21 Asia-Pacific International Symposium on Electromagnetic Compatibility, April 12-16, 21, Beijing, China FDTD Calculation of Lightning-Induced Voltages on an Overhead Two-Wire Distribution Line Toshiki Takeshima #1, Yoshihiro Baba #1, Naoto Nagaoka #1, Akihiro Ametani #1 Jun Takami #2, Shigemitsu Okabe #2, Vladimir A. Rakov #3 # 1 Doshisha University, Kyotanabe, Kyoto 61-321, Japan dtj151@mail4.doshisha.ac.jp, ybaba@mail.doshisha.ac.jp nnagaoka@mail.doshisha.ac.jp, aametani@mail.doshisha.ac.jp # 2 Tokyo Electric Power Company, R&D Center, Yokohama, Kanagawa 23-851, Japan takami.jun@tepco.co.jp, okabe.s@tepco.co.jp # 3 University of Florida,Gainesville, FL32611, USA rakov@ece.ufl.edu Abstract We have calculated lightning-induced voltages on a 68-m long overhead two-wire line using the finite-difference time-domain (FDTD) method for solving Maxwell s equations. The FDTD method employed here uses a three-dimensional nonuniform grid, which is fine (cell side length is.875 m) in the vicinity of overhead wires and coarse (maximum cell side length is 7 m) in the rest of the space. The overhead wires having radii of several millimetres are simulated by placing a wire having an equivalent radius of about.2 m (.23.875 m) in the center of an artificial rectangular prism having a cross-sectional area of (2.875 m) (2.875 m) and the modified electrical constants: low permittivity and high permeability. The inducedvoltage waveform, calculated for the condition that the returnstroke wavefront speed is 13 m/μs, the ground conductivity is 3.5 ms/m, and the grounding resistance ranges from 3 to 75 Ω, agrees well with the corresponding waveform measured by Barker et al. in a rocket-triggered lightning experiment. I. INTRODUCTION In order to optimize lightning protection means of telecommunication and power distribution lines, one needs to know voltages that can be induced on overhead wires by nearby lightning strikes. Lightning-induced voltages on an overhead multi-conductor line have been calculated reasonably accurately using field-to-wire electromagnetic coupling models (e.g., [1]) and approximate expressions for electric fields over lossy ground, such as the Norton approximate expressions [2] and the Cooray-Rubinstein formula [3]. Also, lightning-induced voltages have recently been calculated with a similar accuracy using the method of moments (MoM) (e.g., [4]) and the hybrid electromagnetic/circuit model (HEM) (e.g., [5], [6]). In [4] and [5], in order to consider the effects of lossy ground on electromagnetic fields, the Norton approximate expressions were employed, while in [6] the Cooray-Rubinstein formula was used. More recently, the finite-difference time-domain (FDTD) method [7] has been applied to analyzing lightning-induced voltages [8], [9]. In [8], the three-dimensional (3D) FDTD method was used for examining the influence of a nearby lightning strike object on the voltages induced on a single overhead wire above lossy ground. In [9], the 2D-FDTD method, instead of approximate expressions, was used for evaluating the electric fields over lossy ground, and the FDTD-calculated fields were used for evaluating the lightning-induced voltages with the Agrawal et al. field-towire coupling model [1]. One of the reasons for using the latter, hybrid approach is probably related to a difficulty of representing closely-spaced overhead thin wires in the 3D uniform-grid FDTD method. One of the advantages