在高速數位電路中，為了能更有效利用有限的基板體積，印刷電路勢必往多層化發展，而過孔結構(via)正是將各個訊號層做垂直方向穿層連結。然而由於過孔結構在電路板中為相對細小且不規則形狀的結構，以傳統的時域有限差分法來模擬時，必須將網格切的更細來近似其結構，為此，勢必得花費更多的記憶體與運算時間。本論文為了解決這類問題，提出結合時域有限體積法與時域有限差分法的處理方式，藉由針對細部結構以時域有限體積法網格做細部處理，再令網格以一定比例放大結合外部時域有限差分法處理，而達到縮減網格以節省記憶體並增加運算速度的效果。此外我們將以一種利用串接方式以達寬頻效果的電磁能隙(EBG)結構為基礎，提出另一種藉由移動過孔結構而取代以上方法之中較細小的過孔結構的效果，而達成降低成本的需求。

In high-speed digital circuits, in order to utilize the space of printed circuit boards(PCB) efficiently, the signal via is a heavily used interconnection structure to communicate different signal layers. However, because of vias are small and irregular structure in the PCB. When we try to simulate these problems with traditional FDTD method. We must using more fine grid to approximate the structure, so it will take a lot CPU memory and computing times. In this author, we try to combine FDTD and FVTD method. Take FVTD method in these partial small structure and magnify grid in a ratio. Finally, combine the larger FDTD grid to achieve reducing the numbers of grids that will save CPU memory and raise computing speed. In addition, we will present another solution that shifting via to replace using small size via based on a method that is using cascaded EBG structure achieve broadband effects to cost down.

第一章 序論
1.1 概論...............................................................................................................1
1.2 論文大綱.......................................................................................................2
第二章 時域有限差分法
2.1 介紹...............................................................................................................4
2.2 FDTD公式推導............................................................................................4
2.3 Courant 穩定準則........................................................................................7
2.4 阻抗性電壓源與電阻模擬...........................................................................8
2.5 吸收邊界條件.............................................................................................10
第三章 時域有限體積法
3.1 介紹.............................................................................................................12
3.2 FVTD公式推導..........................................................................................12
3.3 FVTD與FDTD的結合...............................................................................15
3.4 FVTD/FDTD公式修正..............................................................................19
第四章 印刷電路板中過孔結構
4.1 介紹.............................................................................................................20
4.2 二維FVTD模擬.........................................................................................21
4.3 過孔結構模擬.............................................................................................23
4.4 FVTD/FDTD與CFDTD之比較................................................................27
第五章 利用串接方式以達到寬頻效果的蘑菇型EBG結構
5.1 介紹.............................................................................................................38
5.2 蘑菇型EBG結構理論................................................................................38
5.3 改變EBG結構參數對頻帶的影響............................................................40
5.3.1 蘑菇型EBG結構的頻帶...........................................................40
5.3.2 EBG單位結構數量的效應.......................................................41
5.3.3 過孔結構半徑尺寸的效應........................................................45
5.3.4 利用串接方法達成寬頻的EBG結構.......................................46
5.4 新的寬頻EBG結構....................................................................................48
第六章 結論............................................................................................................53

[1] K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagat., Vol. AP-14, pp.302-307, May 1966.
[2] A. Taflove, 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,” IEEE Trans. Antennas and Propagat., Vol. 36, pp.247-256, Feb. 1988.
[3] M. Okoniewski, E. Okoniewska, and M.A. Stuchla, “Three-dimensional subgridding algorithm for FDTD,” IEEE Trans. Antennas and Propagat., Vol.45, pp.422-429, Mar. 1997.
[4] K. Xiao, D. J. Pommerenke, and J. L. Drewniak, “A three-dimensional FDTD subgridding algorithm based on interpolation of current density,” IEEE Int. Sym. Electromagnetic Compatibility, pp.118-123, Aug. 2004.
[5] N.K. Madsen, “Divergence preserving discrete surface integral methods for Maxwell’s curl equations using non-orthogonal unstructured grids,” J. Comput. Phys., 119, pp.34-45, 1995.
[6] R. Holland, V.P. Cable and L.C. Wilson, “Finite-volume time-domain (FVTD) techniques for EM scattering,” IEEE Trans. Electromagn. Compat., Vol. 33, pp.281-294, Nov. 1991.
[7] M. Yang, Y. Chen, and R. Mittra, ”Hybrid finite-difference/finite-volume time-domain analysis for microwave integrated circuits with curved PEC surfaces using a nonuniform rectangular grid, ” IEEE Trans. Microwave Theory and Tech., Vol. 48, No. 6, Jun. 2000.
[8] F. Edelvik, “A survey of finite volume time domain methods for solving Maxwell’s equations, ” Technical Report, Department of Scientific Computing, Uppsala Universitet, 1998.
[9] W. D. Guo, G. H. Shiue, C. M. Lin, and R.B. Wu, “An integrated signal and power integrity analysis for signal traces through the parallel planes using hybrid finite-element and finite-difference time-domain techniques, ” IEEE Trans. Advanced Packaging, Vol. 30, No. 3, Aug. 2007.
[10] E. A. Navarro, N. T. Sangary, and J. Litva, ”Some considerations on the accuracy of the nonuniform FDTD method and its application to waveguide analysis when combined with the perfectly matched layer technique, ” IEEE Trans. Microwave Theory and Tech., Vol. 44, No. 1, Jul. 1996.
[11] W. Yu and R. Mittra, “A conformal finite difference time domain technique for modeling curved dielectric surfaces, ” IEEE Microwave Wireless Compon. Lett., Vol. 11, No. 1, Jan. 2001.
[12] W. Yu and R. Mittra, “Accurate modelling of planar microwave circuit using conformal FDTD algorithm,” IEEE Electronics Lett., Vol. 36, No. 7, Mar. 2000.
[13] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alex′opolous, and E. Yablonovitch, ” High-impedance electromagnetic surfaces with a forbidden frequency band, ” IEEE Trans. Microwave Theory and Tech., Vol. 47, No. 11, Nov. 1999.
[14] F. R. Yang, K. P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuits, ” IEEE Trans. Microwave Theory and Tech., Vol. 47, No. 8, Aug. 1999.
[15] L. Yang, M. Fan, F. Chen, J. She, and Z. Feng, “A novel compact electromagnetic-bandgap (EBG) structure and its applications for microwave circuits, ” IEEE Trans Microwave Theory and Tech., Vol. 53, No. 1, Jan. 2005.
[16] L. Liang, C. H. Liang, L. Chen, and X. Chen, “A novel broadband EBG using cascaded mushroom-like structure, ” Microwave Opt. Technol. Lett., Vol. 50, No. 8, Aug. 2008.
[17] R. T. Remski, “Analysis of photonic bandgap surfaces using ansoft HFSS, ” Microwave J., Vol. 53, pp.190-198, Sep. 2000.