本論文提出一種混合式等效寬頻模型萃取法，利用時域演算法萃取出印刷電路板上之連通柱結構的SPICE相容等效寬頻模型，結合利用全波模擬所得到之頻域響應值萃取出印刷電路板上之不規則傳輸線結構的SPICE相容等效寬頻模型，透過兩種不同萃取方式可以建構出任意尺寸化的寬頻等效模型的元件庫，並可以結合兩種方法所萃取的等校模型來建構出設計著所需求的電路結構。此兩種萃取模型的每個模組分別有最佳化的極點-餘數的形式表示。配合系統化的集總元件萃取技術，可將模組最佳化的極點-餘數分別轉換成相對應的集總電路模型。並從頻域的角度上與全波模擬(3D-FDTD or HFSS)的結果比較，以驗證所提出的混合式等效寬頻模型萃取的正確性。此外，所萃取的模型可以有效的利用現有的商用電路模擬軟體如:Hspice、ADS進行訊號完整性分析與電源完整性分析。

The thesis proposes a hybrid broadband equivalent model extraction method, and our goal is to combine via structure and irregular transmission line in print circuit board for extraction of broadband SPICE-compatible model by using the time domain algorithm and full wave simulation in frequency domain, respectively. We can construct broadband SPICE-compatible macro-model scalable library with two kind of different extraction methods, tow kind of extraction of equivalent model can construct the circuit structure for designer demand.

Every modules of the broadband macro model of the two extraction models are represented by the optimum pole-residue forms. Using a systematic lumped-model extraction technique, all the optimum pole-residue rational functions can be transfered into a corresponding lumped circuit model. The accuracy of Extraction of Broadband Equivalent Models is demonstrated in frequency -domain responses compared with the 3D-FDTD or HFSS simulation. In addition, the extraction model can simulate in commercial tools effectively, ex: Hspice、ADS. Even the model can simulate signal integrality and power integrality in Hspice or ADS.

目錄
目錄…………………………………………………………………………………….………i
圖表索引……………………………………………………………………………………...iii
第一章 序論…………………………………………………………………………………1
1.1 研究背景與方法…………………………………………………………………….1
1.2 論文大綱…………………………………………………………………………….3
第二章 時域反射波型模擬與量測…………………………………………………………4
2.1 FDTD演算法…………………………………………………………………...…………...4
2.1.1 FTDT公式…………………………………………………………………….4
2.1.2 穩定準則………………………………………………………………………..6
2.1.3 吸收邊界………………………………………………………………………..7
2.1.4 FDTD模擬…………………………………………………………………….8
2.2 TDR理論…………………………………………………………………………....9
2.2.1 TDR波形量測……………………………………………………………...…9
2.2.2 TDT波形量測……………………………………………...………………...13
第三章 矩陣束法……………………………………………………………………..……16
3.1矩陣束法(Matrix Pencil Method，MPM) …………………………………….…...16
3.1.1 簡介……………………………………………………..…………………....16
3.1.2 矩陣束法理論…………………………………………..…………………....16
3.2寬頻等校集總元件模型……………………………………..…………..………....20
3.2.1 電抗理論………………………..……………………..…………..………....20
3.2.2 電路等校模型…………………..……………………..…………..………....21
3.2.3 寬頻集總元件模型……………..…………………..…………..…..……......23
3.3寬頻等校模型萃取範例……………..……………………..…………..………......26
3.3.1 雙阜寬頻等校模型…………..……………………..……..……..………......26
3.3.2 等校模型萃取實例…………..……………………..……..……..………......28
第四章 向量擬合法……………....………………………………………………………..29
4.1向量擬合法(Vector Fitting，VF)……………………..............................................33
4.1.1 曲線擬合(curve fitting)…………………………………………………...…...33
4.1.2 極點與留數的形式………………..………………………………………….34
4.1.3 向量擬合法理論……………..………………………………….…………….35
4.2寬頻等校集總元件模型…..…………………………………………….………….37
4.2.1 電路等校模型………………………………………………………………..37
4.3 寬頻等校模型萃取範例………………………………………………………..….42
4.3.1 雙阜寬頻等校模型…………………………………………………………..42
4.3.2 等校模型萃取實例…………………………………………………………..44
4.4 穩定性和被動性………………………………...…………………………………..51
第五章 混合式等效模型萃取……………...….…………………………………………..57
5.1 混合等效模型簡介…………………………………………………..…………….57
5.2混合等效模型範例…………………………………………………………...…….58
5.3 混合等效模型應用…………………………………………………………...……63
第六章 結論………………………….…………………………………………………….66
參考文獻……………………………………………………………………………………..67






