本論文提出一種結合直交表與響應表面模型以設計天線的創新方法—稱為改良式等差田口最佳化法，將其用於π型天線的設計，設計流程和同樣使用連續直交表的等差田口最佳化法類似。  
設計流程大致包括下述的步驟: 1)決定參數的搜尋範圍，2)利用直交表以提供取樣點並進行模擬/實驗，3)並根據模擬/實驗結果將天線設計流程所需的輸入及輸出間之關係建立成響應表面模型(本研究採用MQ函數來建立響應表面模型)，4)針對響應表面模型，以全域搜尋法來搜尋該代參數的最佳解，5) 依循將搜尋範圍以等差逐代縮減的過程，依序進行迭代實驗，以達到收斂得目的。
   本研究提出的改良式等差田口最佳化法捨棄建立響應表以獲致當代最佳位準組合的步驟，改以建立響應表面模型取代，再針對該響應表面模型，使用全域搜尋法來搜尋該代參數的最佳解，這是本論文的主要貢獻。 
　　針對兩種雙頻π型天線，本研究以上述引入響應表面模型概念的改良式等差田口最佳化法進行迭代設計，並將結果與之前應用等差田口最佳化法進行設計所獲致的結果進行比較。研究結果顯示使用響應表面模型確實能更精確地描述天線設計流程的輸入與輸出間之函數關係，因此，經由較少次的迭代實驗(指耗時的HFSS 的function calls)便能獲得符合規格要求的參數組合。

This thesis proposes a novel scheme for the application of antenna designs that integrates the ideas of orthogonal array and response surface model. The proposed scheme is called improved arithmetric Taguchi optimization method, which is applied to design the π-type antennas. The design process is similar to the arithmetric Taguchi optimization method, which also utilizes consecutive orthogonal arrays. In general, the design procedures proposed include the following steps: 1) determine the search ranges of the parameters, 2) the use of orthogonal array to provide the sampling points for simulations/experiments, 3) establish a response surface model for the antenna design process according to the results of simulations/experiments (MQ functions are used in this study for the response surface model), 4) apply a global search method to search the optimal parameters of the response surface model for the generation, 5) reduce the searching range iteratively in an arithmetric way, and repeat the simulations/experiments to achieve convergence.
   It should be noted that the proposed improved arithmetic Taguchi optimization method does not utilize the response table, which is usually used to obtain the best attainable level combination. Instead, it create a response surface model and then employ a global search method to search the optimal parameters of the response surface model for the generation, which is the main contribution of this thesis.
   For two kinds of dual-band π-shaped antenna, this study applies the improved arithmetic Taguchi optimization method to accomplish the iterative design, of which the results are compared with those obtained by previous application through arithmetic Taguchi optimization method. It is concluded that the improved scheme using the response surface model can indeed describe more accurately the input and output relationship of the parameter space for the antenna design.  Therefore, fewer iterations/experiments (as the time-consuming HFSS function call is concerned) are required in order to obtain the parameters that meet the specifications.

