New Scaling Factors of 2-D Isotropic-Dispersion Finite Difference Time Domain (ID-FDTD) Algorithm for Lossy Media

Transcription

1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO., FEBRUARY 8 63 is shown by squares. Each point is computed by averaging reflection coefficient values calculated for each component of the regular H wave propagating in the waveguide. Results of the calculations are compared with analytical calculations ug approximate formulas and experimental results found in literature []. Good coincidence between our data and the results from [] indirectly proves the effectiveness of the proposed method of calculation of the electromagnetic field components on the z-axis. To calculate the electromagnetic field components at r =, the authors also used formulas based on a series polynomial expansion [5], [6]. Unfortunately, calculated H r and E ' values at r =differ significantly from those shown in Fig. 3(a) and (c). Apparently, the number of cells in the azimuthal direction used in the calculations of the authors is too small to obtain sufficient accuracy of the series expansion. It should be noted that the angular dependence of electromagnetic field components can be directly considered in Maxwell s equations () and (3), and the current 3-D problem can be reduced to an equivalent -D one [3] in the rz plane. From (7), (9), and (), it is easy to show that in the -D case for this particular excitation of H wave H rj n+= = ;k+ H'jn+= ; ;k+ Ezjn+ ;k+ =; E'jn+ ;k = Erjn+ ;k (4) where the first lower index corresponds to r and the second to the z coordinate, respectively. Computed distribution of electromagnetic field components in the rz plane as well as reflection coefficient values coincide with those computed for the 3-D model. The authors compare the results of the calculation of electromagnetic field components for -D model obtained by ug the present approach (4) and the methods handling gularity presented in [] and [3] for r = z = : and = c = :8. In [], the updated formulas for calculating H r and E ' at r =were obtained from an integral form of Maxwell equations. Comparing this method with the present approach (4), it was found that the difference between the amplitudes of components H r and E ' calculated at r =was less than 3. Calculated off-axis components coincide with each other to an accuracy of a floating-point number. The other method to overcome the gularity by shifting the grid with respect to the z-axis by r= was proposed in [3]. Ug such a grid, the only component that should be determined at r =is H z but it is zero due to rotational symmetry of the structure. Comparing this method with the present approach (4), it was found that the difference between the amplitudes of component E ' calculated at r =:was less than :7 3. It is seen that the differences in both cases is less than r. On a basis of these calculations, it can be concluded that the proposed method provides sufficient accuracy for the FDTD procedure. The proposed method can also be applied to the electromagnetic waves propagating through a material medium. Conductivity and dielectric properties of the material should be considered in (9) for E z calculation in the same way as for the FDTD scheme in Cartesian coordinates. The method of resolving numerical gularity is also applicable in the construction of higher order accuracy in space FDTD schemes [] for the cylindrical coordinate system. REFERENCES [] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell s equation in isotropic media, IEEE Trans. Antennas Propag., vol. 4, pp. 3 37, May 966. [] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 995. [3] Y. Chen, R. Mittra, and P. Harms, Finite-difference time-domain algorithm for solving Maxwell s equation in rotationally symmetric geometries, IEEE Trans. Microw. Theory Tech., vol. 44, pp , Jun [4] N. Dib, T. Weller, M. Scardeletti, and M. Imparato, Analysis of cylindrical transmission lines with the finite-difference time-domain method, IEEE Trans. Microw. Theory Tech., vol. 47, pp. 59 5, Apr [5] F. Liu and S. Crozier, An FDTD model for calculation of gradientinduced eddy currents in MRI system, IEEE Trans. Appl. Supercond., vol. 4, pp , Sep. 4. [6] A. Trakic, H. Wang, F. Liu, H. S. López, and S. Crozier, Analysis of transient eddy currents in MRI ug a cylindrical FDTD method, IEEE Trans. Appl. Supercond., vol. 6, pp , Sep. 6. [7] G. S. Constantinescu and S. K. Lele, A highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates ug series expansions, J. Comput. Phys., vol. 83, pp , Nov.. [8] K. Fukagata and N. Kasagi, Highly energy-conservative finite difference method for the cylindrical coordinate system, J. Comput. Phys., vol. 8, pp , Sept.. [9] S. I. Baskakov, Basics of Electrodynamics (in Russian). Moscow: Sov. Radio, 973, pp [] Q. Chen and V. Fusco, Three dimensional cylindrical coordinate finite difference time domain analysis of curved slotline, in Proc. nd Int. Conf. Computations in Electromag., Londong, U.K., 994, pp [] G. Z. Aizenberg, V. G. Jampolskij, and O. N. Terioshin, UHF Antennas (in Russian). Moscow: Sviaz, 977, pt., p. 