volume | PIER Journals

Abstract

The stability and numerical error of the extended four-stages split-step finite-difference time-domain (SS4-FDTD) method including lumped inductors are systematically studied. In particular, three different formulations for the lumped inductor are analyzed: the explicit, the semi-implicit, and the implicit schemes. Then, the numerical stability of the extended SS4-FDTD method is analyzed by using the von Neumann method, and the results show that the proposed method is unconditionally-stable in the semi-implicit and the implicit schemes, whereas it is conditionally stable in the explicit scheme, which its stability is related to both the mesh size and the values of the element. Moreover, the analysis of the numerical error of the extended SS4-FDTD is studied, which is based on the Norton equivalent circuit. Theoretical results show that: 1) the numerical impedance is a pure imaginary for the explicit scheme; 2) the numerical equivalent circuit of the lumped inductor is an inductor in parallel with a resistor for the semi-implicit and implicit schemes. Finally, a simple microstrip circuit including a lumped inductor is simulated to demonstrate the validity of the theoretical results.

1. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag., Vol. 14, No. 3, 302-307, May 1966.
doi:10.1109/TAP.1966.1138693 Google Scholar

2. Taflove, A. and S. C. Hagness, "Computational Electrodynamics: The Finite-difference Time-domain Method," Artech House, Boston, MA, 2000. Google Scholar

3. Namiki, T., "A new FDTD algorithm based on alternating-direction implicit method," IEEE Trans. Microwave Theory Tech., Vol. 47, No. 10, 2003-2007, Oct. 1999.
doi:10.1109/22.795075 Google Scholar

4. Zheng, F., Z. Chen, and J. Zhang, "Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method-," IEEE Trans. Microwave Theory Tech., Vol. 48, No. 9, 1550-1558, Sep. 2000.
doi:10.1109/22.869007 Google Scholar

5. Tay, W. C., D. Y. Heh, and E. L. Tan, "GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer," Progress In Electromagnetics Research M, Vol. 14, 177-192, 2010.
doi:10.2528/PIERM10090605 Google Scholar

6. Gan, T. H. and E. L. Tan, "Stability and dispersion analysis for three-dimensional (3-D) leapfrog ADI-FDTD method," Progress In Electromagnetics Research M, Vol. 23, 1-12, 2012.
doi:10.2528/PIERM11111803 Google Scholar

7. Sun, G. and C. W. Trueman, "Approximate Crank-Nicolson schemes for the 2-D finite-difference time-domain method for TEz waves," IEEE Trans. Antennas Propag., Vol. 52, No. 11, 2963-2972, Nov. 2004.
doi:10.1109/TAP.2004.835142 Google Scholar

8. Sun, G. and C. W. Trueman, "E±cient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method," IEEE Trans. Microw. Theory Tech., Vol. 54, No. 5, 2275-2284, May 2006.
doi:10.1109/TMTT.2006.873639 Google Scholar

9. Rouf, H. K., F. Costen, S. G. Garcia, S. Fujino, "On the solution of 3-D frequency dependent Crank-Nicolson FDTD scheme," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 16, 2163-2175, Jan. 2009.
doi:10.1163/156939309790109261 Google Scholar

10. Xu, K., Z. Fan, D. Z. Ding, and R.-S. Chen, "GPU accelerated unconditionally stable Crank-Nicolson FDTD method for the analysis of three-dimensional microwave circuits," Progress In Electromagnetics Research, Vol. 102, 381-395, 2010.
doi:10.2528/PIER10020606 Google Scholar

11. Fu, , W. and E. L. Tan, "Development of split-step FDTD method with higher-order spatial accuracy," Electron. Lett., Vol. 40, No. 20, 1252-1253, Sep. 2004.
doi:10.1049/el:20046040 Google Scholar

12. Fu, W. and E. L. Tan, "Compact higher-order split-step FDTD method," Electron. Lett., Vol. 41, No. 7, 397-399, Mar. 2005.
doi:10.1049/el:20057927 Google Scholar

13. Shibayama, J., M. Muraki, J. Yamauchi, and H. Nakano, "Effcient implicit FDTD algorithm based on locally one-dimensional scheme," Electron. Lett., Vol. 41, No. 19, 1046-1047, Sep. 2005.
doi:10.1049/el:20052381 Google Scholar

14. Do Nascimento, V. E., J. A. Cuminato, F. L. Teixeira, and B.-H. V. Borges, "Unconditionally stable finite-difference time-domain method based on the locally-one-dimensional technique," Proc. XXII Simposio Brasileiro de Telecomunicacoes, 288-291, Campinas, SP, Brazil, Sep. 2005. Google Scholar

15. Liang, F. and G. Wang, "Fourth-order locally one-dimensional FDTD method," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 14-15, 2035-2043, Jan. 2008.
doi:10.1163/156939308787538017 Google Scholar

16. Tan, E. L., "Unconditionally stable LOD-FDTD method for 3-D Maxwell's equations," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 2, 85-87, Feb. 2007.
doi:10.1109/LMWC.2006.890166 Google Scholar

17. Ahmed, I., E. K. Chua, E. P. Li, and Z. Chen, "Development of the three-dimensional unconditionally-stable LOD-FDTD method," IEEE Trans. Antennas Propag., Vol. 56, No. 11, 3596-3600, Nov. 2008.
doi:10.1109/TAP.2008.2005544 Google Scholar

