TRANSMISSION LINES AND WAVEGUIDES. Uniformity along the Direction of Propagation

Transcription

1 TRANSMISSION LINES AND WAVEGUIDES Uniformi along he Direion of Propagaion Definiion: Transmission Line TL is he erm o desribe ransmission ssems wih wo or more mealli onduors eleriall insulaed from eah oher. Definiion: Waveguide The propagaion in a waveguide is generall ensured b suessive refleions on he guide boundaries. These are onduing walls in he ase of mealli WG s. Dieleri WG s and opial fibers uilie he oal inernal refleion. Clasifiaion of he Modes of Propagaion The presene and apsene of longiudial field omponens affe he propagaion behaviour of he modes. Four mode aegories an eis: E Name Mode 0 0 Transverse TEM Eleromagnei 0 0 Transverse TE Eleri TM Magnei 0 0 Hbrid H 0 0 Transverse

2 Classifiaion of Lines and Waveguides A large number of differen sruures an be used o ransmi eleromagnei signals. The an be: ) Eiher open (radiaion an ake plae) or losed (fields enlosed wihin a onduing envelope); ) Eiher homogeneous (one single propagaion medium wihou he ransverse dependene of he maerial properies) or inhomogeneous (several differen propagaion media); 3) Eiher onduorless or proessing one or more onduors. PROPAGATION WITHIN A CLOSED HOMOGENEOUS METALLIC WAVEGUIDE

3 We make he following assumpions: ) The sruure is uniform along he direion of propagaion. ) The ross-seion is arbirar. I an be simpl onneed (hollow WG) or mulipl onneed (several onduors). 3) The maerial whih ompleel fills he WG is isoropi, linear and homogeneous. Is maerial parameers, and do no depend on he posiion wihin he WG, nor upon he ampliude of he signal. 4) The WG does no onain an elerial spae harge 0. Separaion of Mawell s Equaions ino Longiudinal and Transverse Componens The del operaor an be epressed as: aˆ Where is he ransverse del operaor and is given in he Caresian oordinaes b aˆ aˆ

4 Assuming now, ime-harmoni fields wih an j e ime dependene and wave propagaion along he + -ais, he field veors an be wrien as:,,, ˆ, E e ae e,,, ˆ, H h a h e The equaion Will ake he form: X E jh j j ˆ j, ˆ a X e ae, e j j h e j h e aˆ Or j,, j j j ( ˆ ) ˆ X e e X ae e a X ( e e ) j aˆ ˆ X ae e j j jhe jh e aˆ Or,

5 ˆ X e e a X jee e e j j jhe jh e aˆ j j j From he seond url equaion XH E j E we equivalenl ge: j j j ˆ X h e a X j h e h e j j e e j e e aˆ j Equaing now he ransverse and longiudinal pars gives:,, ˆ,, ˆ, ˆ,, ( ) ˆ, X e j h a X h j e a Longiudinal Pars e j e j a X h h j h j a X e Transverse Pars Epanding he operaors, we ge he following salar equaions: e h je jh jh je e h je jh jh je e e h h jh je Above we assumed ha 0 (lossless dialei in a WG).

6 GENERAL SOLUTIONS FOR THE TEM, TE AND TM WAVES The above si equaions an be solved for he four ransverse field omponens in erms of h h e e j e h k j e h k j e h k j e h k e and h as: Where and k. k k TEM WAVES For TEM waves, he fields are, b definiion, purel ransverse (he field veors E and H lie in he plane whih is perpendiular o he direion of propagaion). As a resul, E and H are boh ero. Then. E 0gives:

7 ˆ j., ˆ a e ae, e 0 Or,. e(, ) je 0. e(, ) Similarl, from. h(, ) je. H 0 Sine e 0, h 0 jh, we obain for a TEM wave, we have h. e, 0., 0 Also, from longiudinal pars we wrie X h X e, 0, 0 The equaions e X e, 0., 0 Define a wo-dimensional elerosai problem while h X h, 0., 0 desribe a wo-dimensional magneosai problem.

8 Consider now, e j e jh h jh j e Whih redues o je jh jh je j je j e je j e j Or, From whih we onlude ha In his ase k 0. Sine k j E k E 0 inside he WG and E e e We will have Or j j e 0 e k ee e 0 k e Sine k we ge,

9 e 0 We an wrie e 0, h 0 Boh problems admi as soluions hose of he wo-dimensional Laplae s equaion. For a simpl onneed WG, Laplae s equaion admis he rivial soluion e, 0. The boundar ondiion imposes a onsan poenial on he meal ube. Soluions of Laplae s equaion require ha he poenial be onsan everwhere inside he ube. Thus, an emp hollow WG anno propagae a TEM wave. On a wo-onduor sruure, a TEM wave an propagae, sine differen poenials ma eis on he wo onduors. The volage beween wo onduors an be found as: V e. dl The urren on a onduor an be found from Ampere s Law as: I H. dl Where C, is he ross-seional onour. The wave impedane of a TEM mode is defined as: Z e TEM h