中文摘要 ................................................. I 
英文摘要 ............................................... III 
第一章 序論............................................... 1 
1.1 簡介 ................................................. 1 
1.2 研究背景 ............................................. 1 
1.3 論文架構 ............................................. 5 
第二章 平面雙頻天線設計 .................................. 6 
2.1 單極天線 ............................................. 6 
2.2 倒L 型天線 ........................................... 9 
2.2.1 單極天線與倒L 型天線之特性比較 ..................... 10
2.3 倒F 型天線 ........................................... 13
2.4 針對PIFA 天線窄頻的改良-Π型天線 ...................... 1
2.5 連續直交表 ........................................... 16
第三章 以田口最佳化法進行天線參數最佳化 .................. 24
3.1 簡介 ................................................. 24
3.2 設計天線參數及位準進行連續直交表的迭代 ............... 25
3.2.1 目標值分開討論以及延伸連續直交表的迭代次數 ......... 31
第四章 應用改良式田口最佳化法於天線設計 .................. 43
4.1 簡介 ................................................. 43
4.2 針對天線A 進行直交表實驗及響應表面模型建立 ........... 45
4.2.1 第二次迭代 ......................................... 53
4.2.2 第三次迭代 ......................................... 57
4.2.3 第四次迭代 ......................................... 61
4.2.4 第五次迭代 ......................................... 65
4.3 針對天線B 進行直交表實驗及響應表面模型建立 ........... 70
4.3.1 第二次迭代 ......................................... 75
4.3.2 第三次迭代 ......................................... 75
4.3.3 第四次迭代 ......................................... 83
4.3.4 第五次迭代 ......................................... 87
4.4 實作與量測 ........................................... 93
第五章 結論............................................... 95
參考文獻 ................................................. 97

圖2.1 (a)單極天線 (b)單極天線電流分佈 .................... 6 
圖2.2 (a)偶極天線及其輻射場形 (b)偶極天線有著串聯的電壓與對  
稱的平面 (c)單極天線在無窮大接地面上 ..................... 8 
圖2.3 倒L 型天線的示意圖 ................................. 9 
圖2.4 單極天線彎曲(Bend)形成倒L 型天線的示意圖 ........... 10
圖2.5 不同高度H 與彎曲長度L 的倒L 型天線反射損耗模擬圖 ... 11
圖2.6 不同高度H 與彎曲長度L 的倒L 型天線的輸入阻抗圖 ..... 11
圖2.7 倒F 型天線的示意圖 ................................. 13
圖2.8 π 型天線結構示意圖 ................................. 1
圖2.9 以重疊定理與奇偶模概念分析一對稱系統的示意圖 ....... 15
圖2.10 田口最佳化法流程圖 ................................ 19
圖2.11 改良式田口最佳化法流程圖 .......................... 22
圖3.1 初始Π 型天線結構示意圖 ............................. 2
圖3.2 五次迭代實驗之反射損耗變化圖 ....................... 28
圖3.3 天線A 的五次迭代實驗之反射損耗變化圖 ............... 32
圖3.4 天線B 的五次迭代實驗之反射損耗變化圖 ............... 32
圖3.5 天線A 修改目標頻率之五次迭代實驗反射損耗變化圖 ..... 34
圖3.6 頻率2.4GHz 時的電流分佈圖(天線A) ................... 35
圖3.7 頻率2.64GHZ 時的電流分佈圖(天線A) .................. 35
圖3.8 頻率5.24GHZ 時的電流分佈圖(天線A) .................. 36
圖3.9 參數A 微小變動對S11 參數之影響(天線A) .............. 36
圖3.10 參數B 微小變動對S11 參數之影響(天線A) ............. 37
圖3.11 參數C 微小變動對S11 參數之影響(天線A) ............. 37
圖3.12 天線B 修改目標頻率之五次迭代實驗反射係數幅度變化圖. 38
圖3.13 第1 次～第5 次之迭代實驗反射係數幅度變化圖 ........ 40
圖3.14 第6 次～第10 次之迭代實驗反射係數幅度變化圖 ....... 40
圖3.15 天線B 縮小範圍的五次迭代實驗反射係數幅度變化圖 .... 42
圖4.1 (a) 當C 參數固定時A、B 參數之響應表面模型(b) 當B 參數  
固定時A、C 參數之響應表面模型(c) 當A 參數固定時B、C 參數     
之響應表面模型 ....................................... 50、51
圖4.2 第一次迭代實驗之反射係數幅度變化圖 ................. 52
圖4.3 (a) 當C 參數固定時A、B 參數之響應表面模型(b) B 參數固定
時A、C 參數之響應表面模型(c) A 參數固定時B、C 參數之響應     
表面模型 ............................................. 54、55
圖4.4 第二次迭代實驗之反射係數幅度變化圖 ................. 56
圖4.5 (a) 固定C 為參數最佳解時A、B 參數之響應表?            