圖表索引
圖1-1　 連通柱（via）結構圖，及其適合萃取法……………………………………………2
圖1-2 分解互聯結構示意圖…………………………………..…………………………3
圖2-1 單一空間網格電磁場配置圖………………………………………………..……6
圖2-2 電磁場時間配置圖…………………………………………………………..……6
圖2-3 PML空間示意圖…….…………………………………………………………...7
圖2-4 FDTD運算流程圖…………………………………………………………….……8
圖2-5 訊號上升時間示意圖………………………………………………………………9
圖2-6 時域反射儀示意圖………………………………………………………….……...9
圖2-7 時域反射儀終端有負載示意圖………………………………...……………..….11
圖2-8 時域反射儀寄生電容示意圖……………………………………...….……….….11
圖2-9 時域反射儀寄生電感示意圖…………………………………………..…………12
圖2-10 阻抗非均勻傳輸線結構圖…………………………………………..……………12
圖2-11 TDR量測阻抗非均勻傳輸線電壓波形圖………………………..………………13
圖2-12 TDR量測阻抗非均勻傳輸線阻抗波形圖………………………..………………13
圖2-13 TDR/TDT量測示意圖……………………………………………………………14
圖2-14 不同長度傳輸線的時域穿透波形傳遞延遲結果………………………………14
圖2-15 不同上升時間經過連通柱對的耦合雜訊時域反射波型圖………………...….15
圖3-1 極點分布範圍示意圖……………………………………………………………21
圖3-2　 無損 網路驅動點阻抗圖……………………………………….……..……..22
圖3-3 無損 網路驅動點導納圖……………………………………………...…..…22
圖3-4 有損網路驅動點阻抗…………………………….………………...……………23
圖3-5 有損網路驅動點導納圖……………………………………………………..…..23
圖3-6 待測物輸入終端輸入阻抗反射係數關係圖……………..…….……………….23
圖3-7 有理函數轉換成等效電路的四種型態…….…..………...……………………..26
圖3-8 有理函數萃取出的等效寬頻模型………..……………………………………..26
圖3-9 等校 模型結構示意圖…..…………………………………………...….……..27
圖3-10 時域反射儀量測非均勻傳輸線結構圖………………………………………......28
圖3-11 不連續結構之食欲反射波形穿透波形…….……………………………….........29
圖3-12 近似反射波形以及穿透波形圖…….……………...…………………………......29
圖3-13 不連續結構散色參數比較圖(a)大小(b)相位圖…………………………….……30
圖4-1 實數對極點對應之等效 串聯電路………………………...………..………...39
圖4-2 等效 串聯電路……………………………………………………..….……..39
圖4-3 第一種共軛複數對極點對應之等效 串聯電路…………………..…...……39
圖4-4 第二種共軛複數對極點對應之等效 串聯電路…………………..….……..40
圖4-5 第一種共軛複數對極點對應之等效 並聯電路…………………..….….….40
圖4-6 第二種共軛複數對極點對應之等效 並聯電路…………………..…….…..40
圖4-7 等校 模型..............................................................................................................41
圖4-8 第一個電路結構圖………………………………………………………..………42
圖4-9 向量擬合法進似模擬結構之散色參數(a)大小(b)相位圖…………………….....43
圖4-10 結構一之HFSS模擬與等校模型萃取的(a)大小(b)相位比較圖..........................44
圖4-11 第二個電路結構圖………………………………………………………..………44
圖4-12 結構二之HFSS模擬與等校模型萃取的(a)大小(b)相位比較圖………………..45
圖4-13 第三個電路結構圖..................................................................................................46
圖4-14 結構三之HFSS模擬與等校模型萃取的(a)大小(b)相位比較圖………..………46
圖4-15 第四個電路結構圖……………………………………………………...……..….47
圖4-16 結構四之HFSS模擬與等校模型萃取的(a)大小(b)相位(c)大小局部放大(b) 相位局部放大比較圖……………………………………………………….….…49
圖5-1 矩陣束法萃取流程圖………………………………………………………...…...50
圖5-2 向量擬合法萃取流程圖……………………………………………………...…...51
圖5-3 矩陣束法與向量擬合法之模型結合示意圖……………………………….…….52
圖5-4 三層板連通柱結構側視圖…………………………………………………….….52
圖5-5 三層板連通柱結構之TDR、TDT波型圖……………………………..……...…..53
圖5-6 三層板連通柱結構之(a)大小(b)相位比較圖.........................................................54
圖5-7 三層板連通柱結構側視圖………………………………………………………..54
圖5-8 四層板連通柱結構之(a)大小(b)相位比較圖……………….……………...…….55
圖5-9 PCB (a) 結構圖 (b) 局部結構放大圖………………………………...…...……56
圖5-10 互連性結構示意圖…………….……………………………………….…………57
圖5-11 互連性結構之(a)大小(b)相位比較圖…………………………………..……...…58
表3-1 電路結構內部參數....……………………………………………………………..32
表4-1 第一個電路結構內部參數………………………………………………………..54
表4-2 第二個電路結構內部參數………………………………………………………..55
表4-3 第四個電路結構內部參數………………………………………………………..56
表5-1 時間比較表………………………………………………………………………..65

