[Ismibayli*, 5(3): March, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785

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1 IJERT INTERNATIONAL JOURNAL OF ENGINEERING CIENCE & REEARCH TECHNOLOGY IMULATION OF ELECTROMAGNETIC FIELD FROM MICROWAVE RECTANGULAR WAVEGUIDE TO CIRCULAR IN TRANITION DEVICE E.G.Ismibyli, I.J.Islmov, Y.G.Gziyev Deprtmet of Rdio Egieerig d Telecommuictio Azerbij Techicl Uiversity, Bu, Azerbij DOI: 0.58/zeodo ABTRACT New mthemticl models of electromgetic field of microwve rectgulr d circulr wveguides hve bee estblished i polr coordite system. Ritz d Gleri methods for solvig mthemticl models hve bee pplied d the lgorithms of these methods hve bee pplied to ddress the issue uder cosidertio. As result of solvig mthemticl models curves of evipotesil lies of electromgetic field of microwve rectgulr d roud wveguide hve bee estblished. Built-i curves creted the opportuity to moitor the distributio of electromgetic field of microwve rectgulr d circulr wveguide. The developed methods re differetited by their uiverslity. Thus, this techique c be pplied to the other extreme microwve devices. KEYWORD: microwve, rectgulr wveguide, circulr wveguide, electromgetic fields, Ritz method, Gleri method. INTRODUCTION At preset, microwve devices re widely used i the productio [-3]. Therefore, the improve of costructive, techicl d opertig prmeters d the desig of ew devices which hve more optiml costructive sizes re very importt d ctul both scietific poit of view d i terms of productio [4-7]. For this reso, the itroduced rticle hs bee dedicted to the mthemticl modelig of electromgetic field of microwve rge devices. I the rticle, the structure of microwve rectgulr d roud wveguides is more complicted th of the electromgetic field mthemticl modelig of the E-type d H-type is cosidered. Developed ew lgorithms hve bee bsed o mthemticl poit of view, ccurte d complete [8-0]. Two mi ims re touched whe this rticle is prepred:. For mthemticl modelig of microwve devices, vilble lyticl, umericl methods re lyzed d compred d more effective lgorithm is wored out.. Developed lgorithms re pplied d microwve devices re wored out which re more perfect. [695]

2 ) b) Figure. Trsitio from microwve rectgulr wveguide to circulr () d sptil view (b). MATHEMATICAL MODEL OF ELECTROMAGNETIC FIELD OF MICROWAVE CIRCULAR WAVEGUIDE İf we poit out prevlece of electromgetic field i microwve wveguide with oblst, we c idicte the itegrl equtio tht express the field s follows [, ]: E E J F( x, Eds, () x y here F( x, is give fuctio. We c cll tht s fuctiol, becuse the price of () itegrl depeds o the fuctio We should ccept for clrifyig the physicl ture of the problem, E fuctio is miimum of () fuctiol. Therefore, let's loo t the followig fuctio [3-5]: E( x, ( x,, here is very smll proportio, ( x, is cotiuous fuctio d i oblst the first two differetil of itself is pprochig to zero. Therefore, uder the coditio of 0 J( E ) J( E). o, whe () fuctiol becomes 0, it becomes s miimum price, so, dj ( E ) d 0 0. Let's cout this differetil: J ( E ) d E E F( x, ( E ) ds. d d x x y y, we shll determie: E E F ds 0. x x y y Now if dmit tht there is 0 () If we tur () itegrl, we c write s follows: x x y y x y E E E E ds F ds 0, s [696] E.

