Section 1
Why Footing Resistance Controls Backflashover
This article is Part 12 of the APS EMTP® overhead-line series and Part Two of the backflashover sub-series. Part One introduced the basic BFR (backflashover rate) equation and showed that footing resistance controls the tower voltage rise. This page explains why the high-current impulse resistance \(R_i\) used in lightning studies may be lower than the measured low-current resistance \(R_0\).
In a backflashover study the tower footing resistance is one of the most important quantities. When lightning hits the tower or shield wire, part of the current flows down the tower and into the earth through the footing, raising the base voltage:
\[ V_F = I_i\,R_i \]
Table 1 — Notation for the footing voltage rise equation.
| Symbol | Meaning |
|---|
| \(V_F\) | Footing voltage rise (V, or kV) |
| \(I_i\) | Impulse current into the tower footing (A, or kA) |
| \(R_i\) | Impulse (high-current) footing resistance (\(\Omega\)) |
If \(I_i\) is in kA and \(R_i\) in \(\Omega\), then \(V_F\) is in kV — keep the current and resistance units consistent.
A higher footing resistance gives a higher tower voltage and so a higher stress on the line insulation. The whole chain runs one way:
\[ R_i \uparrow \;\Rightarrow\; V_F \uparrow \;\Rightarrow\; V_{\text{ins}} \uparrow \;\Rightarrow\; \text{BFR} \uparrow \]
So reducing footing resistance is one of the most effective ways to cut the backflashover rate.
Section 2
Measured Resistance vs Impulse Resistance
Table 2 — Resistance / impedance symbols across this series
| Symbol | Meaning |
|---|
| \(R_0\) | Measured low-current footing resistance (this page) |
| \(R_{LF}\) | Low-frequency footing resistance — for this simplified page, the same quantity as the measured \(R_0\) |
| \(R_i\) | High-current impulse resistance in the simplified soil-ionisation model (this page) |
| \(Z_P\) | Impulse impedance, peak electrode voltage over peak injected current (HIFREQ page) |
| IC | Impulse coefficient, \(Z_P/R_{LF}\) (HIFREQ page) |
An ordinary earth-resistance test uses relatively low current, giving the measured footing resistance \(R_0\) (sometimes \(R_n\)). But lightning current is tens to hundreds of kiloamps. During the impulse the soil around the electrode may ionise, changing the grounding behaviour, so the resistance seen by the impulse is \(R_i\):
\[ R_i < R_0 \quad\text{(typically)} \]
High current ionises the soil, making the electrode behave as if it were larger than its physical size — so the impulse resistance is lower than the measured value. Notation note: some references denote the measured low-current value \(R_n\); here \(R_0\) is used consistently for the measured low-current resistance and \(R_i\) for the high-current impulse resistance. In this simplified, concentrated-electrode ionisation model \(R_i\) is typically lower than \(R_0\); do not confuse it with the broader frequency-dependent impulse impedance \(Z_P\) on the HIFREQ frequency-dependent soil page, which also includes high-frequency electrode and soil behaviour and is not always simply lower than the low-frequency resistance.
Ionisation does not excuse poor grounding
A lower \(R_i\) is helpful, but it must not be used to justify a weak footing. A high measured \(R_0\) still indicates a poor grounding system, and even after ionisation the tower voltage may remain high. Soil resistivity also varies with moisture: at the start of a thunderstorm after a dry spell, footing resistance can be high before rain penetrates the soil — so seasonal or worst-case values are often more relevant than a single favourable measurement.
Section 3
Why High Current Reduces Resistance
The electric field in the soil rises with current density, \(E \propto \rho J\). Near the electrode the current density is high, so the field is high. When it exceeds a critical gradient \(E_0\), soil breakdown begins:
\[ E \propto \rho\,J \qquad E_0 \approx 400\ \text{kV/m} \]
Table 3 — Notation for the soil field and ionisation breakdown gradient.
| Symbol | Meaning |
|---|
| \(\rho\) | Soil resistivity |
| \(J\) | Current density in the soil |
| \(E_0\) | Critical soil-ionisation breakdown gradient, ~400 kV/m — the field in the ground, not the air-gap leader-inception gradient used on other pages |
Above \(E_0\), streamers form, soil moisture may evaporate, arcs develop, the resistivity inside the ionised zone falls, and the effective electrode size grows — the ionised zone behaving almost as a perfect conductor.
