**3. Mathematical model of the equivalent circuit of the grounding system in the frequency domain**

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 medium with permittivity *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> <sup>10</sup>−<sup>9</sup> <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>4</sup>*<sup>π</sup>* H/m.

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 to achieve a compromise solution between the two determinants.

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 terminal nodes SEP. We define a branch current *I<sup>k</sup> <sup>b</sup>* , branch voltage *<sup>U</sup><sup>k</sup> <sup>b</sup>* , and leakage current *Ik <sup>s</sup>* on the *k*th (*k* = 1, 2, . . . , *Nl*) segment.

### **3.1. Mathematical model of the grounding system in the frequency domain**

With the above considerations, according to [28]–[31], the electric circuit may be studied using the conventional nodal analysis method [35], resulting in the following equations:

$$\begin{aligned} \mathbf{[\overline{F}]} = [\overline{\mathbf{Y}}] \cdot [\overline{\mathbf{V\_n}}] \end{aligned} \tag{2}$$

The magnetic field intensity (MFI) at any point can be calculated by

*Nl* ∑ *i*=1 ∇ × �

*li*

The study of the performance of the grounding system in the frequency domain has been

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

From [28]–[31], we know that each segment is modeled as a lumped resistance and self-inductance. Mutual inductances or impedances between branch segments' branch

The diagonal elements consists of self impedance and self induction, the other elements belong to mutual induction between a pairs of conductor segments. The formula for self impedance and self induction can be found in [28]–[31], here, we give the formula for

*<sup>G</sup>*¯¯ *<sup>A</sup>*(*r*¯*j*,*r*¯*i*) • ¯*Ibi*

*<sup>G</sup>*¯¯ *<sup>A</sup>*(*r*¯*j*,*r*¯*i*) • ¯*Ibi*

*X*1,1 ... *X*1,*<sup>i</sup>* ... *X*1,*<sup>j</sup>* ... *X*1,*Nl* :::::: : *Xi*,1 ... *Xi*,*<sup>i</sup>* ... *Xi*,*<sup>j</sup>* ... *Xi*,*Nl* : ... : ... : : : *Xj*,1 ... *Xj*,*<sup>i</sup>* ... *Xj*,*<sup>j</sup>* ... *Xj*,*Nl* : ... : ... : : : *XNl* ,1 ... *XNl* ,*<sup>i</sup>* ... *XNl* ,*<sup>j</sup>* ... *XNl* ,*Nl*

(*r*¯*i*)*dti*. (7)

Based on Quasi-Static Complex Image Method

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

397

(8)

 

*dti* • ¯*Ibjidtj* (9)

<sup>4</sup>*<sup>π</sup>* <sup>1</sup> *Rij*

*dtj* (10)

¯¯*I*0, where

*B*¯(*r*¯*j*) =

currents or leakage currents are also included in the model:

 

[**Xq**]*Nl*×*Nl* =

mutual induction:

¯¯*I*<sup>0</sup> is the diagonal unit matrix.

1. The case of [**Xq**]=[**Zb**]: *Xi*,*<sup>i</sup>* = *Ri* + *jωLi*, *Xi*,*<sup>j</sup>* = *Ri*,*<sup>j</sup>* + *jωMi*,*j*.

*Mi*,*<sup>j</sup>* =

�

�

*lj*

For an infinite homogeneous conductivity medium, one has *<sup>G</sup>*¯¯ *<sup>A</sup>*(*r*¯*j*,*r*¯*i*) = *<sup>µ</sup>*

�

�

*lj* 1 *Rij* ¯*Ibi dti* • ¯*Ibj*

*li*

The above integral can be analytically calculated, this formula can be found in [27].

*li*

*Mi*,*<sup>j</sup>* <sup>=</sup> *<sup>µ</sup>* 4*π*

reduced to the computation of [**Zs**] and [**Zb**] matrices.

