**6. Verification of the method**

In this part, our model has been verified through comparison with experimental data from other published papers; the validation of our model will also be discussed.

### **6.1. Verification of the method**

To verify the method proposed in this work, some cases solved by other authors are studied.

In the first case, from [42], which gives some grounding impedance measurement results, the grid used for measurement is a 16-mesh grid of 100 m × 100 m. The earth is modeled by a multilayered earth model. The earth model is given by (a) *ρ*<sup>1</sup> = 25Ω m, *ρ*<sup>2</sup> = 120Ω m, and thickness of the upper earth is 5 m; (b) *ρ*<sup>1</sup> = 120Ω m, *ρ*<sup>2</sup> = 25Ω m, *ρ*<sup>3</sup> = 250Ω m, thickness of the first and second layers of earth are 5 m and 9 m, respectively. The radius of the grid conductors is 0.5 cm, and they are buried at 0.5 m depth in the earth. Here, the conductivity of the copper conductors is *<sup>σ</sup>Cu* = 5.8 × 107 S/m, and the permittivity of earth is set to *ε*<sup>1</sup> = 5. The results can be seen in Table 1.

In the second case, which is from [41], a typical grounding grid configuration can be seen in Fig. 2, which was made of round copper conductors with 50 mm<sup>2</sup> cross section. The grounding grid was buried at 0.5 m depth in two-layer horizontal earth, whose resistivity

<sup>406</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 407


**Table 1.** Comparison with published measurement result: Frequency is 80Hz

14 Computational and Numerical Simulationse

voltage values is defined as [40]:

defined by the following expression [41]:

**6. Verification of the method**

**6.1. Verification of the method**

algorithm, is used, i.e.,

As the frequency response *U*(*f*) is represented by a discrete set of values the integral (41) cannot be evaluated analytically and the discrete Fourier transform, in this case the IFFT

Implementation of this algorithm inevitably causes an error due to discretization and truncation of essentially unlimited frequency spectrum. The discrete set of the time domain

where *n* = 0, . . . , *N*, *F* is the highest frequency taken into account, *N* is the total number of

Finally, the impulse impedance, an also essential parameter in grounding system design, is

*Zc* <sup>=</sup> *<sup>U</sup>*

where *U* is the voltage maximum at the discharge point and *I* is the injected current

In this part, our model has been verified through comparison with experimental data from

To verify the method proposed in this work, some cases solved by other authors are studied. In the first case, from [42], which gives some grounding impedance measurement results, the grid used for measurement is a 16-mesh grid of 100 m × 100 m. The earth is modeled by a multilayered earth model. The earth model is given by (a) *ρ*<sup>1</sup> = 25Ω m, *ρ*<sup>2</sup> = 120Ω m, and thickness of the upper earth is 5 m; (b) *ρ*<sup>1</sup> = 120Ω m, *ρ*<sup>2</sup> = 25Ω m, *ρ*<sup>3</sup> = 250Ω m, thickness of the first and second layers of earth are 5 m and 9 m, respectively. The radius of the grid conductors is 0.5 cm, and they are buried at 0.5 m depth in the earth. Here, the conductivity of the copper conductors is *<sup>σ</sup>Cu* = 5.8 × 107 S/m, and the permittivity of earth

In the second case, which is from [41], a typical grounding grid configuration can be seen in Fig. 2, which was made of round copper conductors with 50 mm<sup>2</sup> cross section. The grounding grid was buried at 0.5 m depth in two-layer horizontal earth, whose resistivity

other published papers; the validation of our model will also be discussed.

*N* ∑ *k*=0

*U*(*t*) = *F*

frequency samples, ∆*f* is sampling interval and ∆*t* is the time step.

magnitude at the time instant when *U* has been reached.

is set to *ε*<sup>1</sup> = 5. The results can be seen in Table 1.

*U*(*t*) = *IFFT*(*U*(*f*)) (41)

*<sup>U</sup>*(*k*∆*f*)*ejk*∆*f n*∆*<sup>t</sup>* (42)

*<sup>I</sup>* (43)

ratio for the upper and the lower soil layers is *ρ*1/*ρ*<sup>2</sup> = 50/20, the upper layer thickness being *H* = 0.6 m. The inject lightning current parameter was set to *T*<sup>1</sup> = 3.5*µ* s, *T*<sup>2</sup> = 73*µ* s and *Im* = 12.1 A, the feed point is at the corner of the grid. For our model, the permittivities of the two layer earth model were set to *ε*<sup>1</sup> = 30*ε*<sup>0</sup> and *ε*<sup>2</sup> = 20*ε*0. The transient SEP can be seen in Fig. 3, which ultimately agreed with the measured curve in Fig. 4 (a) in [41]; meanwhile, the impulse grounding impedance was 2.12 Ω as given by [41], and it is 2.08Ω for our model.

**Figure 2.** Typical grounding grid configuration

### **6.2. Validation of our method**

The maximum frequency of applicability of the method is limited by the quasi-stationary approximation of the electromagnetic fields, which means the propagation effect of the electromagnetic field around the grounding system can be neglected, so

$$e^{-\gamma\_{\ell}R} \approx 0\tag{44}$$

where *γ*<sup>2</sup> *<sup>e</sup>* = *jωµ* (*jωε<sup>e</sup>* + *σe*),*e* = 1, . . . , *Ne*.

For most of the usual electrodes, this may be applied up to some hundreds of kHz.

The quasi-static complex image in this case has two terms, the *αn* and *βn* can be seen below

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

409

The transient SEP at the injection point is given in Fig. 5. From this figure, we can see that the maximum value of the transient SEP at the injection point disagrees with that of the lightning current, the maximum value of the transient SEP at injection point occurs at 12*µ* s,

The distribution of absolute values of grounding impedance dependance |*Z*(*jω*)| on frequency can be seen in Fig. 6. This figure shows that |*Z*(*jω*)| is independent of the frequency below 100 kHz and equal to the low frequency grounding impedance, which

To further discuss the electromagnetic field characteristics along the surface above the grounding grid, the distribution of the electromagnetic field along the surface with three different chosen frequencies (17 kHz, 250 kHz and 800 kHz) have been given in Figs. 7–15. Among these, Figs. 7–9 show the distribution of SEP *ϕ* along the surface, Figs. 10–12 show the distribution of the x-component of the EFI, *Ex*, along the surface, and Figs. 13–15 show

From Figs. 7–9, we know that the ground SEP rise is dependent on the magnitude of the injecting current, the ground SEP rise at 17 kHz is generated by injecting current with (-4.077,-10.964) kA, the ground SEP rise at 250 kHz is generated by injecting current with (-0.289,+ 0.143) kA, and the ground SEP rise at 800 kHz is generated by injecting current

and the maximum value of the lightning current occurs at 10*µ* s.

the distribution of the x-component of the MFI, *Bx*, along the surface.

*αn βn* 1 (0.294, −62.137) ( 9.937, − 0.959) 2 (0.086, +53.354) (22.453, −11.859)

Table 3.

**Table 3.** Quasi-static complex image coefficients

**Figure 4.** Typical grounding system

agrees with the viewpoint of [43].

**Figure 3.** Transient SEP at injection point