Section 4
The Hemisphere Picture
At low current, the current leaves a ground rod through its small physical surface. At high current, ionisation expands the effective boundary, so the current behaves as if it leaves a much larger conducting region — approximated as a hemisphere:
\[ \text{small rod} \;\longrightarrow\; \text{larger ionised conducting hemisphere} \]
Concentrated grounds
A concentrated ground is an electrode within about 15 m of the tower base — ground rods, short counterpoises near the tower, or grouped electrodes at the foundation. They act as local footing electrodes and set how much lightning current can enter earth locally at the struck tower.
Section 5
Hemispherical Electrode Resistance
For a hemisphere of radius \(r\) in uniform soil:
\[ R = \frac{\rho}{2\pi r} \qquad\Longrightarrow\qquad r \uparrow \;\Rightarrow\; R \downarrow \]
This is the whole idea: if ionisation increases the effective radius \(r\), the resistance falls. The hemispherical model is an idealisation for a concentrated electrode in uniform soil — not a physical model of every rod array, counterpoise, foundation or complex tower footing; check the assumptions before applying it (distributed electrodes are covered in the counterpoise sections below and in detail on the HIFREQ page).
Section 6
The Low-Current Equivalent Radius
Represent the measured resistance \(R_0\) by an equivalent hemisphere of radius \(r_0\):
\[ R_0 = \frac{\rho}{2\pi r_0} \qquad\Longrightarrow\qquad r_0 = \frac{\rho}{2\pi R_0} \]
\(r_0\) is not the physical rod radius — it is the equivalent radius that reproduces the same measured resistance in the hemispherical model.
Section 7
The Ionisation Threshold Current Ig
Ionisation starts when the field at the equivalent hemisphere reaches \(E_0\). With hemispherical current flow \(J = I/(2\pi r^2)\) and \(E = \rho J\), set \(E = E_0\) at \(r_0\):
\[ E_0 = \frac{\rho\,I_g}{2\pi r_0^2} \;\Rightarrow\; I_g = \frac{2\pi E_0 r_0^2}{\rho} = \frac{E_0\,\rho}{2\pi R_0^2} \]
Table 4 — Notation for the soil-ionisation threshold current estimate.
| Symbol | Meaning | Typical Value / Note |
|---|
| \(I_g\) | Current at which soil ionisation begins | Used in the impulse-resistance approximation |
| \(E_0\) | Critical soil gradient | Approximately \(400\,\text{kV/m}\) |
| \(\rho\) | Soil resistivity | In \(\Omega\cdot\text{m}\) |
| \(R_0\) | Measured low-current footing resistance | Not necessarily equal to the impulse resistance \(R_i\) |
This expression rests on the hemispherical equivalent-electrode assumption — the measured \(R_0\) is represented by an equivalent hemisphere of radius \(r_0\). \(I_g\) is the transition current between low-current and impulse behaviour: for \(I < I_g\), ionisation is insignificant and \(R_i \approx R_0\); for \(I > I_g\), ionisation occurs and \(R_i < R_0\).
Section 9
Worked Example
With \(E_0 = 400\) kV/m, \(R_0 = 40\,\Omega\), \(I = 100\) kA, \(\rho = 800\,\Omega\cdot\text{m}\):
\[ I_g = \frac{E_0\,\rho}{2\pi R_0^2} = 31.8\ \text{kA} \qquad R_i = 40\sqrt{\frac{31.8}{100}} \approx 40 \times 0.564 \approx 22.6\ \Omega \]
The square-root relation is a simplified CIGRE-style approximation, so the value should not be read as exact. The source quotes ~19.7 Ω for this case; the simplified formula gives a result of the same order (~22.6 Ω), the difference coming from the exact empirical expression and the intermediate assumptions. Either way the engineering result is the same: \(R_i \approx 0.5\,R_0\) — here the impulse resistance is about half the measured value.
