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基于拉格朗日体系的考虑谐波互作用的理论模型，将离散的粒子近似处理为流体，得到电子相位的连续分布函数.对电子相位连续分布函数进行傅里叶一阶展开，并结合贝塞尔母函数关系式，建立了考虑谐波互作用的欧拉非线性理论模型.应用考虑谐波互作用的欧拉非线性理论模型对一支L波段空间行波管和一支C波段空间行波管进行大信号分析，并与拉格朗日理论模型进行对比.结果表明：在增益1 dB压缩点之前，考虑谐波互作用的欧拉非线性理论模型与拉格朗日理论模型十分符合，增益最大误差不超过4%.考虑谐波互作用的欧拉非线性理论模型能够有效的对增益1 dB压缩点之前的谐波进行分析.仿真结果验证了考虑谐波互作用的欧拉非线性理论模型的正确性和有效性.考虑谐波互作用的欧拉非线性理论不但提供了一个谐波快速计算模型，而且为后续研究行波管谐波的产生机理与抑制方法奠定了基础.Traveling wave tube amplifiers are one of the most widely used vacuum electronic devices which are employed in various applications, in the areas of such as radar, wireless communication and electronic countermeasures system. Among traveling wave tubes, space-borne helix traveling wave tubes which are of high power, high efficiency, high reliability, long life and radiation hardened, are extensively used in satellite transmitter, data communication system and global positioning system. With the rapid development of the multiphase digital modulation schemes, communication systems are placing greater demands on the output power, electronic efficiency and nonlinear distortion characteristics of space-borne helix traveling wave tubes. However, the nonlinear beam-wave interaction will lead to the generation of harmonics, and thus reduces the output power and electronic efficiency. The harmonics can also act to create beats with the fundamental wave, and thus generate these beat frequencies which are commonly known as intermodulation products. As a result, the bit-error-rate will be increased and the system performance will be compromised. Therefore, the generation of harmonics is of significant current interest in space-borne helix traveling wave tubes. Understanding this effect provides a strong motivation for nonlinear analysis of a helix traveling wave tube. In this paper, a continuous electron phase distribution is obtained by treating the discrete electron beam as a charge fluid based on the Lagrangian theory. Then, to obtain a nonlinear Eulerian theory considering harmonic interaction, the electron phases in Lagrangian theory have been expanded into a series of harmonic components. Considering the 0th component and 1st component of the electron phases only and integrating over the initial phase distribution with the help of the relation of Bessel function, the nonlinear Eulerian theory considering harmonic interaction is established. The nonlinear Eulerian theory considering harmonic interaction is compared to a Lagrangian theory on a set of traveling wave tube parameters which are based on a single section of L-and C-bands traveling wave tubes. It is found that the nonlinear Eulerian theory considering harmonic interaction agrees accords well with the Lagrangian theory before the saturation effect occurs. But, it begins to make a difference near saturation point where the electron overtaking happens. The maximum error in gain between the nonlinear Eulerian theory considering harmonic interaction and the Lagrangian theory is less than 4% at 1 dB gain compression point. So the present nonlinear Eulerian theory considering harmonic interaction can effectively describe harmonic generation at 1 dB gain compression point. The simulation results validate the correctness and effectiveness of our nonlinear Eulerian theory considering harmonic interaction. In futuristic future efforts, it is hoped that the present nonlinear Eulerian theory considering harmonic interaction may provide insights into the behavioral mechanisms of nonlinear effects in space-borne helix traveling wave tubes.
