**3.3 Rectangular grid**

A rectangular grounding grid was investigated, illustrated in **Figure 11**. It is a <sup>20</sup> � 16 m<sup>2</sup> grid divided into 4 � 4 m<sup>2</sup> squares buried 0*:*5 m deep in the soil. It is the same geometric configuration as the one presented in [29], but the soil parameters used here are different. Similarly, to the previous section, the Alipio-Visacro soil model is used with mean values [26] considering, in one case, a low-frequency conductivity *σ*<sup>0</sup> of 1 mS and 10 mS for the other one.

For this case, the step voltage was calculated in the þ*x-*direction along the line *y* ¼ 8, that is, along the middle of the grid. The current was injected in the grid's corner at *x* ¼ *y* ¼ 0.

The harmonic impedance of that grid is shown in **Figure 12**. Because the total conductor area is bigger and goes further from the injection point, the harmonic impedance of the grid is smaller than the horizontal electrode. This effect is stronger

**Figure 11.** *Simulated grounding grid.*

**Figure 12.** *Harmonic impedance of a grounding grid. (a) Absolute value and (b) phase.*

**Figure 13.**

*Harmonic step voltage above a grounding grid comparing the potential difference* **Δ***u and voltage drop Up***<sup>1</sup>**�*p***2***. (a) f* ¼ 100 *Hz, (b) f* ¼ 305*:*39 *kHz, (c) f* ¼ 0*:*98 *MHz, and (d) f* ¼ 3*:*13 *MHz.*

for less conductive soil. The phase behavior of the impedance, however, is very similar.

The harmonic step voltage is shown in **Figure 13**. For low frequencies, the maxima are near the grid edges and peaks when crossing conductors (namely at *x* ¼ 0, 4, 8, 12, 16, 20). Nearer the injection point, the voltage drop *Up*<sup>1</sup>�*p*<sup>2</sup> is greater than the potential difference Δ*u*. The voltage drop quickly becomes smaller than the potential difference further away from the injection point. The step voltage in a higher conductivity soil is much smaller than that for a low conductivity soil.

The GPR, illustrated in **Figure 14**, is smaller than the one observed for the horizontal conductor due to the smaller harmonic impedance.

The transient step voltage is illustrated in **Figure 15**. There is little coincidence between the potential difference Δ*u* and voltage drop *Up*<sup>1</sup>�*<sup>p</sup>*2. However, the difference between these quantities takes longer to fade for the grid. The voltage drop has

**Figure 14.** *Ground potential rise (GPR) of a grounding grid.*

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

**Figure 15.**

*Transient step voltage above a grounding grid comparing the potential difference Δu and voltage drop Up***<sup>1</sup>**�*p***2***. (a) t* ¼ 0*:*1 *μs, (b) t* ¼ 0*:*2 *μs, (c) t* ¼ 0*:*4 *μs, and (d) t* ¼ 1*:*2 *μs.*

a negative offset due to the nonconservative electric field's direction associated with the conductors' longitudinal currents.

#### **3.4 Three communication towers**

This case is about three communication towers in close proximity to each other. All of them are grounded by 12 � 12 m2 square grids buried 0*:*5 m deep. Each grounding grid is 6 m apart from the next one, as illustrated in **Figure 16**. Four cases are considered: having all the grids connected or isolated from each other while the soil is represented by the Alipio-Visacro model [29] with mean values and lowfrequency conductivity *σ*<sup>0</sup> of 1 mS and 10 mS. The current is always injected at the first grid's lower-left corner (the grids are enumerated 1 to 3, from left to right). These connections are a common practice since it reduces the global grounding impedance for low-frequency phenomena.

**Figure 16.** *Grounding grids from the communication towers.*

**Figure 17.** *Harmonic potential at the lower-left corner of each tower's grounding grid. (a) σ*<sup>0</sup> ¼ 1 *mS and (b) σ*<sup>0</sup> ¼ 10 *mS.*

**Figure 18.** *GPR at the lower-left corner of each tower's grounding grids. (a) σ*<sup>0</sup> ¼ 1 *mS and (b) σ*<sup>0</sup> ¼ 10 *mS.*

The absolute values of the harmonic potentials that appear at the corner of each grid are shown in **Figure 17**. Connecting the grounding grids of the towers has the effect of equalizing their potentials in low frequencies, reducing the potential that arises in the tower that injects the current in the ground, but raising the potential in the other ones. However, after a few MHz, connecting has practically no effect in reducing harmonic potential. This can be explained based on electromagnetic wave theory. With the increase in both frequency and conductivity, the propagation constant has a greater attenuation constant (real part of the propagation constant). Hence, the nearby grounding grids do not guarantee an actual impact on the potential.

These results show that there is an effective length<sup>1</sup> of the conductors, indicating that for some current excitation, the interconnection between grids does not reduce GPR. Thus, adding more or longer conductors to reduce potential has no practical impact in high frequencies and high conductivity soils, although it is very beneficial

<sup>1</sup> The current dispersed to the soil along the grounding electrode shows nonuniform distribution. This nonuniformity is more pronounced at high frequencies. In this case, the attenuation effects are intense. Associated with this attenuation there is a critical electrode length, such that if a longer electrode is considered there will be no additional current dispersion. At this critical length is given the name of effective length, which depends on the soil conductivity and the frequency spectrum of the injected current.