Use the right value for the right tower
For a high-current struck-tower stroke, \(R_i\) is more realistic than \(R_0\) and using \(R_0\) directly can overestimate the BFR. But the reduction must not be applied blindly — it depends on soil resistivity, measured resistance, stroke magnitude, electrode arrangement, ionisation threshold, moisture and geometry. For low current or adjacent-tower currents, \(R_0\) may still be closer to reality. Soil ionisation is physically real, but the simplified \(R_i(I)\) relation is a screening approximation: it is quasi-static and ignores the finite ionisation/de-ionisation time, so the minimum resistance can occur after the current peak. The reduction should therefore not be used as an automatic favourable correction in detailed backflashover studies unless its time dependence and assumptions are justified (see the HIFREQ soil page).
Section 10
Counterpoises
A counterpoise is a horizontal conductor buried (typically ~1 m deep) and tied to the tower base — a horizontal grounding electrode that lowers footing impedance and improves lightning performance.
Why “counterpoise”?
The name came from an early belief that these conductors worked mainly by capacitive coupling to the phase conductors. Later studies showed that coupling is small (typically 3–10%); the real benefit is simply reduced grounding impedance.
Section 11
A Counterpoise Is a Travelling-Wave Element
During the first moments of a surge a counterpoise is not a lumped resistance — it behaves like a transmission line buried in soil. When the tower surge reaches it, waves travel outward at about \(v_c \approx c/3\), with one-way travel time \(T_c = \ell_c/v_c\). The far end is not “seen” immediately:
\[ v_c \approx \tfrac{1}{3}c \]
\[ T_c = \frac{\ell_c}{v_c} \]
\[ t = 0 \;\Rightarrow\; Z_{\text{cp}} \approx Z_c \]
\[ t \approx 2T_c \;\Rightarrow\; Z_{\text{cp}} \approx R_e \]
Table 5 — Counterpoise travelling-wave parameters.
| Symbol | Meaning | Typical Value / Note |
|---|
| \(v_c\) | Propagation velocity along the buried counterpoise | Approximately \(c/3\) |
| \(T_c\) | One-way travel time along the counterpoise | \(T_c = \ell_c / v_c\) |
| \(\ell_c\) | Counterpoise length | Length of one buried conductor branch |
| \(Z_{\text{cp}}\) | Apparent counterpoise impedance during the transient | Changes with time |
| \(Z_c\) | Initial surge impedance of the counterpoise | Typically \(120\text{–}220\,\Omega\), commonly about \(150\,\Omega\) |
| \(R_e\) | Total leakage resistance of the counterpoise | Final value approached after the travelling-wave transient |
At the first instant, the counterpoise is seen mainly as a surge impedance \(Z_c\). After approximately \(2T_c\), the response approaches the total leakage resistance \(R_e\).
So initially the counterpoise looks like a fairly high surge impedance (much higher than its eventual leakage resistance); only after the wave travels to the end and reflects (\(\approx 2T_c\)) does the impedance fall toward \(R_e\). Bewley’s equivalent circuit models this with an initial \(Z_c\), a final \(R_e\) and an inductance chosen so the transition is ~95% complete in \(2T_c\). This delay matters because a lightning front may last only a few microseconds — during the front, a counterpoise can be less effective than its low-frequency resistance suggests. A lower measured \(R_0\) is useful, but it is not the whole lightning-performance story: beyond the effective length, extra counterpoise gives little improvement in peak impulse voltage.
