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The far-field calculation of arbitrarily oriented electric dipole in a stratified confined space is of great significance in analyzing electromagnetic properties in the lightning return stroke, submarine communication, over-the-horizon ground-wave radar, etc. Based on the mirror image method and the far-field approximation of an electric dipole in free space, a three-layer horizontal confined space model with an arbitrarily oriented dipole is established in this work. Through novel vector operations, the expression of the far field generated by an arbitrarily oriented electric dipole in the confined space model is derived, where the direct wave from the source point to the observation point and the waves reflected by the upper boundary and the lower boundary are all comprehensively considered. On this basis, the radiation characteristics of the electric dipole with frequencies of 100 kHz, 6 MHz and 10 MHz at different positions in the Earth-ionosphere waveguide are compared and analyzed, which are taken for example. Three different orientations of electric dipoles,i.e. vertical direction, horizontal direction, and 45º tilt are taken into account and the corresponding radiation patterns are presented. The results show that the radiation characteristics of electric dipoles in the Earth-ionosphere waveguide will change greatly with their frequencies, orientations and positions. For the electric dipole source at the same location, the higher the frequency, the more the number of radiation lobes is. In addition, when the source frequency keeps unchanged, the farther the dipole source is from the bottom boundary, the more the radiated lobes are. The proposed expression can conveniently and accurately consider the direct wave of a dipole source and its primary reflection from the upper interface and the lower interface in a confined space, and can also be further extended to solving the contribution of multiple reflections from the interfaces.
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Keywords:
- electric dipole /
- horizontal layering /
- arbitrary direction /
- mirror image method
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Gu T T 2019 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)
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图 1 电偶极子在水平分层受限空间中的辐射示意图
Figure 1. Radiation diagram of an electric dipole in the horizontal layered confined space.
图 2 下界面上方80 m处频率为100 kHz的电偶极子立体辐射方向图(
$ \theta \in \left({76.5}^{\circ }, {90}^{\circ }\right) $ ) (a) 垂直偶极子; (b) 水平偶极子; (c)${\theta _0}{{ = \pi }}/4, {\text{ }}{\varphi _0}{{ = \pi }}/4$ 的偶极子Figure 2. Radiation pattern of an electric dipole located at 80 m high above the lower interface as the frequency is 100 kHz (
$ \theta \in \left({76.5}^{\circ }, {90}^{\circ }\right) $ ): (a) Vertical dipole; (b) horizontal dipole; (c)${\theta _0}{{ = \pi }}/4, {\text{ }}{\varphi _0}{{ = \pi }}/4$ dipole.
图 3 下界面上方80 m处频率为6 MHz的电偶极子立体辐射方向图 (a) 垂直偶极子; (b) 水平偶极子; (c)
${\theta _0}{{ = \pi }}/4, {\text{ }}{\varphi _0}{{ = \pi }}/4$ 的偶极子Figure 3. Radiation pattern of an electric dipole located at 80 m high above the lower interface as the frequency is 6 MHz: (a) Vertical dipole; (b) horizontal dipole; (c)
${\theta _0}{{ = \pi }}/4, {\text{ }}{\varphi _0}{{ = \pi }}/4$ dipole.
图 4 下界面上方80 m处频率为10 MHz的电偶极子立体辐射方向图 (a) 垂直偶极子; (b) 水平偶极子; (c)
${\theta _0}{{ = \pi }}/4, {\text{ }}{\varphi _0}{{ = \pi }}/4$ 的偶极子Figure 4. Radiation pattern of an electric dipole located at 80 m high above the lower interface as the frequency is 10 MHz: (a) Vertical dipole; (b) horizontal dipole; (c)
${\theta _0}{{ = \pi }}/4, {\text{ }}{\varphi _0}{{ = \pi }}/4$ dipole.
图 5 频率为6 MHz的水平电偶极子于下界面上方不同高度时的立体辐射方向图 (a) h = 25 m; (b) h = 50 m; (c) h = 100 m
Figure 5. Radiation pattern of a horizontal electric dipole located at different heights above the lower interface as the frequency is 6 MHz: (a) h = 25 m; (b) h = 50 m; (c) h = 100 m.