[1] Staiculescu D., Pham A., Laskar J., Consolazio S., Moghe S., “Analysis and performance of BGA interconnects for RF packaging”, IEEE Radio Frequency Integrated Circuits (RFIC) Symposium, 7-9 , pp.131 – 134June 1998.
[2] Chen M., Tai C., Huang Y., Fang L., “Electrical characterization of BGA test socket for high-speed applications”, 4th International Symposium on Electronic Materials and Packaging, 2002, 4-6 Dec. 2002, pp.123 – 126.
[3] Chung C.Y., “A novel package electrical characterization processapplication on plastic BGA packages”, 1st IEEE Electronic Packaging Technology Conference, 8-10 Oct. 1997, pp. 304 – 309.
[4] Lai Y-L., Wu C-L, Chiang K., “RF modeling of ball grid array packages using coupled transmission line model”, 2003 IEEE Radio Ferquency Integrated Circuits Symposim.
[5] K. S. Yee, "Numerical solution of inital boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol.
14, pp. 302-307, May 1966.
[6] 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. Microwave Theory and Techniques., vol. 38, pp. 849-857, July. 1990.
[7] J. P. Berenger, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas and Propagation, vol. 44, pp. 110-117, Jan. 1996.
[8] J. M. Jong and V. K. Tripathi, “Time domain characterization of interconnect discontinuities in high speed circuits,” IEEE Trans. Comp., Hybrids, Manuf Technol., vol. 14, pp. 497-504, Aug. 1992.
[9] Y. Hua and T. K. Sarkar, “Generalized pencil-of function method for extracting poles of an EM System from its tansient response”, IEEE Trans. Antenna Propagat. , vol. 37, no. 2, pp. 229 - 234, February 1989.
[10] T. K.Sarkar and O. Pereira, “Using matrix pencil method to estimate the parameters of a sum of complex exponentials”, IEEE Antennas and Propagat. Magazine,vol.37,no1,pp. 48 - 53. February 1995.
[11] Hsiao-Chen Chang, Tzong-Lin Wu, “Equivalent Circuit Extraction of Embedded High-speed Interconnect by Combining FDTD method and Layer Peeling Technique”, Master Thesis of National Sun Yat-Sen University, 2002.
[12] Chen-Chao Wang, Chih-Wen Kuo, “A Time Domain Approach for Effective Synthesizing of Broadband SPICE-Compatible Models of the Power Delivery Networks with Resonance Effect”, Doctor Thesis of National Sun Yat-Sen University, 2008.
[13] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain
responses by vector fitting,” IEEE Trans. Power Delivery, vol. 14, pp. 1052–1061, July 1999.
[14] G. Antonini, “SPICE equivalent circuits of frequency-domain responses,” IEEE Trans. Electromagnetic Compatibility, vol. 45, pp. 502-512, Aug. 2003.
[15] A. Taflove, Computational Electromagnetic; the Finite-Difference Time Doamin Method. Norwood, MA: Artech House, 1995.
[16] Z. Wu and J. Fang, “Numerical implementation and performance of perfectly matched layer boundary condition for waveguide structures,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2676-2683, Dec. 1995.
[17] R. Mittra and U. Pekel, “A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave and Wireless Components Letters, vol. 5, pp. 84-86, March 1995.
[18] Jian-Sheng Shie, Tzong-Lin Wu, “Characterization and Equivalent Circuit Modeling for Interconnection Structure from Time Domain Measurement,”, Master Thesis of National Sun Yat-Sen University, 2000.
[19] B. Gustavsen, “Computer code for rational approximation of frequency dependent admittance matrices, ” IEEE Trans Power Delivery , Volume: 17,pp. 1093- 1098, Oct 2002.
[20] Chen-Chao Wang, Chih-Wen Ku, Chun-Chih Kuo, Tzong-Lin Wu, “A Time-Domain Approach for Extracting Broadband Macro-π Models of Differential Via Holes,” IEEE Transactions on Advanced Packaging, Vol. 29, No. 4, pp. 789 - pp. 797, Nov. 2006.
[21] T. L. Wu, C. C. Wang, C. C. Kuo, and J. S. Hsieh, “A Novel Time-Domain Method for Synthesizing Broadband Macro-pi Models of Differential Via,” IEEE Microwave and Wireless Components Letters, vol. 15, pp. 378-380, May 2005.
[22] 張智星, MATLAB程式設計與應用,清蔚科技出版, 2000年
[23] B. Gustavsen and A. Semlyen, “Enforcing passivity for admittance matrices
approximated by rational functions,” IEEE Trans. Power Syst., vol. 16,
no. 1, pp. 97–104, Feb. 2001.