FDTD是將馬克斯威爾方程式予以中央差分離散化，配合空間網格之電磁場配置，在時域上分析各種電磁問題的數值方法。在現今電腦系統，速度有愈來愈快的趨勢，當CPU速度達到數百MHz，甚至於GHz頻段時，由於邏輯閘切換時的瞬間電變化，造成IC連接到power/ground電力平面時接腳寄生電感上的電壓突波，這個波動會對整個電力平面的偏壓造成瞬間的壓降，當這個暫態電壓大到足以干擾板子上其他IC的正常操作時，這種現象就稱之為”接地彈跳(Ground bounce)”或同時切換雜訊。本論文運用時域有限差分法研究電力平面因接地彈跳雜訊所造成的共振現象，並且透過FDTD集總電路元件模型，以及克希荷夫表面積分公式之近場與遠場轉換，針對不同的抑制接地反彈對策，分析其對電力平面的信號完整性以及輻射干擾的影響。

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誌謝………………………………………………………………………….. I
論文提要……………………………………………………………………… II
目錄…………………………………………………………………………… III
圖表目錄……………………………………………………………………… V
第一章 序論………………………….……………………………………... 1
1.1研究背景…………………………………………………………….. 1
1.2研究目的…………………………………………………………….. 1
1.3論文大綱…………………………………………………………….. 2
第二章 FDTD演算法………………………………………………………... 4
2.1馬克斯威爾方程式及FDTD公式………………………………….. 4
2.2穩定準則…………………………………………………………….. 8
第三章 吸收邊界條件……………………………………………………….. 10
3.1吸收邊界條件………………………………………………………. 10
3.2 Mur吸收邊界………………………………………………………. 11
3.3 Berenger完美匹配層(Perfectly Matched Layer)………………….. 14
3.3.1 PML理論及PML介質的TE例子………………………... 14
3.3.2 TE平面波在PML介質中的傳播………………………….. 15
3.3.3 PML層的指數時間步階演算法……………………………. 19
3.3.4 PML介質的TM例子………………………………………. 20
3.3.5三維PML吸收邊界的全向量方程式……………………… 21
3.3.6 Berenger’s PML吸收邊界的數值結果…………………….. 22
3.4非分離場完美匹配層(Unsplit_field PML)………………………… 26
3.4.1修改的非分離場PML………………………………………. 26
3.4.2非分離場PML的離散化…………………………………… 29
3.4.3非分離場完美匹配層的數值結果…………………………. 30
第四章 FDTD模擬集總電路元件…………………………………………. 32
4.1 FDTD延伸至電路元件的演算法…………………………………. 32
4.2電阻…………………………………………………………………. 33
4.3阻抗性電壓源………………………………………………………. 34
4.4電容…………………………………………………………………. 37
4.5電感…………………………………………………………………. 39
4.6二極體………………………………………………………………. 40
4.7雙載子接面電晶體…………………………………………………. 42
第五章 次網格及非均勻正交網格…………………………………………. 45
5.1次網格………………………………………………………………. 45
5.1.1基本的輪廓積分內差模型………………………………….. 45
5.1.2狹縫的輪廓-路徑模型………………………………………. 47
5.2非均勻正交網格……………………………………………………. 50
5.2.1非均勻正交網格演算法……………………………………. 50
5.2.2非均勻正交網格演算法數值結果…………………………. 53
第六章 近場與遠場轉換…………………………………………………… 56
6.1 FDTD在電磁輻射問題的分析…………………………….……… 56
6.2克希荷夫表面積分公式…………………………………………… 57
6.3遠場轉換積分面的修正…………………………………………… 60
6.4近場計算與遠場轉換的數值結果………………………………… 62
第七章 高速數位電路之接地彈跳……..………………………………….. 67
7.1接地彈跳雜訊(Chip level)…………………………………………. 67
7.2接地彈跳雜訊(Board level)…………………………………………. 69
7.3 FDTD模擬接地彈跳共振效應……………………………………… 70
7.3.1接地彈跳的電容防護………………………………………… 73
7.3.2隔絕敏感的IC……………………………………………….. 77
7.4接地彈跳的FDTD模擬與實驗結果………………………………. 80
7.5阻抗性電壓源激發與輻射干擾分析………………………………. 87
第八章 結論…………………………………………………………………. 98
參考文獻……………………………………………………………………… 99

[1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. on Ant. and Propag., vol. 14, pp. 302-307, May 1966.
[2] A. Taflove, Computational Electrodynamics – The Finite Difference Time Domain Method, CRC Press, 1992.
[3] J. M. Jong, V. K. Tripathi, and B. Janko, “Modeling and simulation of switching noise with the associated package resonance for high speed digital circuits,” Proc. Electron. Components Technol. , pp. 323-328,1995.
[4] J. M. Wiliamson, M. S. Nakhla, Q.-J. Zhang, and P. Van der Puije, “Ground noise minimization in integrated circuit packages through pin assignment optimization,” IEEE Trans. on Comp., Packag., Manuf. Technol., vol. 19, pt. B, pp. 361-371, May 1996.
[5] J.-G. Yook, L.P. B. Katehi, K.A. Sakallah, R. S. Martin, L. Huang, and T. A. Schreyer, “Application of system-level EM modeling to high-speed digital IC packages and PCB’s,” IEEE Trans. on Microwave Theory and techniques, vol. 45, no. 10, pp. 1847-1856,October 1997.
