Lumped-Parameter Grounding Model

Based on experimental results, Bewley suggested that the lightning behavior of coun-terpoise electrodes could be represented by the simple equivalent circuit presented in Figure 2.40 [91], where

R

• c represents the counterpoise leakage resistance.

R

• _s is a resistor selected so that the high-frequency impedance of the circuit cor-responds to the surge impedance of the counterpoise (Zc).

L

• c is the inductor responsible for the transition from the surge impedance to the low-frequency impedance; its value is dependent on the length of the counterpoise.

Chisholm and Janischewskyj suggest from their experimental results that the apparent impedance of the ground plane as a function of time can be obtained as follows [81]:

⎛ ⎞

=60h ⎜⎝ > =^t h⎟⎠

Z t t

ct c (2.109)

where

h is the height of the tower

c is the velocity of light (=300 m/μs) t is time

tt is the surge travel time of the tower

It is possible to represent the initial surge response of a perfectly grounded tower as an inductance that is a function of tower height. The value of this inductance can be approxi-mated by means of the following expression:

≈ ⎛ ⎞⎜ ⎟⎝ ⎠

f t

t

60 ln t

L t

t (2.110)

where tf is the front time.

R_s

R_s = Z_c – R_c

L_c = 2ℓ(Z_c – R_c) R_c

L_c R_c = ρ

2πℓ ln4ℓa –1

Z_c = 1

ln4ℓa –1 L=

C 2π

μ_o ε

FIGURE 2.40

Equivalent circuit of a counterpoise.

This equivalent inductance is valid for the normal case, in which the front time tf is much greater than the tower travel time tt. Consequently, the modifi ed equivalent circuit of the counterpoise is that shown in Figure 2.41. Longer front times give higher values of average inductance, but the voltage rise from the inductance at the crest current is actually lower.

2.5.4.3 Discussion

A comparison between the performance of three frequency-dependent models for a vertically driven ground rod was presented in [107] a lumped-parameter circuit (see Figure 2.42), a distributed-parameter circuit (see Figure 2.39), and a more rigorous approach based on antenna theory. The main conclusions of the study can be summarized as follows:

The application of the lumped-parameter circuit model is limited to cases where the

•

length of the rod is less than one-tenth the wavelength in earth, which limits the frequency range of the validity of this model to low frequencies. This approach can be used for a preliminary analysis, but keeping in mind that it greatly overes-timates the ground rod impedance at high frequencies.

The approximate distributed-parameter circuit reduces the overestimation of the

•

ground rod impedance at high frequencies in comparison with the RLC circuit.

The best fi t to ground rod impedance was achieved by means of a model based on

•

antenna theory, with a nonuniform distributed-parameter model, whose param-eters can be deduced by curve matching.

The following example analyzes the behavior and validity of different models for a ground rod.

Example 2.4

Consider a ground rod with a length of 9 m and a radius of 10 mm, vertically driven into a homogeneous soil whose resistivity and relative permittivity are respectively 100 Ω m and 10. The goal of this example is to analyze the transient response and the frequency-domain response of this ground rod considering three different modeling approaches: a constant resis-tance, a high-frequency lumped-parameter circuit (Figure 2.42) and a high-frequency distrib-uted-parameter circuit (Figure 2.39).

Table 2.9 shows the parameters calculated for each model. To obtain the high-frequency distrib-uted-parameter representation, the circuit has been built with sections that correspond to 0.5 m of grounding rod.

Figure 2.43 depicts the frequency response of the two high-frequency models. The plots show the relationship of the complex impedance modulus, |Z(jω)|, and the low-frequency resis-tance value, R0, as well as the corresponding phase angles, as a function of the input current frequency.