定為參數最佳解時A、C 參數之響應表面模型(c) A 固定為參數最    
佳解時B、C 參數之響應表面模型 ........................ 58、59
圖4.6 第三次迭代實驗之反射係數幅度變化圖 ................. 59
圖4.7 (a) 固定C 為參數最佳解時A、B 參數之響應表面模型(b)固定 
B 為參數最佳解時A、C 參數之響應表面模型(c)固定A 為參數最     
佳解時B、C 參數之響應表面模型 ........................ 62、63
圖4.8 第四次迭代實驗之反射係數幅度變化圖 ................. 63
圖4.9 (a) 固定C 為第五次迭代參數最佳解時A、B 參數之響應表面  
模型(b)固定B 為第五次迭代參數最佳解時A、C 參數之響應表面     
模型(c)固定A 為第五次迭代參數最佳解時B、C 參數之響應表面     
模型 ................................................. 66、67
圖4.10 第五次迭代實驗之反射係數幅度變化圖 ................ 67
圖4.11 天線A 五次迭代實驗之反射損耗圖 .................... 69
圖4.12 (a) 固定C 參數為第一次迭代最佳解時A、B 參數之響應表面 
模型(b)固定B 參數為第一次迭代最佳解時A、C 參數之響應表面     
模型(c)固定A 參數為第一次迭代最佳解時B、C 參數之響應表面     
模型 ................................................. 72、73
圖4.13 天線B 第一次迭代實驗之反射係數幅度變化圖 .......... 73
圖4.14 (a) 固定C 參數為第二次迭代最佳解時A、B 參數之響應表面 
模型(b)固定B 參數為第二次迭代最佳解時A、C 參數之響應表面     
模型(c)固定A 參數為第二次迭代最佳解時B、C 參數之響應表面     
模型 ................................................. 76、77
圖4.15 天線B 第二次迭代實驗之反射係數幅度變化圖 .......... 77
圖4.16 (a) 固定C 參數為第三次迭代最佳解時A、B 參數之響應表面 
模型(b)固定B 參數為第三次迭代最佳解時A、C 參數之響應表面     
模型(c)固定A 參數為第三次迭代最佳解時B、C 參數之響應表面     
模型 ................................................. 80、81
圖4.17 天線B 第三次迭代實驗之反射係數幅度變化圖 .......... 81
圖4.18 (a) 固定C 參數為第四次迭代最佳解時A、B 參數之響應表面 
模型(b)固定B 參數為第四次迭代最佳解時A、C 參數之響應表面     
模型(c)固定A 參數為第四次迭代最佳解時B、C 參數之響應表面     
模型 ................................................. 84、85
圖4.19 天線B 第四次迭代實驗之反射係數幅度變化圖 .......... 85
圖4.20 (a) 固定C 參數為第五次迭代最佳解時A、B 參數之響應表面 
模型(b)固定B 參數為第五次迭代最佳解時A、C 參數之響應表面     
模型(c)固定A 參數為第五次迭代最佳解時B、C 參數之響應表面                                                                
模型 ................................................. 88、89
圖4.21 天線B 第五次迭代實驗之反射係數幅度變化圖 .......... 89
圖4.22 天線B 五次迭代實驗之反射係數幅度變化圖 ............ 91
圖4.23 π 型天線實體圖 .................................... 94
圖4.24 π 型天線之反射損耗實測與模擬比較圖 ................ 94

表2.1 虛部為零，實部與共振頻率隨高度H 改變時的變化表 ..... 12
表2.2 直交表OA (9,3,3,2) ................................. 17
表3.1 直交表OA (9,3,3,2) ................................. 28
表 3.2 OA(9,3,3,2)在第一次迭代實驗之位準、適應值及訊號雜訊比 
表 3.3 第一次迭代實驗後經由計算所得之響應表 .............. 30
表 3.4 第一次迭實驗後代經由響應表所選取之最佳參數組合 .... 30
表 4.1 天線A 與天線B 第一次迭代使用之直交表 .............. 46
表 4.2 OA(9,3,3,2)在第一次迭代實驗之位準、目標函數值及訊號雜 
訊比 ..................................................... 47
表 4.3 響應表面模型化於第一次迭代之係數C1~C9 ............. 49
表 4.4 各參數於第二次迭代之位準值與搜尋範圍 .............. 53
表 4.5 各參數於第三次迭代之位準值與搜尋範圍 .............. 57
表 4.6 各參數於第四次迭代之位準值與搜尋範圍 .............. 61
表 4.7 各參數於第五次迭代之位準值與搜尋範圍 .............. 65
表 4.8 天線A 五次迭代實驗之個別最佳參數組合 .............. 68
表 4.9 SADD 與HFSS 於五次迭代實驗所獲得之cost value ...... 68
表 4.10 天線B 於第一次迭代各參數位準之位準值與搜尋範圍 ... 70
