THE finite element method (FEM ) has been widely used

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1 Frequencies of propagation of electromagnetic waves in a hexagonal waveguide Arti Vaish Harish Parthasarathy Abstract In this work, cut-off frequencies of propagation of electromagnetic waves in a hexagonal waveguide are calculated using two-dimensional (-D) finite element method. The numerical approach is a stard one involves six finite elements. A new type of hexagonal waveguide structure for the simple homogeneous dielectric case has been considered. The starting point is Maxwell s equations in conjunction to the exponential dependence of the fields on the Z- coordinates. For the homogeneous case, it results in the Helmholtz equations. Finally, finite element method has been used to derive approximate values of the possible propagation constant for each frequency. Keywords Finite-element-method, Variational principle, Eigenvector, Matrix Equation, frequencies of propagation, hexagonal waveguide. I. INTRODUCTION THE finite element method (FEM ) has been widely used over the decades in the analysis of waveguide components. It is because the propagation characteristics of arbitrarily shaped waveguides of is based on a spatial discretization of cross-section [1], [], [3], [6].This approximation allows hling of waveguide cross section geometries which are very similar to the real structures employed in practice. As a consequence, FEM constitutes a promising tool to characterize such problems [7]. Modern phased array radars imply the requirements for polarization agility of wideb array elements. Surface hexagonally poled lithium niobate for two dimensional non-linear interactions in optical waveguide structures has been reported [8], [9]. One possible choice for a radiating element with this property is the hexagonal waveguide. In this paper, a numerically efficient finite-element formulation is proposed to solve waveguides problems. Propagation modes obtained by this formulations may be used to analyse problems involving linear systems of arbitrary complex tensor permittivity permeability. The solution of these eigenvalue problems results in the approximate fields for all components of different eigenmodes in the waveguide which can further be used to obtain the corresponding eigenvalues [1], [11], [1]. A possible comparison of the proposed methodology with the available theoretical results has also been presented herein this paper to claim the accuracy reliability of the solution method. Arti Vaish Division of Electronics Communication Engineering at Manav Rachna International University, NCR INDIA. vaisharti@gmail.com. Division of Electronics Communication Engineering at Netaji Subhash Institute of Technology, University of Delhi, ND-78 Fig. 1. hexagonal cross section of the waveguide II. THE FINITE ELEMENT FORMULATION The basic idea of taking hexagonal cross-section dividing the cross-section in to finite number of elements has been taken from else [], [6]. In this paper, the equilateral hexagonal cross section of the waveguide is divided into a number of finite elements. An element is considered to be first-order triangular in shape. An schematics of a triangular finite element in the hexagonal waveguide is shown in figure 1. Consider a triangle having vertices(x 1,y 1 ),(x,y ),(x 3,y 3 ). We draw a vector u joining (x 1,y 1 ),(x,y ) another vector v joining (x 1,y 1 )(x 3,y 3 ). Let = u = (x x 1 ) +(y y 1 ) (1) = v = (x 3 x 1 ) +(y 3 y 1 ) () The unit vector along the two directions u v are û = u u = (x x 1,y y 1 ) (3) ˆv = v v = (x 3 x 1,y 3 y 1 ) (4) ISSN: Issue 1, Volume 1, January 11

