Modal analysis of waveguide using method of moment

Transcription

1 HAIT Journal of Science and Engineering B, Volume x, Issue x, pp. xxx-xxx Copyright C 27 Holon Institute of Technology Modal analysis of waveguide using method of moment Arti Vaish and Harish Parthasarathy Digital Signal Processing Group, Division of Electronics and Communication Engineering, Netaji Suhas Institute of Technology, Dwarka Sector 3, New Delhi 1175, India Corresponding author: vaisharti@rediffmail.com Received 16 June 27, accepted 15 Novemer 27 Astract Maxwell s equation for a waveguide whose medium has an inhomogeneous dielectric constant is formulated. The resulting modified Helmholtz equation is transformed to a matrix generalized eigen value prolem using the method of moments. A complete numerical solution of Helmholtz equation for nonhomogeneous rectangular waveguide has een presented in this paper. This is implemented in MATLAB and the numerical values for the modes of propagation are otained. Keywords: Method of moment, anisotropic waveguide, propagation modes, eigen-vector, matrix equation. 1 Introduction The method of moments (MOM) is argualy the most developed numerical technique [1] for solving electromegnatic (EM) scattering and radiation prolems. Yet, the application of this method has een limited to resonant and lower frequencies. 1

2 A method of moments technique for the computational analysis of rectangular waveguide is presented elsewhere [1]. The waveguides are used to transmit electromagnetic signals. At high frequencies, this is the only practical way of transmitting electromagnetic radiation. In a waveguide, only special modes are transmitted and thus, the analysis of the eigenvalues and their eigenvectors ecomes very important in the electromagnetic. The other advantage of the waveguides is that they can handle high power with low losses e.g., attenuation loss, transmission loss etc. The electromagnetic fields in these waveguides are given y the Maxwell equations. While these equations are linear, the analytical solutions of them can only e otained in the spacial cases of oundary interface conditions. Due to this, electromagnetic applications have ecome more complex, however, the prediction of their performance, in general, would e important. Therefore, numerical methods have ecome a useful tool [2] to analyze the complex system of equations. Wave equations govern two types of the wave propagation, namely, the propagation of mechanical waves and of the e1ectromagnetic waves. They can often e written either in the second order or in a system of the first order. Most of the literature focus on the numerical solution of the first order wave equations ecause the second order can e transformed into the first order [3]. But there are some cases that the second order wave equations need to e solved. There are several explicit methods with second order and higher order approximations [4, 5], ut all the explicit methods have an upper and limit, for the time step size used. Present method can e used to find the solution of wave-equation for a waveguide having different region of permittivity and also for any shaped waveguide. The availale literature lacks with the numerical solution of the inhomogeneous Maxwell equations, even for a rectangular waveguides [6, 7]. Thus, the present work aims to fulfill this gap in the literature. In this work, Maxwell equations for a rectangular waveguide, whose medium has an inhomogeneous dielectric constant, are formulated. This formulation results in the modified Helmholtz equations, which are transformed into a generalized eigen value prolem using the method of moments. A complete numerical solution of this generalized eigen value prolem for inhomogeneous rectangular waveguide has een otained using MATLAB. The numerical values for modes of the propagation are otained and compared with the practical values. 2

3 2 Prolem statement Assume that the waveguide is rectangular with dimensions a and (Figure 1). The rectangular has een divided into four equal area regions. Let 1 and 2 e the permittivity for the lower ( y /2) regions1( x a/2) and 2 (a/2 x a), respectively. Figure 1: Rectangular waveguide cross section for four regions of permittivity Similarly, 3 and 4 is the permittivity for the upper part (/2 y )of the rectangle, i.e., regions 3 ( x a/2) and4(a/2 x a), respectively. A cross-section for this type of waveguide is shown in Figure 1. The wave equation is given y µ 2 x y 2 + ω2 µ + γ 2 ψ(x, y) = (1) suject to the oundary condition ψ(,y)==ψ(a, y) =ψ(x, ) = ψ(x, ) where ε(= ε.ε r ) and µ(= µ.µ r ) are the permittivity and the permeaility of the medium, respectively. The ε r and µ r are the relative permittivity and permeaility of the medium. The permittivity and the permeaility of the free space are given y ε = F/m (2a) µ = 4π 1 7 H/m (2) 3