4. [] K. L. Shlager and J. B. Schneider, Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes, IEEE Trans. Antennas Propag., vol. 5, pp. 95 4, Apr. 4. New Scaling Factors of -D Isotropic-Dispersion Finite Difference Time Domain (ID-FDTD) Algorithm for Lossy Media Il-Suek Koh, Hyun Kim, and Jong-Gwan Yook Abstract We modify the -D explicit isotropic-dispersion finite-difference time-domain (ID-FDTD) scheme originally proposed for lossless media [], which can drastically improve the accuracy of the -D ID-FDTD scheme for lossy media. The proposed scheme adopts new scaling factors for material properties such as the permittivity, permeability and conductivity. The new scaling factors can generate nearly the exact phase velocity and attenuation factor of a lossy medium for a gle frequency. For the validation of the proposed scheme, two scattering problems are considered. Index Terms Finite-difference time-domain (ID-FDTD) methods, numerical analysis, numerical dispersion. I. INTRODUCTION Since the standard finite-difference time-domain (FDTD) method was proposed by K. Yee, providing many advantages such as low computation complexity, great flexibility, and a simple implementation, the method has been applied to various electromagnetic problems [], [3]. The standard FDTD method was formulated based on the central Manuscript received February 7, 7; revised July 3, 7. This work was supported by the Korea Research Foundation funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) under Grant KRF-6-33-D398. I.-S. Koh is with the Graduate School of Information Technology and Telecommunication, Inha University, Incheon 4-75, Korea ( ikoh@inha.ac.kr). H. Kim and J.-G. Yook are with the Department of Electrical and Electronics Engineering,Yonsei University, Seoul -749, Korea. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier.9/TAP X/$5. 8 IEEE

2 64 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO., FEBRUARY 8 FD approximation for the time and spatial derivatives in Maxwell s equation. Hence, the scheme experiences what is known as numerical dispersion problem, which prevents the method from being applied to large scale or phase-sensitive problems [3]. Several schemes have been proposed to rectify the numerical dispersion problem, as shown in [4]; one of these solutions is based on a weighted summation of two different FD approximations for the spatial derivative []. The resulting -D FDTD scheme is denoted as the isotropic dispersion FDTD (ID-FDTD) scheme, which is formulated and verified for lossless media in []. For lossy media, the dispersion error of the original ID-FDTD scheme may be increased due to the conductivity. Therefore, in this paper, the original ID-FDTD scheme is modified so that it can produce the nearly exact phase velocity and the attenuation factor for a given lossy medium for a gle frequency. For this purpose, the scaling factor of the ID-FDTD scheme is reformulated. In Section II, the new scaling factors are obtained, and various properties of the resulting scheme are investigated. In order to verify the proposed scheme, two scattering examples are considered in Section III. II. FORMULATION In the ID-FDTD scheme two parameters, the weighting factor and the scaling factor, should be determined based on the dispersion relation []. The weighting factor can control the anisotropy of the dispersion of the scheme over the propagation angle. The scaling factor is used to adjust the numerical dispersion to the exact known value. For consistency with a lossless case, the weighting factor for a lossless medium can be used. However, the scaling factor should be reformulated, as in a lossy medium the phase velocity and the magnitude of a field are affected simultaneously by the medium. For a lossless case, one scaling factor is required to scale the permittivity and the permeability of the medium to achieve the exact phase velocity. For a lossy medium, however, an additional scaling factor should be introduced for the conductivity to consider the phase velocity and the attenuation factor simultaneously. In this section, the two required scaling factors are formulated. A. Dispersion Relation for Lossy Media In a lossy medium whose relative permittivity, relative permeability and conductivity are given by " r, r and, respectively, the ID-FDTD scheme for the TM z mode can be represented as [] for the time derivatives and d x;y is the spatial FD approximation defined in []. By ug the Z-transform method [3], the dispersion relation of () can be obtained simply as = c " r!t x j^ x x + y j^ y y j t!t " " j^ y y j^ x x where c is the phase velocity of the free space. ^ x = ^ and ^ y =^. Here, ^ is a numerical wavenumber in the lossy media and is an azimuth angle (propagation angle). is the weighting factor for which the closed expression can be found in []. As () is a complex equation and is a real number, the roots, ^, of () should be a complex number, as expected. For =, () is reduced to that for the lossless media [], for which ^ becomes a pure imaginary number. In this paper, only a square cell is considered, which is x =y =as in the lossless case []. B. Scaling Factors To investigate the fact that the weighting factor for lossless media can provide an isotropic dispersion (phase velocity and attenuation factor) even for a lossy case, the dispersion is plotted as a function of the propagation angle in Fig. for " r = : j:, = cells per wavelength (CPW), and S =:7. Here, S is a Courant number defined by c t=. As seen in the figure, the computed dispersion (uncorrected case) is nearly isotropic, but the phase velocity and the attenuation factor are off the exact values. Hence, to obtain the exact wavenumber, the real part of permittivity, the permeability and the conductivity of the medium are scaled by sc and sc, respectively. As in [], the permittivity and the permeability can be scaled by the same factor of sc. Hence, new permittivity, permeability and conductivity values are given by sc ", sc r, and sc, respectively. The numerical wavenumber can be calculated by () at = over all propagation angles due to the resulting isotropy, which is given by () d t H n x I;J + = t r y d ye n z I;J +!t j t!t " " d t H n y I + t ;J = d xe n z I + rx ;J d t E n+= z (I;J)+ t E n+ " " z (I;J)+E n z (I;J) = t n+= dxh " " y (I;J) x = c j^ " r : (3) For the given material properties, the exact wavenumber, exact, is known [5]. However, exact does not satisfy (3) for real values of ", r and, thus (3) should be modified with the scaling factors, as t " " y dyhn+= x (I;J) ()!t j sc t!t " sc " where f n (I;J) indicates f(ix; Jy; nt), and " and represent the permittivity and the permeability of the free space, respectively. " r = " j" is assumed, and " is related to the conductivity as =!" ". Here, j = p and! is known as the angular frequency. In addition, x and y are the cell sizes of the x, and y directions, respectively, and t is a time step. d t is the central FD approximation = c sc " r j exact : (4) Equation (4) can be decomposed into two real equations by balancing the real and imaginary parts of both side of (4), as all parameters in (4) are real except exact. The equations can then be analytically solved

3 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO., FEBRUARY 8 65 Fig.. Comparison of the dispersion error of the ID-FDTD and Yee schemes as a function of the propagation angles. For this calculation, it is assumed that " = : j:, CPW and S = :7. (a) Normalized attenuation constant. (b) Normalized phase velocity. Fig.. Dispersion error of the ID-FDTD and Yee schemes as a function of the real part of the dielectric constant with three imaginary parts:.,. and.5. CPW and S =:7 are assumed. (a) Attenuation error. (b) Phase error. for the scaling factors, as sc = S p " r!t Re sc = sc " " tan!t t j exact Im j Re j (5) : (6) where Re() and Im() are the real and imaginary parts of a complex number, respectively. For =, sc becomes zero, and sc is reduced to that for the lossless case, as expected []. Fig. shows a comparison of the phase velocity and the attenuation constant of the Yee and ID-FDTD schemes. As seen in the figure, the new scaling factors, (5) and (6) (corrected case), can provide nearly exact results. To examine the effect of the dielectric constant, the difference between the maximum and minimum of the numerical wavenumber along the azimuth angle is plotted as a function of the real part of the dielectric constant in Fig.. For this calculation, CPW and S = :7 are assumed, and three imaginary terms of the dielectric constant are considered:.,. and.5. If the real and imaginary parts of the dielectric constant increase, the difference will slowly increase. However, for a larger dielectric constant, a smaller cell should be required. Thus, CPW is not sufficient for large dielectric constants, but if the CPW is increased, the difference can be reduced. Therefore, the resulting dispersion error can be ignored for practical applications. Fig. 3 shows the wideband performance of the proposed ID-FDTD scheme. For this simulation, all parameters are optimized for 3 CPW. As expected, a null can be observed at N =3, and the bandwidth of the null may be narrow compared to other methods [4]. III. NUMERICAL RESULTS To demonstrate the validity of the proposed ID-FDTD scheme, two scattering problems are considered. For all simulations, the operating frequency is set at 3 MHz; r = and a TM z incidence wave are assumed. The first example is of scattering by an infinitely long -D dielectric circular cylinder whose radius and dielectric constant are 5, and " r = : j:, respectively. Here, is the free

