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Introduction

1.1 Historical Overview of Lightning Electromagnetic-Field and Surge Computations

Lightning return-stroke electromagnetic fields have been calculated using analytical expressions, derived for a vertical lightning channel (e.g., Uman et al. 1975). Effects of finite ground conductivity on lightning electromagnetic fields have also been studied using analytical expressions (e.g., Rachidi et al. 1996). These analytical expressions are still being used. Lightning-induced voltages on an overhead power distribution line or telecommunication line have been calculated using an engineering model of the lightning return stroke (e.g., Uman et al. 1975) and a field-to-wire coupling model (e.g., Rachidi 1993). Horizontal electric fields above a finitely conducting ground, which are needed for calculating lightning-induced voltages, have been evaluated using approximate expressions such as the Cooray–Rubinstein formula (Rubinstein 1996). Note that the Cooray–Rubinstein formula is given in the frequency domain, although its time-domain counterparts also exist. Lightning surges due to a direct lightning strike to an overhead power transmission or distribution line have been analyzed using distributed-circuit simulation methods such as the electromagnetic transients program (EMTP) (Dommel 1969). EMTP and other similar programs are still widely used in lightning surge simulations.

Around 1990, electromagnetic computation methods were first applied to lightning electromagnetic and surge simulations. One of the advantages of electromagnetic computation methods, in comparison with circuit simulation methods, is that they allow a self-consistent full-wave solution for both the transient current distribution in a 3D conductor system and resultant electromagnetic fields, although they are computationally expensive. Podgorski and Landt (1987) applied the method of moments (MoM) in the time domain (Van Baricum and Miller 1972; Miller et al. 1973) to analyze the lightning current along a tall object struck by lightning. Grcev and Dawalibi (1990) applied the MoM in the frequency domain (Harrington 1968) to analyze the surge characteristics of a grounding electrode. Since then, the MoM in the frequency domain has been frequently used in lightning surge simulations (e.g., Baba and Rakov 2008 and references therein).

Tanabe (2001) applied the finite-difference time domain (FDTD) method (Yee 1966), which is one of the electromagnetic computation methods, to studying the surge characteristics of a grounding electrode. Baba and Rakov (2003) used the FDTD method to compute lightning electromagnetic fields. More than 60 journal papers and a large number of conference papers, which use the FDTD method in lightning electromagnetic-field and surge simulations, have been published during the last 15 years (e.g., Baba and Rakov 2014 and references therein). Interest in using the FDTD method continues to grow. The FDTD method is presently the most widely used electromagnetic computation method in lightning electromagnetic-field and surge simulations.

Other electromagnetic computation methods such as the finite-element method (FEM) (e.g., Sadiku 1989), the partial-element equivalent-circuit (PEEC) method (Ruehli 1974), the hybrid electromagnetic model (HEM) (Visacro and Soares 2005), the transmission line matrix or modeling (TLM) method (Johns and Beurle 1971), and the constrained interpolation profile (CIP) method (Takewaki et al. 1985) have been recently applied to analyzing lightning electromagnetic fields and surge simulations (e.g., Yutthagowith et al. 2009 (PEEC); Silveira et al. 2010 (HEM); Smajic et al. 2011 (FEM); Tanaka et al. 2014a (TLM); Tanaka et al. 2014b (CIP)).

In the following section, we briefly introduce each of these electromagnetic computation methods.