4 and ω 2 µ + γ 2 = h 2 Here h 2 is a constant, and the propagation constant γ is given as follow γ = α + jβ Where, α and β are the attenuation constant and the phase constant, respectively. The test function, ψ(x, y), can e chosen as sin or cos function. By solving this equation, one can find the admissile values of propagation constant (γ). 3 The method of moments analysis To apply the method of moments, the electromagnetic field is expressed via the potential orthonormalized modal function ψ(x, y)as follow ψ(x, y) = m,n=1,2 NX m,n=1 C mn sin( mπx a )sin(nπy ) (3) Sustitution of equation (3) into equation (1) gives NX [( mπ a )2 +( nπ )2 ]C mn sin( mπx a )sin(nπy ) + ω 2 µ ( x, y) + γ 2 N X m,n=1,2 NX m,n=1,2 C mn sin( mπx a sin( mπx a )sin(nπy ) )sin(nπy )= (4) After multiplying equation (4) y sin( rπx a )sin(sπy ) and integrating over the cross-section of the waveguide, we get ( a 4 )[(rπ a )2 +( sπ )2 ]C rs NX Z a Z + ω 2 µc mn (x, y)sin( mπx a )sin(rπx a )sin(sπy ) 4 m,n=1,2 sin( nπy a )dx.dy + γ2 4 C rs = (5)

5 where r, s =1, 2 N. Let D e the diagonal matrix of size N 2 N 2 whose [N(r 1)+s, N(r 1) + s] th entry is a 4 [(rπ a )2 )+( sπ )2 )] and let A[r, s/m, n] = Z a Z x,y sin( nπy )sin(rπx a )sin(sπy )sin(mπx a )dx.dy Now, we reak this integration into four parts (of the rectangle, see Fig. 1) A[r, s/m, n] = Z a/2 Z /2 Z a Z /2 a/2 Z a/2 Z Z a a/2 /2 Z /2 1 sin( nπy )sin(rπx a )sin(sπy 2 sin( nπy )sin(rπx a )sin(sπy 3 sin( nπy )sin(rπx a )sin(sπy 4 sin( nπy )sin(rπx a )sin(sπy )sin(mπx a )dx.dy )sin(mπx a )dx.dy )sin(mπx a )dx.dy )sin(mπx a )dx.dy Let A[r, s/m, n] =I1[r, s/m, n]+i2[r, s/m, n]+i3[r, s/m, n]+i4[r, s/m, n] where I1[r, s/m, n] = a 1 4π 2 sin(r m)π/2sin(s n)π/2 sin(r m)π/2sin(s + n)π/2 [ (r m)(s n) (r m)(s + n) sin(r + m)π/2sin(s n)π/2 sin(r + m)π/2sin(s + n)π/2 + ] (r + m)(s n) (r + m)(s + n) for r 6= m, s 6= n (6) andifr=mands=nthen I1[r, s/m, n] = 1 a 16 (7) 5

6 and I2[r, s/m, n] = a 2 4π 2 sin(s n)π/2sin(r + m)π/2 sin(s n)π/2sin(r m)π/2 [ (s n)(r + m) (s n)(r m) sin(r + m)π/2sin(s + n)π/2 sin(s + n)π/2sin(r m)π/2 + ] (r + m)(s + n) (s + n)(r m) for r 6= m, s 6= n and if r=m and s=n then Similarly, I2[r, s/m, n] = 2 a 16 (8) (9) I3[r, s/m, n] = a 3 4π 2 sin(r m)π/2sin(s + n)π/2 sin(r m)π/2sin(s n)π/2 [ (r m)(s + n) (r m)(s n) sin(r + m)π/2sin(s + n)π/2 sin(r + m)π/2sin(s n)π/2 + ] (r + m)(s + n) (r + m)(s n) for r 6= m, s 6= n (1) and if r=m and s=n then and I3[r, s/m, n] = 3 a 16 (11) I4[r, s/m, n] = a 4 4π 2 sin(s + n)π/2sin(r + m)π/2 sin(s n)π/2sin(r + m)π/2 [ (s + n)(r + m) (s n)(r + m) sin(r m)π/2sin(s + n)π/2 sin(s n)π/2sin(r m)π/2 + ] (r m)(s + n) (s n)(r m) for r 6= m, s 6= n (12) and if r=m and s=n then 6 I4[r, s/m, n] = 4 a 16 (13)

7 Now I[r, s/m, n] =I1[r, s/m, n]+i2[r, s/m, n]+i3[r, s/m, n]+i4[r, s/m, n] (14) Sustituing the values of I 1, I 2, I 3 and I 4 into equation (14). If r6= mand s6=n thenweget I[r, s/m, n] = a( ) 4π sin(s 2 n)π/2 sin(s + n)π/2 s n s + n sin(r m)π/2 sin(r + m)π/2 r m r + m (15) andifr=mands=nthen I[r, s/m, n] = a 16 [ ] (16) Here we assume that 1 =4, 2 =2, 3 =1, 4 =3 and =4 (17) Now y sustituting the value of equation (17) into equations (15) and (16), we have I[r, s/m, n] = a sin(s n)π/2 sin(s + n)π/2 π 2 s n s + n sin(r m)π/2 sin(r + m)π/2 r m r + m for r 6= m, s 6= n (18) and at r=s and m=n we have I[r, s/m, n] = a (19) 4 Now, let A denote an N 2 N 2 matrix whose [N(r 1) + s, N(m 1) + n] th entry equals A[r, s/m, n], then the system of linear equation will e expressed as ω 2 µac + Dc + γ 2 ( a )c = (2) 4 Here ω 2 µa + D = B (21) 7

