Why the Most Common Characterization of a Ground Rod May Not Work for Lightning
In 1997, an experiment at the Camp Blanding center for lightning testing  challenged the predominant view that ground rods are essentially resistive. What that experiment found was that the waveshapes of lightning currents in a building grounding system and those entering the electrical circuits of the building were considerably different. That was at odds with IEC 61312-1:1995  assertions that they should be the same. The conclusion was that, for lightning, the ground rod had an impedance with a reactive component in addition to the resistive one.
So how do we take into account the impedance effects for lightning? Well, it turns out not to be so simple. Professor Leonid Grcev, who with his students has conducted extensive studies of grounds, has found that a simple modeling of a ground rod as an R-L-C circuit doesn’t give correct results, due to surge propagation effects which cause a deviation from the low frequency behavior during the fast-transient period. So the challenge is to determine what this deviation is.
Considering normal grounds (those not chemically treated or otherwise enhanced), Grcev has shown that they can be characterized in terms of effective length and impulse coefficient (IC) . The IC is the ratio of peak voltage across an actual ground rod to the peak voltage across a purely resistive ground rod in response to a surge. It shows how the impedance of the ground rod affects the expected peak voltage due to a surge relative to what it would have been if the ground rod were purely resistive.
The first thing to consider is the ground rod effective length leff, which is the maximum length of the ground electrode for which the impulse coefficient is equal to one. leff will be used later in the discussion of the IC (which is what we really want).
To calculate leff, Grcev  has developed the relation:
ρ = soil resistivity in ohm-m and T1 is the zero-to-peak rise time of the lightning current pulse. MIL‑HDBK‑419 Table 2.3  shows a range for average soil resistivity of 1 to 500 ohm-m. CIGRE TB549 Table 3.5  shows a range of front durations of 1.1 µsec for the average subsequent stroke to 18 µsec for the maximum first stroke. Considering those values, the ρT1 product could reasonably range from 1 to over 1000 ohm-m-µsec. We can use those values in equations (2) and (3) to make a plot of leff vs. ρT1, as shown in Figure 1. Both slower rise-time and higher soil resistivity lead to a longer effective ground-rod length.
If the length s of the ground rod is less than leff (see Figure 1), the ground rod is primarily resistive, with some capacitive effect. If the length of the ground rod is greater than leff, the ground rod will have inductive effects. So which effect do we have, and what is the consequence of that effect? Well, that’s what the IC determines. Grcev  has proposed the relation:
where A = Z/R is the impulse coefficient, Z is the effective impedance, R is the ground rod resistance, α is calculated from equation (2), and β is calculated from equation (3).
For A > 1, the ground rod has an effective series inductance in addition to its resistance. In this case, the peak voltage will be A times bigger than it would have been if the ground rod were purely resistive.
For A < 1, the ground rod has an effective parallel capacitance in addition to its resistance. In this case, the peak voltage will be A times lower than it would have been if the ground rod were purely resistive.
From equation (4) the effect of the ground rod reactance can be calculated. As an illustration, take the four cases of ρT1, = 100, 300, 1000, and 10,000, and use equation (4) to plot the impulse coefficient A vs. length of the rod. Ground rods with a low ρT1 product have a high impulse coefficient, whereas ground rods with a high ρT1 product have a low impulse coefficient, as shown in Figure 2.
Figure 3 is a replot of Figure 2 for ground rods of a length normally used (≤ 10 m).
For ground rods ≤ 10 m, the low value of the impulse coefficient means that the peak voltage across the ground rod will be less than would be calculated for a purely resistive ground rod. For example, for a common 2 m rod, the ratio of peak voltage to the peak voltage across a purely resistive ground rod is in the range of 0.2 to 0.4, depending on the ρT1 product. The voltage across the ground rod as a surge decays is determined primarily by the resistance of the ground rod. So as the surge decays, the effect of the ground rod reactance dies away (remember that the impulse coefficient is relevant only during the rise-time period).
Current Flowing in the Ground Rod
The peak voltage developed across the ground rod is given by:
where Irod is the peak current captured by the ground rod, and Z is the ground rod impedance.
To calculate Irod we need to calculate the fraction of the lightning current Imax captured by the ground rod. IEEE Std 142  shows that 99% of the current flowing in the ground rod is captured in a volume having a radius of twice a ground rod length, s. Figure 4 illustrates this situation, where d is the distance from the lightning strike point to the edge of a cylinder representing the ground rod outer effective extent.
