如何正確且有效率的模擬高速的電子電路，並考慮其產生的輻射效應、近場效應以及電路間的耦合效應，一直是一個重要的課題。在利用時域有限差分法分析集總元件時，傳統上所使用的等效性電流源法必須在每一個時間步階裡不斷的疊代求解，因此會需要較多的計算時間。在本論文中，我們提出了一種利用集總元件的電壓-電流關係式(導納或阻抗)求解的方法，避免了在每一個時間步階裡不斷求解的過程，因此可以有較好的計算效率。雖然目前亦有許多類似利用元件的電壓-電流關係式求解的方法，但卻僅僅只適用於單埠元件。而我們所提出的方法可以適用於多埠的集總元件。在此論文中，藉由與等效性電流源法的比較，我們驗證了此方法的正確性。此外，亦在論文中驗證了此方法的穩定性。

Developing full-wave simulators for high-frequency circuit simulation is a topic many researchers have investigated. Generally speaking, methods invoking analytic pre-processing of the device’s V-I relations (admittance or impedance) are computationally more efficient than methods employing numerical procedure to iteratively process the device at each time step. For circuits providing complex functionality, two-port or possibly multi-port devices whether passive or active, are sure to appear in the circuits. Therefore, extensions to currently available full-wave methods for handling one-port devices to process multi-port devices would be useful for hybrid microwave circuit designs. In this dissertation, an efficient scheme for processing arbitrary multi-port devices in the finite-difference time-domain (FDTD) method is proposed. The device’s admittance is analytically pre-processed and fitted into one grid cell. With an improved time-stepping expression, the computation efficiency is further increased. Multi-port devices in the circuit can be systematically incorporated and analyzed in a full-wave manner. The accuracy of the proposed method is verified by comparison with results from the equivalent current-source method and is numerically stable.

Abstract ii
Contents iii
List of Figures v
1. Introduction 1
1.1 Research Motivations 1
1.2 Finite-Difference Time-Domain Analysis of Lumped Devices 3
1.3 Overview 7
2. Finite-Difference Time-Domain Method 9
2.1 Basic Conception 9
2.2 Numerical Stability Condition 15
2.3 Lumped Circuit Elements 17
2.4 Equivalent Current-Source Method (ECSM) 20
2.5 Other Pre-Processing Approaches for Integrating Lumped Devices into the FDTD Method 24
2.6 Near-to-Far-Field Transformation 40
3. An Efficient Scheme for Lumped Multi-Port Devices 45
3.1 Derivation from the Admittance 45
3.2 An Example of the Proposed Approach 54
3.3 Derivation from the Impedance 57
4. Numerical Validation 65
4.1 Passive Devices 65
4.2 Active Devices 72
4.3 Influence on Radiation with Mounted Active Devices 77
5. Conclusion 82
Appendix 84
References 87
Vita 93
Publication List 95

[1] J. Zhang and Y. Wang, “FDTD analysis of active circuits based on the S-parameters,” in Proc. Asia-Pacific Microwave Conference, Dec. 1997, vol. 3, pp. 1049-1052.
[2] 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.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] D. M. Pozar, Microwave Engineering. New York: Wiley, 1998.
[26] 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.
[27] 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.
[28] J. Choma, Electrical Networks: Theory and Analysis. New York: Wiley, 1985.
[29] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. 14, no. 3, pp. 302-307, May 1966.
[30] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed: Artech House 2000.
[31] 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.
[32] 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.
[33] 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.
[34] 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.
[35] 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.
[36] 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.
[37] 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.
[38] J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Principles Algorithms and Applications. New York: Macmillan, 1992.
[39] 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.
[40] D.M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antennas Propagat., vol. 40, no. 10, pp. 1223–1230, Oct 1992.
[41] G.E. Carlson, Signal and linear system analysis. Wiley, 1998.
[42] 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.
[43] 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.
[44] 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.
[45] 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.
[46] 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.
[47] 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.
[48] 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.
[49] 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.
[50] 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.
[51] 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.
[52] T. K. Ishii, Practical Microwave Electron Devices. New York: Academic, 1990.