Section 1
Why Impulse Resistance Matters
In lightning backflashover studies the tower footing resistance is one of the most important parameters. When a stroke terminates on a tower or shield wire, part of the current flows into earth through the footing, and the tower voltage rises roughly as:
\[ V_F = I_i\,R_i \]
Table 1 — Notation for the tower footing-voltage relation under lightning.
| Symbol | Meaning |
|---|
| \(V_F\) | Footing-voltage component |
| \(I_i\) | Current flowing through the footing path |
| \(R_i\) | Impulse (high-current) footing resistance |
| \(R_0\) | Measured low-current / low-frequency footing resistance |
The key point is that the value you measure is not the value the lightning sees. Under a high-current impulse the soil ionises and the effective resistance falls:
\[ R_i \neq R_0, \qquad R_i < R_0 \]
This can strongly affect the calculated critical current \(I_c\) and the backflashover rate, so it must be handled carefully. This is a self-study companion note: it explains the grounding model behind \(R_i\), the impulse resistance used in lightning-performance and backflashover calculations.
What this page teaches
- why \(R_i\) can be lower than the measured \(R_0\);
- how soil ionisation expands the effective electrode size;
- how the hemisphere model leads to the ionisation current \(I_g\);
- why the Weck form is used for practical estimates;
- how ground rods and counterpoises behave differently;
- why mutual resistance and electrode spacing matter;
- how impulse resistance affects the backflashover rate.
Section 2
Low-Current Resistance versus Impulse Resistance
Low-current resistance \(R_0\) is the value measured in field tests — fall-of-potential, clamp-on (where applicable) or low-frequency methods. It represents the grounding system under normal conditions, before any high-current soil breakdown. It is useful, but it does not always represent what a high-magnitude lightning impulse sees.
Impulse resistance \(R_i\) applies during the stroke, when the injected current is tens to hundreds of kiloamps. The soil field can exceed the critical gradient \(E_0\); soil breakdown and ionisation occur around the electrode; the effective electrode grows; and the resistance falls:
\[ I \uparrow \;\Rightarrow\; \text{ionisation zone expands} \;\Rightarrow\; \text{effective electrode size} \uparrow \;\Rightarrow\; R_i \downarrow \]
Using \(R_0\) directly in a backflashover calculation may therefore overestimate the tower voltage and the BFR.
Section 3
The Physical Mechanism of Soil Ionisation
Near the electrode the current density \(J\) is high, so the soil field \(E = \rho J\) is high. When it reaches the critical gradient \(E_0\), soil breakdown begins:
\[ E = \rho J, \qquad \text{breakdown when } E \ge E_0 \]
Table 2 — Notation for the soil field that triggers ionisation.
| Symbol | Meaning |
|---|
| \(E\) | Soil electric field / voltage gradient |
| \(\rho\) | Soil resistivity |
| \(J\) | Current density |
The process then cascades — streamers form, moisture evaporates, arcs develop and the local resistivity drops, so the effective electrode grows:
\(I \uparrow\)
\(E \uparrow\)
\(E \ge E_0\)
Soil ionisation
Effective radius increases
\(R_i < R_0\)
The simplified model treats the ionised zone as a conducting extension of the electrode.
Section 4
The Critical Soil Gradient E0
The simplified CIGRE / Weck approach uses a critical soil gradient as a general engineering value for rods and tower footings:
\[ E_0 \approx 400\ \text{kV/m} \qquad (\text{other work: } E_0 \approx 1000\ \text{kV/m}) \]
An assumption, not a constant
\(E_0 = 400\) kV/m is an engineering assumption for the simplified method, not an exact material constant. Different researchers have used different values (Oettle, for example, proposed \(\approx 1000\) kV/m), so results should be read with that uncertainty in mind.
Section 5
The Hemisphere Electrode Model
At very high current the ionised zone becomes more spherical, so the rod or footing is approximated as a hemispherical electrode, which has a simple resistance:
\[ R = \frac{\rho}{2\pi r} \qquad R_0 = \frac{\rho}{2\pi r_0} \qquad R_i = \frac{\rho}{2\pi r} \]
Table 3 — Notation for the physical and ionised hemisphere radii.
