IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 51, NO. 4, NOVEMBER 2009 975 Lightning-Induced Voltage Over Lossy Ground by a Hybrid Electromagnetic Circuit Model Method With Cooray–Rubinstein Formula Peerawut Yutthagowith, Student Member, IEEE, Akihiro Ametani, Fellow, IEEE, Naoto Nagaoka, Member, IEEE, and Yoshihiro Baba, Member, IEEE Abstract—This paper presents the calculation of lightning- induced voltages over lossy ground, generated by a lightning strike to a flat ground and a tall object. A hybrid electromagnetic circuit model method adopting an approximation formula, i.e., Cooray– Rubinstein expression, is employed in this paper. A comparison of the simulation results with experimental data shows that the Cooray–Rubinstein formula is still good enough for the calcula- tion of a lightning-induced voltage. Electric fields calculated by the proposed method, and by equations derived by Master and Uman and a conventional dipole technique, are compared. The two approaches yield the same total electrical fields but different components of electrostatic, induction and radiation. Index Terms—Cooray–Rubinstein formula, hybrid electromag- netic circuit model (HECM) method, lightning-induced voltage, lossy ground. I. INTRODUCTION T HE HYBRID electromagnetic circuit model (HECM) method is formulated by current sources that have elec- tromagnetic coupling among the elements, with each one of them represented by a source of transversal and longitudinal currents. This same kind of formulation was first proposed by Visacro and Portela [1], who developed a frequency-domain model to address the transient response of grounding electrodes, using Fourier transform. This model employs electric scalar and magnetic vector potentials for taking account of electromag- netic coupling among elements that represent the current path, but it is formulated in terms of circuit quantities, voltages, and currents. Later, this method was named the so-called hybrid electromagnetic (HEM) model, detailed by Visacro and Soares [2], to address the simulation of general lightning-related engineering problems, such as overvoltage developed by direct lightning strikes and voltages induced by nearby strikes [3], [4]. Based on the same formulation, Salari and Portela presented [5] an approach [frequency-domain electromagnetic transient program (FDETP)] to represent time-dependent effects such as the one associated to nonlinear elements (e.g., diodes, switches, transformers, surge arresters, etc.). Besides, both models, HEM Manuscript received April 28, 2009; revised August 3, 2009. First published September 18, 2009; current version published November 18, 2009. The authors are with Doshisha University, Kyoto 610-0321, Japan (e-mail: eti1107@mail4.doshisha.ac.jp; aametani@mail.doshisha.ac.jp; nnagaoka@ mail.doshisha.ac.jp; ybaba@mail.doshisha.ac.jp). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2009.2029702 and FDETP, allow representing concentrate elements (R, L, G, and C). An HECM method can be applied to calculate electromag- netic fields, voltages, and currents along conductors and also induced voltages, as described in the Appendixes. The first advantage of this method is that it can incorporate electrical components based on a circuit theory in the frequency domain, such as RLC elements, transmission lines (TLs), cables, single-phase transformers, switches, etc. The second is that the composition matrix in this method depends on the configuration of a considered system and a medium, and does not depend on the sources. Disadvantage of this method is that it is time-consuming when a lossy soil effect is included. Even if the soil effect can be taken into account by using the image method, it is not a simple task in the case of a lossy ground. The ground conductivity plays an important role for lightning electromagnetic fields and line parameters. The latter can be ne- glected if the line length is less than 2 km [6]. For calculation of the electromagnetic fields generated by a lightning channel, the ground conductivity significantly affects the horizontal electric field. The computation of a lossy ground effect can be modeled by using well-known Sommerfeld integrals and can be im- plemented by the Norton approximation for a numerical so- lution. The results by the approximate formula show good agreement with exact solutions, but the approximation is still time-consuming. Rubinstein proposed an approximate formula, which is called the Cooray–Rubinstein formula [7], in which a horizontal electric field at specific height is composed of the two terms given in (1). The first one is the electric field com- puted by using the perfect-ground assumption and the second is the horizontal magnetic field multiplied by the ground surface impedance on the assumption of perfect ground at ground level. The validity of this formula is examined in comparison with an exact solution by Wait [8] and Shoory et al. [9]. The formula in (1) was validated at close range, 100–1000 m, and for ground conductivities, 1 and 10 mS/m [7] E aρ (ρ,z,jω)= E aρp (ρ,z,jω) −  jω σ 2 + jωε 2 H aφp (ρ, 0,jω) (1) which are a typical case affecting the insulation of a distribu- tion system and a telecommunication system. An error of the calculated horizontal electric field by the Cooray–Rubinstein 0018-9375/$26.00 © 2009 IEEE