2 any point(x, y) inside this triangle can be represented as so WSEAS TRANSACTIONS on COMPUTERS (x,y) = (x 1,y 1 )+u.û+v.ˆv = (x 1,y 1 )+ u(x x 1,y y 1 ) + v(x 3 x 1,y 3 y 1 ) (x x 1 ) = u(x x 1 ) (5) (y y 1 ) = u(y y 1 ) (6) Equations (5) (6) are two linear equations for the variable u v solving them gives us u, v as linear functions of x, y. The area measure is given by x 1 y 1 1 x y 1 x 3 y 3 1 Thus we find that a b = V 1 V c V 3 a = V 1(y y 3 )+V (y 3 y 1 )+V 3 (y 1 y ) b = V 1(x x 3 )+V (x 3 x 1 )+V 3 (x 1 x ) (1) (11) c = V 1(x.y 3 x 3.y )+V (x 3.y 1 x 1 y 3 )+V 3 (x 1.y x.y 1 ) (1) ds(u,v) = u v du.dv u v = sin(α) = x.y 3 x 3.y +x 3.y 1 x 1.y 3 +x 1.y x.y 1 (13) thus for x,y we have here angle α between the vectors u v defined as cos(α) = u.v. = (x x 1 )(x 3 x 1 )+(y y 1 )(y 3 y 1 ). (7) The integral of a function can be evaluated as I(φ) = 1 if φ = 1 then we get φ[x 1 + u(x x 1 ) y 1 + u(y y 1 ) ]sinα.dudv (8) I(φ) =. sinα which is the correct formula for the area of the triangle. Suppose we write for V(x,y) = ax+by +c x,y with as the area bounded by the triangle. a,b,c are chosen so that V at the vertices are given, i.e., (9) V(x,y) = ax+by +c = V 1 φ 1 (x,y)+v φ (x,y)+v 3 φ 3 (x,y) φ 1 (x,y) = (y y 3 )x+(x x 3 )y +(x.y 3 x 3.y ) (14) φ (x,y) = (y 3 y 1 )x+(x 3 x 1 )y +(x 3.y 1 x 1.y 3 ) (15) φ 3 (x,y) = (y 1 y )x+(x 1 x )y +(x 1.y x.y 1 ) (16) The following two integrals occur when one uses the finite element method first I 1 = V(x,y) dx.dy second I = V dx.dy TABLE I NODAL COORDINATES OF THE FINITE ELEMENT MESH OF FIGURE Thus, V(x 1,y 1 ) = V 1 V(x,y ) = V V(x 3,y 3 ) = V 3 S.No. Element no. Coordinates 1 Element 1 (.,.), (1.,1.73), (.,.) Element (.,.), (1.,-1.73), (.,.) 3 Element 3 (.,.), (-1.,-1.73), (1.,-1.73) 4 Element 4 (.,.), (-.,.), (-1.,-1.73) 5 Element 5 (.,.), (-1.,1.73), (-.,.) 6 Element 6 (.,.), (-1.,1.73), (1.,1.73) ISSN: Issue 1, Volume 1, January 11

3 T [ ] a(x x 1 )+b(y y 1 ) d [ ] 1 a(x3 x 1 )+b(y 3 y 1 ) uvdudv () T 3 [ ] a(x3 x 1 )+b(y 3 y 1 ) v dudv (3) Fig.. A finite element mesh (6 elements 7 nodes). The numbers shown in circles represent the elements. Now I 1 = V (x,y) dx.dy = (ax+by +c) dx.dy (17) By substituting the value of (x,y) in terms of (x i,y i )in equation(17), we get I 1 = V(x,y) dx.dy [a(x 1 + u(x x 1 ) )+b(y 1 + u(y y 1 ) )+c] dudv (18) The use of method of variable separation for u v results in the following I 1 = V(x,y) sinα d1 dx.dy = [u ( a(x x 1 )+b(y y 1 ) )+v ( a(x 3 x 1 )+b(y 3 y 1 ) )+c ] (19) c = ax 1 +by 1 +c Equation (19) can be written as I 1 = V(x,y) dx.dy = T 1 +T +T 3 +T 4 +T 5 +T 6 Here T 1 [ a(x x 1)+b(y y 1) ] u dudv () (1) T 4 T 5 T 6 [ C (a(x x 1 )+b(y y 1 )) [ C (a(x 3 x 1 )+b(y 3 y 1 )) C dudv The above integration for first element is given as follows I 1 = V(x,y) dx.dy = v v 1 v v v 1 ( 1.73)v v 1 ( 1.73)v v v 3 ] ududv (4) ] vdudv (6) v v 1v 3 (7) After calculating the above integrals for each element in the above stated manner, we will find the sum of these integrals over the elements in which we have divided the cross-section. Here we have divided the cross-section into 6 elements (Fig. ). Summation of these integrals will result in a matrix B of size 7 7. Here, III. CALCULATION OF INTEGRAL V V dudv = The Jacobian J is given by ( X ) X J = U V = Y U ) ( ) ] V + dxdy dxdy = Jdudv (9) Y V ( x x 1 y y 1 x 3 x 1 y 3 y 1 ). (3) (5) (8) ISSN: Issue 1, Volume 1, January 11