of the use of the FDTD method in analyzing lightning-induced voltages is that it yields electromagnetic fields in the presence of lossy ground directly in the time domain without using any approximate formula in the frequency domain. Another advantage is that it does not necessarily need a field-to-wire coupling model. One of the disadvantages is that it is computationally expensive. In this paper, we calculate lightning-induced voltages on a 68-m long overhead two-wire line using the FDTD method for different return-stroke wavefront speeds, 6, 13, and 2 m/μs, ground conductivity values,.35, 3.5, and 35 ms/m, and grounding resistance values, 3, 5, and 75 Ω. The employed FDTD method uses a three-dimensional nonuniform grid, which is fine (cell side length is.875 m) in the vicinity of overhead wires and coarse (maximum cell side length is 7 m) in the rest of the space. The overhead thin wires are simulated using arbitrary-radius-wire representations, which place a wire having an equivalent radius of about.2 m (.23.875 m) in the center of an artificial rectangular prism having a cross-sectional area of (2.875 m) (2.875 m) and the modified electrical constants: low permittivity and high permeability. We compare the FDTDcalculated waveforms of lightning-induced voltages with the corresponding waveforms measured by Barker et al. [11] in a rocket-triggered lightning experiment. II. METHODOLOGY Fig. 1 (a) shows the experimental configuration, including a 682-m long two-wire test distribution line and a rocket- 978-1-4244-5623-9/1/$26. 21 IEEE 1317
triggered lightning channel located 145 m away from the distribution line, that was employed by Barker et al. [11]. Fig. 1 (b) shows plan and side views of its representation for FDTD calculations. The two horizontal wires at heights 7.875 m and 6.125 m are parallel to the y axis. The working volume of 425 m 966 m 56 m for FDTD calculations is divided nonuniformly into rectangular cells, and is surrounded by six planes of Liao s second-order absorbing boundary condition [12] to avoid reflections there. The side length in y direction of all the cells is 7 m (constant). x- and z-directed sides of the cells are not constant:.875 m in the vicinity of the two horizontal (y-directed) wires, and the side length increases gradually (to 1.75, 3.5, and 7 m) with increasing distance from the wires. The total number of cells in the working volume is 8 138 111 = 1,225,44. The 7.6-mm-radius upper and 5.9-mm-radius lower horizontal wires are represented by using the thin-wire representation of Noda and Yokoyama [13], which places a wire having an equivalent radius of about.2 m (.23.875 m) in the center of an artificial rectangular prism having a cross-sectional area of (2.875 m) (2.875 m) and the modified electrical constants: low permittivity and high permeability. At the ends of the horizontal wires, the upper wire is connected to the lower wire via a 455-Ω resistor. At P1, P9 (located approximately at the middle of the distribution line), and P15, the lower wire is connected via a vertical wire (down conductor) to the surface of the ground (the thickness from the ground surface to the bottom absorbing boundary is set to 7 m) having relative permittivity ε r =1 and conductivity, σ =.35, 3.5 or 35 ms/m, via a vertical wire in series with a lumped resistor of R g =3, 5 or 75 Ω. Note that the ground conductivity was not given by Barker et al. [11], but the grounding