**A. Appendix: Electromagnetic modeling for determining potential difference, voltage drop, and step voltage**

After determining the current distributions in the grounding electrodes, as presented in Section 2, it is possible to calculate the total electric field, the potential difference, and voltage drop. This Appendix presents a brief quantification of these quantities.

#### **A.1 Total electric field** *E* ! **Total and its components**

Knowing *IT* and *IL*, it is possible to calculate the total electric field (*E* ! Total) at a generic point in soil, as well as its conservative (E! C) and nonconservative (*E* ! *NC*) components. Here, it is called conservative electric field component, the part associated with the transversal current, that is, the one that contemplates the electromagnetic field's divergence nature. On the other hand, here it is called nonconservative electric field component, the part associated with the longitudinal current, that is, the one that contemplates the curling nature of the electromagnetic field.

#### **A.1.1 Conservative electric field component (***E* ! *C***)**

To illustrate, consider a point current source *IT*�*<sup>p</sup>*, located in the soil. This current generates an electric scalar potential *VP* given by:

$$V\_P = \frac{I\_{T-p}}{4\pi[\sigma + j\alpha\nu]} \frac{e^{-\gamma r}}{r} \tag{4}$$

where *r* is the distance between the source point and the observation point *P*.

The electric field associated with this current source (*E* ! *<sup>C</sup>*) is straightforwardly obtained (at any generic point *P*, into the ground) by the gradient of *VP* [27]:

$$
\overrightarrow{E}\_C = -\overrightarrow{\nabla} \mathbf{V}\_P = \frac{I\_{T-p}}{4\pi[\sigma + j\alpha\varepsilon]} \frac{(\mathbf{1} + \eta r) \, e^{-\gamma r}}{r^2} \overrightarrow{a}\_\varepsilon \tag{5}
$$

where *a* ! *<sup>e</sup>* is the unit vector that defines the direction of *E* ! *C*.

The electric field generated by the total transverse current *IT* from a given segment of length *ℓ* ¼ *L=N* (see **Figure 5**) can be determined by the cumulative effect of point current sources along the segment, considering a continuous and

uniform current distribution with linear density *ITi=Li*. Then, the electric field *E* ! *Ci* due to these currents is given by [27]:

$$\overrightarrow{E}\_{\rm Ci} = \frac{1}{4\pi[\sigma + j\alpha\epsilon]} \int\_{\ell\_i} \frac{I\_{\rm Ti}\left(1 + \gamma r\_i\right)e^{-\gamma r\_i}}{L\_i r\_i^2} d\ell\_i \,\overrightarrow{a}\_{\epsilon} \tag{6}$$

where *ri* is the distance between the infinitesimal element *dℓ<sup>i</sup>* and point *P*; *a* ! *ei* is the unit vector that defines the direction of *E* ! *Ci*.

With similar reasoning, it is possible to determine the conservative electric field component associated with the transverse current of the image segment *E* ! *Ci*�image.

$$\overrightarrow{E}\_{Ci-\text{image}} = \frac{1}{4\pi[\sigma + j\alpha\epsilon]} \int\_{\ell\_i} \frac{\Gamma\_r I\_{Ti}\left(1 + \chi r\_{i-\text{image}}\right) e^{-\gamma r\_{i-\text{image}}}}{L\_i r\_{i-\text{image}}^2} d\ell\_{i-\text{image}} \overrightarrow{a}\_{\epsilon-\text{image}} \tag{7}$$

where *ri*�image is the distance between the in the infinitesimal element *dℓ<sup>i</sup>*�image and point *P*; *a* ! *<sup>e</sup>*�image is the unit vector that defines the direction of *E* ! *Ci*�image and Γ*<sup>r</sup>* is the reflection coefficient given by [19, 20] (see **Figure 5**):

$$\Gamma\_r = \frac{\sigma + jo(\varepsilon - \varepsilon\_0)}{\sigma + jo(\varepsilon + \varepsilon\_0)} \tag{8}$$

By considering the system linear and applying the superposition theorem, one can determine the total conservative component *E* ! *Ci*�Total associated with the segment *i* (see **Figure 5**):

$$
\overrightarrow{E}\_{\text{Ci-Total}} = \overrightarrow{E}\_{\text{Ci}} + \overrightarrow{E}\_{\text{Ci-image}} \tag{9}
$$