3 or div( E) ds ( E F) ds 0. (3) Due to Ostrogrdsi theorem, we c write: E div( E) ds dl,, here C - the limited cotour with the surfce - the orml directed to cotour It should be oted tht becuse prmeter is pprochig to zero log C cotour, we receive div ( E) ds 0, ( E F) ds 0. Acquired expressio does ot deped o fuctio d uder the followig coditio is true: E E E F( x,. x y o it c be cocluded tht, () the miimum of itegrl (4) is equivlet to the solutio of the Poisso equtio. It should be oted tht () itegrl is miimum d it is ecessry tht this is equivlet to the solutio of the equtio (4). I prticulr, whe F ( x, 0 () itegrl tes the form of itegrl of Dirichlet [5-9]. Therefore, the solvig of the Lplce equtio is equivlet with determitio of miimum fuctio of itegrl of Dirichlet. I geerl the determitio of the miimum fuctio of () itegrl is fulfilled by pplyig the methods of Ritz d Gleri. Therefore, let's pply these methods for modelig of electromgetic field of microwve rectgulr d circulr wveguides. APPLICATION OF THE METHOD OF RITZ AND THE GALERKIN It should be ccepted tht E( x, is exct solutio of the problem d miimum of () fuctiol is J( U) m. If we re ble to fid out fuctio of E ( x, which meet the coditios of the border for E, the built E fuctio will be pproched to geuie solutio of. sequece of the E. E fuctio, ie, Thus, ccordig to the method of Ritz issue: E F( x, y,,,..., ). () if we put C. (4) E O the other hd, if we hve determied the J( E ) m, the this sequece will ssemble up to the settlemet of the E fuctio will deped o the followig prmeters i the treted E fuctio isted of E fuctio d fulfill itegrtio opertio, the the result will be fuctio of the (,,..., ) prmeters. o tht, J( E ) J(,,..., ). The issue is correlted with ppoitig of miimum of J fuctiol, the followig equtios: prmeters should be s the [697]

4 J 0 (,,..., ). It should be oted tht the lim J( E ) J( E) m. E fuctios re full uder the followig feture: Accordig to the method of Ritz, issue of miimiztio should be soughted i the form of lier fuctios which deped o E l prmeters: ( x,. (6) It should be oted tht E prmeter fulfills the homogeeous boudry coditios log C cotour. Otherwise, you should fid y of the E ( x, E( x, y ). E ( x, y Therefore, it is ecessry to iclude the ew ) fuctio, this fuctio must crry out the followig coditio: V ( x, E ( x, V( x, E ( x,. o tht, V F E. If ( x, 0 is log the cotour C, the you c write the followig s fuctio. This fuctio must fulfill the followig coditio: [698] fuctiol system:, x, 3 y, 4 x, 5 xy, 6 y. If F ( x, 0 is log the cotour the we c choose ( x, fuctio s follows [0-5]: ( x, F( x,. At the sme time the solvig of (4) Poisso equtio will be (6) i the form s follows: E E J( E ) F E ds F ds x y x s s C,, (7) s s here s s ds, x x y y s F ds. (8) s The system of equtios c be writte s follows which defie prmeters: J( E ) ss 0 (,,..., ), (9) or J ( E) E E F ds 0 (,,... ). (0) x x y y

5 We should poit out solvig of (9) or (0) system with. The we will receive bsis o the method of Ritz : E ( x, ( x,. () Accordig to the method of Gleri, solvig of the problem is soughted i the form of lier fuctios deped o which. But i this cse the equtios will be i other form of for their determitio. I this cse, if we do coversios i the (0) equtio, we will receive: E E E E E E F ds ds F ds x x y y x x y y x y s s E E E dl F 0, ds x y c here - C is outer orml towrds the cotour. If the sought solutio pys homogeeous boudry coditios, the cotour C. I this cse, we will receive: fuctio will be equl to zero log the E E F 0 (,,..., ). ds x y If the equtio is i the form of LE ( ) 0, the ccordig to the method of Gleri pprochig to () re s follows: E ( x, ( x,, L( E) ds 0 (,,..., ). (3) Now, we should solve mthemticl models for specific device. Oe side of the device is circulr wveguide s show i figure d the other side is the rectgulr wveguide which oe of them wor i rtios H type, other wors o H0 type wves. Rdius of circulr wveguide is R, side of rectgulr wveguide is (figure ). The itesity of electric field is E 0 i circulr wveguide. The issue should be resolved with itegrtio of equtio E 0. First of ll, we should me boudry coditios s homogeeous form. Becuse of this, we should receive s follows: E V E. At the sme time, we choose Efuctio so tht pproches to E 0 i circulr prt of the wveguide, to zero i rectgulr sectio. Let's loo t the issue i (, ) polr system. I this cse, we c write E fuctio s follows: F( ) E E0, R F ( ) here [699]