1.2 Overview of Existing Electromagnetic Computation Methods

1.2.1 Method of Moments

The MoM in the time domain (Van Baricum and Miller 1972; Miller et al. 1973) has been used to analyze responses of thin-wire conducting structures to external transient electromagnetic fields. The entire conducting structure is modeled by a combination of cylindrical wire segments whose radii are much smaller than the wavelengths of interest. The so-called electric-field integral equation for a perfectly conducting thin wire in air (shown in Figure 1.1) is given below, assuming that current I and charge q are confined to the wire axis (thin-wire approximation) and that the boundary condition on the tangential electric field on the surface of the wire (this field must be equal to zero) is fulfilled:

Figure 1.1 Thin-wire cylindrical segment for method of moment (MoM)-based computation. Current is confined to the wire axis, and the tangential electric field on the surface of the wire is set to zero.

where C is an integration path along the wire axis; Einc denotes the incident electric field that induces current I; r and t denote the observation location (a point on the wire surface) and time, respectively; r′ and t′ denote the source location (a point on the wire axis) and time, respectively; R = r − r′; s and s′ denote the distance along the wire surface at r and that along the wire axis at r′, respectively; ŝ and ŝ′ denote unit vectors tangential to path C in Eq. (1.1) at r and r′, respectively; μ0 is the permeability of vacuum; and c is the speed of light. By numerically solving Eq. (1.1), which is based on Maxwell’s equations, the time-dependent current distribution along the wire structure excited by external field is obtained.

The thin-wire time domain (TWTD) code (Van Baricum and Miller 1972), developed at the Lawrence Livermore National Laboratory, is based on the MoM in the time domain. One of the advantages of the time-domain MoM is that it can incorporate nonlinear effects such as the lightning attachment process (Podgorski and Landt 1987) or the back-flashover at an overhead-power transmission line tower struck by lightning (Mozumi et al. 2003), although it does not allow lossy ground and wires buried in lossy ground to be incorporated. Other representative applications of the time domain MoM to lightning electromagnetic-field or surge simulations are found in Moini et al. (1998, 2000), Kato et al. (1999), Kordi et al. (2002, 2003), Pokharel and Ishii (2007), and Bonyadi-Ram et al. (2008).

The MoM in the frequency domain (Harrington 1968) has been widely used in analyzing responses of thin-wire conducting structures to incident electromagnetic fields. In order to obtain the time-varying responses, Fourier and inverse Fourier transforms are employed. The electric-field integral equation derived for a perfectly conducting thin wire in air, as shown in Figure 1.1, in the frequency domain is given by

where ω is the angular frequency, μ0 is the permeability of vacuum, and ε0 is the permittivity of vacuum. Other quantities in Eq. (1.2) are the same as those in Eq. (1.1). Current distribution along the thin-wire conducting structure can be obtained by numerically solving Eq. (1.2). Note that triangular and/or rectangular patches based on a surface-current formulation could also be used in the MoM in the frequency domain.

This method allows lossy ground and wires in lossy ground to be incorporated into the model. The numerical electromagnetic codes such as NEC-2 (Burke and Poggio 1980) and NEC-4 (Burke 1992), developed at the Lawrence Livermore National Laboratory, are based on the MoM in the frequency domain. Representative applications of the MoM in the frequency domain to lightning electromagnetic-field or surge simulations are found in Grcev and Dawalibi (1990), Baba and Ishii (2000, 2003), Pokharel et al. (2003, 2004), Shoory et al. (2005, 2010), Geranmayeh et al. (2006), Pokharel and Ishii (2007), Miyazaki and Ishii (2008a, 2008b), Sheshyekani et al. (2008), Aniserowicz and Maksimowicz (2011), Khosravi-Farsani et al. (2013), and Miyamoto et al. (2015). The MoM is the second most frequently used electromagnetic computation method in lightning electromagnetic-field and surge simulations.

1.2.2 Partial-Element Equivalent-Circuit Method

The PEEC method (Ruehli 1974) provides a full-wave solution to Maxwell’s equations. The method is applicable to both time (Wang et al. 2010) and frequency domains. A significant difference from the MoM is that the conductor system subject to analysis is transformed into its equivalent circuit. Although the PEEC method is based on exact field theory, it was originally developed not for electromagnetic-field computations but for the analysis of interconnect and packaging structures. In the 1990s, field retardation, external field excitation, and the treatment of dielectric materials were incorporated (Ruehli and Heeb 1992). This method has been recently employed in lightning-surge simulations (e.g., Yutthagowith et al....