impedance values were measured at P1, P9, and P15 at a frequency of 4 khz, and ranged from 3 to 75 Ω. Since the cross-section of the cells closest to the vertical wire is not a square (.875 m 7 m), we cannot employ the thinwire representation of Noda and Yokoyama [13]. Instead, we use the representation of Railton et al. s [14]. The radius of the down conductors is set to 7.5 mm. The vertical lightning channel is represented by a vertical array of current sources [15]. Each current source is activated by the arrival of an upward-propagating return-stroke wavefront whose speed is v = 6, 13 or 2 m/μs. Note that lightning return-stroke wavefront speed was not measured by Barker et al. [11]. The time increment is set to 1.3 ns. (a) III. ANALYSIS AND RESULTS Fig. 2 shows the waveform of lightning channel-base current measured by Barker et al. [11], and its approximate waveform used for the FDTD calculations. Fig. 3 (a) shows FDTD-calculated waveforms of lightning-induced voltages between the upper wire and the ground surface at P9 for the return-stroke wavefront speed v = 13 m/μs, the grounding resistance value R g = 5 Ω, and different values of ground conductivity σ =.35, 3.5, and 35 ms/m. Fig. 3 (b) shows FDTD-calculated waveforms for R g = 5 Ω, σ = 3.5 ms/m, and different values of v = 6, 13, and 2 m/μs. Fig. 3 (c) shows FDTD-calculated waveforms for v = 13 m/μs, σ = 3.5 ms/m, and different values of R g = 3, 5, and 75 Ω. In each part of Fig. 3, the voltage waveform measured at P9 by Barker et al. [11] is also shown. 1 2 3-5 (b) Fig. 1. (a) Experimental configuration employed by Barker et al. [11], including a 682-m long two-wire test distribution line and a rocket-triggered lightning channel located 145 m away from the distribution line and (b) plan and side views of its representation for FDTD calculations. Current [ ka ] -1-15 -2-25 Approximated Fig. 2. Waveform of a lightning channel-base current measured by Barker et al. [11], and its approximate waveform used for FDTD calculations. 1318
Induced voltage [ kv ] Induced voltage [ kv ] Induced voltage [ kv ] 7 6 5 4 3 2 1 σ =.35 ms/m 3.5 ms/m 35 ms/m -1 1 2 3 7 6 5 4 3 2 1 (a) -1 1 2 3 7 6 5 4 3 2 1 (b) v = 2 m/μs 13 m/μs 6 m/μs R g = 75 Ω -1 1 2 3 5 Ω 3 Ω (c) Fig. 3. FDTD-calculated waveforms of lightning-induced voltage at point P9 for (a) v=13 m/μs, R g =5 Ω, σ =.35, 3.5 or 35 ms/m, (b) R g =5 Ω, σ =3.5 ms/m, v=6, 13 or 2 m/μs, and (c) v=13 m/μs, σ =3.5 ms/m, R g =3, 5 or 75 Ω. Also shown is the measured voltage waveform. It is clear from Figs. 3 (a), (b), and (c) that the magnitude of lightning-induced voltage increases with decreasing the ground conductivity, increasing the return-stroke wavefront speed, and increasing the grounding resistance. When σ =3.5 ms/m and v=13 m/μs are assumed, the FDTD-calculated waveform agrees best with the corresponding measured waveform. This conclusion also agrees with that of Ren et al. [9] who used the 2D-FDTD calculations with the Agrawal et al. field-to-wire coupling model, and that of Yutthagowith et al. [6] who used the HEM calculations with the Cooray-Rubinstein formula. Figs. 4 (a) and (b) show waveforms of vertical electric field and horizontal magnetic field, respectively, calculated for σ = 3.5 m/ms and v= 13 m/ms. The corresponding waveforms measured by Barker et al. [11] are also shown. The calculated field waveforms agree reasonably well with the measured ones. Vertical electric