時域有限差分法 ( Finite-Difference Time-Domain,，FDTD )，是將馬克斯威爾方程式 ( Maxwell’s Equation )以二階中央差分法離散化，配合空間網格上之電磁場配置，在一個有限體積之計算空間內，以電生磁磁生電的跳步(leapfrog)方式，一步步計算出空間中電磁場的分佈情形。不同於頻域分析的方法(如Finite Element Method )，在頻域計算中需一個一個頻率點去分析，故當欲觀察的頻率極其寬頻時，所花費的時間將非常長；而使用時域的分析法，其好處為分析出來的數據只要透過傅利葉轉換即可得到完整的頻域響應，而不用再重新計算。

由於現今的套裝軟體，並無法將場論與電路模擬做一兩者兼具的結合，故微波電路的模擬與實做，常會因考量不夠週全而產生誤差。而時域有限差分法除了也是全波分析的技巧，經過延伸，如連結到SPICE或加入S參數等，更可在模擬結構中包含集總元件、非線性元件或主動元件等。故本論文中，我們提出了一種高效率處理複雜集總元件之時域有限差分方法，避免了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，因此有較好的計算效率，並且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。

The finite-Difference Time Domain method (FDTD) derives the discrete form of the Maxwell’s equations with second-order central difference with the electromagnetic distribution of the Yee space lattice, and computes the value of the electric field and magnetic field in the simulation space using leapfrog for time derivatives. This method is different from the frequency domain method which needs to analyze its value individually (ex. Finite Element method). The frequency domain method needs to take a long time for analyzing the response on each spectrum point when the bandwidth is very wide. The advantage of time domain analysis is to obtain the complete frequency response from the simulation value through Fourier Transform method.

It’s difficult to combine the electromagnetic analysis with the lumped circuit simulation in current simulation CAD. Thereby the performance of the simulation result and the practical implementation always causes error. The FDTD method is the full-wave algorithm which can also simulate the lump element, nonlinear element or active element in simulation space by linking to SPICE or S-parameter. In this dissertation, an efficient scheme for processing arbitrary one-port devices in the finite-difference time-domain (FDTD) method is proposed. 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. The accuracy of the proposed method is verified by comparison with results from the equivalent current-source method and is numerically stable.

