在高速數位系統中，由於操作頻率的提高，使得高速電路與印刷電路板間的訊號完整性的問題愈趨重要。要完整分析訊號完整性的問題，可以將主動及被動元件及整個結構一起放入全波模擬器裏分析。時域有限差分法便是其中一個較方便的全波模擬方法，其原因是，時域有限差分法可直接與電路的SPICE模型做結合。如何更有效率的將集總電路加入時域有限差分法便是一個重要的課題。本論文利用控制理論及Crank-Nicolson 法，可將任意的單埠或多埠線性電路有效率的結合入時域有限差分法。並提出一種將二極體模型加入時域有限差分法而不需減少時間步階的新方法。在本論文中，藉由與等效性電流源法及ADS的模擬結果比較，可驗我們方法的正確性。此外，亦在論文中討論了此方法的穩定度。

In a high-speed digital system, issues of the signal integrity (SI) associated with the high-speed circuit and printed circuit boards (PCB) become more important as the operating frequency increases. To obtain accurate results of SI problems in the high-speed digital systems, circuit devices (active or passive) are simulated in a full-wave electromagnetic simulator to analyze the interaction between PCB and circuits. The finite-difference time-domain (FDTD) method is one of the most useful methods of full-wave electromagnetic simulation for analyzing passive and active circuits on a PCB. This dissertation presents two efficient schemes for processing arbitrary lumped devices and a novel scheme for modeling diodes into FDTD method. For arbitrary linear two-terminal circuits, an efficient formulation that is based on control theory is presented. A novel FDTD approach that is based on Crank-Nicolson method is proposed to incorporate multi-port circuits into FDTD. The accuracy of the proposed approaches is confirmed by comparing results obtained using them with those of the equivalent current source method (ECSM) and Agilent’s commercial software, ADS. A novel FDTD approach that efficiently incorporates nonlinear devices such as diodes by solving a quadratic equation in each time step to update the electric field is developed. The novelty of this proposed method is that the quadratic equation always has real solutions. The stability of the proposed method is numerically demonstrated and the accuracy is verified by comparison with ECSM and ADS.

Contents

Abstract ii
Contents iii
List of Figures v
List of Tables vii


1. Introduction 1
1.1 Motivations for Research 1
1.2 Finite-Difference Time-Domain Analysis of Lumped Element Network 3
1.3 Overview 5
2. Finite-Difference Time-Domain Method 7
2.1 FDTD Method 7
2.2 Lumped Circuit Elements 14
2.3 Equivalent Current-Source Method (ECSM) 17
2.4 Other Approaches for Incorporating Lumped Circuits into the FDTD method 21
2.5 Stability Condition 36
2.6 A New Stability Theorem for FDTD Method 37
2.7 Convolution Perfectly Matched Layer Absorbing Boundary 41
3. Two Efficient Schemes for Processing Arbitrary Complicated Lumped Devices in the FDTD Method 44
3.1 An Improved FDTD Formulation for General Linear Lumped Microwave Circuits Based on Control Theory 44
3.2 Numerical Validation I 51
3.3 An Efficient Scheme for Multi-Port Devices by Crank-Nicolson Method 56
3.4 Numerical Validation II 58
4. A Novel Scheme for Modeling Diodes into FDTD Method 63
4.1 Modeling Diodes into FDTD Method 63
4.2 Stability Analysis 69
4.3 Numerical Validation 73
5. Conclusion 79
References 81
Vita 90
Publication List 91

References

[1] R. Senthinathan and J. L. Prince, “Simultaneous switching ground noise calcul- ation for packaged CMOS devices,” IEEE J. Solid-State Circuits, vol. 26, pp. 1724-1728, Nov. 1991.
[2] S. V. den Berghe, F. Olyslager, D. de Zutter, J. d. Moerloose, and W. Temmerman, “Study of the ground bounce caused by power plane resonances,” IEEE Trans. Electromag. Compat., vol. 40, pp. 111-119, May 1998.
[3] A. R. Djordjevic and T. K. Sarkar, “An investigation of delta-I noise on integrated circuits,” IEEE Trans. Electromag. Compat., vol. 35, pp. 134-147, May 1993.
[4] T. L. Wu, Y. H. Lin, J. N. Hwang, and J. J. Lin, “The effect of test system impedance on measurements of ground bounce in printed circuit boards,” IEEE Trans. Electromag. Compat., vol. 43, pp. 600-607, Nov. 2001.
[5] K. Ren, C. Y. Wu, and L. C. Zhang, “The restriction on delta-I noise along the power/ground layer in the highspeed digital printed circuit board,” in Proc. IEEE Int. Symp. Electromagnetic Compatibility, Colorado, USA, Aug. 1998, pp. 511-516.
