Hybrid Neural Network/Modal Method Modeling of Uniaxial Waveguide Discontinuities

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1 70 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 Hyrid Neural Networ/Modal Method Modelig of Uiaxial Waveguide Discotiuities M. Yahia* +#, J. W. Tao, H. Bezia #3, M. N. delrim +#4 + Uité de Recherche Modélisatio, alyse et Commade des Systèmes MCS # Ecole Natioale d géieurs de Gaès Rue Omar Elhatta 609 Gaès Tuisia *Tel: ; Fax: ; [email protected] [email protected] [email protected] Laoratoire Plasma et Coversio d Eergie LPLCE Ecole Natioale Suérieure d'electrotechique, d'electroique, d'formatique, d'hydraulique et des Télécommuicatios, rue Camichel 307 Toulouse cedex Frace [email protected] stract-we roose a hyrid eural etwor/modal method comuter-aided desig (CD) tool to the aalysis of uiaxial discotiuities i rectagular waveguides. The eural etwor is traied usig the cotiuity coditios of the trasverse electric ad magetic fields i the discotiuity lae. Ulie the covetioal modal methods, the erformaces of the roosed euromodel are cotrolled y the oerator ad ca outerform the classical modal methods i CPU time. The arallel ature of the roosed hyrid CD tool maes it a iterestig solutio for arallel imlemetatio i hardware ad software. dex Terms- Uiaxial discotiuities, modal methods, artificial eural etwors, hyrid methods.. NTRODUCTON The artificial eural etwors (NNs) are iformatio rocessig tools isired y the aility of the huma rai to lear ad to geeralize y astractio []. fter the traiig, M. Yahia et al, "Uiaxial Discotiuities alysis Usig the rtificial Neural Networs ", teratioal Coferece o Sigals, Circuits ad Systems, 6-8 Nov. 009, Djera. the NNs roduce accurate ad istataeous resoses. Cosequetly, they are alied i diverse scietific fields (image rocessig, seech rocessig, medical, root, atter recogitio ). Their use i electromagetic is i exasio esecially i microwave circuits []. The uiaxial discotiuities i waveguides are widely utilized for the desig of umerous assive microwave comoets for satial alicatios (e.g., filters, dilexers ad multilexers, olarizers, etc). [3], authors roosed a hyrid fiite elemet method (FEM) ad NNs CD tool to solve secod order artial derivative rolems. The weights of the NNs are traied with the odal fiite elemet system equatios. They successfully alied their euromodel to the aalysis of umerous waveguide cofiguratios. lso, the similar CD tools that comie FEM ad NNs are give i [4 7]. [8], the solutios to electromagetic comatiility rolems are give y comiig vector fiite elemet method with NNs. However, whe cosiderig simle discotiuities i waveguides, the modal method (e. g. multimodal variatioal method (MVM) [9], the mode matchig (MM) [0], etc.) are the most aroriate CD tools used for such devices. They outerform satial meshig techique (i. e. JMOT SRMT

2 7 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 fiite elemet [] ad fiite differece []) i CPU time ad memory storage. this aer, we sustitute the fiite elemet formulatio i [3-8] y the modal oe. The NN is traied usig the cotiuity coditios of the trasverse electric ad magetic fields i the discotiuity lae. The euromodel is alied to the aalysis of well-ow discotiuities. ll simulatios are develoed i Matla o a PC (tel CPU.66 GHz with -GB RM).. THE PROPOSED NEUROMODEL. Geeral rchitecture of the NNs. UNXL WVEGUDE DSCONTNUTES Fig.. rchitecture of the NNs (a) () Fig.. Uiaxial discotiuity etwee two rectagular waveguides. Fig. 3. Structure of a euro Fig.. resets a uiaxial discotiuity etwee two rectagular waveguides. the S discotiuity surface, the cotiuity coditios of the trasverse electric ad magetic fields ca e exressed as follows: ζ ζ N ( a + ) e ( a + ) e 0 N = m m m m = m= = N N = m am m hm B a + h = m= = () ( ) ( ) 0 B () e h N N B a + N a N ζ ζ a + N a + N B N N e h N ad N are the umers of modes i the waveguides ad, resectively. i i am ad m eig the icidet ad reflected wave amlitudes of the m th eigemode at the discotiuity lae i th i th waveguide, resectively. i i em ad hm eig the m th modal electric ad magetic asis fuctios i the i th waveguide, resectively. i m ad B i m are two comlex arameters where the i i ratio B eig the modal imedace. m m ut layer Outut layer Hidde layer Fig. 4. The roosed euromodel architecture. Geerally, NNs have two comoets: The rocessig elemets called euros (fig. ). JMOT SRMT