4 66 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO., FEBRUARY 8 Fig. 3. Dispersion error of the ID-FDTD and Yee schemes as a function of cell size. All parameters are optimized at N =3and S =:7. Four dielectric constants are considered: : i:, : j:5, j: and j:5. (a) Attenuation error. (b) Phase error. Fig. 4. Scattered electric field by a large dielectric circular cylinder calculated by the ID-FDTD scheme, the Yee scheme, and the exact eigen-series solution. For this calculation, " = :j:, and S = :7 are assumed, and frequency is set to be 3 MHz. (a) Real part. (b) Imaginary part. space wavelength. Fig. 4 shows plots of the real and imaginary parts of the scattered field calculated by four methods: the ID-FDTD scheme with the weighting factors for lossless and lossy cases, the standard FDTD scheme and the exact eigen-series solution [5]. The observation points are located along a line, (; ) in the forward direction, as seen in Fig. 4. For this simulation, a cell size of = is used for the ID-FDTD scheme, and S is fixed at.7 for both the ID-FDTD and the Yee schemes. As seen in the figure, the accuracy of the Yee method is not satisfactory even for the smaller cell (=), but the ID-FDTD method can generate very accurate results compared with the exact result over the whole comparison region. It was also observed that the accuracy of the ID-FDTD scheme with the weight factor of the lossless case is degenerated for even the slightly lossy medium of this cylinder. This occurs because the size of the cylinder is large enough for the scattered field to be greatly affected by the phase velocity and the attenuation factor of the medium. The second example is of scattering by an infinitely long inhomogeneous square cylinder. The scatterer is shown in Fig. 5. It has three layers with a total size of 5 5. The two outer layers have the same dielectric constant of "r =: j:5 and the same thickness of. The core layer has a dielectric constant of "r =: j: and a thickness of 3. The scattered field is calculated by the four methods: the two ID-FDTD schemes, the standard FDTD scheme, and method of moments (MoM) with a pulse basis function as shown in Fig. 5. For this simulation, the cell size is fixed at = for all FDTD methods. As expected, the accuracy of the ID-FDTD scheme with the new scaling factors is superior to those of the other FDTD methods compared with that of MoM with cell sizes of = and =5. The small discrepancy between the results of the ID-FDTD scheme and MoM may have resulted from the fact that the observation line was very close to the scatterer, caug the accuracy of the pulse basis function in the MoM method and that of the ID-FDTD scheme to be unsatisfactory for this type of very near region. Therefore, the error may be reduced if a smaller cell is used, as can be observed in Fig. 5.

5 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO., FEBRUARY 8 67 APPENDIX The proposed -D ID-FDTD update equations for lossy media are explicitly given by E n+ z (I;J) = C E n z (I;J) + C H n+= y I + n+= ;J Hy I ;J H n+= x I;J + + H n+= x I;J +C 3 H n+= y I + n+= ;J+ Hy I ;J+ + H n+= y I + n+= ;J Hy I ;J H n+= x I +;J + + H n+= x I +;J H n+= x I ;J+ + H n+= x I ;J (A.) where C = (" " t)=(" " + t), C = ( )t=(" " + t), and C 3 =:5t=(" " + t) H n+= x I;J+ = H n= x I;J + C 4 (E n z (I;J +) E n z (I;J)) C 5(E n z (I +;J+)E n z (I +;J) H n+= y I + n= ;J = Hy I + ;J + E n z (I;J+)E n z (I;J)) (A.) Fig. 5. Scattered electric field by an inhomogeneous square cylinder calculated by the ID-FDTD scheme, the Yee scheme, and MoM with a pulse basis function. The cylinder has three dielectric layers. For this calculation, " =:j:, " = : j:5, and S = :7 are assumed, and frequency is set to be 3 MHz. (a) Real part. (b) Imaginary part. IV. CONCLUSION In this paper, the -D ID-FDTD scheme for a lossless medium proposed in [] is modified for application to a lossy medium. The proposed -D ID-FDTD scheme has the same weighting factor as that of the lossless case for consistency, but the scaling factor is analytically reformulated. The resulting dispersion is nearly isotropic, and the exact phase velocity and attenuation factor can be achieved for a given medium. As other properties of the proposed scheme, such as the stability condition and the computational complexity, are independent on the scaling factor, the same results for a lossless case can be applied to a lossy case. To investigate the validity of the proposed ID-FDTD scheme, two scattering examples were considered. The first of these was of scattering by a large lossy dielectric cylinder, and the second was of an inhomogeneous square cylinder. For the two cases, the proposed ID-FDTD scheme produced very accurate results. + C 4 (E n z (I +;J) E n z (I;J)) + C 5 (E n z (I +;J+)E n z (I;J+) + E n z (I +;J)E n z (I;J)) (A.3) where C 4 = ( :5)t= r, C 5 = :5t= r.itis easily observed that at each iteration, 7 multiplication and 4 addition/ subtraction operators are required when the coefficients of C C 5 are pre-calculated and saved. REFERENCES [] I. Koh, H. Kim, J. Lee, J. Yook, and C. Pil, Novel explicit -D FDTD scheme with isotropic dispersion and enhanced stability, IEEE Trans. Antennas Propag., vol. 54, no., pp , Nov. 6. [] K. Yee, Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media, IEEE Trans. Antennas Propag., vol. 4, no. 3, pp. 3 37, May 966. [3] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, nd ed. Norwood, MA: Artech House, 995. [4] K. L. Shlager and J. B. Schneider, Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithm, IEEE Trans. Antennas Propag., vol. 5, no. 3, pp , Mar. 3. [5] N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics. Boston, MA: Artech House, 99.