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Keywords:
- traveling wave tube /
- harmonic interaction /
- Eulerian /
- nonlinear theory
[1] Abe D K, Levush B, Antonsen Jr T M, Whaley D R 2002 IEEE Trans. Plasma Sci. 30 1053
[2] Whaley D R, Armstrong C M, Gannon B, Groshart G 1998 IEEE Trans. Plasma Sci. 26 912
[3] Abe D K, Levush B, Antonsen Jr T M, Whaley D R 2002 Proceedings of the Vacuum Electronics Conference Monterey, CA, USA, April 25-25, 2002 p312
[4] Katz A 2009 Microwave Magazine IEEE 2 37
[5] Qiu J, Abe D, Antonsen Jr T M, Danly B G, Levush B 2002 Proceedings of the Microwave Symposium Digest Monterey USA, April 25-25, 2002 p457
[6] Qiu J X, Abe D K, Antonsen Jr T M, Danly B G 2003 IEEE Trans. Microwave Theory Tech. 51 1911
[7] Lau Y Y, Chernin D P, Wilsen C, Gilgenbach R M 2000 IEEE Trans. Plasma Sci. 28 959
[8] Bai A Y, Zou C M, Mo Y L 1996 Journal of University of Electronic Science and Technology of China 25 43 (in Chinese)[白安永, 邹长民, 莫元龙 1996 电子科技大学学报 25 43]
[9] Mo Y L, Xie Z L 1996 Journal of University of Electronic Science and Technology of China 25 625 (in Chinese)[莫元龙, 谢仲怜 1996 电子科技大学学报 25 625]
[10] Dionne N J 1970 IEEE Trans. Electron Dev. 17 365
[11] Li B, Yang Z H, Li J Q, Zhu X F, Huang T, Jin X L, Hu Q, Hu Y L, Xu L, Ma J J, Peng W F, Liao L, Xiao L, He G X 2009 IEEE Trans. Electron Dev. 56 919
[12] Li B, Li J Q, Hu Q, Hu Y L, Xu L, Huang T, Jin X L, Zhu X F, Yang Z H 2014 IEEE Trans. Electron Dev. 61 1735
[13] Hao B L, Xiao L, Liu P K, Li G C, Jiang Y, Yi H X, Zhou W 2009 Acta Phys. Sin. 58 3118 (in Chinese)[郝保良, 肖刘, 刘濮鲲, 李国超, 姜勇, 易红霞, 周伟 2009 物理学报 58 3118]
[14] Hu Y L, Yang Z H, Li J Q, Li B, Gao P, Jin X L 2009 Acta Phys. Sin. 58 6665 (in Chinese)[胡玉禄, 杨中海, 李建清, 李斌, 高鹏, 金晓林 2009 物理学报 58 6665]
[15] Li J Q, Mo Y L 2006 Acta Phys. Sin. 55 4177 (in Chinese)[李建清, 莫元龙 2006 物理学报 55 4177]
[16] Chernin D, Antonsen Jr T M, Levush B, Whaley D R 2001 IEEE Trans. Electron Dev. 48 3
[17] Duan Z Y, Gong Y B, Wei Y Y, Wang W X 2008 Chin. Phys. B 17 2484
[18] Li B, Yang Z H 2003 Chin. Phys. 12 1235
[19] Booske J H, Converse M C 2004 IEEE Trans. Plasma Sci. 32 1066
[20] Datta S 1998 Inter. J. Electron. 85 377
[21] Datta S, Reddy S, Jain P, Basu B 1999 Inter. J. Infr. Mill. Waves 20 483
[22] Datta S K 2000 Inter. J. Electron. 87 89
[23] Datta S K, Jain P K, Narayan R, Basu B N 1999 IEEE Trans. Electron Dev. 46 420
[24] Whlbier J G, Booske J H, Dobson I 2004 IEEE Trans. Plasma Sci. 32 1073
[25] Whlbier J G, Booske J H, Dobson I 2002 IEEE Trans. Plasma Sci. 30 1063
[26] Whlbier J G, Dobson I, Booske J H 2002 Phys. Rev. E 66 056504
[27] Whlbier J G, Booske J H 2004 Phys. Rev. E 69 066502
[28] Hu Y L 2011 Ph. D. Dissertation (Chengdu:University of Electronic Science and Technology of China) (in Chinese)[胡玉禄 2011 博士学位论文 (成都:电子科技大学)]
[29] Hu Y L, Yang Z H, Li J, Li B 2015 Proceedings of the Vacuum Electronics Conference (IVEC) Beijing, April 27-29, 2015 p1
[30] Dong C F, Zhang P, Chernin D, Lau Y Y 2015 IEEE Trans. Electron Dev. 62 4285
[31] Antonsen Jr T M, Levush B 1998 IEEE Trans. Plasma Sci. 26 774
