在本論文中我們將探討時域有限差分法(Finite Difference Time Domain，FDTD)應用於平行板電容的分析，經由驅動方程式建立準靜電荷的特性來研究平行電容板的結構，由準靜電荷穩定後的電場值來計算出平行金屬板間的電壓，然後再將電荷總量除以電壓即可獲得電容值，並可將平行板電容的特性分析應用於感測器上。
    我們設計三種不同結構的電容，來探討線性度的問題，同時利用FR4板金屬以蝕刻方式實際製作電容和FDTD模擬結果做比較；因邊緣效應產生出的非線性問題，會影響到電容式感測器的準確度。我們同時透過實驗法及遺傳演算法( genetic algorithm )進一步求出示波器輸入阻抗及同軸電纜線的輸入電容，再透過電容分壓原理，即可求出未知平行電容板的電容值。電容感測器不僅能做距離的量測，可經由實驗和FDTD模擬來找出不同介電常數的待測物材料。

In this thesis, the application of the finite-difference time-domain (FDTD)method for the parallel-plate capacitor are investigated. Through the use of the transparent current source, the parallel-plates are charged such that the transient and static E fields can be simulated using the FDTD update equation. The static E field is used to calculate the voltage across the parallel-plates. The capacitance is obtained directly by dividing the charge over the voltage.  The study of the characteristics for the parallel plate capacitor is make to emphasis on the linearity analysis for the possible sensor application.
There are three capacitor structures investigated for linearity analysis. The capacitors are fabricated by using FR4 board, of which the capacitances are compared with the FDTD simulation results.  The non-linearity due to the fringing effect would degrade the performance of a capacitor sensor.  The simulated linearity curves of certain structure obtained in this thesis are different from the literature. To confirm the simulated results, the capacitors are fabricated and measured carefully by using a digital oscilloscope.  To increase the measurement accuracy we setup a de-embedding procedure for the calibration of the connecting cable and the digital oscilloscope.  Furthermore, the genetic algorithm (GA) is employed to find out input impedance of oscilloscope and the capacitance of the probe.  Finally, through a simple voltage dividing rule, we can resolve the capacitance for the capacitor under test. On the other hand, the capacitive sensors are also applied to find the dielectric property between the parallel plates.

目   錄
第一章    序 論...	1
1.1研究動機.......	1
1.2 論文大綱......	3
第二章	有限時域差分法....	4
2.1 簡介...................4
2.2 馬克斯威爾方程式.......	4
2.3 Yee單胞的解析方法......	6
2.4 FDTD的演算法..	.........8
2.5 單胞的尺寸大小和Courant穩定準則...12
2.6 吸收邊界條件....................	.13
2.7 激發源(exciting source)	..........14
 第三章	電容感測器線性度分析........16
3.1 簡介..............16
3.2 電容結構設定..	....17
3.3 電容線性度比較	....21
3.3.1有guard ring的平板電容線性度分析........	21
3.3.2有guard ring的平板電容電場的分佈........	25
3.3.3針形平板電容線性度分析	......31
3.3.4針形平板電容電場的分佈	......36
第四章	電容量測實驗............39
4.1 簡介...................39
4.2示波器輸入阻抗效正......	39
4.2.1單頻法校正同軸線電容 (1).......	41
4.2.2多頻法校正示波器輸入阻抗(1)....	44
4.2.3單頻法校正同軸線電容 (2).......	48
4.2.4多頻法校正示波器輸入阻抗(2)....	49
4.3有guard ring的平板電容實驗.......	56
4.4未知材質的介電常數量測...........	58
第五章 結論與展望...................	61
附錄一 遺傳演算法基本概述...........	62
參考文獻.............................73