3 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 7 The coectios etwee euros are called lis or syases. Each li has a weight coefficiet (w i ) (fig. ). Each euro receives iuts (stimuli x i ) from eighorig euros coected to it, rocess the iformatio (f) ad geerate a outut (y) (fig. 3). Neuros that receive stimuli from outside the NN costitute the iut layer. Neuros whose oututs are used outside the NN form the outut layer. Neuros whose iuts ad oututs are i the NN costitute the hidde layer. There are differet ids of NNs. this aer, we are iterested i the multilayer ercetro eural etwor (MLPNN) []. B. licatio of the hyrid CD Tool to the alysis of Waveguide Discotiuities. Equatios () ad () ca e exressed i a MLPNN model (fig. 4) where f is the idetity fuctio ad is equal to zero (fig. 3). The architecture of the euromodel is detailed as follows: ut layer: the umer of euros i the iut layer is equal to N +N. Hidde layer: the umer of euros i the hidde layer is equal to the oe i the iut layer. Outut layer: we tae two euros which geerate ζ ad ζ. The waveguide is excited y the fudametal mode TE 0. Cosequetly, a =0 ad a =0 excet a () =. fig. 4, the uows are ad. They will e comuted iteratively i the traiig rocess. To get a solutio to the equatios () ad (), the oututs of the roosed euromodel must geerate ζ = 0 ad ζ = 0. To attai this ed, we emloy the least mea squares (LMS) algorithm [3]. We miimize simultaeously ad. The arameters ζ ζ ad are adjusted usig the followig iterative exressios: = µ* ζ (3) = µ* ζ (4) * eig the comlex cojugate, is the gradiet oerator ad µ eig the ste arameter where 0 < < µ. ζ d ζ * = ζ = ζ B h d = B * N * ( ) ( ) a + B B a + h h = N * ( ) ( ) a + B B a + h h = (5) = µ B (6) * d ζ * ζ = ζ = ζ e d (7) = N = * ( a + ) e e + ( a + ) ( a + ) e e + ( a + ) N * = µ (8) =.. eig the ier roduct defied as follows. e ( x,y) e ( x,y) = e ( x,y) e ( x,y) dxdy (9) x,y S The traiig rocess rogresses as follows: i. Comute the ier roduct etwee modes, choose the value of µ, iitialize radomly ad ad choose a covergece threshold ψ. ii. djust ad usig the exressios (6) ad (8). Comute the relative error η etwee two successive iteratios ad + as: η = 00 (0) iii. Reeat ste ii while η ψ. iv. Comute the reflectio ad the trasmissio coefficiets of the discotiuity which are exressed as follows. JMOT SRMT

4 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 73 () a () () = () S = () a () () = () S = V. RESULTS this sectio, we aly the hyrid model to the aalysis of various discotiuities i rectagular waveguides. The erformaces are comared to various modal methods. The modes i the waveguides i oth side of the discotiuity are ordered y ascedat cut-off frequecies.. E-H Discotiuity Fig. 7. S ad S of the studied umerical methods for the E-H discotiuity. Fig. 5. E-H Discotiuity. Fig. 8. Phases of the S aramerers of the studied umerical methods for the E-H discotiuity. Tale : CPU time of the studied umerical methods for the E-H discotiuity MM MVM Proosed Neuromodel CPU time (s) Fig. 6. Variatio of η, ζ ad ζ accordig to the umer of iteratios N for f =.5 GHz ad µ = We cosider a E-H discotiuity etwee a WR6 waveguide (a=5.799 mm, =7.899 mm) ad a WR90 waveguide (a'=.86 mm, '=0.6 mm) (fig. 5). We tae 50 modes i the waveguide ad 96 modes i the waveguide. Fig. 6 dislays the variatio of η, ζ ad accordig to the umer of iteratios N for ζ JMOT SRMT

5 74 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 f =. 5GHz GHz ad µ = We oserve that,η, ζ ad ζ simultaeously decrease with the icrease i the umer of iteratios N. Ulie the covetioal modal methods, the erformaces of roosed euromodel are cotrolled y the oerator y choosig the value of µ.this feature maes our euromodel more versatile tha the covetioal modal methods. We tae N=80 iteratios ad µ = which gives η < % for all studied frequecies. Fig. 7 dislays the variatio of S ad S i the cosidered frequecy ad. We oserve that, the roosed euromodel coverges to the same solutios rovided y the MVM ad the MM. However, the roosed euromodel is the most raid (see tale ). Fig. 8 dislays the variatio of the hases ϕ ij of the S arameters i the cosidered frequecy ad. The solutios rovided y the roosed euromodel agree well with those roduced y the modal methods. B. E-Discotiuity Fig. 9 dislays a E-discotiuity (a=.86 mm, =6 mm ad '=0 mm). We tae 50 modes i the waveguide ad 88 modes i the waveguide. We set N=60 iteratios ad µ = which -3 rovides η <.70 % for all cosidered frequecies. Tale : CPU time for the studied umerical methods for the E-discotiuity MM MVM Proosed Neuromodel CPU time (s) Fig. 0. S ad S of the studied umerical methods for the E-discotiuity. Fig. 0 dislays the variatio of S ad S i the cosidered frequecy ad. We ca see that all studied umerical methods coverge to the same solutio. However, tale shows that, the roosed euromodel is the most raid. We ca otai more raid solutios if we tae less iteratios ut the recisio of the solutio will e reduced. C. H-Discotiuity Fig. 9. E-Discotiuity Fig.. H-discotiuity JMOT SRMT