Finally, the total conservative component *E* ! *<sup>C</sup>*�Total is obtained *via* the sum of all the conservative field components associated with the transverse currents in the *N* segments:

$$
\overrightarrow{E}\_{\text{C-Total}} = \sum\_{i=1}^{N} \left( \overrightarrow{E}\_{\text{Ci}} + \overrightarrow{E}\_{\text{Ci-image}} \right) \tag{10}
$$

#### **A.1.2 Nonconservative electric field component (***E* ! *NC***)**

Consider the longitudinal current *ILi* flowing along segment *i* (see **Figure 5**). This current source generates a magnetic vector potential *A* ! *Pi* at a generic point *P* of the medium given by:

$$
\overrightarrow{A}\_{Pi} = \frac{\mu\_0}{4\pi} \int\_{\ell\_i^{\ell\_i}} I\_{Li} \frac{e^{-\gamma r\_i}}{r\_i} d\overrightarrow{\ell}\_i \tag{11}
$$

where: *dℓ* ! *<sup>i</sup>* is the length vector element of segment *i*, which defines the direction of *A* ! *Pi*.

In the frequency domain, the associated nonconservative electric field *E* ! *NCi* is given by [27]:

*Recent Topics in Electromagnetic Compatibility*

$$\overrightarrow{E}\_{\rm NCi} = -\text{jo}\overrightarrow{A}\_{\rm Pi} = -\text{jo}\frac{\mu\_0}{4\pi} \int\_{\mathcal{E}\_i} I\_{Li} \frac{e^{-\gamma r\_i}}{r\_i} d\vec{\ell} \,\tag{12}$$

Similarly, the component corresponding to the image segments *E* ! *NCi*�image is given by [27]:

$$\overrightarrow{E}\_{\text{NCi-image}} = -\text{jo}\overrightarrow{A}\_{\text{Pi-image}} = -\text{jo}\frac{\mu\_0}{4\pi} \int\_{\ell\_i'} I\_{Li} \frac{e^{-\gamma r\_{i-\text{image}}}}{r\_{i-\text{image}}} d\overrightarrow{\ell} \,\textsubscript{i-\text{image}} \tag{13}$$

In this case, the image method is applied as shown in [21], where the reflection coefficient is equal to the unit, that is, Γ*<sup>r</sup>* ¼ 1.

Finally, for the total nonconservative component of the electric field *E* ! *NC*�Total, Eq. (14) is obtained.

$$
\overrightarrow{E}\_{\text{NC}-\text{Total}} = \sum\_{i=1}^{N} \left( \overrightarrow{E}\_{\text{NC}i} + \overrightarrow{E}\_{\text{NC}i-\text{image}} \right) \tag{14}
$$

#### **A.1.3 Total electric field (***E* ! **Total)**

The total electric field *E* ! Total is determined by the vector sum of Eqs. (10) and (14):

$$
\overrightarrow{E}\_{\text{Total}} = \sum\_{i=1}^{N} \left( \overrightarrow{E}\_{\text{C-Total}} + \overrightarrow{E}\_{\text{NC-Total}} \right) \tag{15}
$$

#### **A.2 Potential difference, voltage drop, and step voltage**

In several cases, potential difference, voltage drop, and step voltage are more practical parameters. However, these definitions depend on the electric field's nature. For instance, in studying personal protection or electromagnetic compatibility, these parameters may be more interesting and applicable. Here follows a brief definition of each one of them.

### **A.2.1 Potential difference (Δ***u***) – path independent**

The potential difference corresponds to the line integral of *E* ! *<sup>C</sup>*�Total, which depends only on the start (*p*1) and end (*p*2) points, not depending on the integration path. Thus, it is defined according to Eq. (16).

$$
\Delta \mathfrak{u} = -\int\_{p\_1}^{p\_2} \overrightarrow{E}\_{C-\text{Total}} \cdot d\overrightarrow{\ell} = \mathfrak{u}\_2 - \mathfrak{u}\_1 \tag{16}
$$