| Symbol | Meaning |
|---|
| \(r_0\) | Low-current (physical) equivalent radius |
| \(r\) | Ionised effective radius |
As ionisation grows the effective radius increases (\(r_0 \to r\), with \(r > r_0\)), so \(R_i < R_0\). For hemispherical flow the current density and field at radius \(r\) are:
\[ J = \frac{I}{2\pi r^2}, \qquad E = \rho J = \frac{\rho I}{2\pi r^2} \]
Setting \(E = E_0\) gives the ionised radius, which grows with current (\(I\uparrow \Rightarrow r\uparrow\)):
\[ r = \sqrt{\frac{\rho I}{2\pi E_0}} \]
Section 6
The Ionisation Current Ig
Ionisation only dominates once the current is large enough to produce the critical gradient at the original electrode surface. That threshold is \(I_g\):
\[ I_g = \frac{E_0\,\rho}{2\pi R_0^2} \]
Table 4 — Notation for the soil-ionisation threshold current estimate.
| Symbol | Meaning |
|---|
| \(I_g\) | Current at which soil ionisation begins |
| \(E_0\) | Critical soil gradient |
| \(R_0\) | Low-current footing resistance |
The sensitivities are intuitive: \(R_0 \uparrow \Rightarrow I_g \downarrow\) (a poor electrode ionises sooner) and \(\rho \uparrow \Rightarrow I_g \uparrow\). \(I_g\) marks where the high-current ionisation effect becomes important.
Section 7
The Weck Simplified Method
Real tower footings and rods do not behave like ideal hemispheres at the start of ionisation. Weck's simplified method was fitted to measured impulse-resistance data and adopted in the CIGRE procedure for concentrated earth electrodes. The practical form is:
\[ R_i = \frac{R_0}{\sqrt{1 + I/I_g}}, \qquad I_g = \frac{E_0\,\rho}{2\pi R_0^2} \]
Table 5 — Notation used in the Weck impulse-resistance formula.
| Symbol | Meaning |
|---|
| \(R_i\) | Impulse (high-current) footing resistance |
| \(R_0\) | Measured low-current footing resistance |
| \(I\) | Lightning current considered |
| \(I_g\) | Ionisation current / current scale for soil breakdown |
| \(E_0\) | Critical soil gradient |
| \(\rho\) | Soil resistivity |
For \(I \gg I_g\), the expression approaches the high-current limiting form \(R_i \approx R_0\sqrt{I_g/I}\). This has the correct engineering behaviour at both ends:
Table 6 — How the Weck formula behaves at low and high currents.
| Regime | Approximation | Result |
|---|
| Low current (\(I \ll I_g\)) | \(1 + I/I_g \approx 1\) | \(R_i \approx R_0\) |
| High current (\(I \gg I_g\)) | \(1 + I/I_g \approx I/I_g\) | \(R_i \approx R_0\sqrt{I_g/I}\) |
So the familiar square-root expression is the high-current limiting form — on log-log axes the \(R_i\)–\(I\) line has a negative half-slope — and should not be applied blindly below \(I_g\):
\[ R_i \approx R_0\sqrt{\frac{I_g}{I}} \quad (I > I_g), \qquad R_i \approx R_0 \quad (I < I_g) \]
Section 8
Worked Example
Table 7 — Input values assumed for the worked impulse-resistance example.
| Input | Value |
|---|
| \(E_0\) | 400 kV/m |
| \(\rho\) | 800 Ω·m |
| \(R_0\) | 40 Ω |
| \(I\) | 100 kA |
\[ I_g = \frac{E_0\,\rho}{2\pi R_0^2} = \frac{400000 \times 800}{2\pi \times 40^2} \approx 31.8\ \text{kA} \]
\[ R_i = \frac{R_0}{\sqrt{1 + I/I_g}} = \frac{40}{\sqrt{1 + 100/31.8}} \approx 19.7\ \Omega \]
Result
\(R_i \approx 19.7\,\Omega \approx 0.5\,R_0\) — the effective resistance during the high-current impulse is about half the measured low-current value.
Section 9
Why This Matters for Backflashover
The footing voltage is \(V_F = I_i R_i\). Using the measured \(R_0\) directly gives \(V_F = I_i R_0\), which is too high; using the impulse value \(R_i < R_0\) lowers \(V_F\), which raises the current needed to flash over and lowers the rate:
\[ R_i \downarrow \;\Rightarrow\; V_F \downarrow \;\Rightarrow\; I_c \uparrow \;\Rightarrow\; \text{BFR} \downarrow \]
Ionisation helps, but it is not a substitute for grounding
Soil ionisation can reduce the effective resistance during a stroke, but a high measured \(R_0\) still indicates a weak grounding system. Even with \(R_i < R_0\), the impulse resistance may remain high enough to create a dangerous tower voltage rise — soil ionisation must not be used to justify poor grounding design.