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

(and extensively used in grounding design) in steady-state operations. In timedomain analysis, it is possible to notice this phenomenon in the first-time steps. Take **Figure 18(a)**, for instance. Considering Grid 1, until around 0*:*5 μs, the curves associated with the connected system and the isolated one are overlapped. However, after that point, the values start to drift away from each other. This is less pronounced in higher conductivity media; see **Figure 18(b)**, for instance.

As mentioned before, connecting the grids has the downside of raising the GPR in the other grids as all of them become equipotential. Thus, it is of utmost importance to properly design the grounding grid, taking into account the equipment that will be connected in these grids as well as the possibility of protecting such elements from unexpected transferred potential. For example, consider **Figure 18(a)**. If a fast current strikes Grid 1 and Grid 2 has sensible electronics equipment connected to it, the potential rise on this equipment will be around three times higher if the grids are connected, leading to a faulted equipment.

As expected from the analysis of previous cases, the harmonic step voltage in the þ*x*-direction calculated by scalar potential difference Δ*u* and calculated by voltage drop *Up*<sup>1</sup>�*p*<sup>2</sup> has the same numerical values for low frequencies but differ in higher frequencies, as illustrated in **Figures 19** and **20**. That difference is greater when the grids are connected because the current can then travel farther from its injection point and, therefore, cause a stronger electric field above the other grids.

The transient step voltage in the þ*x-*direction is shown in **Figures 21** and **22**. In the first moments, while the injection current is rising, the step voltage has highfrequency components, and the step voltage is zero, far from the first grid.

The step voltage calculated by the voltage drop *Up*<sup>1</sup>�*p*<sup>2</sup> has a negative offset compared to the potential difference Δ*u* because of the current direction in the

#### **Figure 19.**

*Harmonic step voltage on the ground surface along the line y* ¼ **6 m** *comparing the potential difference Δu and voltage drop Up***<sup>1</sup>**�*p***<sup>2</sup>** *above the tower's grounding grids for soil with σ***<sup>0</sup>** ¼ **1** *mS. (a) f* ¼ 100 *Hz, (b) f* ¼ 305*:*39 *kHz, (c) f* ¼ 0*:*98 *MHz, and (d) f* ¼ 3*:*13 *MHz.*

**Figure 20.**

*Harmonic step voltage on the ground surface along the line y* ¼ **6 m** *comparing the potential difference Δu and voltage drop* **Up1**�**p2** *above the tower's grounding grids for soil with σ***<sup>0</sup>** ¼ **10** *mS. (a) f* ¼ 100 *Hz, (b) f* ¼ 305*:*39 *kHz, (c) f* ¼ 0*:*98 *MHz, and (d) f* ¼ 3*:*13 *MHz.*

#### **Figure 21.**

*Transient step voltage on the ground surface along the line y* ¼ **6 m** *comparing the potential difference Δu and voltage drop* **Up1**�**p2** *above the tower's grounding grids for soil with σ***<sup>0</sup>** ¼ **1** *mS. (a) t* ¼ 0*:*1 *μs, (b) t* ¼ 0*:*2 *μs, (c) t* ¼ 0*:*4 *μs*, *and (d) t* ¼ 1*:*2 *μs.*

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

**Figure 22.**

*Transient step voltage on the ground surface along the line y* ¼ **6 m** *comparing the potential difference Δu and voltage drop Up***<sup>1</sup>**�*p***<sup>2</sup>** *above the tower's grounding grids for soil with σ***<sup>0</sup>** ¼ **10** *mS. (a) t* ¼ 0*:*1 *μs, (b) t* ¼ 0*:*2 *μs, (c) t* ¼ 0*:*4 *μs*, *and (d) t* ¼ 1*:*2 *μs.*

conductors, particularly in the conductors that connect the grids. The nonconservative component of the electric field from the currents in the conductors that connect the grids has a �*x-*direction.

### **4. Discussions and conclusions**

When making an analysis or project, it is essential to be aware of the used mathematical model's simplifications (and limitations thereof). It is often desirable to reduce computational times, usually through mathematical approximations, which restrains the model's applicability. For instance, the results and modeling are presented here to consider all conductors as thin wires.

One approximation that is often made in electromagnetic compatibility and grounding projects is to consider voltage drop (integral effect of the electric field along a given path) as equal to the scalar electric potential difference. However, this will only be true for low-frequency phenomena. As it was shown as an example in the previous sections, the step voltage calculated by integrating the total electric field is different from the scalar potential difference (i.e., considering only the conservative component of the field).

Simplifying the voltage calculation along a path by computing only the scalar potential difference is desirable because that significantly reduces computational time. Unfortunately, that simplification cannot be made except for some lowfrequency phenomena. The simulations presented here demonstrate that the voltage drop can be higher or lower than the electric potential difference, depending on the studied case. Hence, more careful approach should be made when calculating the voltage due to lightning, and the total electric field should be integrated.

A common practice is to connect multiple grounding structures that are close to each other. However, the results have shown that connecting multiple grounding structures may not always be beneficial, depending on the intention. In low conductivity soils, that connection has the advantage of reducing the ground potential rise (GPR) of a structure when it is subject to current discharges, but it inadvertently raises the GPR in the other structures. It also has the consequence of raising the step voltage near those other grounding structures. Therefore, the common practices and standard recommendations should be inquired if they are the best for a given project and intention.
