**2. Mathematical modeling of the equivalent circuit**

As commented before, for an accurate assessment of the transient response of the grounding system, a wideband representation is commonly required. The frequency range of lightning-related phenomena goes from a few hertz up to tenths of MHz. Thereby, in the frequency domain, a detailed representation is warranted by using approaches such as the Finite Element Method (FEM) [7] or the Method of Moments (MoM) [8–10]. It is also worth noticing that there are methods that properly solve this numerical issue by solving Maxwell equations directly in the time domain, such as the so-called Finite Differences Time Domain (FDTD) [11, 12]. However, these approaches usually take even more computational time and do not consider the frequency dependence of the soil. In this scenario, considering full-wave frequency domain techniques, there are essentially two approaches to do so. The first one relies on solving Maxwell equations (or the associated vector and scalar potentials) numerically, while the second one applies circuit theory approximation whenever possible. For the latter, there are two approaches in the literature that, although developed independently, share a "common kernel." The first one is called partial element equivalent circuit (PEEC) [13, 14] and has focused mainly on electromagnetic compatibility issues. The second one is the so-called hybrid electromagnetic model (HEM) [10] and was initially developed to analyze underground bare conductors such as the ones found in electrical grounding. It is important to highlight that the so-called HEM is a particular case of the PEEC in the frequency domain, considering only cylindrical conductors.

Given that both approaches lead to an equivalent circuit derived from the behavior of the electromagnetic fields, these methods can be understood as hybrid models. Regardless of the adopted approach, one needs to divide the conductors

**Figure 3.** *Horizontal grounding electrode: Real physical system.*

involved in several shorter segments to assume a uniform electric current through it and another uniform current, leaving the segment into the surrounding media. Thus, all methods present a heavy computational burden, demanding an improvement of their numerical performance.

In general, hybrid models consist of segmenting the grounding systems into small segments and estimating the electric current that circulates along each segment (here called longitudinal current or *IL*) and the current that flows from the electrode to the soil (here called transversal current or *IT*). Since these currents are not known previously, it is necessary to apply the MoM to estimate these parameters. To clarify this procedure, consider, as an example, the horizontal grounding electrode shown in **Figure 3**. In this simple case, there are only two semi-infinite media: air and soil. The electrode, with length *L*, radius *a* is buried at depth *d* in a linear, isotropic, and homogeneous soil, with electrical resistivity *ρ* (electrical conductivity *σ* ¼ 1*=ρ*), electrical permittivity *ε*, and magnetic permeability *μ*. In general, according to data [15], the magnetic permeability of the soil is close to that of the vacuum (*μ*≈ *μ*0). The air has *ρ*air ! ∞ (*σ*air ! 0), *ε*air ≈*ε*0, and *μ*air ≈*μ*0.

The first step is to divide the electrode into *N* segments with length *ℓ* ¼ *L=N* to solve the problem. Then, the electromagnetic coupling is calculated for each pair of elements considering the contribution of both *IT* (1) and *IL* (2). The numerical values are obtained considering a simplification of the traditional MoM, that is, considering a piecewise pulse base function (more details about this basis function can be found in [16]). At this point, there are two approaches:


$$u\_{mn} = \frac{I\_{Tn}}{4\pi[\sigma + j\alpha\varepsilon]L\_nL\_m} \int\_{L\_m} \int\_{L\_n} \frac{e^{-rr}}{r} d\ell\_n d\ell\_m \tag{1}$$

$$
\Delta U\_{mn} = -j\alpha\mu \frac{I\_{Ln}}{4\pi} \int\_{L\_m} \int\_{L\_n} \frac{e^{-\gamma r}}{r} d\vec{\ell}\_n \cdot d\vec{\ell}\_m \tag{2}
$$

As in PEEC, HEM considers that both *IT* and *IL* do not vary along the electrode, i. e., it is uniform for each segment. Since the currents do not vary along the electrode, it is possible to represent a linear system concentrating half of each *ITn* in each node that composes a particular segment and considering Kirchhoff Current Law (KCL) in each of these nodes, i.e., considering a pi-equivalent system, similar to the one illustrated in **Figure 4(a)**. Another possibility is to consider a T-equivalent circuit (similar to the one illustrated in **Figure 4b**) and apply the KCL.

*Modeling Grounding Systems for Electromagnetic Compatibility Analysis DOI: http://dx.doi.org/10.5772/intechopen.100454*

**Figure 4.**

*Equivalent circuits obtained to apply the circuit theory in PEEC-type simulation. (a) PEEC pi-equivalent circuit and (b) PEEC T-equivalent circuit.*

**Figure 5.** *Horizontal grounding electrode: corresponding model.*

Furthermore, the interface must be considered. Hence, a classical solution in hybrid models is using the so-called "Image Methods" (IM) [19–21]. **Figure 5** illustrates the equivalent physical system obtained by applying the IM already considering the electrode subdivided into *N* segments and considering uniform currents in each element.

Additionally, it is important to comment that two main bottlenecks guarantee computational inefficiency:


Moreover, these two computationally intense tasks are to be performed at every frequency sample. To overcome such problems, there are some technical propositions in the literature. In [22], the possibility of increasing each segment length is discussed to reduce the matrices'size, leading to a reduction in the computational burden. In [23], it is presented an alternative to the problem by presenting an average exponential term, that is, approximating the *<sup>e</sup>*�*γ<sup>r</sup>* ð Þ *<sup>=</sup><sup>r</sup>* as an average value along the electrode, this leads to the necessity of solving the numerical integral only once for each segment. In [24, 25], it is proposed to use the first term in the MacLaurin series expansion of the integrand to obtain a closed form approximation. Since it is necessary to solve the system just once in both cases, it reduces the computational time. Note that these approximations have presented a feasible alternative in most cases (considering that the inject current has a limit frequency band below the 10 MHz). If the frequency spectrum is superior to 10 MHz, these approximations should be avoided. The Appendix presents a relationship between electromagnetic fields and more practical parameters (potential difference, voltage

drop, and step voltage). If the reader is not familiar with the potential/voltage concepts, the authors recommend that the reader goes to the Appendix.