[6] T. L. Wu, S. T. Chen, J. N. Huang, and Y. H. Lin, “Numerical and experimental investigation of radiation caused by the switching noise on the partitioned DC reference planes of high speed digital PCB,” IEEE Trans. Electromagn. Compat., vol. 46, pp. 33-45, Feb. 2004.
[7] Eric Bogatin, Signal and Power Integrity - Simplified, Prentice Hall PTR, 2010.
[8] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. 14, pp. 302-307, May 1966.
[9] J. Zhang and Y. Wang, “FDTD analysis of active circuits based on the S-parameters,” in Proc. Asia-Pacific Microwave Conference, vol. 3, pp. 1049-1052, Dec. 1997.
[10] J. A. Pereda, 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. Microwave Theory Tech., vol. 47, no. 6, pp. 943-949, Jun. 1999.
[11] C. Emili, F. Alimenti, P. Mezranotte, L. Roselli, and R. Sorrentino, “Rigorous modeling of packaged Schottky diodes by the nonlinear lumped network (NL2N)-FDTD approach,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 12 , pp. 2277-2282, Dec. 2000.
[12] X. Ye and J. L. Drewniak, “Incorporating two-port networks with S-parameters into FDTD,” IEEE Microwave and Wireless Components Lett., vol. 11, no. 2, pp. 77-79, Feb. 2001.
[13] H.E.A. El-Raouf, W. Yu, and R. Mittra, “Application of the Z-transform technique to modelling linear lumped loads in the FDTD,” Microwaves, Antennas and Propagation, IEE Proceedings, vol. 151, no. 1, pp. 67-70, Feb. 2004.
[14] A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits and Systems, vol. CAS-22, no. 9, pp. 735-742, Sep. 1975.
[15] A. Ramadan and A. S. Omar, “A new algorithm for the reconstruction of band-pass signals,” IEEE AP-S International Symposium Digest, vo1. 1, pp. 292-295, Jul. 2001.
[16] W. Sui, D. A. Christensen, and C. H. Durney, “Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements,” IEEE Trans. Microwave Theory Tech., vol. 40, no. 4, pp. 724-730, Apr. 1992.
[17] Y.-S. Tsuei, A.C. Cangellaris, and J.L. Prince, “Rigorous electromagnetic modeling of chip-to-package (first-level) interconnections,” IEEE Trans. Advanced Packaging, vol. 16, no. 8, pp. 876-883, Dec. 1993.
[18] B. Toland, B. Houshmand, and T. Itoh, “Modeling of nonlinear active regions with the FDTD method,” IEEE Microwave Guided Wave Lett., vol. 3, no. 9, pp. 333–335, Sep. 1993.
[19] B. Toland, J. Lin, B. Houshmand, and T. Itoh, “FDTD analysis of an active antenna,” IEEE Microwave Guided Wave Lett., vol. 3, no. 11, pp. 423–425, Nov. 1993.
[20] M. Piket-May, A. Taflove, and J. Baron, “FD-TD modeling of digital signal propagation in 3-D circuits with passive and active loads,” IEEE Trans. Microwave Theory Tech., vol. 42, no. 8, pp. 1514-1523, Aug. 1994.
[21] P. Ciampolini , P. Mezzanotte , L. Roselli , and , R. Sorrentino ,“Accurate and efficient circuit simulation with lumped-element FDTD technique,” IEEE Trans. Microwave Theory Tech., vol. 44, no. 12, pp. 2207-2215, Dec. 1996.
[22] C. C. Wang and C. W. Kuo, “An efficient scheme for processing arbitrary lumped multiport devices in the finite-difference time domain method,” IEEE Trans. Microw. Theory Tech, Vol. 55, No. 5, pp. 958-965, May 2007.
[23] V.A. Thomas, K.-M. Ling, M.E. Jones, B. Toland, J. Lin, and T. Itoh, “FDTD analysis of an active antenna,” IEEE Microwave Guided Wave Lett., vol. 4, no. 9, pp. 296–298, Sep. 1994.
[24] V. A. Thomas, M. E. Jones, M. Piket-May, A. Taflove, and E. Harrigan, “The use of SPICE lumped circuits as sub-grid models for FDTD analysis,” IEEE Microwave Guided Wave Lett., vol. 4, pp. 141–143, May 1994.
[25] P. Ciampolini, P. Mezzanotte, L. Roselli, D. Sereni, R. Sorrentino, and P. Torti, “Simulation of HF circuits with FDTD technique including non-ideal lumped elements,” IEEE MTT-S International Symposium Digest, vol. 2, pp. 361-364, May 1995.
[26] C. N. Kuo, B. Houshmand, and T. Itoh, “FDTD analysis of active circuits with equivalent current source approach,” IEEE AP-S International Symposium Digest, vol. 3, pp. 1510-1513, Jun. 1995.
