本研究旨在將基於均勻介質理論的混合參雜模型導入多層介質架構中，進一步分析氣泡效應對材料電磁參數與吸波特性的影響。隨著 5G 技術快速發展，電磁波吸收材料在抑制設備間電磁干擾（EMI）方面的重要性日益提升。然而，現有文獻多著重於各類有效介質模型的推演與應用，卻少有針對氣泡作為內含物對吸波性能影響的系統性探討。本研究彌補此一缺口，透過結合混合參雜模型與氣泡參數，提供一套可應用於多層材料設計的分析框架，作為未來吸波材料優化的重要依據。
在模型驗證方面，本研究選取兩篇具代表性的文獻作為對照，針對不同頻率下的模擬結果進行分析與比對。透過程式化實作多層介質理論與不同混合參雜模型(Maxwell–Garnett、Bruggeman 與 Looyenga)，並將模擬結果與文獻中的預測數據進行交叉驗證，以確認模型推導與程式碼執行的正確性。此外，採用最小平方法計算方均根誤差（RMSE、NRMSE），用以量化各模型在不同條件下之預測準確度，提升整體分析的可靠度。
最後為模擬氣泡對吸波材料電磁特性的影響，引入混合參雜模型，假設氣泡於材料中均勻分布，作為低介電與低磁導率的內含物，改變整體有效參數。以吸波理論為基礎，結合 Maxwell–Garnett、Bruggeman 與 Looyenga 等混合參雜模型，推估氣泡所造成的等效參數變化，並系統性分析氣泡體積比例對複數介電常數、磁導率與反射損耗之影響，整理出不同模型在氣泡含量推估上的傾向性，此研究成果可作為未來在吸波材料設計中考量氣泡效應時，進行更實用且精確分析之依據。

This study aims to integrate hybrid mixing models, based on effective medium theory, into multilayer structures to further analyze the influence of air bubbles on the electromagnetic parameters and absorption performance of materials.However, existing literature mostly focuses on the derivation and application of various effective medium models, with limited discussion on the systematic impact of air bubbles as inclusions on absorption performance. This research addresses this gap by combining hybrid mixing models with bubble-related parameters to establish an analytical framework applicable to multilayer absorber design, providing a valuable reference for future material optimization.
In the simulation, air bubbles are assumed to be uniformly distributed within the material and modeled as low-permittivity and low-permeability inclusions, altering the overall effective electromagnetic properties. Based on wave absorption theory, this study incorporates Maxwell–Garnett, Bruggeman, and Looyenga models to estimate the variations in effective parameters caused by bubbles. A systematic analysis is conducted on how bubble volume fraction affects the complex permittivity, permeability, and reflection loss. The normalized root mean square error (NRMSE) is used to evaluate model accuracy, and the tendencies of different models in estimating bubble content are summarized. The findings provide a practical and precise basis for incorporating bubble effects in the future design and analysis of wave-absorbing materials.

[1] S.-H. Kim, S.-Y. Lee, Y. Zhang, S.-J. Park, and J. Gu, "Carbon-Based Radar Absorbing Materials toward Stealth Technologies," Advanced Science, vol. 10, no. 32, p. 2303104, 2023, doi: https://doi.org/10.1002/advs.202303104.
[2] A. El Assal, H. Breiss, R. Benzerga, A. Sharaiha, A. Jrad, and A. Harmouch, "Toward an Ultra-Wideband Hybrid Metamaterial Based Microwave Absorber," Micromachines, vol. 11, no. 10, p. 930, 2020. [Online]. Available: https://www.mdpi.com/2072-666X/11/10/930.
[3] D. Micheli et al., "Broadband Electromagnetic Absorbers Using Carbon Nanostructure-Based Composites," IEEE Transactions on Microwave Theory and Techniques, vol. 59, no. 10, pp. 2633-2646, 2011, doi: 10.1109/TMTT.2011.2160198.
[4] Z. Liu et al., "A new quantitative analysis method for electromagnetic energy dissipation in microwave absorption materials," Journal of Magnetism and Magnetic Materials, vol. 516, p. 167332, 2020/12/15/ 2020, doi: https://doi.org/10.1016/j.jmmm.2020.167332.
[5] S. J. Orfanidis, Electromagnetic Waves and Antennas. Rutgers University, 1996, p. 1413.
[6] G. V. Morozov, R. G. Maev, and G. W. F. Drake, "Exact analytic solution of the problem of reflection of an electromagnetic wave from a two-layer periodic dielectric structure," Quantum Electronics, vol. 28, no. 11, p. 977, 1998/11/30 1998, doi: 10.1070/QE1998v028n11ABEH001368.
[7] C. Camacho-Peñalosa and J. D. Baños-Polglase, "On the Definition of Return Loss [Measurements Corner]," IEEE Antennas and Propagation Magazine, vol. 55, no. 2, pp. 172-174, 2013, doi: 10.1109/MAP.2013.6529339.
[8] B. Dai, Y. Qi, M. Song, B. Zhang, N. Wang, and Y. Dai, "Facile synthesis of core–shell structured C/Fe3O4 composite fiber electromagnetic wave absorbing materials with multiple loss mechanisms," The Journal of Chemical Physics, vol. 157, no. 11, 2022, doi: 10.1063/5.0121257.
[9] J. R. Truedson, K. D. McKinstry, P. Kabos, and C. E. Patton, "High‐field effective linewidth and eddy current losses in moderate conductivity single‐crystal M‐type barium hexagonal ferrite disks at 10–60 GHz," Journal of Applied Physics, vol. 74, no. 4, pp. 2705-2718, 1993, doi: 10.1063/1.354665.