Section 12
Several Shorter Counterpoises Beat One Long One
A total length of 300 m can be one 300 m run, two 150 m, three 100 m or four 75 m. The total is the same, but splitting it into shorter branches gives better lightning performance:
- Lower initial surge impedance: \(N\) parallel branches give \(Z_{c,\text{eq}} \approx Z_c/N\).
- Shorter travel time: each branch has a smaller \(T_c = \ell_c/v_c\), so the leakage resistance is reached sooner.
- Faster impedance reduction: the transient moves from surge impedance to leakage resistance more quickly.
A note of caution on data
Many historical counterpoise tests used currents below ~100 A, so they miss high-current soil ionisation. A suggested refinement is to split the counterpoise into ~30 m sections and apply concentrated-electrode impulse equations to each (with travel time between sections) — but this is not fully implemented in standard BFR methods. Counterpoise impulse behaviour is more uncertain than concentrated rod behaviour.
Section 13
Ground Rods in Practice
Tower grounding can take many forms — butt-wrap on a wood pole, a base plate, a concrete augered footing, a rebar cage tied to the leg, extra copper outside the footing, driven rods, or counterpoises. Ground rods (vertical electrodes) suit lower-resistivity soils or where deeper layers are less resistive:
- Driven depth typically 2–6 m in uniform soil — deeper to reach a lower-resistivity layer.
- Multiple rods reduce resistance, but not linearly: overlapping dissipation zones mean diminishing returns, so usually 3–5 rods with ≥ 3 m spacing.
- Rod diameter barely affects grounding resistance — length, soil resistivity, spacing, number and access to low-resistivity layers matter far more.
- Under high current the ionised diameter can be several metres, so spacing of about 5 m gives better effectiveness than the usual low-current rule.
Section 14
Counterpoises in Practice
Counterpoises suit high-resistivity soils, where rods cannot reach a low-resistivity layer and a long horizontal run contacts more soil volume:
- Individual length usually limited to about 50 m in uniform soil — longer runs give diminishing returns, especially for impulse.
- Parallel branches spaced about 10 m, normally limited to about 3 per side to avoid mutual interference and waste.
- Buried ~1 m deep (below ploughing); use copperweld or similar — aluminium is not recommended (corrodes/disappears in soil). Mechanical durability and corrosion resistance matter, not just resistance.
Seasonal soil & the constant-resistivity assumption
Soil resistivity rises in dry weather, so footing resistance is highest at the start of a storm before rain penetrates — \(\text{dry soil} \Rightarrow R_0 \uparrow \Rightarrow R_i \uparrow \Rightarrow \text{BFR}\uparrow\), and flashovers can be more probable early in a storm. The rod/counterpoise equations assume uniform resistivity, so they are for planning and preliminary design; crews are given a target resistance and rod/counterpoise limits, and the final layout is adjusted from field measurement.
Section 15
Ground Rods vs Counterpoises
Table 6 — Practical comparison of the two grounding types.
| Item | Ground Rods | Counterpoises |
|---|
| Orientation | Vertical | Horizontal |
| Best for | Lower-resistivity soils / lower layers | High-resistivity soils |
| Typical length | 2–6 m (deeper if needed) | ~50 m per branch |
| Main limitation | Mutual effects between rods | Travelling-wave delay & mutual effects |
| Impulse behaviour | Simplified ionisation equations available | More uncertain under high current |
| Spacing | ≥ 3 m (~5 m better with ionisation) | ~10 m between branches |
| Practical number | Often 3–5 rods | ~3 branches per side |
| Watch out for | Rod diameter has little electrical effect | One long branch < several shorter ones |
Section 16
How It Feeds BFR, and the Memory Map
The critical current depends strongly on footing resistance: with \(V_{\text{ins}} \approx (1-C)I_i R_i + (\text{tower surge})\), a lower \(R_i\) means a given current produces less stress, so a higher current is needed to flash over — \(R_i \downarrow \Rightarrow I_c \uparrow\). Since \(\text{BFR} = 0.6\,N_L\,P(I \ge I_c)\), \(I_c \uparrow \Rightarrow \text{BFR}\downarrow\). That is why grounding is central to lightning design. The simplified \(R_i\) prepared here is used inside the CIGRE backflashover method, where the critical current, front time and footing resistance are solved iteratively — this page only prepares \(R_i\) and does not repeat that workflow.
