**1. Introduction**

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392 Computational and Numerical Simulations

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For lightning protection design, an exact evaluation of transients electromagnetic field within a complex grounding system has fundamental importance. In fact, the earth electrodes constitute a fundamental part of the electrical apparatus in both industrial and civil structures. Grounding systems should have a suitable configuration in order to avoid serious hazard to humans, and to preserve electrical insulation in electrical and electronic equipment and installations. Moreover, in electrical power installations, the shape and dimensions of the earth termination system, as a part of a lightning protection system, are more important than the specific value of the earth resistance, in order to disperse the lightning current into the earth without causing dangerous overvoltages.

Pioneering but comprehensive work on this subject was conducted in the first half of the twentieth century, which is summarized by Sunde in the well known reference book [1]. Important pioneering work is described also in [2] and [3]. More recent work is summarized in [4]. Recently, computerized analysis methods have been developed based on different approaches, for example, on circuit theory [5]–[8], transmission line theory [9]–[15], electromagnetic field theory [16]–[23], and hybrid method [24]–[31].

Hybrid method has been developed from conventional nodal analysis, which combines the electrical circuit method and the electromagnetic field method. It has been proved to have combined the strong points of both the two methods. Dawalibi earlier discussed how the hybrid method came out of the electromagnetic field method in [24] and further discussed the hybrid method in [25], however, the hybrid method was based on quasi-static electromagnetic field theory, and only discusses steady grounding problem in the frequency domain. Meliopoulos also discusses the hybrid method based on quasi-static electromagnetic

©2012 Li et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 Li et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

field theory in [26]; Huang & Kasten developed a new hybrid model to calculate the current distribution in both the grounding system and the metallic support conductors, while considering the voltage drop along the grounding system conductors [27]. However, the model was also based on quasi-static electromagnetic field theory, meanwhile, leakage currents and network currents within the grounding system are separately considered in the calculation, their mutual coupling influence is neglected, and the capacitive coupling effect of the earth is also neglected. Otero, Cidras & Alamo developed a hybrid method to calculate the current distribution in a grounding system [28] within which the mutual inductive and capacitive coupling influence among these current flowing and leaking along the conductor is considered. However, only a uniform half infinite earth model is considered. This hybrid method was combined with the FFT, and so the transient response from the grounding system was obtained. These confines in the frequency domain promoted the development of a novel mathematical model in [29]–[32], which introduced the quasi-static complex image method (QSCIM) for calculating the current distribution in a grounding system buried in both horizontal and vertical multilayered earth models in the frequency domain. However, the hybrid method can be further developed to numerically calculate the transient response from a grounding system buried in multilayered earth model.

if *i*(*t*) represents the injected current at a point in the grounding system, and *x*(*t*) denotes an

Numerical Calculation for Lightning Response to Grounding Systems Buried in Horizontal Multilayered Earth Model

where *<sup>F</sup>* and *<sup>F</sup>*−<sup>1</sup> are the Fourier and inverse Fourier transforms, respectively, *<sup>W</sup>*(*jω*) is the

• The earth comprises horizontal multilayered media, and the air media are homogenous and occupy half-spaces with a common horizontal plane boundary between the air and

• The earth and the grounding electrodes exhibit linear and isotropic, arbitrary

• The grounding system is assumed to be made of cylindrical metallic conductors with arbitrary orientation. However, they are assumed to be subject to the thin-wire approximation, i.e., the ratio of the length *l* of the conductor segment to its radius *r*

• Energization occurs by the injection of a current impulse of arbitrary shape produced by an ideal current generator with one terminal connected to the grounding system, and the

**3. Mathematical model of the equivalent circuit of the grounding system**

A set of interconnected cylindrical thin conductors placed in any position or orientation makes up a network to form the grounding system. The grounding network's conductors are assumed to be completely buried in a conductive *Ne*-layer media (earth) with conductivity *σe* and permittivity *<sup>ε</sup><sup>e</sup>* = *<sup>ε</sup>re* ·*ε*<sup>0</sup> (here *<sup>e</sup>* = 1, · · · · ··, *Ne*). The air is assumed to be a non-conductive

The proposed methodology is based on the study of all the inductive, capacitive and conductive couplings between the different grounding system conductors. First, the electrode is divided into *Nl* pieces of segments that can be studied as elemental units, where the discrete grounding system has *Np* nodes. A higher segmented rate of the electrode can enhance the model's accuracy but increases its computational time. Therefore, it is necessary

The grounding network is energized by injection of single frequency currents at one or more nodes. In general, we consider that a sinusoidal current source of value *Fj* is connected at the *j*th (*j* = 1, 2, . . . , *Np*) node. A scalar electric potential (SEP) *Vj* of *j*th node on the grounding network referring to the infinite remote earth as zero SEP is defined. In the same way, we define an average SEP *Uk* on *k*th (*k* = 1, 2, . . . , *Nl*) segment. If the segments are short enough, it is possible to consider *Uk* as approximately equal to the average of the *k*th segment's two

<sup>36</sup>*<sup>π</sup>* F/m. All media have permeability *<sup>µ</sup>* <sup>=</sup> *<sup>µ</sup>*<sup>0</sup> <sup>=</sup> <sup>10</sup>−<sup>7</sup>

*<sup>b</sup>* , branch voltage *<sup>U</sup><sup>k</sup>*

<sup>4</sup>*<sup>π</sup>* H/m.

*<sup>b</sup>* , and leakage current

other to the ground at infinity. The influence of the connecting leads is ignored.

*<sup>x</sup>*(*t*) = *<sup>F</sup>*−1*W*(*jω*) · *<sup>F</sup>*[*i*(*t*)] (1)

Based on Quasi-Static Complex Image Method

http://dx.doi.org/10.5772/57049

395

observed response, then

earth.

is *<sup>l</sup>*

*Ik*

characteristics.

**in the frequency domain**

medium with permittivity *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> <sup>10</sup>−<sup>9</sup>

transfer function, and *ω* is the angular frequency.

The physical model is based on the following assumptions.

*<sup>r</sup>* <sup>≫</sup> 1. In practice, a ratio of about 10 is satisfactory.

to achieve a compromise solution between the two determinants.

terminal nodes SEP. We define a branch current *I<sup>k</sup>*

*<sup>s</sup>* on the *k*th (*k* = 1, 2, . . . , *Nl*) segment.

Once the multilayered earth model is adopted, the Green's function of a point source will contain an infinite integral for the Bessel function, a complex image method based on Maclaurin's infinite series expansion have been studied in [33] and [34], which has brought up problem about the convergence of the infinite Maclaurin's series. To avoid this convergence problem, QSCIM is introduced to dealt with the infinite integral, which uses finite exponential terms (usually just 3–4 terms) through the Matrix Pencil approach instead of Maclaurin's series to quickly calculate the Green's function.

In this paper, based on previous works [28]–[31], combined with the FFT, a novel and accurate mathematical model is developed for calculating the harmonic wave currents of lightning currents distribution along the grounding system buried in multilayered earth model in the frequency domain, within which not only the conducting effect of the harmonic wave currents leaking into the soil, but also capacitive and inductive effects between different layers of soil have been considered. Both leakage currents and network currents within the grounding system and their mutual coupling are considered in the calculation. The earth is modeled by a multilayered earth model. To accelerate the calculation, QSCIM and closed form of Green's function were introduced, and the mutual inductive and conductive coefficient have analytical formulae so as to avoid numerical integration.

The maximum frequency of applicability of the method is limited by the quasi-static approximation of the electromagnetic fields. For the usual electrodes, it may be applied up to some hundreds of kHz.