6 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 75 roosed euromodel icrease with the icreasig of CPU time. The euromodel is more raid tha the modal methods ad rovides the same accuracy. REFERENCES Fig.. S ad S of the studied umerical methods for the H-discotiuity. Tale 3: CPU time for the studied umerical methods for the H-discotiuity. MM MVM Proosed Neuromodel CPU time (s) Fig. dislays a H-discotiuity etwee two rectagular waveguides (a=.86 mm, a =5.9 mm ad =0.6 mm). We tae 50 modes i the waveguide ad 69 modes i the waveguide. We set N=60 iteratios ad µ =0. which -5 rovides η < % for all cosidered frequecies. Fig. dislays the variatio of S ad S i the cosidered frequecy ad. The solutios rovided y the umerical methods are similar. However, tale 3 shows agai that the roosed euromodel is the most raid. V. CONCLUSON We roosed a hyrid eural etwor/modal method comuter-aided desig (CD) tool to the aalysis of uiaxial discotiuities i rectagular waveguides. The cotiuity equatios of the tagetial electric ad magetic fields are writte i a eural model. We utilized the LMS algorithm to trai the NN weightig coefficiets. The results show that the erformaces of the [] C. Christodoulou ad M. Georgiooulos, licatios of Neural Networs i Electromagetics, rtech House teas ad Proagatio Lirary 00. [] J. E. Rayas-Sáchez, EM-Based Otimizatio of Microwave Circuits Usig rtificial Neural Networs: The State-of-the-rt, EEE Tras. Microwave Theory Tech., vol. 5, o., , 004. [3] P. Ramouhalli, L. Uda ad S. S. Uda, Fiite- Elemet Neural Networs for Solvig Differetial Equatios, EEE Tras. Microwave Theory Tech., vol. 6, 38 39, Nov [4] J. Taeuchi ad Y. Kosugi, Neural etwor reresetatio of fiite elemet method, Neural Netw., 994, 7, (), [5] H. Yamashita, N. Kowata, V. Cigosi ad K. Kaeda, Direct solutio method for fiite elemet aalysis usig Hofield eural etwor, EEE Tras. Mag., 995, 3, (3), [6] R. Siora, J. Siora, E. Cardelli ad T. Chady, rtificial eural etwor alicatio for material evaluatio y electromagetic methods, EEE t. Joit Cof. Neural Networs, vol. 6, , 999, Washigto, DC. [7] F. Guo, P. Zhag, F. Wag ad X. Guayuaqiu, Fiite elemet aalysis-ased Hofield eural etwor model for solvig oliear electromagetic field rolems, EEE t. Joit Cof. Neural Networs, vol. 6, , 999, Washigto, DC. [8] M.S. l Salameh ad E.T. l Zuraiqi, Solutios to electromagetic comatiility rolems usig artificial eural etwors reresetatio of vector fiite elemet method, ET Microw. teas Proag., vol., o. 4, , 008. [9] J. W. Tao ad H. Baudrad, Multimodal variatioal aalysis of uiaxial waveguide discotiuities, EEE Tras. Microwave Theory Tech., vol. 39, o. 3, , Mar. 99. [0]. Wexler, Solutio of Waveguide discotiuities y Modal alysis, EEE Tras. Microwave Theory Tech., vol. 5, , Set JMOT SRMT

7 NTERNTONL JOURNL OF MCROWVE ND OPTCL TECHNOLOGY, VOL.6, NO., MRCH 0 76 [] J. Ji, The Fiite Elemet Method i Electromagetics, Joh Wiley & sos, Secod editio, New Yor 000. []. Taflove, Comutatioal Electromagetics: The Fiite-Differece Time-Domai Method, Norwood, M: rtech House, 995 [3] S. V. Vaseghi, dvaced Sigal Processig ad Digital Reductio, Wiley ulishers 996. JMOT SRMT