Equation Summary
Footing voltage
\(\displaystyle V_F = I_i\,R_i\)
Hemispherical electrode
\(\displaystyle R = \frac{\rho}{2\pi r}\)
Equivalent radius
\(\displaystyle r_0 = \frac{\rho}{2\pi R_0}\)
Soil field
\(\displaystyle E = \rho J = \frac{\rho I}{2\pi r^2}\)
Ionisation current
\(\displaystyle I_g = \frac{E_0\,\rho}{2\pi R_0^2}\)
Impulse resistance
\(\displaystyle R_i = R_0\sqrt{\frac{I_g}{I}}\ (I > I_g)\)
Counterpoise travel time
\(\displaystyle T_c = \frac{\ell_c}{v_c},\ \ v_c \approx \tfrac{c}{3}\)
Backflashover rate
\(\displaystyle \text{BFR} = 0.6\,N_L\,P(I \ge I_c)\)
When the simplified method is acceptable
The simplified impulse-resistance method is acceptable for first estimates, teaching, sensitivity checks and simplified backflashover-rate calculations. It is not sufficient for complex tower footings, substation grids, multilayer soil, nearby buried metallic systems, transferred-voltage assessment or safety-critical conclusions — those need the physical electrode and frequency-dependent soil modelling on the HIFREQ page.
Key messages
- The measured \(R_0\) is not always the right value: at high current, soil ionisation gives an impulse resistance \(R_i < R_0\) (often \(\approx 0.5\,R_0\)).
- Concentrated grounds (rods, within ~15 m) follow a hemispherical ionisation model: \(R = \rho/2\pi r\), \(I_g = E_0\rho/2\pi R_0^2\), \(R_i = R_0\sqrt{I_g/I}\).
- Counterpoises are travelling-wave elements: an initial surge impedance \(Z_c\) (~150 Ω) falling to the leakage resistance \(R_e\) only after \(\approx 2T_c\).
- Several shorter counterpoises beat one long one (lower \(Z_c/N\), shorter \(T_c\), faster settling).
- Rods suit low-resistivity soil; counterpoises suit high-resistivity soil; mutual effects, ionised spacing (~5 m), seasonal dryness and field measurement all matter.
- The design goal: \(R_i \downarrow \Rightarrow V_{\text{tower}} \downarrow \Rightarrow I_c \uparrow \Rightarrow \text{BFR}\downarrow\).
References
References
The standards, technical brochures, key papers and reference works behind this page.
IEEE Std 1243-1997, IEEE Guide for Improving the Lightning Performance of Transmission Lines. New York, NY, USA: IEEE, 1997.
CIGRE Working Group 33.01, Guide to Procedures for Estimating the Lightning Performance of Transmission Lines, Technical Brochure 63. Paris, France: CIGRE, 1991.
CIGRE Working Group C4.23, Procedures for Estimating the Lightning Performance of Transmission Lines – New Aspects, Technical Brochure 839. Paris, France: CIGRE, 2021.
CIGRE Working Group C4.33, Impact of Soil-Parameter Frequency Dependence on the Response of Grounding Electrodes and on the Lightning Performance of Electrical Systems, Technical Brochure 781. Paris, France: CIGRE, 2019.
S. Visacro, “The use of the impulse impedance as a concise representation of grounding electrodes in lightning protection applications,” IEEE Transactions on Electromagnetic Compatibility, vol. 60, no. 5, pp. 1602–1605, Oct. 2018.
A. R. Hileman, Insulation Coordination for Power Systems. New York, NY, USA: Marcel Dekker, 1999.
J. A. Martinez-Velasco, Ed., Power System Transients: Parameter Determination. Boca Raton, FL, USA: CRC Press, 2010.