Section 10
Ground Rods as Electrodes
For a vertical rod of length \(L\) and radius \(r_0\) in uniform soil, a common low-current estimate is:
\[ R_0 \approx \frac{\rho}{2\pi L}\left[\ln\!\left(\frac{4L}{r_0}\right) - 1\right] \]
The exact form depends on the assumptions, but the sensitivities are clear: \(L \uparrow \Rightarrow R_0 \downarrow\); \(\rho \uparrow \Rightarrow R_0 \uparrow\); and \(r_0 \uparrow\) gives only a small reduction (diameter matters little). Depth has diminishing returns — for \(\rho = 200\,\Omega\cdot\text{m}\) and a \(\sim 13\) mm rod, gains become small beyond about 6 m in uniform soil.
Longer rods help most when they reach better soil
If a deeper layer has lower resistivity, longer rods remain beneficial. Otherwise, in uniform soil, length alone gives diminishing returns — the benefit comes from reaching better soil, not just from being longer.
Section 11
Multiple Rods and Mutual Resistance
Several rods in parallel reduce resistance, but not proportionally, because their current fields overlap (mutual resistance):
\[ R_n > \frac{R_1}{n} \qquad (\text{mutual effects present}) \]
Closer rods overlap more, so the benefit of extra rods falls. In the studied example, beyond about four rods the additional improvement is limited, and a spacing of about 5 m (rods in a circle) is recommended where high-current effectiveness matters. Rod diameter has only an insignificant effect — it enters resistance only through \(\ln(4L/r_0)\):
Length, spacing and resistivity beat diameter
Rod length, spacing and soil resistivity matter far more than rod diameter. Choose diameter mainly for mechanical strength, corrosion allowance, installation method and durability — not to lower resistance.
Section 12
Counterpoises as Horizontal Electrodes
A counterpoise is a buried horizontal conductor connected to the footing — effectively a horizontal ground rod (and a rod is effectively a vertical counterpoise). For the same length in uniform soil their resistances can be similar, but their use differs:
Table 8 — Comparison of vertical ground rods against horizontal counterpoises.
| Item | Ground Rods | Counterpoises |
|---|
| Orientation | Vertical | Horizontal |
| Best use | Reaching lower-resistivity layers | Improving grounding over long horizontal length |
| Main limitation | Mutual effects if rods are close | Travelling-wave delay and mutual effects |
| Key design variable | Depth / spacing | Length / spacing / burial depth |
| Typical practical note | Diameter has small electrical effect | Several shorter branches can outperform one long branch |
Mutual effects matter for close counterpoises too. For two 50 m counterpoises buried \(\sim 1\) m deep in \(\rho = 1000\,\Omega\cdot\text{m}\) soil, the mutual effect is significant for spacing below about 20 m. Transmission lines can often achieve such spacing across the right-of-way; lower-voltage or mountainous routes may not.
During a fast lightning surge a counterpoise behaves differently from a lumped resistance. At the first instant it is not seen as its final leakage resistance — it initially behaves as a travelling-wave conductor with surge impedance \(Z_c\). Only after the wave has propagated to the far end and reflected does the response approach the total leakage resistance \(R_e\):
\[ v_c \approx \frac{c}{3} \qquad T_c = \frac{\ell_c}{v_c} \]
\[ t = 0 \;\Rightarrow\; Z_{\text{cp}} \approx Z_c \qquad t \approx 2T_c \;\Rightarrow\; Z_{\text{cp}} \approx R_e \]
Table 9 — Notation for the counterpoise travelling-wave transient response.
| 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 |
High-current counterpoise modelling is less certain
High-current counterpoise behaviour is less certain than concentrated-electrode behaviour, because many historical counterpoise tests used low currents and did not fully capture soil ionisation along the conductor.
Section 14
Dynamic Models and Footing Inductance
Table 10 — Researcher contributions to dynamic footing and inductance models.
| Investigation | Contribution |
|---|
| Liew & Darveniza | Time- and current-dependent model with ionisation time constant \(T_1\) and deionisation time constant \(T_2\) (ionisation is not instantaneous) |
| Oettle | Normalised estimating curve; suggested \(E_0 \approx 1000\) kV/m (vs the Weck/CIGRE 400 kV/m) |
| Chisholm & Janischewskyj | Equations for the normalised range, plus an estimate of footing inductance |
Footing inductance can add to the transient voltage during fast-front currents, even when ionisation has lowered the resistance:
\[ V_L = L_f\,\frac{dI}{dt}, \qquad \frac{dI}{dt}\uparrow \;\Rightarrow\; V_L \uparrow \]
The Liew–Darveniza dynamic refinement and the inductive component are usually neglected in routine simplified BFR methods, but they explain why very fast fronts can still produce significant footing voltage.