[27] C. N. Kuo, V. A. Thomas, S. T. Chew, B. Houshmand, and T. Itoh, “Small signal analysis of active circuits using FDTD algorithm,” IEEE Microwave Guided Wave Lett., vol. 5, no. 7, pp. 216-218, Jul. 1995.
[28] C. N. Kuo, R. B. Wu, B. Houshmand, and T. Itoh, “Modeling of microwave active devices using the FDTD analysis based on the voltage-source approach,” IEEE Microwave Guided Wave Lett., vol. 6, no. 5, pp. 199-201, May 1996.
[29] C. N. Kuo, B. Houshmand, and T. Itoh, “Full-wave analysis of packaged microwave circuits with active and nonlinear devices: an FDTD approach,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 5, pp. 819-826, May 1997.
[30] Min Chen, Sion Teck Chew, and T. Itoh, “Nonlinear analysis of a microwave diode mixer using the extended FDTD,” IEEE MTT-S International Symposium Digest, vol. 1, pp. 67-70, Jun. 1997.
[31] V. S Reddy and R. Garg, “An improved extended FDTD formulation for active microwave circuits,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 9, pp. 1603-1609, Sep. 1999.
[32] J.-Y. Lee, J.-H. Lee, and H.-K. Jung, “Linear lumped loads in the FDTD method using piecewise linear recursive convolution method,” IEEE Microwave and Wireless Components Lett., vol. 16, no. 4, pp. 158-160, Apr. 2006.
[33] T.-L. Wu, S.-T. Chen, and Y.-S. Huang, “A novel approach for the incorporation of arbitrary linear lumped network into FDTD method,” IEEE Microwave and Wireless Components Lett., vol. 14, no. 2, pp. 74-76, Feb. 2004.
[34] Z. Shao and M. Fujise, “An improved FDTD formulation for general linear lumped microwave circuits based on matrix theory,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 7, pp. 2261-2266, Jul. 2005.
[35] O. Gonzalez, J.A. Pereda, A. Herrera, and A. Vegas, “An extension of the lumped-network FDTD method to linear two-port lumped circuits,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 7, pp. 3045-3051, Jul. 2006.
[36] A. C. Cangellaris and R. Lee, “On the accuracy of numerical wave simulations on finite methods,” J. Electromagn. Waves Applicat., vol. 6, no. 12, pp. 1635-1653, 1992.
[37] K. L. Shlager, J. G.Maloney, S. L. Pay, and A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propagat., vol. 41, no. 12, pp. 1732-1737, Dec. 1993.
[38] J. B. Schneider, “FDTD dispersion revisited: Faster-than-light propagation,” IEEE Microwave and Guided Wave Lett., vol. 9, no. 2, pp. 54-56, Feb. 1999.
[39] K. L. Shlager and J. B. Schneider, “Relative accuracy of several finite-difference time-domain schemes,” IEEE AP-S International Symposium Digest, vol. 1, pp. 168-171, Aug. 1999.
[40] J. B. Schneider and R. J. Kruhlak, “Inhomogeneous waves and faster-than-light propagation in the Yee FDTD grid,” IEEE AP-S International Symposium Digest, vol. 1, pp. 184-187, Aug. 1999.
[41] R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propagat., vol. 40, no. 11, pp. 1297–1301, Nov. 1992.
[42] R. M. Joseph, S. G. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett., vol. 16, no. 18, pp. 1412–1414, 1991.
[43] J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles Algorithms and Applications. New York: Macmillan, 1992.
[44] J.W. Schuster, R.J. Luebbers, and T.G. Livernois, “Application of the recursive convolution technique to modeling lumped-circuit elements in FDTD simulations,” Proc. IEEE Antennas and Propagation Society Int. Symp., vol. 4, pp. 1792–1795, 1998.
[45] D.M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antennas Propagat., vol. 40, no. 10, pp. 1223–1230, Oct 1992.
[46] G.E. Carlson, Signal and linear system analysis. Wiley, 1998.
[47] D. F. Kelley and R. J. Luebbers, “Piecewise linear convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propagat., vol. 44, no. 6, pp. 792–797, Jun. 1996.
[48] R. J. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat., vol. 32, no. 3, pp. 222–227, Aug. 1990.
[49] Z. H. Shao and G. W. Wei, “DSC time-domain solution of Maxwell’s equations,” J. Comput. Phys., vol. 189, no. 2, pp. 427–453, Aug. 2003.
[50] Z. H. Shao, Z. X. Shen, Q. He, and G. Wei, “A generalized higherorder finite difference time domain method and its application in guidedwave problems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 856–861, Mar. 2003.
[51] C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propagat., vol. 37, no. 11, pp. 1181-1192, Sep. 1989.