6 ,, cos , o, cos 4 4 F( ) 3,, si ,. si 4 4 I this cse, let's solve the followig equtio for the fuctio V V E. If we cosider, we will receive: V (, ). The the solvig of the equtio will be i pproximtio s follows: E E V. Accordig to () testimoy we c write for the determitio of the coefficiets mds meds 0 ( m,,..., ), or /4 cos /4 cos d d d E d 0. m m 0 R 0 R uder homogeeous boudry coditios: We c write the followig fuctios uder symmetry coditio of field s fuctio zero i computig regio: (, ) ( R) F( ), (, ) (, ) cos, 3(, ) (, ) si, 4 4(, ) (, ) cos si.... : which pproches to I cocrete cse let's loo t solvig of the fuctio for st pproximtio. At this time we shll get for clcultio of the rtio of : / 4 / cos / 4 / cos d d d Ed 0 0 R 0 R. We should cosider for Lplce to be s follows i polr coordite system: [700]

7 ( ). The we will get rtio of :, p(,, 4 p 0,54 p ) here 0,58 3,73 p 6,36 p 3,p 0,54 p 0,734 p REULT AND DICUION We will receive for itesity i the first pproximtio itesity: E cos ( )( p), ( ). E 0 p cos 4 4 cos, R p. R Jery curve complies with prt of the evipotesil lie i squre 0 / i figure. Whe p 0, 5 becomes the clcultio is crried out. Clerly, the first pproximtio is o-stisfctory for tig dvtge. Therefore, the followig is ccepted for the ext pproximtio. E E, ) (, ), here ( cos is the sme,. This type of solutio hs bee chose for simplifyig the clcultios d coected i (4) equtio. At the sme time,, d 3 fuctios hve bee pirs of fctors hve bee te. Thus, if we detied fctor of the, we would hve clculted the itervl. This would lso me difficult solvig of the issue. We will receive the system of followig equtios for determiig rtios of d : ds ds Eds, ds ds E ds. [70]

8 Figure. Distributio of electromgetic field from microwve rectgulr wveguide to circulr i trsitio devices By logy, if the clcultio is show bove, we will receive: p) ( p) ( ), ( 3 p ( p) ( p) 3( p ), R here p d ( p) (0,5 3,73p 6,36p 3,p 0,5p 0,73p ), ( p) (,93 9,5p,08p 9,39p 5,08p ), p 3( p) p(,,43p 0,5p ), ( p ) (0,08,7 p,43 p,7 p,3 p 0,47 p ), ( p ) (,09 3,94 p 5,07 p 5,6 p,89 p 0,78 p ), 3( p) p(0,5 0,5p 0,43p ). If we ssume p 0, 5, we will receive: 0,5, 0,3, 0, 6, 3 0,38, 0,7, 3 0,8, 0, 708 0,508,. Thus, the solutio sought is s follows: E F ( F)( R) (0, 708 0,508cos ). E0 R F Evipotesil lies of the field determiig by this formul hve bee show with curves by complete lies i figure. CONCLUION. New mthemticl models of electromgetic field of microwve rectgulr d circulr wveguides hve bee estblished i polr coordite system. [70]

