**3. Case studies and results**

Some case studies were selected to show the response in both frequency and time domains. For the frequency domain analysis, harmonic analysis is performed by injecting a 1*A* current in every frequency. In the time domain, a double exponential function with unit amplitude was injected to simulate a 0*:*1*=*50 μs wave, illustrated in **Figure 6** and given by (3), in *A*. The investigated quantities are the harmonic impedance, GPR, and step voltage along a 1 � m straight line in the þ*x*direction, calculated both by the potential difference Δ*u* and by the voltage drop *Up*<sup>1</sup>�*<sup>p</sup>*2, as defined in the Appendix. Note: the �*z-*direction is the one considered downward, that is, deeper in the soil.

$$i(t) = e^{-2 \times 10^4 \text{ t}} - e^{-10^\circ \text{ t}} \tag{3}$$

#### **3.1 Horizontal electrode**

The first investigation is of a horizontal electrode, 15 m long with a 7 mm radius. The conductor starts at *x* ¼ 2*:*5 m and ends at *x* ¼ 17*:*5 m. A current of 1*A* is injected in the *x* ¼ 2*:*5 m point of the conductor. Two cases are considered: a soil which lowfrequency conductivity *σ*<sup>0</sup> is 1 mS and other which it is 10 mS. In both cases, the Alipio-Visacro soil model is used with mean values [26].

This case is very similar to the one presented by Alipio et al. [27] but has an important difference: The soil models used therein were both with constant parameters and one presented by Portela et al. [28]. Therefore, a difference in their results is expected because the Portela soil model overestimates the soil conductivity in higher frequencies compared to the Alipio-Visacro soil model used here.

The harmonic impedance is shown in **Figure 7**. According to these results, in the low-frequency spectrum, it is mainly resistive, thus being approximately modeled by a resistor. However, up to a few megahertz, its capacitive and inductive

**Figure 6.** *Injected current for time-domain analysis.*

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

**Figure 7.** *Harmonic impedance of a horizontal electrode. (a) Absolute value and (b) phase.*

#### **Figure 8.**

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

natures start to play an important role. These pieces of information can be seen in both module and phase values. This impact is even more pronounced in low conductivity soil.

The harmonic step voltage absolute value is shown in **Figure 8**. In low frequencies, the potential difference Δ*u* coincides with the voltage drop *Up*<sup>1</sup>*<sup>p</sup>*2. Then, after a few kHz, they begin to differ significantly. This difference is accentuated in higher frequencies. Hence, it is important to consider the nonconservative component of the electric field for high-frequency phenomena.

The GPR is greater for low conductivity soils for the time-domain simulation, as shown in **Figure 9**.

The transient step voltage, illustrated in **Figure 10**, shows a difference between the potential difference Δ*u* and the voltage drop *Up*<sup>1</sup>*<sup>p</sup>*2, which is greater in the first

**Figure 9.** *GPR of a horizontal electrode.*

**Figure 10.**

*Transient step voltage above a horizontal electrode comparing the potential difference* **Δ***u and voltage drop Up***<sup>1</sup>***<sup>p</sup>***2***. (a) t = 0.1 us, (b) t = 0.2 us, (c) t = 0.4 us and (d) t = 1.2 us.*

time steps. This difference diminishes when the high-frequency components of the injected current go to zero in later time steps.

#### **3.2 Vertical rod**

Another simple case that was investigated is of a vertical rod, but its harmonic impedance and GPR are very similar to that of a horizontal electrode. Moreover, because the longitudinal current is downward in the *z*-direction, the

nonconservative electric field only has a component in the *z*-direction. Therefore, the potential difference Δ*u* is equal to the voltage drop *Up*<sup>1</sup>�*<sup>p</sup>*2, independently of the frequency. Hence, this case is omitted.
