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研究生:洪士哲
研究生(外文):Shih-Che hung
論文名稱:表面擇頻元件之數值研究
論文名稱(外文):Numerical Study of Frequency Selective Surface by Spectral Galerkin Method
指導教授:張建成張建成引用關係
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:應用力學研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2005
畢業學年度:93
語文別:中文
論文頁數:93
中文關鍵詞:表面擇頻元件頻率撰擇表面週期性陣列頻域動差法頻域阻抗法
外文關鍵詞:Frequency selective surfaceperiodic arrayspectral moment methodspectral immitance approach
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表面擇頻元件 (Frequency Selective Surface) 為空間中二維的週期性結構. 由於它的頻譜會在某個頻率造成全反射 (補貼型結構) 和全穿透 (孔洞型結構), 因此就像一個濾波器 (filter)一樣. 多年來, 此元件已經有許多學術上的研究. 而除了學術上的研究, 表面擇頻元件更是已經在電磁波中的許多頻段得到應用. 如在微波頻段, 它可幫助雷達罩 (radomes) 和天線系統 (antenna systems) 的設計; 在紅外線和次微米頻段, 可用來做極化器 (polarizers) 和分波器 (beam splitter); 在遠紅外線頻段, 其可應用於太陽能收集表面 (solar selective surfaces)等等.

在本文中, 我們引用頻域Galerkin的方法來分析該元件的散射問題. 首先, 我們會將問題分成入射波傳播和金屬散射兩個部份以便處理兩種不同特性的問題. 在金屬散射的問題中, 我們用了Floquet’s Theorem, 使得週期性結構內的表面感應電流得以用傅立葉級數來表示, 並使無限延伸的計算域縮減至一個單位單元 (a unit cell). 帶入在金屬表面的邊界條件後, 我們可由已知的入射場和頻域的格林函數求得未知的表面感應電流, 並由此分析其穿透和反射的現象. 為了分離TM (transverse-magnetic) 和TE (transverse-electric) 入射波, 我們運用位勢能的方法來假定入射波. 另外, 我們還使用Itoh的頻域阻抗法 (spectral domain immitance approach) 來求得頻域的格林函數.

為了讓我們的程式在計算的形狀上擁有自由度, 我們採用子空間基礎函數 (Subdomain Basis Function) 做為表面感應電流的展開基底. 在未知數目很大的時候, 我們運用共軛梯度演算法, 以迭代的方式來求未知矩陣. 而運算上, 我們更採用快速傅立葉轉換來加速我們的運算速度.

這篇論文著重在薄金屬屏幕的表面擇頻元件 (Thin-screen Frequency Selective Surface). 影響該元件頻率響應的參數非常多. 我們將探討不同形狀, 不同尺度的金屬屏幕, 不同入射角度, 包括補貼型和孔洞型的元件結構. 另外, 我們還探討了阻抗及介電質所造成的影響. 此外, 我們還應用了多次的傳輸線理論來完成了多層的表面擇頻元件結構. 同時, 我們也用參考文獻裡已知的結果來驗證我們的理論, 計算, 及電腦程式的正確性.
Frequency Selective Surfaces (FSSs) are a kind of periodically two-dimensional structures. Since it exhibits total reflection (patch FSSs) and total transmission (aperture FSSs) in certain frequency, there are a lot of researches in recent years. In addition to researches, there are many applications, like radomes and antennas system in microwave region or polarizers and beam splitters in the infrared and submillimeter region, and so on.

In this thesis, spectral Galerkin method is used to solve the scattering problem of FSSs. We will divide the physical problems into problems of incident-propagating and induced current scattering. On the problem of scattering, Floquet’s Theorem is used to express induced surface current density in Fourier series, and reduce computation domain into a unit cell. After applying the boundary condition at the location of conduction screen, we can calculate out the unknown induced surface current density distribution by using known incident and spectral Green’s function. And then, we can use that to analyze reflection and transmission. In order to decouple TE (transverse-electric) and TM (transverse-magnetic) waves, we use vector potential approach to suppose incident. In addition, spectral domain immitance approach is used to obtain spectral Green’s function.

In computation, in order to let our shapes free, subdomain basis functions are used to be basis of expansion of induced surface current. In addition, in the cases of big unknown matrices, conjugate gradient method is used. And we also use Fast Fourier Transform (FFT) to speed up our computation.

We stress on thin-screen frequency selective surfaces in this thesis. Many arguments affect its frequency responses. We are going to discuss several different kinds of numerical cases, including different shapes, different scales of conducting screen, patch and aperture, and different incident angles. Further, the FSSs with impedance and dielectrics, and multilayered-FSSs are considered. At the same time, we present some reference to verify the accuracy of our computations.
目錄

誌 謝 i

ABSTRACT ii

摘要 iii

第一章 序論 1

第二章 物理問題和定理 5
2.1 基本概念 5
2.2 單層表面擇頻元件 (Free-standing FSSs) 6
2.2.1 完美導體 7
2.2.2 非完美導體 12
2.2.3 小結 12
2.3 多層表面擇頻元件 13
2.3.1 多層結構的入射場 13
2.3.2 多層表面擇頻元件的頻域格林函數 17
2.3.2.1 頻域阻抗法 (Spectral Domain Immitance Approach) 18
2.3.2.2 傳輸線 (transmission line) 19
2.4 小結 23

第三章 運算方程式 24
3.1 動差法 (Method of Moment) 24
3.1.1 動差法的討論與運用 24
3.1.2 屋頂基礎函數 (Roof-top Basis Function) 25
3.1.3 運算方程式 28
3.2 增加運算速度 33
3.2.1 快速傅立葉轉換 (Fast Fourier Transform) 33
3.2.2 共軛梯度演算法 (Conjugate Gradient Algorithm) 34

第四章 反射與穿透係數 36
4.1 反射係數 36
4.2 穿透係數 41
4.3 小結 42

第五章 數值結果 44
5.1 幾何形狀 44
5.1.1 正方形, 長方形, 和十字形 44
5.1.2 補貼型和孔洞型表面擇頻元件 45
5.1.3 不同金屬比例 54
5.2 斜向入射 55
5.3 阻抗 62
5.3.1 垂直入射 62
5.3.2 斜向入射 67
5.4 介電質 67
5.4.1 介電質厚度的影響 67
5.4.2 介電常數的影響 69
5.5 多層導體屏幕表面擇頻元件 (multi-screen FSSs) 69

第六章 未來的工作 77

附錄 A Floquet’s Theorem 79

附錄 B 傅立葉級數倒晶格 (Reciprocal Lattice)的推導 82

附錄 C 特徵阻抗的推導 84

附錄 D 快速傅立葉轉換 87

參考文獻 90
參考文獻

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