9 . Ritz d Gleri methods for solvig mthemticl models hve bee pplied d the lgorithms of these methods hve bee developed for solvig the issue. 3. As result of the settlemet of the developed mthemticl models, curves of evipotesil lies of electromgetic field of microwve rectgulr d circulr wveguides hve bee estblished. 4. Built-i curves hve bee creted the opportuity to moitor to the distributio of electromgetic field of microwve rectgulr d circulr wveguides. 5. The developed methods re differetited by their uiverslity. Thus, this techique c be pplied to the other extreme microwve devices. REFERENCE. Y.E. Elmozze, hfi Lotfollh. Numericl solutio of couplig betwee two collier prllel-plte wveguides. IEEE Trs. Microwve Theory d Tech., 3(), , T.P. Brto. Electromgetic-field clcultios for irregulrly shped, lyered cylidricl prticles with focused illumitio. Appl. Opt., 36, 3-39, V. Iomot, P. Alotto. Mgetic field computtio i medi with hysteresis. Rev. roum. sci. tech. er. Electrotech. et eerg. 4(3), 9-96, K. Dishitulu Klluri. Eletromqetic Wves. Mterils d computtio with Mtlb. CR Press Lodo-New Yor, 835p., N.O. Mtthew diu. Numericl Techiques i Electromgetic with Mtlb. CR Press Lodo-New Yor, 707p., Muteu Iri. A method for umericlly solvig Mxwell's equtios i three dimesio. Politeh. Uiv Buchrest. 56(-4), -38, Miller Edmud K., Bure Gerld J. Low frequecy computtiol electromgetics for te lysis. Proc. IEEE. 80(), 4-43, Islm J. Islmov. Applictio of fiite elemets method by clcultig electric field of hollow rectgulr wveguide. IATED-005, Cd, Clgry, 6-3, Islm J. Islmov. Numericl method for lysis of electromgetic field of microwve wveguide. Itertiol Coferece o Ate Theory d Techiques. eptember 9-, evstopol, Urie, 45-50, E.G. Ismibeyli, I.D. Islmov. The Alysis of Mgetic Devices I Frequecy Directors of Teledymometrig ystems. The Third Itertiol cietific Prcticl Coferece "Khzreftgsytg-98", November, Bu, 0-3, Elshd Ismibyli, Islm Islmov, Yusif Gziyev. Modelig of Aisotropic Rectgulr Wveguide Prtilly Embedded i Aisotropic ubstrte. Jourl of Multidiscipliry Egieerig ciece d Techology (JMET), Vol., Issue, Germy, 53-57, 05.. Che Bo Xue, Hm Hiromi, Iwmur Koy. J. Opt. oc. Amer. 9(8), , H. Cory, A. Novic-Cohe, D. Levy. Vritiol expressio for propgtio log closed rectgulr chiro wveguide. Ates d Propg., 43(), 74-78, B.N. Ds.,.B. Chrbrty. Alysis of pir of dielectric coted bove dielectric substrte. IEE Proc. Microwves, Ates d Propg.43(), 67-7, A. Lrer, I. Doets,. Bryzglo. The semi iversio method for cylidricl microwve structures. Jour. of electromgetic Wves d Applictio, 0(6), , 996. [703]

10 AUTHOR BIBLIOGRAPHY E.G.Ismibyli Doctor of techicl scieces, professor. ice 97 he hs bee worig t the Deprtmet "Rdio Egieerig d telecommuictios" of Azerbij Techicl Uiversity. Hs more th 04 published scietific wors i the form of rticles, jourls, theses d tutorils. Reserch iterests: Alysis d sythesis of microwve telecommuictio devices. I.J.Islmov Ph.D., ssocite professor. ice 993 he hs bee worig t the Deprtmet "Rdio Egieerig d telecommuictios" of Azerbij Techicl Uiversity. Hs more th 55 published scietific wors i the form of rticles, jourls, theses d tutorils. Reserch iterests: Alysis d sythesis of microwve telecommuictio devices. Y.G.Gziyev eior Lecturer. ice 99 he hs bee worig t the Deprtmet "Rdio Egieerig d telecommuictios" of Azerbij Techicl Uiversity. Hs more th 64 published scietific wors i the form of rticles, jourls, theses d tutorils. Reserch iterests: Alysis d sythesis of microwve telecommuictio devices. [704]