FINITE DIFFERENCE APPROACH TO WAVE GUIDE MODES COMPUTATION

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1 FINITE DIFFERENCE ROCH TO WVE GUIDE MODES COMUTTION Ing.lessando Fani Elecomagneic Gou Deamen of Elecical and Eleconic Engineeing Univesiy of Cagliai iazza d mi, 93 Cagliai, Ialy

2 SUMMRY Inoducion Finie Diffeence aoach o he mode evaluaion fo an elliic waveguide. The use of D elliical gid allows o ake eacly ino accoun he elliical bounday. s a consequence, we ge an high accuacy,wih a educed comuaional buden, since he esuling mai is highly sase; Sandad Finie diffeence comuaion of waveguide modes equies wo diffeen gids, one fo TE and anohe fo TM modes, because he bounday condiions ae diffeen. We oose and assess hee use of a single gid.

3 SUMMRY finie-diffeence echnique o comue Eigenvalues and mode disibuion of non sandad waveguide (and aeue) is esened. I is based on a mied mesh (caesian-ola) o avoid disceizaion of cuved edges, and is able o give an accuacy comaable o FEM echniques wih a educed comuaional buden. new geneal scheme fo he FD aoimaion of he Lalace oeao, based on a non-egula disceizaion, is discussed hee. I allows o ake ino accoun in he FD scheme he bounday condiions, and heefoe allows o use he eac shae of he bounday. s a consequence, he field disibuion deails can be moe accuaely modeled.

4 INTRODUCTION n accuae knowledge of he cu-off fequency and field disibuion of waveguide modes is imoan in many waveguide oblems. The same ye of infomaion is necessay in he analysis wih he mehod of momens (MOM) of hick-walled aeues. Indeed, hese aeues can be consideed as waveguide, and he modes of hese guides ae he naual basis funcions fo he oblem. a fom some simle geomeies, mode comuaion canno be done in closed foms, so ha suiable numeical echniques mus be used. oula echnique fo cu-off fequency and field disibuion evaluaion is Finie Diffeence (FD),.i.e, diec disceizaion of he eigenvalue oblem. This allows a simle and vey affecive evaluaion, also because he oblem is educed o he comuaion of he eigenvalues and eigenvecos of an highly sase mai

5 INTRODUCTION The sandad fou-oin FD aoimaion of he Lalace oeao, howeve, canno be used fo moe comle geomey since i equie a egula (ecangula) disceizaion gid, and heefoe a bounday wih all sides aallel o he ecangula aes. Theefoe cicula and elliic boundaies ae yically elaced by sai case aoimaion. im of his esenaion is o develo, and assess, a geneal scheme fo he FD aoimaion of he Lalace oeao, based on a egula ola and elliic gid.

6 DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE 3 4 Sandad FD disceizaion in Caesian coodinaes fo a ecangula cell : leads o he aoimaion of he Lalace oaao ( ) [ ] 3 4 y y y y

7 DESCRITION OF THE TECNIQUE OLR FRMEWORK Le use conside a cicula waveguide. oh TE and TM modes can be found fom a suiable scala eigenfuncion φ, soluion of he Helmohz equaion: k () n wih he bounday condiion fo TE mode, fo TM mode

8 Conside he cell aound oin enclosed oins CD. Remembe he fom of he Lalacian in ola coodinaes: ( ) ( ) ( ) ( ) C Using a second ode Taylo aoimaion fo oins and C, and summing: ( ) ( ) C C D DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE OLR FRMEWORK OLR FRMEWORK ()

9 using he same ocedue fo oins and D we ge: The aoimaion of he lalacian becomes: C D ( ) ( ) D D and DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE OLR FRMEWORK OLR FRMEWORK ( ) ( ) ( ) ( ) ( ) ( ) D C

10 CENTER OINT CENTER OINT Fo he cene oin we inegae () ove a disceizaion cell ds k ds Use of Gauss Theoem gives: Γ F F S n ds k dl i i.e Γ F k dl n (3) whee F Γ is he cell bounday, F S is he cell suface and is evaluaed a he disceizaion node. ( ) π N q q (4) N N-