The angle θ subtended by the ground rod is given by:
Note that the arcsin is not defined for arguments greater than 1, so there are two cases for equation (6): Case 1 where d ranges from 2s to infinity, and case 2 where d ranges from 2s to 0.
For case 1, if the arcsin is in degrees, then the fraction f1 of the lightning current Imax captured by the ground rod is:
For case 2, if the fraction f2 of the lightning current Imax captured by the ground rod is:
Combining equations (7) and (8), Irod = Imax (f1 + f2), which is:
Remember that in calculating Irod, the first term in equation (9) is only valid for d greater than 2s, and the second term is only valid for d less than 2s.
The peak voltage is calculated from equation (5). The effective impedance Z of the ground rod to be used in equation (5) can be calculated from Dwight’s  equation multiplied by A:
where a is the radius of the ground rod.
Substituting equations (9) and (10) in equation (5):
As an example of the calculation of Vpeak, consider a 12 kA 4.5/77 subsequent surge from TB549  impinging on a 10 m rod 5/8 inches in diameter in the soil of 50 ohm-cm, 200 ohm-cm 600 ohm-cm, and 3000 ohm-cm.
For these cases, Figure 5 shows how Vpeak changes due to a decrease in ground-rod current capture with increasing distance.
Applicability of the Peak Voltage Calculation
Now a word about the applicability of the foregoing analysis. In the region near the lightning strike point, the ground resistivity ρ is highly variable. In particular, soil breakdown can happen when the electric field overcomes the soil ionization gradient . Soil ionization occurs when the electric fields at the ground electrode surface become greater than the ionization threshold of approximately 300 kV/m . In this case, in the region surrounding the current striking point, local transverse discharges start from the lightning strike point and stop at the points where the electric field drops below the critical breakdown strength. An illustration of this point is shown in Figure 6.
The literature on lightning shows that the streaks in Figure 6 are places where the ground is ionized. A circle of radius r0 can be put around this area. The size of r0 is determined by both the magnitude of the lightning current and ρ. In Figure 6, r0 appears to be about 6 m, but that may or may not be typical. In any case, to avoid the area where ρ is highly variable, d should generally exceed 2r0.
With the foregoing discussions in mind, different lightning waveforms, different ρ, and different ground rod lengths will result in different peak voltages from those shown in Figure 5.
The usual assumption that ground rods are purely resistive is actually not what is observed in the case of lightning. Particularly for the relatively short ground rods commonly used, during the rise-time period the ground rods look like an impedance with a significant capacitive component. The result is that for these commonly used ground rods, the peak voltage due to a lightning strike is generally significantly lower than would be the case for a purely resistive ground rod. Whether the peak voltage is higher or lower than for a purely resistive ground rod depends on a number of variables, including the surge waveform, the ground resistivity, the length of the ground rod, and the distance the observer is from the lightning strike point. The peak voltage across the ground rod can be calculated, based on estimates of these variables.
- V. A. Rakov et al, “Direct Lightning Strikes to the Lightning Protective System of a Residential Building: Triggered-lightning Experiments,” IEEE Transactions on Power Delivery, vol. 17, no. 2 (April 2002).
- IEC Standard 61312-1:1995, Protection Against Lightning Electromagnetic Impulse- Part 1: General Principles.
- L. Grcev, “Impulse Efficiency of Ground Rods,” IEEE Transactions on Power Delivery, vol. 24, no. 1 (January 2009), 441-451.
- H. B. Dwight, “Calculation of Resistances to Ground,” Transactions of the American Institute of Electrical Engineers, vol. 55 (1936), 1319-1328.
- IEEE Std 142-1991, IEEE Recommended Practice for Grounding of Industrial and Commercial Power Systems.
- MIL‑HDBK‑419, Military Handbook Grounding, Bonding, and Shielding for Electronic Equipments and Facilities, Volume 1 of 2 Volumes on Basic Theory, January 1982.
- Cigre TB549, Lightning Parameters for Engineering Applications, August 2013.
- A. Geri, “Behaviour of Grounding Systems Excited by High Impulse Currents: The Model and Its Validation,” IEEE Transactions on Power Delivery, vol. 14, no. 3, July 1999.
- L. Grcev and V. Arnautovski, Proceedings of 24th International Conference on Lightning Protection (ICLP’98), Birmingham, UK, 14-18 September 1998, vol. 1, pp. 524-529