10 Z e TEM h The propagaion of a TEM mode on a homogeneous mulionduor line onl depends on he propagaion medium; i is independen of he geomer and he line dimensions. The TEM mode an propagae a an frequen on a mulionduor. For a TEM mode, we have, TE WAVES Z, aˆ X e, h TEM E 0 ( b definiion), H 0. Then, h h e e j h k j h k j h k j h k Where, h saisfies:

11 k 0, h k k The soluions of he above equaion are onl found for pariular values of, when we appl he boundar k ondiions. These are he eigenvalues of he TE mode problem. The ransverse wavenumber is speified b he guide ross-seion (shape and sie) and b he ransverse disribuion of he fields for he mode onsidered; i is independen of he medium filling he guide. In a homogeneous guide, he ransverse wave number is alwas real and posiive. k The ransverse fields an ompal be wrien as: j j e, aˆ Xh,, h, h k k The TE wave impedane is: Z E E TE H H k Sine is frequen dependen, ZTE depends on he frequen. TE waves an be suppored inside losed onduors, as well as beween wo or more onduors.

12 TM WAVES E 0 (b definiion), H 0. Then, h h e e Where j e k j e k j e k j e k e saisfies: k 0, e k k The ransverse fields an ompal be wrien as: j j e e h a X e k,,,,, ˆ k The TM Wave Impedane: Z TM E E H H k Frequen dependen TM waves an be suppored inside hollow onduors, as well as beween wo onduors.

13 RECTANGULAR WAVEGUIDES Hollow WG s are ommonl used as TL s a frequenies above 5GH. Compared o oaial lines; WG s have he following advanages: ) Higher power handling apabili ) Lower loss per uni lengh 3) A simpler, lower os mehanial sruure 4) The refleions aused b he flanges used in onneing WG seions is usuall less han ha assoiaed wih oaial onneors. The disadvanages are: ) Lager ross-seional dimensions ) Lower usable bandwidh A large varie of omponens suh as ouplers, deeors, isolaors, aenuaors and sloed lines are ommeriall available for various WG bands from GH o over 0GH. The hollow WG s an suppor TM and TE modes bu no TEM modes.

14 Convenionall, he longer side of he WG is loaed along he -ais, so a b. TE MODES E 0 b definiion and h saisfies: k h, 0...(*) The omplee epression for H, is:,,, H h e j And k k wih k. A Separaion of Variables Soluion for h, Assume h, X ( ) Y( ) Subsiuing his ino (*) and dividing eah erm b XY :

15 d X d Y k 0 X d Y d This equaion an be saisfied for all and onl if: d X d Y k 0 0 k X d Y d Wih: The general soluion for h(, ) k k k an hen is wrien as: h (, ) Aos k Bsin k Cos k Dsin k The Boundar Condiions: We mus have: e, 0 a 0 and b e, 0 a 0 and a Bu we had, e j h j h e k k Therefore i is neessar ha: We mus have

16 h 0 a 0 and b h 0 a 0 and a Or Ak sin k Bk os k 0 a 0 and a B 0for k Whih gives, 0 k a m m Similarl, and 0,,,... Ak sin k a 0 whih gives Ck sin k Dk os k 0 a 0 and bwhih gives D 0 for k 0 and Whih gives, k b n n 0,,,... Ck sin k b 0 m n So, k k m 0,,,... n 0,,,... a b

17 Then, m n h, Amn os os a b And m n H,, Amn os os e a b Where A mn j : Consan depending on he eiaion srengh. The oher field omponens are found as: j m n E namn os sin e k b a b j m n E mamn sin os e k a a b j m n H mamn sin os e k a a b j m n H namno s sin e k b a b The propagaion onsan k k j j j j Sine m n k k k a b

18 m n k a b m n a b Consider wo ases of ineres: A) k k If he frequen f is high enough so ha, for a given se of values of a, b, m and n, k m n f a b hen is real. Real orresponds o propagaion. wavenumber. Eah mode has a uoff frequen k f mn is he uoff given b: f mn k m n a b Le v =Phase veloi for an unbounded medium filled wih maerial having and. Then, f mn m n v m n a b a b The mode wih he lowes uoff frequen is alled he dominan mode.

19 Sine a>b, he lowes f 0 v a a a TE 0 mode is he dominan mode. Sine E, E, H and f ours for m= and n=0. So, H are all ero for m=n=0, here is no 00 TE mode. B) k k In his ase beomes purel imaginar, n m j k jq a b, q is real. Then, he j jq q erm beomes e e whih orresponds o he aenuaion of fields eponeniall. Suh modes are nonpropagaing or evanesen. Noe ha his aenuaion is no assoiaed wih he dissipaive losses. j e The Wave Impedane: Z E E TE H H k k, i) When k k f f, and ZTE are boh real.

20 ii) When k k f f, imaginar. Z and Z TE are boh purel k k a a TE0 0 The guide wavelengh: g For a propagaing mode k k so k and g g k m n k a b Le,, k, uoff wavelengh. Then, g k k k k Or,

21 g / g v/ f / / f f, v We have he following relaionship: g For he dominan mode: k, a, so / a a g a The phase veloi: v p k k k v p f k

22 p v p v f f