It can be seen that the impedance is frequency independent and equal to the low- frequency resistance up to a frequency of about 100 kHz. For higher frequencies, the lumped-parameter model exhibits inductive behavior whereas the distributed-parameter model becomes capacitive

L_av

R_c

R_s L_c

FIGURE 2.41

Modifi ed equivalent circuit of a counterpoise.

for frequencies close to 1 MHz. The frequency until which the electrode shows a resistive behav-ior increases as the dimension of the electrode decreases and the soil resistivity increases [108]. A resistive–capacitive behavior can be advantageous since the whole impedance is equal or smaller than the low-frequency resistance. Usually capacitive behavior is typical for electrodes with small dimensions buried in highly resistive soils; otherwise, the grounding electrode behavior is mostly inductive [109].

Figure 2.44 shows the transient response of the ground rod for two input current pulses with different front time.

For a slow front-time current pulse (left plots in Figure 2.44), the transient voltage response

•

at the feeding point is very similar with the two high-frequency models and very close to the response that would result with a constant resistance model, although the transient impedance may reach very high values with both high-frequency models. Since such cur-rent pulse does not have signifi cant frequency content above 100 kHz, except during the front time, the voltage response is not substantially infl uenced by the inductive part of the harmonic impedance (Figure 2.43), although the voltage response precedes in both cases the current pulse during the current front time, as in a typical inductive behavior. The tran-sient impedance starts from a high value, but it settles down to the low-frequency resistance value during the rise of the current.

The transient voltages for a faster front-time current pulse (right plots in Figure 2.44) show

•

some differences with respect to those obtained with the previous current pulse. Since the new current pulse has signifi cant frequency content above 100 kHz, the new response exhibits very different behavior, as predicted from the frequency responses. The transient impedance in both models starts again from high values and settles down to a value close to the low-frequency resistance value after the current peak. The voltage response in both models is oscillatory and reaches its peak value before the current pulse peak, as for a typi-cal inductive behavior. However, the peak value of the transient voltage is very different with the two high-frequency models. The transient voltage obtained with the lumped-pa-rameter RLC model reaches a large peak value, while this value is a little bit larger than the value obtained with a constant-resistance model if the distributed-parameter model is used.

That is, the RLC circuit overestimates the ground rod impedance at high frequencies, and the approximate distributed-parameter circuit reduces the overestimation, as mentioned in the previous discussion (Section 2.5.4.3).

When the ground electrode exhibits inductive behavior at high frequencies and the current pulse has enough high frequency content, it can be concluded that voltage peak will occur

R

L

C

R = ρ

2πℓ ln 4ℓa –1

C = 2πεℓ a ln 4ℓ –1 L =

a ln 4ℓ –1 2π

μ₀ℓ

FIGURE 2.42

High-frequency lumped equivalent circuit of a ground electrode.

1.E+01

Example 2.4: Frequency response of the ground rod. (a) High-frequency lumped-parameter model, (b) high-frequency distributed-parameter model.

TABLE 2.9

Example 2.4: Parameters of Grounding Models

Grounding Model Parameters

Note: Rod length = 9 m; rod radius = 10 mm; soil resistivity = 100 Ω m; relative soil permit-tivity = 10; n = 18.

during the rise of the current pulse; however, after few microseconds, the transient process is fi nished and the transient impedance settles down to the low-frequency resistance. An impor-tant result of this analysis is that the use of the impulse impedance, as defi ned in Equation 2.107 can mislead to wrong conclusions; that is, although the impulse impedance can exhibit very high values, the transient voltage peak of an accurate high-frequency model for a ground rod might not be much higher than the voltage peak obtained with a constant-resistance model.

Interestingly, the use of the conventional impedance, as defi ned in Equation 2.108, may be more appropriate for this application. The values of this impedance for the cases analyzed in this example were as follows:

Slow front-time current pulse:

•

Constant resistance model Z = 12.71 Ω

High-frequency lumped-parameter model Z = 12.68 Ω High-frequency distributed-parameter model Z = 12.67 Ω Fast front-time current pulse:

•

Constant resistance model Z = 12.71 Ω

High-frequency lumped-parameter model Z = 48.82 Ω High-frequency distributed-parameter model Z = 14.17 Ω