18. Kong, Y.-D. and Q.-X. Chu, "Reduction of numerical dispersion of the six-stages split-step unconditionally-stable FDTD method with controlling parameters," Progress In Electromagnetics Research, Vol. 122, 175-196, 2012.
doi:10.2528/PIER11082512 Google Scholar

19. Chu, Q. X. and Y. D. Kong, "High-order accurate FDTD method based on split-step scheme for solving Maxwell's equations," Microwave. Optical Technol. Lett., Vol. 51, No. 2, 562-565, Feb. 2009.
doi:10.1002/mop.24100 Google Scholar

20. Chu, , Q. X. and Y. D. Kong, "Three new unconditionally-Stable FDTD methods with high-order accuracy," IEEE Trans. Antennas Propag., Vol. 57, No. 9, 2675-2682, Sep. 2009.
doi:10.1109/TAP.2009.2027045 Google Scholar

21. Kong, Y. D. and Q. X. Chu, "Unconditionally-stable FDTD methods with high-order accuracy in two and three dimensions," IET Microwaves, Antennas Propag., Vol. 4, No. 10, 1605-1616, Nov. 2010.
doi:10.1049/iet-map.2009.0222 Google Scholar

22. Kong, Y. D. and Q. X. Chu, "High-order split-step unconditionally-stable FDTD methods and numerical analysis," IEEE Trans. Antennas Propag., Vol. 59, No. 9, 3280-3289, Sep. 2011.
doi:10.1109/TAP.2011.2161543 Google Scholar

23. Pereda, J. A., F. Alimenti, P. Mezzanotte, L. Roselli, and R. Sorrentino, "A new algorithm for the incorporation of arbitrary linear lumped networks into FDTD simulators," IEEE Trans. Microw. Theory Tech., Vol. 47, No. 6, 943-949, Jun. 1999.
doi:10.1109/22.769330 Google Scholar

24. Chen, Z.-H. and Q.-X. Chu, "FDTD modeling of arbitrary linear lumped networks using piecewise linear recursive convolution technique," Progress In Electromagnetics Research, Vol. 73, 327-341, 2007.
doi:10.2528/PIER07042002 Google Scholar

25. Su, D. Y., D.-M. Fu, and Z.-H. Chen, "Numerical modeling of active devices characterized by measured S-parameters in FDTD," Progress In Electromagnetics Research, Vol. 80, 381-392, 2008.
doi:10.2528/PIER07120902 Google Scholar

26. Silva-Macedo, J. A., M. A. Romero, and B.-H. V. Borges, "An extended FDTD method for the analysis of electromagnetic field rotators and cloaking devices," Progress In Electromagnetics Research, Vol. 87, 183-196, 2008.
doi:10.2528/PIER08101507 Google Scholar

27. Thiel, W. and L. P. B. Katehi, "Some aspects of stability and numerical dissipation of the finite-difference time-domain (FDTD) technique including passive and active lumped elements," IEEE Trans. Microw. Theory Tech., Vol. 50, No. 9, 2159-2165, Sep. 2002.
doi:10.1109/TMTT.2002.802330 Google Scholar

28. Kung, F. and H. T. Chuah, "Stability of classical finite-difference time-domain (FDTD) formulation with nonlinear elements --- A new perspective," Progress In Electromagnetics Research, Vol. 42, 49-89, 2003.
doi:10.2528/PIER03010901 Google Scholar

29. Pereda, J. A., A. Vegas, and A. Prieto, "Study on the stability and numerical dispersion of the FDTD technique including lumped inductors," IEEE Trans. Microw. Theory Tech., Vol. 52, No. 3, 1052-1058, Mar. 2004.
doi:10.1109/TMTT.2004.823589 Google Scholar

30. Fu, W. and E. L. Tan, "Unconditionally stable FDTD technique including passive lumped elements," International RF and Microwave Conference Proceedings, 277-280, Putrajaya,Malaysia, Sep. 2006. Google Scholar

31. Fu, W. and E. L. Tan, "Unconditionally stable ADI-FDTD method including passive lumped elements," IEEE Trans. , Vol. 48, No. 4, 661-668, Nov. 2006. Google Scholar

32. Chu, Q. X. and Z. H. Chen, "Numerical dispersion of the ADI-FDTD technique including lumped models," IEEE MTT-S Int. Mcro. Sym. Dig., 729-732, Honolulu, Unite States, Jun. 2007. Google Scholar

33. Lin, G. and Y. G. Zhou, "Extending the three-dimensional LOD-FDTD method to lumped load and voltage source with impedance," International Conference on Microwave Technology and Computational Electromagnetics, 341-343, Beijing, China,2009. Google Scholar

34. Lee, J. and B. Fornberg, "A split step approach for the 3-D Maxwell's equations," J. Comput. Appl. Math., Vol. 158, 485-505, Mar. 2003.
doi:10.1016/S0377-0427(03)00484-9 Google Scholar

35. Pereda, J. A., L. A. Vielva, A. Vegas, and A. Prieto, "Analyzing the stability of the FDTD technique by combining the von Neumann method with the Routh-Hurwitz," IEEE Trans. Microw. Theory Tech., Vol. 49, No. 2, 377-381, Feb. 2001.
doi:10.1109/22.903100 Google Scholar

Dr. Yong-Dan Kong

Prof. Qing-Xin Chu

Prof. Rong-Lin Li