開口式共振腔之研究常應用於磁旋管、磁旋振盪管中，與閉口式共振腔不同的是其有明確的邊界條件，因此我們運用程式做數值模擬其解析解來探討其不同的物理意義。先前的研究主要有兩種，一是在時域下，分析基本結構或是改變不同的共振腔結構（加入不匹配負載結構）、材料性質後，解析及探討不同的品質因子以及共振頻率的關係；二是在頻域下，注入不同頻率的電磁波後，看其反射係數與其注入頻率的關係。
本論文主要結果是發展自先前的基礎研究，在時域基礎模型中，定義電動力學中的Poynting vector在一個週期中的z方向上平均值為開口式共振腔功率流的淨值P_net，也改變共振腔的結構及材料性質，運用數值模擬分析電磁波能流大小在數值上受複數影響的樣態，並在文中給予合理的物理解釋及探討。

Researches of Open Cavities are commonly used in gyrotrons and gyromonotron oscillators. Different from the enclosed cavities, it has well-defined geometrical boundary conditions. There are mainly two kinds in previous researches. One is under Time-Domain, analyzing and discussing the relation of different quality factor Q and resonant frequency with the basic structure or changing the structure or the material of the cavities. The other is under Frequency-Domain, injecting electromagnetic waves with different frequency. Then, observe and discuss the relation of coefficient and the injected frequency.
The main result of this thesis is developed from the previous researches of Time-Domain. Basic from the knowledges of Electrodynamics, we denote the average value of the Poynting vector in the z-direction among a period as the net wave power flowing. Also, changing the structure, analyze the power flow of the wave with numerical simulations. Lastly, we give reasonable physical explanations in this thesis.

Contents
口試委員審定書………………………………………………………………………ii
誌謝……………………………………………………………….…………………..iii
中文摘要……………………………………………………………………………....v
Abstract…………………………………...………….………………….……………vi
1 Introduction……………………………...…………………………………………1
2 Time-domain models…………………….…………………………………………5
2.1 Dispersion relation of waveguide and standing wave in cavity………………..5
2.2 Assumptions and theory of the field basic equations…………………………..6
2.3 Energy flow and power loss in the waveguide…………………………………12
2.4 Field profile and the numerical results………………………………………...14
2.5 Relation between quality factor and the mode number………………………19
2.6 Effect of changing the length of the simple structure…………………………20
2.7 Effect of changing the taper angle of the simple structure……………………24
2.8 A slightly mismatched load at the output wave guide…………………………28
2.9 The effect of different position of the mismatched load…………………...…..30
2.10 The periodic results of three TE modes with mismatched load..……..……..39
3 Power flow analysis under time model….…………..…………………………..41
3.1 Time averaged net wave power, Pnet………………………………………...….41
3.2 Time averaged forward and backward wave powers, Pfwd and Pbwd…………51
3.3 Power flow analysis on cavities with different resistivity…………………..…58
3.4 Power flow analysis on cavities with New structure…………………………..68
4 Conclusion………………………………...………………………………………73
Bibliography…………………………………………………………………………..74
Appendix……………………………………………………………………..….….…75
A The Runge-Kutta method………………………………………………………..75
B Muller’s method…………………………………………………………………..79
C Wave Equation of circular structure…………………………………………….82
D Power flow functions of circular structure…………………………….….….…85