表 4.11 天線B 於第二次迭代中各參數位準之位準值與搜尋範圍 . 75
表4.12 天線B 於第三次迭代中各參數位準之位準值與搜尋範圍 .. 79
表 4.13 天線B 於第四次迭代中各參數位準之位準值與搜尋範圍 . 83
表 4.14 天線B 於第五次迭代中各參數位準之位準值與搜尋範圍 . 87
表 4.15 天線B 於第五次迭代之最佳參數組合 ................. 90
表 4.16 第五次迭代中SADDE 與HFSS 模擬所獲得之cost value .. 90

[1]S. Ahn and H. Choo , “A Systematic Design Method of On-Glass Antennas Using Mesh-Grid Structures,” IEEE Transactions on Vehicular Technology, Volume 59 , Issue 7, Pages 3286-3293, 2010.
[2]A. Andujar, J. Anguera and C. Puente , “A systematic method to design broadband matching networks,” 2010 Proceedings of the Fourth European Conference on Antennas and Propagation (EuCAP), pp. 1-5, 2010.
[3]O. Lopez, L.G. de Vicuna, M. Castilla, M. Lopez and J. Majo, “A systematic method to design sliding mode surfaces by imposing a desired dynamic esponse”, IECON '98. Proceedings of the 24th Annual Conference of the IEEE, vol.1, pp.381-384, 1998.
[4]M. Moosazadeh, A.M. Abbosh, and Z. Esmati, “Design of compact planar ultrawideband antenna with dual-notched bands using slotted square patch and pi-shaped conductor-backed plane,” IET on Microwaves, Antennas & Propagation, Volume 6 , Issue 3, pp.290-294, 2012.
[5]W.A. Swart, and J.C. Olivier, “Numerical synthesis of arbitrary discrete arrays,” IEEE Transactions on Antennas and Propagation, Volume 41, Issue 8, pp.1171-1174, 1993.
[6]A. Boag, A. Boag, E. Michielssen and R. Mittra, “Design of electrically loaded wire antennas using genetic algorithms,” IEEE Transactions on Antennas and Propagation, Volume 44, Issue 5, 1996.
[7]Anguera, J.; Puente, C.; Borja, C., “A procedure to design wide-band electromagnetically-coupled stacked microstrip antennas based on a simple network model”,IEEE Antennas and Propagation Society International Symposium, 1999, Volume 2, pp. 944 –947.
[8]N. Jin, and Y. Rahmat-Samii, “Parallel particle swarm optimization and finite- difference time-domain (PSO/FDTD) algorithm for multiband and wide-band patch antenna designs,” IEEE Transactions on Antennas and Propagation, Volume 53, Issue 11, pp. 3459-3468, 2005.
[9]D. Staiculescu, N. Bushyager, A. Obatoyinbo, L.J. Martin, and M.M. Tentzeris, “Design and optimization of 3-D compact stripline and microstrip Bluetooth/WLAN balun architectures using the design of experiments technique,” IEEE Transactions on Antennas and Propagation, Volume 53,Issue 5, pp. 1805-1812, 2005.
[10]Wei-Chung Weng; Fan Yang; Elsherbeni, A.Z., “ Linear Antenna Array Synthesis Using Taguchi's Method: A Novel Optimization Technique in Electromagnetics,” IEEE Transactions on Antennas and Propagation, Volume 55, Issue 3, Part 1, pp. 723 – 730, 2007.