图 6 频率为6 MHz的垂直电偶极子放置在不同空间环境时的辐射方向图
Figure 6. Radiation pattern of a vertical electric dipole located at different space environments at a frequency of 6 MHz.
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[1] 李凯 2010 分层介质中的电磁场和电磁波 (杭州: 浙江大学出版社) 第21—85页
Li K 2010 Electromagnetic Fields and Waves in Layered Media (Hangzhou: Zhejiang University Press) pp21–85 (in Chinese)
[2] 吴静, 周志为, 闫旭 2015 64 194101
Google Scholar
Wu J, Zhou Z W, Yan X 2015 Acta Phys. Sin. 64 194101
Google Scholar
[3] 张歆, 童昱泽, 田志颖, 王金洪, 姚泽 2020 69 248401
Google Scholar
Zhang X, Tong Y Z, Tian Z Y, Wang J H, Yao Z 2020 Acta Phys. Sin. 69 248401
Google Scholar
[4] 王宏磊 2016 博士学位论文 (西安: 西北工业大学)
Wang H L 2016 Ph. D. Dissertation (Xi’an: Northwestern Polytechnical University) (in Chinese)
[5] Zeng H R, He T, Li K 2021 IEEE Trans. Antennas Propag. 69 5870
Google Scholar
[6] Sommerfeld A 1909 Ann. Phys. 28 665
[7] Nazari M E, Huang W M 2020 IEEE Trans. Antennas Propag. 68 1181
Google Scholar
[8] Wang J H, Li B 2017 IEEE Trans. Antennas Propag. 65 2707
Google Scholar
[9] Wait J R 1966 IEEE Trans. Antennas Propag. 14 790
Google Scholar
[10] King R W P, Sandler S S 1994 IEEE Trans. Antennas Propag. 42 382
Google Scholar
[11] Zhang H Q, Pan W Y 2002 Radio Sci. 37 1
Google Scholar
[12] Collin R E 2002 Electromagentic Wave Theory (Beijing: Higher Education Press) pp219–328
[13] 洪清泉, 仲伟博, 余燕忠, 蔡植善, 陈木生, 林顺达 2012 61 160302
Google Scholar
Hong Q Q, Zhong W B, Yu Y Z, Cai Z S, Chen M S, Lin S D 2012 Acta Phys. Sin. 61 160302
Google Scholar
[14] 潘威炎 2004 长波超长波极长波传播 (成都: 电子科技大学出版社) 第40—80页
Pan W Y 2004 Long Wave Ultra Long Wave Very Long Wave Propagation (Chengdu: University of Electronic Science and Technology Press) pp40–80 (in Chinese)
[15] 陈聪, 李定国, 蒋治国, 刘华波 2012 61 244101
Google Scholar
Chen C, Li D G, Jiang Z G Liu H B 2012 Acta Phys. Sin. 61 244101
Google Scholar
[16] He L N, He T, Li K 2020 Int. J. Antenn. Propag. 2020 1
Google Scholar
[17] 朱秀芹, 潘威炎, 官伯然 2009 电波科学学报 24 71
Google Scholar
Zhu X Q, Pan W Y, Guan B R 2009 Chin. J. Radio Sci. 24 71
Google Scholar
[18] 顾婷婷 2019 博士学位论文 (杭州: 浙江大学)
Gu T T 2019 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)
[19] Gao Y, Di Q Y, Wang R, Fu C M, Liang P F, Zheng F H 2021 IEEE Trans. Geosci. Remote Sens. 60 1
Google Scholar
[20] Zhao S F, Zhang X M, Zhao Z Y, Shen X H 2014 Ann. Geophys. 32 194
Google Scholar
[21] 葛德彪, 魏兵 2012 61 050301
Google Scholar
Ge D B, Wei B 2012 Acta Phys. Sin. 61 050301
Google Scholar
[22] Cao L, Wei B, Ge D B 2013 Waves Random Complex Media 23 446
Google Scholar
[23] Shen N, Wei B, Yin W K 2019 Results Phys. 14 102388
Google Scholar
[24] 葛德彪, 魏兵 2011 电磁波理论 (北京: 科学出版社) 第245—261页
Ge D B, Wei B 2011 Electromagnetic Wave Theory (Beijing: Science Press) pp245–261 (in Chinese)
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