**3.2. Computation of** [**Zb**] **and** [**Zs**] **matrices**

$$\mathbf{E}\left[\overline{\mathbf{Y}}\right] = \left[\overline{\mathbf{K}}\right]^t \cdot \left[\overline{\mathbf{Z}}\right]^{-1} \cdot \left[\overline{\mathbf{K}}\right] + \left[\overline{\mathbf{A}}\right] \cdot \left[\overline{\mathbf{Z}}\right]^{-1} \cdot \left[\overline{\mathbf{A}}\right]^t \tag{3}$$

where [**F**] is an *Np* × 1 vector of external current sources; [**Z***b*] is the *Nl* × *Nl* branch mutual induction matrix of the circuit including resistive and inductive effects, which gives a matrix relationship between branch currents [**Ib**]; [**Zs**] is an *Nl* × *Nl* mutual impedance matrix, which gives a matrix relationship between the average SEP [**U**] and leakage currents [**Is**] through the rapid Galerkin moment method [38]. Both [**A**] and [**K**] are incidence matrices, which are used to relate branches and nodes. There are rectangular matrices of order *Nl* × *Np*, for whose elements we refer to [28]–[31].

The vector of nodal SEP [**Vn**] may be obtained by solving 2. The average SEP [**U**], leakage current [**Is**], branch voltage [**Ul**], and branch current [**Ib**] can also be calculated [28]–[31].

Once the branch currents and leakage currents are known, the SEP at any point can be calculated by

$$\varphi(\vec{r}\_{\dot{j}}) = \sum\_{i=1}^{N\_l} \int\_{l\_i} G\_{\Phi}(\vec{r}\_{\dot{j}}, \vec{r}\_i) \cdot \frac{I\_{s\_i}(\vec{r}\_{\dot{i}})}{l\_i} dl\_i \tag{4}$$

where *Gϕ*(*r*¯*j*,*r*¯*i*) is the scalar Green's function of a monopole in the multilayered earth model. The vector magnetic potential (VMP) *A* at any point can be calculated by

$$\bar{A}(\vec{r}\_{\dot{j}}) = \sum\_{i=1}^{N\_l} \int\_{l\_i} \bar{\bar{G}}\_A(\vec{r}\_{\dot{j}}, \vec{r}\_i) \bullet \bar{I}\_{b\_l}(\vec{r}\_i) dt\_i. \tag{5}$$

Here, *<sup>G</sup>*¯¯ *<sup>A</sup>*(*r*¯*j*,*r*¯*i*) is the dyadic Green's function of a dipole in the multilayered earth model, which will be introduced later.

The electrical field intensity (EFI) at any point can be calculated by

$$\bar{E}(\bar{r}\_{\bar{l}}) = -\sum\_{i=1}^{N\_{\bar{l}}} j\omega \int\_{l\_{\bar{l}}} \bar{\bar{G}}\_{A}(\bar{r}\_{\bar{l}}, \bar{r}\_{\bar{l}}) \bullet \bar{I}\_{b\_{\bar{l}}}(\bar{r}\_{\bar{l}}) dt\_{\bar{l}} - \sum\_{i=1}^{N\_{\bar{l}}} \nabla \int\_{l\_{\bar{l}}} \mathsf{G}\_{\theta}(\bar{r}\_{\bar{l}}, \bar{r}\_{\bar{l}}) \cdot \frac{I\_{\mathsf{s}\_{\bar{l}}}(\bar{r}\_{\bar{l}})}{l\_{\bar{l}}} dl\_{\bar{l}}.\tag{6}$$

<sup>396</sup> Computational and Numerical Simulations Numerical calculation for lightning response to grounding systems buried in horizontal multilayered earth model based on quasi-static Numerical Calculation for Lightning Response to Grounding Systems Buried in Horizontal Multilayered Earth Model Based on Quasi-Static Complex Image Method http://dx.doi.org/10.5772/57049 397

The magnetic field intensity (MFI) at any point can be calculated by

$$\bar{B}(\vec{r}\_{\dot{j}}) = \sum\_{i=1}^{N\_l} \nabla \times \int\_{l\_i} \bar{\bar{G}}\_A(\vec{r}\_{\dot{j}}, \vec{r}\_i) \bullet \bar{I}\_{b\_l}(\vec{r}\_i) dt\_i. \tag{7}$$

The study of the performance of the grounding system in the frequency domain has been reduced to the computation of [**Zs**] and [**Zb**] matrices.