field [ kv/m ] Horizontal magnetic field [ A/m ] -1 1 2 3 d = 11 m -1-2 -3-2 -4-6 Calculated for σ = 3.5 ms/m and v = 13 m/μs (a) -1 1 2 3 d = 11 m Calculated for σ = 3.5 ms/m and v = 13 m/μs (b) Fig. 4. FDTD-calculated waveforms of (a) vertical electric field and (b) horizontal magnetic field at horizontal distance 11 m from the lightning channel for σ =3.5 ms/m and v=13 m/μs. Also shown are the measured vertical electric and horizontal magnetic field waveforms. IV. CONCLUSIONS We have calculated lightning-induced voltages on an overhead two-wire line using the FDTD method. The FDTD method employed here uses a 3D nonuniform grid, which is fine in the vicinity of overhead wires and coarse in the rest of the space. The overhead wires having radii of several millimetres are simulated using thin-wire representations. The FDTD-calculated waveform of lightning-induced voltage for the condition that the return-stroke wavefront speed is 13 1319
m/μs, the ground conductivity is 3.5 ms/m, and the grounding resistance ranges from 3 to 75 Ω, agrees well with the corresponding rocket-triggered lightning voltage waveform measured by Barker et al. [11]. This indicates that lightninginduced voltages on realistic multi-wire distribution lines can be analyzed reasonably accurately using the FDTD method without invoking a field-to-wire coupling model. REFERENCES [1] M. Paolone, F. Rachidi, A. Borghetti, C. A. Nucci, M. Rubinstein, V. A. Rakov, and M. A. Uman, Lightning electromagnetic field coupling to overhead lines: theory, numerical simulations, and experimental validation, IEEE Trans. Electromagn. Compat., vol. 51, no. 3, pp. 532-547, Aug. 29. [2] K. A. Norton, The propagation of radio waves over the surface of the earth and in the upper atmosphere, part II --- the propagation from vertical, horizontal, and loop antennas over a plane earth of finite conductivity, Proc. IRE, vol. 25, no. 9, pp. 123-1236, 1937. [3] M. Rubinstein, An approximate formula for calculation of the horizontal electric field from lightning at close, intermediate, and long ranges, IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 531-535, Aug. 1996. [4] R. K. Pokharel, M. Ishii, and Y. Baba, Numerical electromagnetic analysis of lightning-induced voltage over ground of finite conductivity, IEEE Trans. Electromagn. Compat., vol. 45, no. 4, pp. 651-656, Nov. 23. [5] F. H. Silveira, S. Visacro, J. Herrera, and H. Torres, Evaluation of lightning-induced voltages over a lossy ground by the hybrid electromagnetic model, IEEE Trans. Electromagn. Compat., vol. 51, no. 1, pp. 156-16, Feb. 29. [6] P. Yutthagowith, A. Ametani, N. Nagaoka, and Y. Baba, Lightninginduced voltage over lossy ground by a hybrid electromagnetic circuit model method with Cooray-Rubinstein formula, IEEE Trans. Electromagn. Compat., vol. 51, no. 4, Nov. 29. [7] K. S. Yee, Numerical solution of initial boundary value problems in solving Maxwell s equations in isotropic media, IEEE Trans. Anntennas and Propat., vol. 14, no. 3, pp. 32-37, Mar. 1966 [8] Y. Baba, and V. A. Rakov, Voltages induced on an overhead wire by lightning strikes to a nearby tall grounded object, IEEE Trans. Electromagn. Compat., vol. 48, no. 1, pp. 212-224, Feb. 26. [9] H.-M. Ren, B.-H. Zhou, V. A. Rakov, L.-H. Shi, C. Gao, and J.