 
List of Figures
1 Simple configuration of an open cavity……………………………………..…..….1
2 Circular cross-section of the open cavity………………………………………..….8
3 Amplitude of field profile |f(z)| as a function of z in three TEmnl modes……..…...15
4 Phase angle of field profile Φ(z) as a function of z in three TEmnl modes…....…16
5 Cross-section view of the closed cavity with the dimension given in table 1..……17
6 |f(z)| of the open cavity (red line) and the closed cavity (black dash line).…….….17
7 Relation between quality factor and the mode number………………….….……..20
8 Relation between changing length and the quality factor……….……….…….…..21
9 Schematic group velocity in the cavity………………………….……….………..22
10 The relation between changing length and the resonance frequency…..………...23
11 The corresponding relation between quality factor and the resonance frequency under changing length…………………………………………………..………..24
12 The relation between changing taper angle and the quality factor…….…...……25
13 The relation between changing taper angle and the quality factor…….………...26
14 The simple structure with a slightly mismatched load (iris)…………………….28
15 Schematic figure of the cavity with the mismatched load, which is seen as two
cavities for convenient explanations……………………………………………..29
16 Field profile |f(z)| as a function of z in three TEmnl modes with iris…….….…..29
17 Resonant frequency via the length of the 2nd cavity with the mismatched….…..30
18 Quality factor via the length of the 2nd cavity with the mismatched………….…31
19 Five points among half-period of the varying quality factor……………….…….32
20 Field profile |f(z)| of five points among half-period of the varying quality factor..33
21 Frequency and quality factor simultaneously via the length of the “cavity 2”…..34
22 Four points at four continuous wave crest of the varying quality factor…………35
23 Field profile |f(z)| at four continuous wave crest of the varying quality factor…..35
24 Plot four field profile in Figure 20 separately……………………………………36
25 Quality factor via the length of the 2nd cavity with the mismatched of three
different height……………………………………………………………………38
26 Resonant frequency via the length of the 2nd cavity with the mismatched of three
TE modes…………………………………………………………………………39
27 Quality factor via the length of the 2nd cavity with the mismatched of three TE
modes………………………………………………………………………..……40
28 Net wave power in the simple structure…………………………………………..46
29 Net wave power of log scale in the simple structure……………………………...47
30 Net wave power of TE112 mode in the simple structure……………….……..…...47
31 Net wave power of TE113 mode in the simple structure……………….……..…...48
32 Net wave power and field profile of TE112 mode in the simple structure…………49
33 Net wave power and field profile of TE113 mode in the simple structure…………49
34 Net wave power of three TE modes in the simple structure………………………50
35 Forward and backward wave power compared with net power of TE111 mode…..54
36 Forward and backward wave power compared with net power of TE112 mode…..56
37 Forward and backward wave power compared with net power of TE113 mode…..56
38 Reflection coefficient varies with the frequency………………………………….57
39 Reflection coefficient varies with the frequency over cutoff frequency………….57
40(a) Configuration of an cavity with “Cu-Perfect Conductor-Cu” resistivity………59
40(b) An cavity with “Perfect Conductor-Cu-Perfect Conductor” resistivity………..59
41(a) Net wave power in the “Cu-PC-Cu” structure…………………………………60
41(b) Net wave power in the “PC-Cu-PC” structure…………………………………61
42(a) Absolute value of net wave power in the “Cu-PC-Cu” structure……………...61
42(b) Absolute value of net wave power in the “PC-Cu-PC” structure……………...62
43(a) Forward and backward wave power in the “Cu-PC-Cu” structure…………….62
43(b) Forward and backward wave power in the “PC-Cu-PC” structure……………63
44 The section structure with all the same lossy wall………………………………..64
45 Net wave power of the all copper resistivity structure……………………………65
46 Forward and backward wave power of the all copper resistivity structure……….65
47 Schematic diagrams of three-sections structure of cavities with Pnet……………..66
48 New structure of the open cavity………………………………………………….68
49 Net wave power of TE111 mode in the new structure……………………………..69
50 field profile of TE111 mode in the new structure…………………………………..69
51 Forward and backward wave power in the new structure……………….……..…70
52 Forward and backward wave power under a relatively small scale………………71
53 Net wave power of different taper angle in the new structure…………………….71
54 A trial step to calculate the midpoint of the interval……………………….……..77
55 The starting approximations p0, p1, and p2 for Muller’s method, and the differences
h0 and h1…………………………………………………………………………...79




 
List of Tables
1 dimensions of the open cavity…………………………………………….…….…..7
2 The resonant frequency and effective cavity length of two cavities………….…...18
3 Quality factor and resonant frequency of three TE modes………………….….….27
4 Dimensions of the open cavity with the slightly mismatched load………………..28
5 Information for choosing five length values among half-period……………….….33
6 Information for another group of changing length of “cavity 2”……………….…37
7 Information for changing the height of the iris……………………………………38
8 Dimensions of the open cavity with different section of resistivity……………….60
9 Information for three similar structure of cavities…………………………………66
10 Dimensions of the new structure………………………………………………….68
11 Information of different taper angle with new structure of TE111 mode…………..72

[1] K. R. Chu, “Time-Domain Analysis of Open Cavities” (2013)
[2] J. D. Jackson, “Classical Electrodynamics,” John Wiley & son, Inc., p.p. 264-265, 353-374 (1998)
[3] C. L. Hung, Y. C. Tsai, and K. R. Chu, Fellow, “A study of Open-End Cavities by the Field-Energy Method,” IEEE Transactions on Plasma Science, Vol. 26, No. 3 (1998)
[4] 陳冠文, “Power flow analysis,” (2017)
[5] 王俊霖, “開口式共振腔之能流分析,” 國立台灣大學碩士論文 (2018)
[6] 盧姿穎, “開口式共振腔頻譜特性之研究,”國立台灣大學碩士論文 (2015)
[7] 王宗強,”新型開口式共振腔之數值模擬與分析,” 國立台灣大學碩士論文(2017)
[8] William H. Press, Brian P. Flanner, Saul A. Teukolsky and William T. Vetterling, “Numerical Recipes in C: The Art of Scientific Computing, “Chapter 16
[9] John H. Mathews and Kurtis K. Fink, “Numerical Methods Using Matlab,“ chaper 2