[52] R. J. Luebbers, K. S. Kunz, M. Schneider, and F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propagat., vol. 39, no. 4, pp. 429-433, Apr. 1991.
[53] M. J. Barth, R. R. McLeod, and R. W. Ziolkowski, “A near and far field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Applicat., vol. 6, no. 1, pp. 5-18, 1992.
[54] I. J. Craddock and C. J. Railton, “Application of the FDTD method and a full time-domain near-field transform to the problem of radiation from a PCB,” Electron. Lett., vol. 29, no. 23, pp. 2017-2018, Nov. 1993.
[55] J. De Moerloose and D. De Zutter, “Surface integral representation radiation boundary condition for the FDTD method,” IEEE Trans. Antennas Propagat., vol. 41, no. 7, pp. 890-896, Jul. 1993.
[56] O. M. Ramahi, “Near-field and far-field calculation in FDTD simulations using Kirchhoff surface integral representation,” IEEE Trans. Antennas Propagat., vol. 45, no. 5, pp. 753-759, May 1997.
[57] H. H. Su , C. W. Kuo and T. Kitazawa, “An efficient algorithm for the incorporation of diodes into FDTD method,” IEEE Antennas and Propagation Society International Symposium (APSURSI), 2010.
[58] H. H. Su , C. W. Kuo and T. Kitazawa, “A Novel Approach for Modeling Diodes into FDTD Method,” Progress in Electromagnetics Research Symposium (PIERS), 2011.
[59] F. Kung and H. T. Chuah, “Modeling of Diode in FDTD Simulation of Printed Circuit Board,” J. of Electromagn. Waves and Appl., vol.16, pp. 99-110, 2002.
[60] O. Gonzalez, J.A. Pereda, A. Herrera, A. Grande, and A. Vegas “Combining the FDTD Method and Rational-Fitting Techniques for Modeling Active Devices Characterized by Measured S -Parameters ,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 477–479, July. 2007
[61] F. Kung and H. T. Chuah, “Stability of Classical Finite-Difference Time-Domain (FDTD) Formulation with Nonlinear Elements- A New Perspective,” Progress In Electromagnetics Research, PIER 42, pp. 49-89, 2003.
[62] Pereda, J. A., L. A. Vielva, A. Vegas., “Analyzing the stability of the FDTD technique by combining the von Neumann method with the Routh-Hurwitz criterion,” IEEE Trans. Microwave Theory Tech., Vol. 49, 377–381, Feb. 2001.
[63] 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. Microwave Theory Tech., Vol. 50, 2159–2165, Sep. 2002.
[64] Remis, R. F., “On the stability of the finite-difference time-domain method,” J. of Computational Physics, Vol. 163, No. 1, 249–261, Sep. 2000.
[65] Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole Mathematics Series, 1989.
[66] Scheinerman, E. R., Invitation to Dynamical Systems, Prentice Hall, 1996.
[67] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed: Artech House 2000.
[68] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computat. Phys., vol. 114, pp. 185-200, 1994.
[69] S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas and Propagat., vol. 44, no. 12, pp. 1630-1639, Dec. 1996.
[70] Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat., vol. 43, pp. 1460 –1463, Dec. 1995.
[71] J. P. Berenger, “An effective PML for the absorption of evanescent waves in waveguides,” IEEE Microwave and Guided Wave Lett., vol. 8, no. 5, May 1998.
[72] J. P. Berenger, “Evanescent waves in PML's: origin of the numerical reflection inwave-structure interaction problems,” IEEE Trans. Antennas Propagat., vol. 47, no. 10, pp. 1497-1503, Oct. 1999.
[73] E. Bécache, P. G. Petropoulos, and S. D. Gedney, “On the long-time behavior of unsplit perfectly matched layers,” IEEE Trans. Antennas Propagat., vol. 52, no. 5, pp. 1335-1342, May 2004.
[74] J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microwave and Optical Tech. Lett., vol. 27, no. 5, pp. 334-339, Dec. 2000.
[75] J. Choma, Electrical Networks: Theory and Analysis. New York: Wiley, 1985.
[76] G. Massobrio and P. Antogenetti, Semiconductor Device Modeling with SPICE, 2nd edition, McGraw-Hill, 1993.
[77] J. Millman and C. C. Halkias, Integrated Electronics, McGraw-Hill, 1971.
[78] T. K. Ishii, Practical Microwave Electron Devices. New York: Academic, 1990.
[79] D. M. Pozar, Microwave Engineering. 4th edition, New York: Wiley, 2012.
[80] M. F. Golnaraphi and B. C. Kuo, Automatic Control Systems, 9th ed: John Wiley&Sons, Inc., 2010.
[81] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. New York: Oxford University Press, 1985