-H. Yang, Analysis of lightning-induced voltages on overhead lines using a 2-D FDTD method and Agrawal coupling model, IEEE Trans. Electromagn. Compat., vol. 5, no. 3, pp. 651-659, Aug. 28. [1] A. K. Agrawal, H. J. Price, and H. H. Gurbaxani, Transient response of muliticonductor transmission lines excited by a nonuniform electromagnetic field, IEEE Trans. Electromagn. Compat., vol. 22, no. 2, pp. 119-129, May 198. [11] P. P. Barker, T. A. Short, A. R. Eybert-Berard, and J. P. Berlandis, Induced voltage measurements on an experimental distribution line during nearby rocket triggered lightning flashes, IEEE Trans. Power Del., vol. 11, no. 2, pp. 98-995, Apr. 1996. [12] Z. P. Liao, H. L. Wong, B. -P. Yang, and Y. -F. Yuan, A transmitting boundary for transient wave analysis, Science Sinica, Series A, vol. 27, no. 1, pp. 163-176, 1984. [13] T. Noda, and S. Yokoyama, Thin wire representation in finite difference time domain surge simulation, IEEE Trans. Power Del., vol. 17, no. 3, pp. 84-847, Jul. 22. [14] C. J. Railton, D. L. Paul, and S. Dumanli, The treatment of thin wire and coaxial structures in lossless and lossy media in FDTD by the modification of assigned material parameters, IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 654-66, Nov. 26. [15] Y. Baba, and V.A.Rakov, On the transmission line model for lightning return stroke representation, Geophys. Res. Lett, vol. 3, no. 24, 2294, doi:1.129/23gl 1847, Dec. 23. 132
易迪拓培训 专注于微波 射频 天线设计人才的培养网址 :http://www.edatop.com 射频和天线设计培训课程推荐 易迪拓培训 (www.edatop.com) 由数名来自于研发第一线的资深工程师发起成立, 致力并专注于微波 射频 天线设计研发人才的培养 ; 我们于 26 年整合合并微波 EDA 网 (www.mweda.com), 现已发展成为国内最大的微波射频和天线设计人才培养基地, 成功推出多套微波射频以及天线设计经典培训课程和 ADS HFSS 等专业软件使用培训课程, 广受客户好评 ; 并先后与人民邮电出版社 电子工业出版社合作出版了多本专业图书, 帮助数万名工程师提升了专业技术能力 客户遍布中兴通讯 研通高频 埃威航电 国人通信等多家国内知名公司, 以及台湾工业技术研究院 永业科技 全一电子等多家台湾地区企业 易迪拓培训课程列表 :http://www.edatop.com/peixun/rfe/129.html 射频工程师养成培训课程套装该套装精选了射频专业基础培训课程 射频仿真设计培训课程和射频电路测量培训课程三个类别共 3 门视频培训课程和 3 本图书教材 ; 旨在引领学员全面学习一个射频工程师需要熟悉 理解和掌握的专业知识和研发设计能力 通过套装的学习, 能够让学员完全达到和胜任一个合格的射频工程师的要求 课程网址 :http://www.edatop.com/peixun/rfe/11.html ADS 学习培训课程套装该套装是迄今国内最全面 最权威的 ADS 培训教程, 共包含 1 门 ADS 学习培训课程 课程是由具有多年 ADS 使用经验的微波射频与通信系统设计领域资深专家讲解, 并多结合设计实例, 由浅入深 详细而又全面地讲解了 ADS 在微波射频电路设计 通信系统设计和电磁仿真设计方面的内容 能让您在最短的时间内学会使用 ADS, 迅速提升个人技术能力, 把 ADS 真正应用到实际研发工作中去, 成为 ADS 设计专家... 课程网址 : http://www.edatop.com/peixun/ads/13.html HFSS 学习培训课程套装该套课程套装包含了本站全部 HFSS 培训课程, 是迄今国内最全面 最专业的 HFSS 培训教程套装, 可以帮助您从零开始, 全面深入学习 HFSS 的各项功能和在多个方面的工程应用 购买套装, 更可超值赠送 3 个月免费学习答疑, 随时解答您学习过程中遇到的棘手问题, 让您的 HFSS 学习更加轻松顺畅 课程网址 :http://www.edatop.com/peixun/hfss/11.html `
易迪拓培训 专注于微波 射频 天线设计人才的培养网址 :http://www.edatop.com CST 学习培训课程套装该培训套装由易迪拓培训联合微波 EDA 网共同推出, 是最全面 系统 专业的 CST 微波工作室培训课程套装, 所有课程都由经验丰富的专家授课, 视频教学, 可以帮助您从零开始, 全面系统地学习 CST 微波工作的各项功能及其在微波射频 天线设计等领域的设计应用 且购买该套装, 还可超值赠送 3 个月免费学习答疑 课程网址 :http://www.edatop.com/peixun/cst/24.html HFSS 天线设计培训课程套装套装包含 6 门视频课程和 1 本图书, 课程从基础讲起, 内容由浅入深, 理论介绍和实际操作讲解相结合, 全面系统的讲解了 HFSS 天线设计的全过程 是国内最全面 最专业的 HFSS 天线设计课程, 可以帮助您快速学习掌握如何使用 HFSS 设计天线, 让天线设计不再难 课程网址 :http://www.edatop.com/peixun/hfss/122.html 13.56MHz NFC/RFID 线圈天线设计培训课程套装套装包含 4 门视频培训课程, 培训将 13.56MHz 线圈天线设计原理和仿真设计实践相结合, 全面系统地讲解了 13.56MHz 线圈天线的工作原理 设计方法 设计考量以及使用 HFSS 和 CST 仿真分析线圈天线的具体操作, 同时还介绍了 13.56MHz 线圈天线匹配电路的设计和调试 通过该套课程的学习, 可以帮助您快速学习掌握 13.56MHz 线圈天线及其匹配电路的原理 设计和调试 详情浏览 :http://www.edatop.com/peixun/antenna/116.html 我们的课程优势 : 成立于 24 年,1 多年丰富的行业经验, 一直致力并专注于微波射频和天线设计工程师的培养, 更了解该行业对人才的要求 经验丰富的一线资深工程师讲授, 结合实际工程案例, 直观 实用 易学 联系我们 : 易迪拓培训官网 :http://www.edatop.com 微波 EDA 网 :http://www.mweda.com 官方淘宝店 :http://shop369289.taobao.com 专注于微波 射频 天线设计人才的培养易迪拓培训官方网址 :http://www.edatop.com 淘宝网店 :http://shop369289.taobao.com