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[1] Abe D K, Levush B, Antonsen Jr T M, Whaley D R 2002 IEEE Trans. Plasma Sci. 30 1053
[2] Whaley D R, Armstrong C M, Gannon B, Groshart G 1998 IEEE Trans. Plasma Sci. 26 912
[3] Abe D K, Levush B, Antonsen Jr T M, Whaley D R 2002 Proceedings of the Vacuum Electronics Conference Monterey, CA, USA, April 25-25, 2002 p312
[4] Katz A 2009 Microwave Magazine IEEE 2 37
[5] Qiu J, Abe D, Antonsen Jr T M, Danly B G, Levush B 2002 Proceedings of the Microwave Symposium Digest Monterey USA, April 25-25, 2002 p457
[6] Qiu J X, Abe D K, Antonsen Jr T M, Danly B G 2003 IEEE Trans. Microwave Theory Tech. 51 1911
[7] Lau Y Y, Chernin D P, Wilsen C, Gilgenbach R M 2000 IEEE Trans. Plasma Sci. 28 959
[8] Bai A Y, Zou C M, Mo Y L 1996 Journal of University of Electronic Science and Technology of China 25 43 (in Chinese)[白安永, 邹长民, 莫元龙 1996 电子科技大学学报 25 43]
[9] Mo Y L, Xie Z L 1996 Journal of University of Electronic Science and Technology of China 25 625 (in Chinese)[莫元龙, 谢仲怜 1996 电子科技大学学报 25 625]
[10] Dionne N J 1970 IEEE Trans. Electron Dev. 17 365
[11] Li B, Yang Z H, Li J Q, Zhu X F, Huang T, Jin X L, Hu Q, Hu Y L, Xu L, Ma J J, Peng W F, Liao L, Xiao L, He G X 2009 IEEE Trans. Electron Dev. 56 919
[12] Li B, Li J Q, Hu Q, Hu Y L, Xu L, Huang T, Jin X L, Zhu X F, Yang Z H 2014 IEEE Trans. Electron Dev. 61 1735
[13] Hao B L, Xiao L, Liu P K, Li G C, Jiang Y, Yi H X, Zhou W 2009 Acta Phys. Sin. 58 3118 (in Chinese)[郝保良, 肖刘, 刘濮鲲, 李国超, 姜勇, 易红霞, 周伟 2009 物理学报 58 3118]
[14] Hu Y L, Yang Z H, Li J Q, Li B, Gao P, Jin X L 2009 Acta Phys. Sin. 58 6665 (in Chinese)[胡玉禄, 杨中海, 李建清, 李斌, 高鹏, 金晓林 2009 物理学报 58 6665]
[15] Li J Q, Mo Y L 2006 Acta Phys. Sin. 55 4177 (in Chinese)[李建清, 莫元龙 2006 物理学报 55 4177]
[16] Chernin D, Antonsen Jr T M, Levush B, Whaley D R 2001 IEEE Trans. Electron Dev. 48 3
[17] Duan Z Y, Gong Y B, Wei Y Y, Wang W X 2008 Chin. Phys. B 17 2484
[18] Li B, Yang Z H 2003 Chin. Phys. 12 1235
[19] Booske J H, Converse M C 2004 IEEE Trans. Plasma Sci. 32 1066
[20] Datta S 1998 Inter. J. Electron. 85 377
[21] Datta S, Reddy S, Jain P, Basu B 1999 Inter. J. Infr. Mill. Waves 20 483
[22] Datta S K 2000 Inter. J. Electron. 87 89
[23] Datta S K, Jain P K, Narayan R, Basu B N 1999 IEEE Trans. Electron Dev. 46 420
[24] Whlbier J G, Booske J H, Dobson I 2004 IEEE Trans. Plasma Sci. 32 1073
[25] Whlbier J G, Booske J H, Dobson I 2002 IEEE Trans. Plasma Sci. 30 1063
[26] Whlbier J G, Dobson I, Booske J H 2002 Phys. Rev. E 66 056504
[27] Whlbier J G, Booske J H 2004 Phys. Rev. E 69 066502
[28] Hu Y L 2011 Ph. D. Dissertation (Chengdu:University of Electronic Science and Technology of China) (in Chinese)[胡玉禄 2011 博士学位论文 (成都:电子科技大学)]
[29] Hu Y L, Yang Z H, Li J, Li B 2015 Proceedings of the Vacuum Electronics Conference (IVEC) Beijing, April 27-29, 2015 p1
[30] Dong C F, Zhang P, Chernin D, Lau Y Y 2015 IEEE Trans. Electron Dev. 62 4285
[31] Antonsen Jr T M, Levush B 1998 IEEE Trans. Plasma Sci. 26 774
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