誌謝.................................................................................................................................i
中文謫要........................................................................................................................ii
英文謫要.......................................................................................................................iii
目錄....………………………………………………………………………………...iv
圖表目錄……………………………………………………………………………...vi
第一章 序論…………………………………………………………………………1
1.1 概述…………………………………………………………………………1
1.2 論文大綱……………………………………………………………………3
第二章 FDTD演算法………………………………………………………………4
2.1 FDTD公式推導……………………………………………………………4
2.2 Courant穩定準則………………….………………………………… ……7
2.3 激發源……………………………. ………………………………… ....….7
2.3.1 取代源…………...………………………………………………….8
2.3.2 附加源…………………….…………………….…………………..8
2.3.3 阻抗性電壓源…………….…………………….…………………..8
2.4 吸收邊界……….…………………………………………………………...9
2.4.1 Mur一階吸收邊界….…………………….……...…..…………….9
2.5 導体與介電係數…………………………..………………………………11
2.6 散射參數…………………………………..………………………………11
第三章 集總元件模擬………………………………………………..……………12
3.1 集總元件演算法………………………………………………….……….12
3.1.1 電阻………………………………………………………………..12
3.1.2 電容………………………………………………………………..14
3.1.3 電感…………………………………………….………………….15
3.1.4 阻抗性電壓源……………………………………………………..15
3.2 等效電源法………………………………………………………………..20
3.2.1 等效電流源法……………………………….…………………….20
3.2.2 等效電壓源法……………………………………………………..24
3.2.3 模擬LC串聯電路…………………………………………………27
3.3 其他集總元件併入FDTD演算法的前置處裡方法………………....….29
3.3.1 片段線性遞迴摺積法………………………………………..……29
3.3.2 矩陣理論法…………………………………………..……………34
3.3.3 ㄧ種擴充到雙埠網路的方法……………………………………..37
第四章 ㄧ種高效率以處理集總元件之FDTD方法………………………….....41
4.1 線性動態系統的狀態變數分析……………………………………..……41
4.1.1 轉移函數…………………………………………………………..42
4.1.2 動態方程式………………………………………………………..43
4.1.3 轉移函數的分解…………………………………………………..44
4.2 轉移函數分解法應用至FDTD演算法……………………….………….51
4.3 方法上的比較……………………………………………………………..57
4.3.1 CCF/OCF方法 VS. 集總元件和遞迴摺積法……………… . …57
4.3.2 CCF/OCF方法 VS. 等效性電流源法…………………………...57
4.4 數值驗證與模擬…………………………………………………………..60
4.4.1 模擬一個串聯RLC電路………………………………………….60
4.4.2 模擬一個複雜集總電路…………………………………………..61
4.4.3 模擬蕭基二極體等效模型與帶通濾波器………………………..63
4.5 數值效率比較……………………………………………………………..67
第五章 結論………………………………………………………………………..68
附錄A. 解多重根之牛頓法推導…………………………………………………..69
參考文獻……………………………………………………………………………..76
圖2.1 FDTD單位空間網格的電磁埸空間配置…………………………..…………6
圖2.2 FDTD電磁場演算順序圖………………………………….….……..6
圖2.3 有限體積之計算空間示意圖…………………………………………………9
圖2.4 入射到吸收邊界的電磁波…………………………………………………..10
圖2.5介質與空氣交界處之示意圖..………………...…………………………11
圖3.1 阻抗性電壓源在網格上之示意圖…………........................………………..16
圖3.2微帶線饋入矩形patch天線(平面波激發).......………………………….…..17
圖3.3微帶線饋入矩形patch天線(阻抗性電壓源激發)…………………………..18
圖3.4採用不同激發源的反回損耗 之比較………………..……………………18
圖3.5 在時間步階40，240，440，640時，矩形patch天線的 場散佈情形(平面源激發)…………………………...……………………………………………..…19
圖3.6 在時間步階40，240，440，640時，矩形patch天線的 場散佈情形(阻抗性電壓源激發)…………………………...……………………………………..…19
圖3.7等效電流源法概念圖………………………….………….………………….20
圖3.8等效電流源法之等效電路示意圖………….…….……….……………...….22
圖3.9等效電流源法的計算流程圖………………………..………………………..23
圖3.10等效電壓源法概念圖……….………………………………………………24
圖3.11等效電壓源法之等效電路示意圖………………………………………….26
圖3.12等效電壓源法的計算流程圖..…………………………...………………….27
圖3.13 LC串聯電路示意圖……………………..…………………………………27
圖3.14 LC串聯電路之Return Loss 比較圖…………….…………...……………28
圖3.15片段線性方法電場圖形……………….………………………………...…..30
圖3.16電場時間反轉和時間位移圖…………………………..……………….…...30
圖4.1各種描述系統方法之彼此關係…………………………..………….….……42
圖4.2轉移函數表示法…………………………………….………………….……..43
圖4.3 CCF/OCF和等效電流源法之等效電路示意圖……………………………..57
圖4.4串聯RLC電路折返損耗(Return Loss)參數比較圖……...……………….…60
圖4.5串聯RLC電路折返損耗(Return Loss)相位的比較圖…………….……..….61
圖4.6複雜集總電路折返損耗(Return Loss)參數的比較圖…………………….….63
圖4.7複雜集總電路折返損耗(Return Loss)相位的比較圖…………………….….63
圖4.8蕭基二極體等效模型電路圖…………………………………………………64
圖4.9帶通濾波電路………………………………………………………....………64
圖4.10蕭基二極體的折返損耗(Return Loss)參數比較圖…………………………65
圖4.11蕭基二極體的折返損耗(Return Loss)相位比較圖…………………………65
圖4.12帶通濾波電路的折返損耗(Return Loss)參數比較圖………………………66
圖4.13帶通濾波電路的折返損耗(Return Loss)相位比較圖………………………66

表4.1模擬lump-element方法之電壓-電流關係式……..…………………………57
表4.2各種模擬lump-element方法之優缺點比較表…....…………………………59
表4.3 CCF/OCF與等效性電流源執行速度效率比較表..........……………………67

[1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propagat., vol.14, No.3, pp.300-307, May 1966.