[6] F. Y. Yuan, “Analysis of power/ground noises and decoupling capacitors in printed circuit board systems,” IEEE International Symposium on Electro-magnetic Compatibility, pp. 425-429, Jun, 1997.
[7] A. R. Djordjevic and T. K. Sarkar, “An investigation on Delta-I noise on integrated circuits,” IEEE Trans. on Electromagn. Compat., vol. 35, pp. 134-147, May 1993.
[8] Mur, G., “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equation,” IEEE Trans. on Electromagnetic Compatibility, vol. 23, 1981, pp. 377-382.
[9] Berenger, J.-P., “Aperfectly matched layer for free-space simulation in finite-difference computer codes,” submitted to Annales des Telecommunications, 1994.
[10] R. Mittra, and Perkel, “A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave and Guide Wave Lett., vol. 5, no. 3, pp. 84-86, March 1995.
[11] J. C. Veihl, R. Mittra, “An efficient implementation of Berenger’s perfectly matched layer (PML) for finite-difference time-domain mesh truncation,” IEEE Microwave and Guide Wave Lett., vol. 6, no. 2, pp.94-96, February 1996.
[12] 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. on Microwave Theory and Techniques, vol. 42, no. 8, pp. 1514-1523, August 1994.
[13] A. Talflove, K. R. Umashankar, B. Berker, F. Harfoush, Kane. S. Yee, “Detailed FD-TD Analysis of electromagnetic fields penetrating narrow slots and lapped joints in thick conducting screens,” IEEE Trans on Antennas and Propagations, vol. 36, no. 2, pp. 247-255, February 1988.
[14] Tulintseff, A., “The finite-difference time-domain method and computer program description applied to multilayered microstrip antenna and circuit configurations,” AMT:336.5-92-041, Technical Report, Jet Propulsion Laboratory, May 1992.
[15] O. M. Ramahi, “Near- and Far-Field calculations in FDTD simulations using Kirchhoff surface integral representation,” IEEE Trans. on Antennas and Propagation, vol. 45, no. 5, pp. 753-759, May 1997.
[16] 邱志龍, “時域有限差分法之遠場轉換分析與應用,” 中山大學碩士論文, 1998.
[17] A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scatting problems using the time dependent Maxwell’s equation,” IEEE Trans Microwave Theory Tech., vol. MTT-23, pp.623-630, August 1975.
[18] Z. Wu and J. Fang, “Numerical implementation and performance of perfectly matched layer boundary condition for waveguide structures,” IEEE Trans. on Microwave Techniques, vol. 43, pp. 26762222, no. 12, December 1995.
[19] D. M. Sheen, S. M. Ali, M. D. Abouzahra, and J. A. Kong, “Application of the three-dimensional finite-difference time-domain method to the analysis of planar microstrip circuits,” IEEE Trans on Microwave Theory and Techniques, vol. 38, no. 37, pp.849-856, July 1990.
[20] R. J. Luebbers, K. S. Kunz, M. Schneider, and F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. on Antenna Propagation, vol. 39, pp. 429-433, April 1991.
[21] Taflove, A., K. R. Umashankar, B. Beker, F. Harfoush, and K. S. Yee, “Detailed FD-TD analysis of electromagnetic fields penetrating narrow slots and lapped joints in thick conducting screens,” TEEE Trans. on Antennas and Propagation, vol. 36, pp247-257, 1988.
[22] Umashankar, K. R., A. Taflove, and B. Beker, “Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity,” IEEE Trans. on Antennas and Propagation, vol. 35, pp. 1248-1257, 1987.
[23] Monk, P., and E. Suli, “A convergence analysis of Yee’s scheme on non-uniform grids,” SIAM J. Numerical Analysis, vol. 31, pp. 393-412, 1994.
[24] Monk, p., “Error estimates for Yee’s method on non-uniform grids, “IEEE Trans. on Magnetics, vol. 30, pp. 3200-3203, 1994.
[25] 謝金明, “高速數位電路設計暨雜訊防治技術,” 惠普科技公司電子測試儀器教育訓練中心, pp. 4-2~4.10, 1997.
[26] K. Ren, C.Y. Wu, C. Z. Lin, “The restriction on delta-I noise along the power/ground layer in the high-speed digital printed circuit board,” IEEE 0-7803-5015-4, pp. 511-516, 1998.
[27] S. V. den Berghe, F. Olyslager, D. De Zutter, J. De Moerloose, and Temmerman, “Study of the ground bounce caused by power plane resonances,” IEEE Trans. on Electromagnetic Compatibility, vol. 40, no. 2, pp. 111-119, May 1998.
[28] J. V. Bladel, Electromagnetic Fileds. New York: Hemisphere, 1985.
[29] D. M. Pozar, “Microwave engineering,” Addison Wesley Publishing Company, pp. 234-240, 1990.
[30] C. Wu, K.-L. Wu, Z.-Q. Bi, and J. Litva, “Accuate characterization of planar printed antennas using finite-difference time-domain method,” IEEE Trans. on Antennas and Propagation, vol. 40, no. 5, pp. 526-533, May 1992.
[31] V. Costa, R. Preatoni, S. Caniggia, “Investigation of EMI on multiplayer printed board: -noise and power supply decoupling,” IEEE 0-7803-3207-5, pp. 436-321, 1996.