5 4at mode gpropagatinonly theThus Hz 105 4106 and Hz 103 1052 103 2 mode Dominant Sol 10 99 TETE 9 2 8 10 0120 a c ff a c f cc c 45 V m 44283 5 4 3 1 3774 105 2 105 1211 1 42 1 sin1 and sin c W1112001211dissipatedPower b W1211102 1102 1 Np m 105 75 dB m 105 a 0 2 222 0 2 0 2 0 00 00 in 2 0 0 0 0 0 loadin 0092 038 01075 5232 loadin 32 3 E E f fabE dxdyHEP x af fE Hx a EE PP eeePP c c ba xy c xy l 46 Cavity Resonators At UHF and higher frequencies we look to an enclosure a cavity used as the resonant circuit 優點 very high Q 47 Rectangular Cavity Resonators Choosing the z axis as the direction of propagation in actuality the existence of conducting end walls at z 0 z d gives rise to multiple reflections and sets up standing waves A three symbol mnp subscript is needed to designate a TM or TE standing wave pattern in a cavity resonator 48 其transverse場與TMmn 似 唯在longitudinal z方向 軸上會 有ej z反射波存在 故可由TMmn之場分 式 保 其x y項次 並增加與z相關項次 z d p zH H E zEE z d p E E d z EE yxz yxmn yxyx cos TM sin 0 0 因此應為作微分所以未對係 並無但 作微分曾對與之場分 式可看出又由 項須有處為必須在又 TMmnp Modes Hz 0 49 222222 22 2 0 2 0 2 0 2 0 2 0 2 or 1 cos sincos cos cossin sin cossin 1 sin sincos 1 cos sinsin d p b n a mu f d p b n a m b n a m h z d p y b n x a m E a m h j zyxH z d p y b n x a m E b n h j zyxH z d p y b n x a m E d p b n h zyxE z d p y b n x a m E d p a m h zyxE z d p y b n x a m EzyxE mnpmnp y x y x z 50 TEmnp Modes Ez 0 似 似TMmnp之推導之推導 z d p y b n x a m H d p b n h zyxH z d p y b n x a m H d p a m h zyxH z d p y b n x a m H a m h j zyxE z d p y b n x a m H b n h j zyxE z d p y b n x a m HzyxH y x y x z cossincos 1 coscossin 1 sincossin sinsincos sincoscos 0 2 0 2 0 2 0 2 0 Different modes having the same resonant frequency Degenerate modes If m n p 為0 TMmnp 與TEmnp 有相同共 振頻 degenerate modes 51 共振腔之excitation 選擇電場最大位置 依所設計之mode而定 藉由small probe 將電磁能 耦合入共振腔中 使用loop antenna 則選擇最強磁通耦合位置 z d x a HH z d x a H d a H z d x a H aj E z x y sincos cossin sinsin mode TE 0 0 0 101 可於Ey最大位置 上 下面 之中心位置放置probe 52 Example 9 8 Determine the dominant modes and their frequencies for a a b d b a d b c a b d Sol for TMmnp modes m n 可為0 但p可為0 否則Ez 0 dominant mode the lowest resonant frequency is TM110 for TEmnp modes m n 可同時為0 且p 可為0 否則Hz對x y微分 0 其他場分 為0 dominant mode are TE011 TE101 222 2 d p b n a mu f 53 a c f anddba da c f bda ba c f dba 2 is modes degenerate theseoffrequency resonant The TE TE TM For c modedominant theis TE 11 2 isfrequency resonant lowest the For b modedominant theis TM 11 2 isfrequency resonant lowest the For a 110 101011110 101 22 101 110 22 110 之共振頻 相同 54 2 freq thisod period onein dissipatedEnergy freq resonant aat storedenergy averageTime 2 Lf P me P WW Q WWW Q L Quality Factor of Cavity Resonator The quality factor or Q of a resonator like that of any