8 Now equation (2) can e written as Bc + λc = (22) now the admissile values of γ are given y r λk γ k = ±2 (23) a k =1, 2, 3 N 2. where λ k : k =1, 2, 3 N 2 are the eigen values of the matrix B = ω 2 µa + D (24) 4 Simulation results and discussion The simulations have een carried out for the frequency, ω =2π rad/sec and for N =8, 16, 24. The values of γ has een calculated from the eigen values of the matrix, B = ω 2 µa + D. The approximate theoretical values of the modes are also calculated y averaging the values of the permittivities throughout the waveguide cross-section. The attenuation constant is very low in the calculated values, which is always desirale. Tale 1 shows a comparison etween the theoretical and calculated values. The two values (theoretical and calculated) are seen to e very close to each other in Tale 1. This close agreement in etween these values clearly show a validity of the present solution procedure. In this work, we have considered the structure with rectangular crosssection. However, this method is not limited to any structure and can e applied to any kind of structure, as shown in the Fig. 2. Similar procedure canefollowedtofind the solution of the other (non-rectangular) structures. In summary, we have started y writing down the modified Helmholtz equation for the propagation of TM modes in a rectangular waveguide with piecewise constant permittivity. The solution ψ(x, y) has een expanded in terms of sine waves. Then the method of moments is applied to reduce the original infinite dimensional prolem to a finite dimensional matrix generalized eigenvalue prolem. The size of the matrices depend on the numer of sine-waves (N) used in the the expansion. As this numer of sine-waves grows, the size of the matrices also grow. In this paper, the propagation modes are otained using MATLAB for three sample values of N (see Tale 1). Numerical simulations show that the modes converge quite rapidly when N>15. It would e quite difficult to otain a theoretical solution of the modes as a function of N. This would involve determining the ehavior 8

9 Tale 1: Comparison etween the theoretical and calculated values of γ. γ(theoretical values) γ(calculated values) N=8 N=16 N= i i i i 933.5i i i i 943.6i i i i 956.9i i i i 956.9i i i i 958.5i i i i 962.6i 937.2i i 964.8i 962.6i i i i 11.6i i 957.4i i 16.4i i i i i i i i i i i i 141.1i i i i 141.1i i i i i i i i i i i i of the eigenvalues of the matrix as a function of its size. Specifically one would need to address a prolem of the following kind. Let 1 and 2 e two linear partial differential operators and φ n n = 1, 2, 3 n is a sequence of test functions. Then determine the ehavior of the solution, of the equation as N increases where µµz A N = f N (λ) =det(a N λb N )= D φ n (x, y) 1 φ m (x, y)dxdy and µµz B N = φ n (x, y) 2 φ m (x, y)dxdy As far as known to us, none of the existing methods can e used to solve such type of prolem. 9

10 Figure 2: Different possile shapes of waveguides in which present method can e applied to otain the propogation modes. 5 Conclusion In this work, electromagnetic wave propagation analysis has een carried out numerically. The Maxwell equations have een formulated for a inhomogeneous rectangular waveguide. The resulting modified Helmholtz equations are transformed into a generalized eigen values prolem using the method of moments. The eigen values of the system of linear equations are otained y using MATLAB. These eigen values are used to otain the wave propagation coefficient (γ). A good correspondence was seen etween the theoretical and practical values. Attenuation is very low and the propagation is high. We could calculate the modes y using any numerical technique like finite element method, finite difference method, ut the reason for using method of moment here is due to it s simplicity. Microwave elements 1

11 constructed using waveguide sections are pretty common. When allowance is made for variale dielectric constant then one attains a greater degree of freedom leading to a wide set of circuit parameters. In other words, a wider range of impedances can e generates using the same waveguide section y appropriately adjusting the distriution of the dielectric inside it. The authors gratefully acknowledge Prof. Raj Senani for his constant encouragement and provision of facilities for this research work. References [1] R.F. Harrington, Field Computation y Moment Methods (Macmillan, New York, 1968). [2] M.N.O. Sadiku, IEEE Transactions on Education 32, 85 (1989). [3] J.W. Thomas, Numerical Partial Differential Equations (Springer- Verlag, New York, 1998). [4] W.F. Ames, Numerical Methods for Partial Differential Equations, 3rd Ed. (Academic Press, Boston, 1986). [5] G.C. Cohen, Higher-order numerical methods for transient wave equations (Springer, New York, 22). [6] L. Tsang, J.A. Kong, K.H. Ding, and C.O. Ao, Scaftering of Electromagnetic Waves (Wiley Interscience, New York, 21). [7] K.S. Yee, IEEE Trans. Antennas and Propagation 14, 32 (1996). 11