11 Fo TE mode OUNDRY OINT OUNDRY OINT ( ) ( ) ( ) ( ) ( ) C C Then:

12 NUMERICL RESULT COMRISON ETWEEN OUR FD CODE ND NLITIC RESULTS FOR TE MODES IN CIRCULR GUIDE WVE k (naliic) k (FIT) (Ou FD code) k Relaive eo % % % % % %

13 DESCRITION OF THE TECNIQUE ELLITIC FRMEWORK Le use conside a elliic waveguide. oh TE and TM modes can be found fom a suiable scala eigenfuncion, soluion of ()

he em in backes eanded eacly as in a ecangula gid: DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE ELLITIC FRMEWORK ELLITIC FRMEWORK ssuming a egula sacing on he coodinae lines, wih se

14 he em in backes eanded eacly as in a ecangula gid: DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE ELLITIC FRMEWORK ELLITIC FRMEWORK ssuming a egula sacing on he coodinae lines, wih se and leing he eigenvalues equaion () can be eessed us: (6), v, u ( ) v q u q, ( ) q k v u v u c sin sinh ( ) ( ) ( ) ( ) ( ) ( ) v u u v u v v u D C ( ) ( ) ( ) ( ) ( ) ( ) v u u v u v v u D C

15 FOCUS If L e L i and L e Finally, conside he foci of he elliical shae gid S S S i n dl k S ds k [( ) L ( ) L ] C E is he aea of he cell, and E L I ae half he lengh of he ac of he ellise and of he ac of he hyebola esecively. h u v ( u, v) h ( u, v) v a sinh u sin v I L, u v u u v hu,vdv hu, hu, 4 L i u v u v u v hv u, du hv, hv, 4

16 NUMERICL RESULT COMRISON ETWEEN OUR FD CODE ND COMMERCIL FIT CODE FOR TE MODE IN ELLITIC WVEGUIDE. k (FIT) k (Ou FD code) Relaive eo % % % % % % %

17 TM MODES Since he fundamenal mode is a TE, hese modes ae he mos ineesing. TM modes can, howeve, be comued in a likely way, aking ino accoun he diffeen bounday condiions. This was done using a gid diffeen fom TE one. This migh be fine fo he calculaion of modes of micowave guiding sucues, bu fo some alicaions (analysis by he mehod of momens of aeue, Mode maching) would be much moe useful he TE gid. Then we eloed he ossibiliy of using a single gid fo boh TE and TM modes.

18 OUNDRY OINT OUNDRY OINT C ( ) ( ) Using a second ode Taylo aoimaion fo Q: Q Q Q ounday condiion: Recalling ha: We can comue:

19 OUNDRY OINT OUNDRY OINT C ( ) ( ) ( ) 3 3 Q The esul is:

20 Fo TM modes: OUNDRY OINT OUNDRY OINT ( ) ( ) ( ) ( ) ( ) C C

21 NUMERICL RESULT COMRISON ETWEEN OUR FD CODE ND ND NLITIC RESULTS FOR TM MODES IN CIRCULR WVE GUIDE k (naliic) k (Ou FD code) Relaive eo % % % % % %

22 DESCRITION OF THE TECNIQUE Fo he all oin we inegae () ove a disceizaion cell Use of Gauss Theoem gives: n i.e dl Γ F k ds k Γ F i (3) whee ds n dl k S F ds is evaluaed a he disceizaion node. SF is he cell suface and

23 The aoimaion of he lalacian becomes: L L L L S S S D C S OINTS ETWEEN CRTESIN OINTS ETWEEN CRTESIN ND OLR GRID ND OLR GRID

24 The aoimaion of he lalacian becomes: ( ) ( ) ( ) n n n... 3 π CENTER OINT CENTER OINT

NUMERICL RESULT D5,X5,a,, k kh kcs ehfs% ecs%.78.8.8.55.5.455.454.45.68.68.489.455.453.7959.8438.556.538.537.357.378.64.653.