[10] T. Hang et al., "Wheat-like Ni-coated core–shell silver nanowires for effective electromagnetic wave absorption," Journal of Colloid and Interface Science, vol. 649, pp. 394-402, 2023/11/01/ 2023, doi: https://doi.org/10.1016/j.jcis.2023.06.087.
[11] R.-B. Yang and W.-F. Liang, "Microwave properties of high-aspect-ratio carbonyl iron/epoxy absorbers," Journal of Applied Physics, vol. 109, no. 7, 2011, doi: 10.1063/1.3536340.
[12] Y. Qian et al., "High performance epoxy resin with efficient electromagnetic wave absorption and heat dissipation properties for electron packaging by modification of 3D MDCF@hBN," Chemical Engineering Journal, vol. 441, p. 136033, 2022/08/01/ 2022, doi: https://doi.org/10.1016/j.cej.2022.136033.
[13] B. T. L. Byoung Taek Lee and H. C. K. Ho Chul Kim, "Prediction of Electromagnetic Properties of MnZn Ferrite-Silicone Rubber Composites in Wide Frequency Range," Japanese Journal of Applied Physics, vol. 35, no. 6R, p. 3401, 1996/06/01 1996, doi: 10.1143/JJAP.35.3401.
[14] R. Zhang, Q. Zhang, C. Guo, X. He, Z. Wu, and T. Wen, "Bubbles in Transformer Oil: Dynamic Behavior, Internal Discharge, and Triggered Liquid Breakdown," IEEE Transactions on Dielectrics and Electrical Insulation, vol. 29, no. 1, pp. 86-94, 2022, doi: 10.1109/TDEI.2022.3148484.
[15] M. H. Nisanci, F. d. Paulis, M. Y. Koledintseva, J. L. Drewniak, and A. Orlandi, "From Maxwell Garnett to Debye Model for Electromagnetic Simulation of Composite Dielectrics—Part II: Random Cylindrical Inclusions," IEEE Transactions on Electromagnetic Compatibility, vol. 54, no. 2, pp. 280-289, 2012, doi: 10.1109/TEMC.2011.2162845.
[16] V. A. Markel, "Introduction to the Maxwell Garnett approximation: tutorial," J. Opt. Soc. Am. A, vol. 33, no. 7, pp. 1244-1256, 2016/07/01 2016, doi: 10.1364/JOSAA.33.001244.
[17] T. Liu, Y. Pang, M. Zhu, and S. Kobayashi, "Microporous Co@CoO nanoparticles with superior microwave absorption properties," Nanoscale, vol. 6 4, pp. 2447-54, 2014.
[18] J. R. Torczynski, S. L. Ceccio, and P. R. Tortora, "The Equivalent Electrical Permittivity of Gas-Solid Mixtures at Intermediate Solid Volume Fractions," 2005.
[19] A. M. Jayannavar and N. Kumar, "Generalization of Bruggeman's unsymmetrical effective-medium theory to a three-component composite," Physical Review B, vol. 44, no. 21, pp. 12014-12015, 12/01/ 1991, doi: 10.1103/PhysRevB.44.12014.
[20] D. Y. Kim, Y. C. Chung, T. W. Kang, and H. C. Kim, "Dependence of microwave absorbing property on ferrite volume fraction in MnZn ferrite-rubber composites," IEEE Transactions on Magnetics, vol. 32, no. 2, pp. 555-558, 1996, doi: 10.1109/20.486547.
[21] H. Looyenga, "Dielectric constants of heterogeneous mixtures," Physica, vol. 31, no. 3, pp. 401-406, 1965/03/01/ 1965, doi: https://doi.org/10.1016/0031-8914(65)90045-5.
[22] D. J. Bergman, "Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric Constant of a Two-Component Composite Material," Physical Review Letters, vol. 44, no. 19, pp. 1285-1287, 05/12/ 1980, doi: 10.1103/PhysRevLett.44.1285.
[23] 黃楷杰, "MATLAB在多層吸波材料之分析方法," 碩士, 電機工程學系, 國立成功大學, 台南市, 2024. [Online]. Available: https://hdl.handle.net/11296/q9b868
[24] J. V. Mantese, A. L. Micheli, D. F. Dungan, R. G. Geyer, J. Baker‐Jarvis, and J. Grosvenor, "Applicability of effective medium theory to ferroelectric/ferrimagnetic composites with composition and frequency‐dependent complex permittivities and permeabilities," Journal of Applied Physics, vol. 79, no. 3, pp. 1655-1660, 1996, doi: 10.1063/1.361010.
[25] 陳韋佑, "適用於12GHz羰基鐵/鎳鋅鈷鐵氧體雙層吸波材料之開發," 碩士, 電機工程學系, 國立成功大學, 台南市, 2023. [Online]. Available: https://hdl.handle.net/11296/hmvb78
[26] Y. B. Feng, T. Qiu, and C. Y. Shen, "Absorbing properties and structural design of microwave absorbers based on carbonyl iron and barium ferrite," Journal of Magnetism and Magnetic Materials, vol. 318, no. 1, pp. 8-13, 2007/11/01/ 2007, doi: https://doi.org/10.1016/j.jmmm.2007.04.012.
[27] R. B. Yang, S. D. Hsu, and C. K. Lin, "Frequency-dependent complex permittivity and permeability of iron-based powders in 2–18 GHz," Journal of Applied Physics, vol. 105, no. 7, 2009, doi: 10.1063/1.3068039.