[11]Lin-Yu Tseng; Tuan-Yung Han, “An Evolutionary Design Method Using Genetic Local Search Algorithm to Obtain Broad/Dual-Band Characteristics for Circular Polarization Slot Antennas” IEEE Transactions on Antennas and Propagation,Volume 58, Issue 5, pp.1449-1456, 2010.
[12]N. Jin, and Y. R. Samii, “Parallel Particle Swarm Optimization and Finite-Difference Time-Domain(PSO/FDTD) Algorithm for Multiband and Wide-Band Patch Antenna Designs,” IEEE Transactionson Antennas and Propagation, vol. 53, No. 11, Nov 2005, pp. 3459-3468.
[13]G. Kiziltas, D. Psychoudakis, J. L. Volakis, and N. Kikuchi, “Topology Design Optimization ofDielectric Substrates for Bandwidth Improvement of a Patch Antenna”. IEEE Transactions onAntennas and Propagation, vol. 51, No. 10, Oct 2003, pp. 2732-2743.
[14]L. C. T. Chang and W. D. Burnside, “An Ultrawide-Bandwidth Tapered Resistive TEM HornAntenna,” IEEE Transactions on Antennas and Propagation, vol. 48, no. 12, pp. 1848–1857, Dec.2000.
[15]M. N. Jahromi, “Novel Wideband Planar Fractal Monopole Antenna,” IEEE Transactions onAntennas and Propagation, Dec. 2008, vol. 56, No. 16, pp. 3844–3849.
[16]A. V. Vorobyov, A. G. Yarovoy, L. P. Ligthart, “An UWB antenna size reduction technique,” IEEEInternational conference on ultra-wideband, 2008, vol. 1, pp. 121–124.
[17]A. Godard, V. Bertrand, J. Andrieu, M. Lalande, B. Jecko, M. Brishoual, S. Colson, R. Guillerey,“Size reduction and radiation optimization on UWB antenna”, IEEE, 2008.
[18]A. Lewis, G. Weis, M. Randall, A. Galehdar and D. Thiel, “Optimising Efficiency and Gain of SmallMeander Line RFID Antennas using Ant Colony System,” , IEEE Congress on EvolutionaryComputation, 2009, pp. 1486–1492.
[19]Q. Dong, S. Kan, L. Qin and Z. Huang, “Sequencing Mixed Model Assembly Lines Based on aModified Particle Swarm Optimization Multi-objective Algorithm,” Proceedings of the IEEEInternational Conference on Automation and Logistics, Aug. 2007, pp. 2818–2823.
[20]Mutlu, A.A., Rahman, M., "Statistical methods for the estimation of process variation effects oncircuit operation", Electronics Packaging Manufacturing, IEEE Transactions on, On page(s): 364 -375 Volume: 28, Issue: 4, Oct. 2005
[21]D. S. Boning and P. K. Mozumder, "DOE/OPT: A system for design of experiments, responsesurface modeling, and optimization using process and device simulation," IEEE Trans. Semicond.Manuf., vol. 6, no. 2, pp.233 -244 1994.
[22]K. K. Low and S. W. Director "An efficient macromodeling approach for statistical IC processdesign", IEEE Int. Conf. CAD, ICCAD-88, pp.16-19, 1988.
[23]P. Cox , P. Yang , S. S. Mahant-Shetti and P. Chatterjee "Statistical modeling for efficient parametricyield estimation of MOS VLSI circuits", IEEE Trans. Electron Devices, vol. ED-32, no. 2,pp.471-478, 1985
[24]T. J. Sanders , K. Rekab , F. M. Rotella and D. P. Means "Integrated circuit design for manufacturingthrough statistical simulation of process steps", IEEE Trans. Semiconductor Manufacturing, vol. 5,no. 4, pp.368-372, 1992.
[25]S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, SpringerSeries in Statistics, Springer-Verlag, New York, 1999.
[26]G. E. P. Box, H.W. Hunter and J.S. Hunter, Statistics for Experimenters: An Introduction to Design,Data Analysis, and Model Building, John Wiley and Sons, 1978.