25 NUMERICL RESULT D5,X5,a,, k kh kcs ehfs% ecs% R R mm, 6.37mm

26 GRHICS MODO FONDMENTLE TE I MODO SUERIORE TE II MODO SUERIORE TE III MODO SUERIORE TE

NUMERICL RESULT D 4 mm, 6.37mm, h.6 MODI K (HFSS) K (num) Modo fond TE.44885.

27 NUMERICL RESULT D 4 mm, 6.37mm, h.6 MODI K (HFSS) K (num) Modo fond TE h modo su TE modo su TE D 3 modo su TE modo su TE

28 GRHICS MODO FONDMENTLE TE I MODO SUERIORE TE II MODO SUERIORE TE III MODO SUERIORE TE

29 DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE ( ) [ ] 3 4 y y y y 3 4 Sandad FD disceizaion in Caesian coodinaes fo a ecangula cell : leads o he aoimaion of he Lalace oaao Ou inees is o use iegula gids

30 DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE conside a non sandad disceizaion We ae an looking fo: ( ) i i i Using a second ode Taylo aoimaion we ge: i i i i i i i y y y y y y whee all deivaives of φ ae comued a he samling oin, and (i, yi) he osiion of he i h oin w.. oin.

31 Theefoe () The i ae linea combinaion of he unknown coefficiens i. DESCRITION OF THE TECNIQUE DESCRITION OF THE TECNIQUE To ge he Lalace oeao we equied 5 34 () which is a linea sysem in he i. Fo eamle is equal o: ( ) y y y i i i 5 4 3

32 OUNDRY OINT Fo a bounday oin, bounday condiion δφ/δn can be eessed as: 5 5 y whee, ae he comonen of a veco nomal o he bounday. Sysem () fo a bounday oin is modified uning in o accoun bounday condiion (3). (3) 3 4

33 EXMLE OF OUNDRY CONDITION EXMLE OF OUNDRY CONDITION 3 4 ( ) 4 ( ) ( )

34 NUMERICL RESULT To assess ou FD echnique wih vaiable gid, we have analyzed a idged waveguide wih aezoidal idges and ecangula aeue. e d e d a b c a b c

35 NUMERICL RESULT TE Mode of idge waveguide aezoidal aeue wih e.55 mm TE mode K (FD) K (CST ) I II.6.6 III IV

36 NUMERICL RESULT TE Mode of idge waveguide ecangula aeue wih e.55 mm TE mode K (FD) K (CST ) I.4.6 II III IV

37 NUMERICL RESULT TE mode.55 mm.5 mm.55 mm 3.5 mm 3.55 mm 4.5 mm 4.55 mm I II III IV TE Mode of idge waveguide aezoidal aeue and ecangula aeue, inceases e I II III IV

38 NUMERICL RESULT FD is able o comue in he maimum field in he waveguide. Comaed he maimum value of he filed idged waveguide wih aezoidal idges and ecangula idges. One oblem of he idge waveguide is he educed owe caabiliy.

39 CONCLUSION new FD aoach o he comuaion of he modes of cicula and elliic waveguide has been descibed. Using an elliical cylindical gid, i akes eacly ino accoun he cuved bounday. oh TE and TM can be comued eihe on diffeen gids o on he same gid. The yical sase mai obained by he FD allows an effecive comuaion of he eigenvalues, wih a vey good accuacy, as shown by ou ess. fuhe significan imovemen in in he comuaional seed can be obained using aallel achieue. n iegula gid FD aoach in he vaiable gid o he comuaion of he all modes of he waveguide has been descibed. The yical sase mai obained by he FD allows an effecive comuaion of he eigenvalues, wih a vey good accuacy, as shown by ou ess visible on acs. fo hee significan imovemen in in he comuaional seed can be obained using aallel achieue.

40 TH N K Y O U F O R Y O U R TTE N TIO N

Lecture 22 Electromagnetic Waves

Lecture 22 Electromagnetic Waves Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should