4 Now Finally Here dx.dy = (x x 1 )(x 3 x 1 ). du.dv (31) V dudv = ) ( ) ] V + (x x 1 )(x 3 x 1 ) dudv (3) ) ( ) ] V + = a +b (33) After substituting all the values in above equation integrate, we get WSEAS TRANSACTIONS on COMPUTERS V dudv =.57798v v 1 v v.57814(v 1 v 3 +v v 3 v 3 ) (34) Here v 1,v, v 3 v n are the nodal potential. Solution of integration of V dudv for all the 6 element, computed in same manner will result in a matrix A of size 7 7. Here V is the vector of vertex nodal field values. Solution of this matrix will give the eigen values.these eigen values are the propagation frequencies of the waveguide.using above method we can calculate the propagation frequencies of a waveguide of any type of cross-section. V. SIMULATION RESULTS In order to validate the procedure, the computed result is compared with those obtained from the theoretical analysis. Table compares the Eigenvalues of the fundamental frequencies of propagation of the hexagonal waveguide with the theoretical computed. MATLAB software has been used here for the simulations [13]. TABLE II EIGENVALUES OF THE FUNDAMENTAL FREQUENCIES OF PROPAGATION OF THE HEXAGONAL WAVEGUIDE S. No. eigen value(this work) eigen value(theoretical) i i IV. FINDING EIGEN VALUES OF THE MATRIX Now the process of finding eigen values of the matrix is as follows. (V T AV k V T BV) when minimized over V gives the quadratic form defined by V dxdy k V dxdy (35) here δ ( V, V )dxdy = (,δ V)dxdy = δv Vdxdy (36) δ V dxdy = V δv dxdy (37) Now from equation (33) δv Vdxdy k δv.vdxdy = (38) or δv( +k )Vdxdy = (39) ( +k )Vdxdy = (4) equation 38 when evaluated approximately using the finite element method gives AV k BV = (41) (A k B)V = (4) det(a k B) = (43) VI. CONCLUSION In this paper an advantageous finite-element- method for the hexagonal cross-sectional waveguide problem has been developed by which complex propagation characteristics may be obtained for arbitrarily shaped waveguide. The extension to higher order elements is straightforward. By suitable modifications of the method it is possible to treat other types of waveguides as well, e.g. dielectric waveguides with impedance walls open unbounded dielectric waveguides properly treating the region of infinity. VII. ACKNOWLEDGEMENT The authors gratefully acknowledge Prof. Raj Senani, Ar. Rajesh Ayodhyawasi Dr. Ram Prakash Bharti for their constant encouragement provision of facilities for this research work. REFERENCES [1] P.P.Silvester R.L.Ferrari, Finite Elements for Electrical Engineers, (Second edition),cambridge University Press, New York,199. [] M.N.O.Sadiku, Numerical Techniques in Electromagnetics, CRC press, 6. [3] Vaish Arti Harish Parthasarathy, Numerical computation of the modes of electromagnetic wave propagation in a non-homogeneous rectangular waveguide, IEEE Conference, INDICON-6,15-17 September6. [4] Arti Vaish Harish Parthasarathy, Analysis of a rectangular Waveguide using Finite Element Method, Progress in Electromagnetic Research C, page , 8. [5] Arti Vaish Harish Parthasarathy, Wave propagation in a waveguide having anisotropic permittivity using method of moments, International Journal of Tomography Statistics, Fall 9, Vol. 1, No. F9. [6] M.N.O.Sadiku, Elements of Electromagnetics, Oxford university press, 7. ISSN: Issue 1, Volume 1, January 11

5 [7] Y. Xu R.G. Bosisio, An efficient method for study of general bianisotropic waveguides, IEEE Trans. Microwave Theory Tech., vol. 43, pp , Apr [8] A. C. Busacca, C. L. Sones, R. W. Eason, S. Mailis, K. Gallo, R. T. Bratfalean N.G. Broderick Surface hexagonally poled lithium niobate waveguides, Optoelectronics research centre,. [9] A. Szameit, Y. V. Kartashov, V. A. Vysloukh, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tnnermann, F. Lederer, L. Torner Angular surface solitons in sectorial hexagonal arrays, Optics letters, Vol. 33, No. 13, July 1, 8. [1] E.K.Miller (Ed.), Computational electromagnetics-a selected reprint volume, IEEE press,. [11] J.M.Jin, The finite element method in Electromagnetics, nd ed.,wiley, New York,6. [1] M. Koshiba K. Inoue, Simple efficient finite-element analysis of microwave optical waveguides, IEEE Trans. Microwave Theory Tech., vol. 4, pp , Feb [13] MATLAB(8) The Mathworks Worldwide.[Online] Available Arti Vaish received M.Tech. degree in Microwave engineering from the Rajiv Ghi Prodyogiki Vishwavidyalaya, Bhopal, INDIA, in 4, is currently working towards the Ph.D. degree. Her research interests include electromagnetic wave propagation in inhomogeneous anisotropic waveguide structures, numerical techniques in electromagnetic field problems design analysis of different types of microstrip patch antenna. She is working as Assistant Professor at Manav Rachna International University, Faridabad. Prof. Harish Parthasarathy received his B.Tech. degree in electrical engineering from IIT,Kanpur, in 199 his Ph.D. degree from IIT, Delhi in He completed his post doctoral programme from Indian Institute of Astrophysics in He has worked as an assistant professor at IIT, Bombay as a visiting faculty at IIT, Kanpur. Currently he is working as a Professor at Netaji Subhash Institute of Technology, New Delhi. His research interests include numerical techniques in electromagnetics, group representation theory stochastic processes. He has authored several books research papers on electromagnetics, signal processing, engineering mathematics physics. ISSN: Issue 1, Volume 1, January 11