[27]J. Denes, and A.D. Keedwell, Latin squares and their applications, New York-London: AcademicPress, 1974.
[28]Alvarez, A.R.; Abdi, B.L.; Young, D.L.; Weed, H.D.; Teplik, J.; Herald, E.R., “Application ofstatistical design and response surface methods to computer-aided VLSI device design,” IEEETransactions on Computer-Aided Design of Integrated Circuits and Systems, Volume 7, Issue 2, pp.272 – 288, 1988.
[29]Ramberg, J.S.; Sanchez, S.M.; Sanchez, P.J.; Hollick, L.J., “Designing simulation experiments:Taguchi methods and response surface metamodels,” Proceedings of Simulation Conference, 1991,pp. 167 - 176.
[30]Malik, Z.; Rashid, K., “Comparison of optimization by response surface methodology withneurofuzzy methods,” IEEE Transactions on Magnetics, Volume 36, Issue 1, Part 2, pp. 241 – 257,2000.
[31]Gillon, F.; Brochet, P., “Screening and response surface method applied to the numericaloptimization of electromagnetic devices,” IEEE Transactions on Magnetics, Volume 36, Issue 4, Part1, pp. 1163 – 1167, 2000.
[32]Hsien-Chie Cheng; Wen-Hwa Chen; I-Chun Chung, “Integration of Simulation and response surfacemethods for thermal design of multichip modules,” IEEE Transactions on Components andPackaging Technologies, Volume 27, Issue 2, pp. 359 – 372, 2004.
[33]Yanli Zhang; Houjian He; Chang SeopKoh, “ An adaptive response surface method combined with(1+λ) evolution algorithm and its application to optimal design of electromagnetic devices,” The 5thIEEE Conference on Industrial Electronics and Applications (ICIEA), 2010, pp. 2216 – 2220.
[34]Kuo-Hao Chang; Hong, L.J.; Hong Wan , “Stochastic trust region gradient-free method (strong) – anew response-surface-based algorithm in simulation optimization,” 2007 Simulation Conference, pp.346 – 354.
[35]Pan Seok Shin; Sung Hyun Woo; Chang SeopKoh , “An Optimal Design of Large Scale PermanentMagnet Pole Shape Using Adaptive Response Surface Method With Latin Hypercube SamplingStrategy,” IEEE Transactions on Magnetics, Volume 45, Issue 3, pp. 1214 – 1217, 2009.
[36]Bo Ping Wang; Zhen Xue Han; Leon Xu; Reinikainen, T., “A novel response surface method fordesign optimization of electronic packages,” Proceedings of the 6th International Conference onThermal, Mechanical and Multi-Physics Simulation and Experiments in Micro-Electronics andMicro-Systems, 2005 (EuroSimE 2005), pp. 175 - 181.
[37]Wu, A.; Wu, K.Y.; Chen, R.M.M.; Shen, Y., “Parallel optimal statistical design method withresponse surface modelling using genetic algorithms,” IEE Proceedings of Circuits, Devices andSystems, Volume 145, Issue 1, pp. 7 – 12, 1998.
[38]Khawas, A.; Banerjee, A.; Mukhopadhyay, S., “A Response Surface Method for Design SpaceExploration and Optimization of Analog Circuits,” 2011 IEEE Computer Society Annual Symposiumon VLSI (ISVLSI), pp. 84 – 89, 2011.
[39]M. Farina, and J. K. Sykulski, “Comparative Study of Evolution Strategies Combined withApproximation Techniques for Practical Electromagnetic Optimization Problems”, IEEE Trans. onMagn., vol. 37, no. 5, pp 3216-3220, Sept. 2001.
[40]D. R. Jones, “A taxonomy of global optimization methods based on response surfaces”, Journal ofGlobal Optimization, vol 21, pp. 345-374, 2001.
[41]孫茂棋，直交表用於π型平面天線之設計，碩士論文，淡江大學電機工程學系，2012年6月
[42]孫積賢，使用隨機是最佳畫法於二為散射體之電磁影像研究，博士論文，淡江大學電機工程學系，2011年6月