                  圖  目  錄
圖2.1  FDTD的Yee單胞..........................6
圖2.2  電磁場的時間分配圖..........................7
圖2.3  分析的空間...................7
圖2.4  一次差分.........................9
圖3.1  平行電容板模擬結圖...........................19
圖3.2  電流源的電流I與金屬板上累積的電荷Q隨時間t
       的變化圖.......................................21
圖3.3  有guard ring的平板電容結構( 、 、
 、 且 )..................22
圖3.4  利用兩種不同結構計算的電容值倒(1)..........23
圖3.5  線性變異量的圖示說明.............................24
圖3.6  兩種不同結構方法線性變異量計算(1).............25
圖3.7  有guard ring的平板電容及一般平板電容下金屬板上的
面電荷密度分佈(由內到外)..................26
圖3.8  下板金屬上電場 ，沿著(y, z)=(50, 0)..........28
圖3.9  下板金屬下面一格處的電場 ，沿著(y, z)=(50, -1)....28
圖3.10 下板金屬上的電場 的分佈，沿著(y, z)=(50, 0)....29
圖3.11  With Guard電容電荷分佈示意圖..................29
圖3.12  With Guard電容變化圖(1)...................30
圖3.13  With Guard電容變化圖(2)......................31
圖3.14 針形平板電容結構.................32
圖3.15 針形平板電容針腳排列方式................32
圖3.16 利用兩種不同結構計算的電容值倒數(2).............33
圖3.17 兩種不同結構方法線性變異量計算值(2)............34
圖3.18 兩種不同結構上板震動的示意圖.....................34
圖3.19 電容上板震動的線性變異量計算值................35
圖3.20 pin腳高度不同時的線性變異量計算值...............36
圖3.21 沿著P1 P2 P3 三點連線上的的面電荷密度的圖示..............37
圖3.22 Capacitor with pins及Simple capacitor電容在d= 
下金屬板上面電荷密度分佈....................37
圖3.23 Capacitor with pins及Simple capacitor電容在d= 
下金屬板上面電荷密度分佈..............38
圖4.1  實驗結構圖....................................39
圖4.2  同軸電纜線上某一小段的等效電路........40
圖4.3  儀器內部等效電路............................40
圖4.4  不同的SMD電容響應圖............................42
圖4.5  LCR meter量出的電容值...........................44
圖4.6  以SMD標準電容的例子，目標函數收斂情形............46
圖4.7  以SMD電容為標準電容實驗數值Vo及GA 的
計算數值Vo...............................................47
圖4.8  平板標準電容setup結構....................48
圖4.9  以保麗龍平板電容為標準電容，目標函數的收斂情形.......50
圖4.10 以保麗龍平板電容為標準電容實驗數值Vo及GA 的
計算數值Vo..........................................51
圖4.11 示波器等效 模型...........................52
圖4.12  pi model 目標函數的收斂情形..............53
圖4.13  以pi model實驗數值Vo及GA 的計算數值Vo.....54
圖4.14  LCR meter量出的探棒電容值.....................56
圖4.15  利用兩種不同結構的電容值倒數....................58
圖4.16  介質為紙的電容值倒數..............................60
圖A-1  遺傳演算法流程....................................72

              表  目  錄
表3.1  兩種不同平板電容結構的計算電荷值...............27
表3.2  Capacitor with pins較Simple capacitor減少
電荷外溢的比例...............................38
表4.1  以15顆100pf SMD電容的電壓量測值..............43
表4.2  以SMD電容為標準電容實驗數值Vo及GA 的
計算數值Vo.................................46
表4.3  以保麗龍平板電容為標準電容實驗數值Vo及GA 
的計算數值Vo....................................50
表4.4  以pi model實驗數值Vo及GA 的計算數值Vo.........54
表4.5  不同電容及模型的各項參數比較................55
表4.6  比較不同方法求出一般簡單平板電容的電容值
(中間介電質為保麗龍 ).......................57
表4.7  比較不同方法求出有guard ring平板電容的電容值
(中間介電質為保麗龍 ).........................57
表4.8  比較不同方法量測介質為紙的電容值............59
表4.9  比較不同方法量測介質為FR4板基材的電容值......60

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