[2] A. Taflove, Computational Electrodynamics The Finite-Difference Time-Domain Method, 1995.

[3] 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 plannar microstrip circuits ,” IEEE Trans. Microwave Theory Tech. , vol. 38 , pp. 849-857 , July 1990.

[4] W. Sui , D. A. Christensen , and Carl H. Durney , “Extending the Two-Dimensional FDTD Method to Hybrid Electromagnetic System with Active and Passive Lumped Elements,” IEEE Transactions on Microwave Theory Tech. , vol. 40 , NO. 4 , April 1992 .

[5] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations”, Electromagnetic Compatibility, IEEE Transactions on Volume: 23, Nov 1981 ,pp. 377 –382

[6] W. Sui , D. A. Christensen , and Carl H. Durney , “Extending the Two-Dimensional FDTD Method to Hybrid Electromagnetic System with Active and Passive Lumped Elements ,” IEEE Transactions on Microwave Theory Tech. , vol. 40 , NO. 4 , April 1992 .

[7] M. P. May , A. Taflove , and J. Baron , “FD-TD Modeling of Digital Signal Propagation in 3-D Circuits with Passive and Active Loads ,” IEEE Transactions on Microwave Theory and Techniques , vol. 42 , No. 8 , August 1994 .

[8] B. Toland, B.Houshmand, and T. Itoh, “Modeling of nonlinear active regions with the FDTD method,” accepted for publication in IEEE Microwave and Guided Wave Letters.

[9] Vincent A. Thomas, Michael E. Jones, Melinda Piket-May, Allen Taflove, and Evans Harrigan, “The use of SPICE lumped circuits as sub-grid models for FDTD analysis.” accepted for publication in IEEE Microwave and Guided Wave Letters, Vol. 4, No. 5, May 1994.

[10] Chien-Nan Kuo , Ruey-Beei Wu , Bijan Houshmand , and Tatsuo Itoh ,”Modeling of Microwave Active Devices Using the FDTD Analysis Based on the Voltage-Source Approach ,” IEEE Microwave and Guided Wave Letters , Vol. 6 , No. 5 , pp. 199-201 , May 1996.

[11] Chien-Nan Kuo , Bijan Houshmand , and Tatsuo Itoh ,”FDTD analysis of active circuits with equivalent current source approach , ” in 1995 IEEE AP-S Int. Symp. Dig. , Newport Beach , CA , June 1995 , pp. 1510-1513 .

[12] David F. Kelley, Raymind J. Luebber, “Piecewise Linear Recursive Convolution for Disperisve Media Using FDTD,” IEEE Trans. Antennas and Propagat., vol. 44,NO.6, pp. 792 -797, June. 1996.

[13] Jung Tub Lee, Jeong Hae Lee, and Hyun Kyo Jung, “Linear Lumped Loads in the FDTD Method Using Piecewise Linear Recursive Convolution Method,” IEEE Microwave and wireless components letters, Vol. 16,NO.4, pp. 158-160. , Apr. 2006

[14] Raymond Luebbers, Forrest P. Hunsberger, Karl S. Kunz, Ronald B. Standler, and Michael Schneider, “A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials,” IEEE Transactions on electromagnetic compatibility., vol.32, pp. 222-227, August 1990.

[15] 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.

[16] 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.

[17] 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.

[18] 楊受陞,江東昇,自動控制,第二版,儒林圖書股份有限公司,1996

[19] Tzong-Lin Wu, Sin-Ting Chen and Yi-Shang Huang,“A Novel Approach for the Incorporation of Arbitrary Linear Lumped Network Into FDTD Method ,” IEEE. Microwave and wireless components letters , vol. 14 , NO.2,pp. 74-76 , Ferbuary 2004.

[20] Benjamin C. Kuo, Automatic Control Systems. ,Wiley, seventh edition ,1995.