Section 15
Practical Design Interpretation
The footing voltage \(V_F = I_i R_i\) falls if \(R_i\) or \(I_i\) falls. Lightning current cannot be controlled, so design targets the grounding impedance. Practical levers:
- improve the tower footing electrode design;
- add ground rods, or counterpoises in high-resistivity soil;
- increase spacing between parallel electrodes to cut mutual effects;
- reach lower-resistivity soil layers where they exist;
- target the worst (high-resistance) towers first;
- use line surge arresters where grounding improvement is ineffective.
Table 11 — Design cautions when improving tower footing grounding.
| Caution | Why |
|---|
| Don't rely only on \(R_0\) | Lightning performance depends on impulse behaviour — use \(R_i\) where appropriate |
| Don't overvalue many rods | Diminishing returns from mutual coupling |
| Don't place rods too close | Overlapping current fields and ionisation zones |
| Don't assume uniform soil | Real soil is layered, seasonal and moisture-dependent |
| Don't treat \(E_0\) as exact | 400 kV/m is a simplified-method value, not a universal constant |
Section 16
Common Misunderstandings
Table 12 — Common footing-resistance misconceptions and their corrections.
| Misunderstanding | Correct Interpretation |
|---|
| “\(R_0\) is always the resistance during lightning” | \(R_i\) may be significantly lower than \(R_0\) due to ionisation |
| “Low \(R_i\) means poor grounding is acceptable” | A high \(R_0\) still indicates a weak footing and may still give high tower voltage |
| “Doubling rod diameter cuts resistance a lot” | Diameter has a small effect; length, spacing and \(\rho\) dominate |
| “Four rods give a quarter of one rod's resistance” | Mutual resistance prevents ideal division: \(R_4 > R_1/4\) |
| “A counterpoise is a simple lumped resistance” | During fast surges it behaves partly as a travelling-wave element — the far end is not seen immediately |
Section 17
Summary and Memory Map
Equation Summary
Footing voltage
\(\displaystyle V_F = I_i R_i\)
Hemisphere resistance
\(\displaystyle R = \frac{\rho}{2\pi r}\)
Soil field (hemisphere)
\(\displaystyle E = \frac{\rho I}{2\pi r^2}\)
Ionisation current
\(\displaystyle I_g = \frac{E_0\,\rho}{2\pi R_0^2}\)
Weck impulse resistance
\(\displaystyle R_i = \frac{R_0}{\sqrt{1 + I/I_g}}\)
High-current limit
\(\displaystyle R_i \approx R_0\sqrt{\frac{I_g}{I}}\)
Ground-rod resistance
\(\displaystyle R_0 \approx \frac{\rho}{2\pi L}\Big[\ln\tfrac{4L}{r_0} - 1\Big]\)
Inductive component
\(\displaystyle V_L = L_f\frac{dI}{dt}\)
Why grounding controls backflashover
The whole chain reduces to one design relationship:
\[ R_i \downarrow \;\Rightarrow\; V_F = I_i R_i \downarrow \;\Rightarrow\; I_c \uparrow \;\Rightarrow\; P(I \ge I_c) \downarrow \;\Rightarrow\; \text{BFR} \downarrow \]
Memory map. Lightning current enters the footing → soil gradient rises → \(E \ge E_0\) → ionisation starts → effective electrode grows → \(R_i < R_0\) → \(V_F = I_i R_i\) falls → \(I_c\uparrow\) → \(\text{BFR}\downarrow\).
Key messages
- The measured \(R_0\) is not the resistance the lightning sees; soil ionisation gives \(R_i < R_0\) (often \(\approx 0.5\,R_0\)).
- The Weck form \(R_i = R_0/\sqrt{1 + I/I_g}\) tends to \(R_0\) at low current and to \(R_0\sqrt{I_g/I}\) at high current; \(I_g = E_0\rho/2\pi R_0^2\).
- \(E_0 = 400\) kV/m is an engineering assumption (others use \(\approx 1000\)) — not a universal constant.
- Rod length, spacing and soil resistivity dominate; diameter barely matters; multiple rods give diminishing returns (\(R_n > R_1/n\)).
- Counterpoises suit high-resistivity surface soil; mutual effects matter below \(\sim 20\) m spacing; during fast surges they act partly as travelling-wave elements.
- Ionisation helps but does not excuse poor grounding; footing inductance can still raise the voltage on fast fronts. The objective stays: \(R_i\downarrow \Rightarrow V_F\downarrow \Rightarrow I_c\uparrow \Rightarrow \text{BFR}\downarrow\).
