Section 1
Why the Simple Estimate Is Not Enough
This article is Part 13 of the APS EMTP® overhead-line series and the third article in the backflashover sub-series. Part One introduced the backflashover mechanism and Part Two the impulse footing resistance. This page explains how the CIGRE method calculates the critical current \(I_c\), which is then used in the BFR equation.
The first-estimate backflashover rate, \(\text{BFR} = 0.6\,N_L\,P(I \ge I_c)\), is a good start — but it effectively assumed a single phase and left out several real effects:
- the power-frequency voltage on each phase, and the three-phase system;
- the different coupling factors and tower voltages opposite each phase;
- the actual non-standard waveform across the insulator;
- the statistical, current-dependent front time \(t_f\);
- corona on the shield wire;
- the iterative nature of the impulse footing resistance \(R_i\).
The CIGRE method adds these to make the calculation realistic. The hard part is never the BFR formula — it is computing the critical current \(I_c\) correctly.
Table 1 — Symbols used on this page
| Symbol | Meaning |
|---|
| BFR | Backflashover rate (flashovers per 100 km-year) |
| \(N_L\) | Lightning strokes to the line (per 100 km-year) |
| \(I_c\) | Critical stroke current (smallest over the three phases) |
| \(\text{CFO}_{\text{NS}}\) | Non-standard critical flashover voltage |
| \(K_{PF}\) | Equivalent power-frequency factor (hand-calculation approximation) |
| \(R_i\) | Impulse footing resistance (current-dependent) |
| \(K_{sp}\) | Span / adjacent-tower correction factor |
| \(C_A, C_B, C_C\) | Coupling factors (shield wire to phases A, B, C) |
| \(K_{TA}, K_{TB}, K_{TC}\) | Tower-voltage coefficients at the phase positions |
| \(K_{TT}\) | Tower-top / ground-wire coefficient |
| \(t_f\) | Front time (time-to-crest) |
Section 2
Power-Frequency Voltage Is Already There
Before the stroke arrives, the phases already carry AC voltage \(V_{LN}\sin(\omega t)\). So the insulator sees the surge plus the instantaneous power-frequency voltage:
\[ V_{\text{ins,total}} = V_{\text{surge}} + V_{\text{power-frequency}} \]
Depending on the instant of the stroke, the AC voltage can add to or subtract from the surge stress — so the same surge with a different phase angle gives a different flashover probability.
Section 3
Three Phases, Three Critical Currents
Each phase has a different instantaneous AC voltage:
\[ V_A = V_{LN}\sin(\omega t), \quad V_B = V_{LN}\sin(\omega t - 120^\circ), \quad V_C = V_{LN}\sin(\omega t + 120^\circ) \]
At any instant one phase may be near a crest, one negative, one near zero, so the surge stress differs per phase. A line flashover is counted if any phase flashes over, so the controlling phase is the one with the lowest critical current:
\[ I_c = \min\bigl(I_{cA},\ I_{cB},\ I_{cC}\bigr) \]
Section 4
The Insulation Voltage on Each Phase
For each phase the surge voltage across the insulation is the tower-side voltage minus the coupled phase voltage, scaled by the adjacent-tower factor — with the AC voltage added:
\[ V_{IA} = (K_{TA} - C_A K_{TT})\,K_{sp}\,I + V_{LN}\sin(\omega t) \]
\[ V_{IB} = (K_{TB} - C_B K_{TT})\,K_{sp}\,I + V_{LN}\sin(\omega t - 120^\circ) \]
\[ V_{IC} = (K_{TC} - C_C K_{TT})\,K_{sp}\,I + V_{LN}\sin(\omega t + 120^\circ) \]
Table 2 — Notation for the per-phase insulation-voltage equations.
| Symbol | Meaning |
|---|
| \(V_{IA}, V_{IB}, V_{IC}\) | Insulation voltage across phases A, B and C |
| \(K_{TA}, K_{TB}, K_{TC}\) | Tower-voltage coefficients at the phase locations |
| \(C_A, C_B, C_C\) | Coupling factors from shield wire to phases A, B and C |
| \(K_{TT}\) | Tower-top / ground-wire coefficient |
| \(K_{sp}\) | Span / adjacent-tower correction factor |
| \(V_{LN}\) | Crest line-to-neutral power-frequency voltage |
| \(I\) | Lightning stroke current |
The bracket \((K_{TA} - C_A K_{TT})\) is the insulation-stress coefficient: tower-side voltage minus coupled phase-conductor voltage. The controlling phase is the one with the lowest critical current, because flashover of any one phase is counted as a line backflashover. Keep the units consistent: \(C_A\) and \(K_{sp}\) are dimensionless, while \(K_{TA}\) and \(K_{TT}\) are tower-voltage coefficients in volts per unit stroke current (impedance units), so \((K_{TA}-C_A K_{TT})\,K_{sp}\,I\) is a voltage.
Section 5
Which Phase Controls?
Setting the phase insulation voltage equal to \(\text{CFO}_{\text{NS}}\), for example \(V_{IA}=\text{CFO}_{\text{NS}}\), gives that phase's critical current:
\[ I_{cA} = \frac{\text{CFO}_{\text{NS}} - V_{LN}\sin(\omega t)}{(K_{TA} - C_A K_{TT})\,K_{sp}} \]
The controlling phase is not obvious. On a vertical double-circuit tower \(K_{TC} < K_{TB} < K_{TA}\) suggests phase A is worst — but coupling can run the other way (\(C_C < C_B < C_A\)), where lower coupling means a higher stress, making phase C worst. The instantaneous AC angle then shifts the answer again.
The exact phase-angle method
Divide the cycle into steps (e.g. 12 angles \(0^\circ, 30^\circ, \dots, 330^\circ\)); at each, compute \(I_{cA}, I_{cB}, I_{cC}\), take the smallest, and the corresponding BFR; then average: \(\;\text{BFR} = \tfrac{1}{12}\sum_{n=1}^{12} \text{BFR}_{\max,n}\). Accurate, but tedious by hand. (In a 115 kV example, phases A and C dominate — yet phase B, despite higher coupling, still flashes often because the AC voltage makes it critical at some angles: coupling alone does not pick the flashover phase.)
Section 6
The Power-Frequency Factor KPF
For hand calculation, the AC effect is bundled into one factor \(K_{PF}\) that reduces the CFO by an equivalent power-frequency contribution. \(K_{PF}\) is an equivalent power-frequency factor used to approximate the effect of the instantaneous three-phase AC voltage at the lightning instant — it is not a physical multiplier tied to one specific phase. It is a hand-calculation approximation of the instantaneous AC contribution and should not replace the phase-angle method where a detailed calculation is required:
\[ I_c \approx \frac{\text{CFO}_{\text{NS}} - K_{PF}\,V_{LN}}{(K_{TA} - C_A K_{TT})\,K_{sp}} \qquad V_{LN} = \sqrt{2}\,\frac{V_{LL}}{\sqrt{3}} \]
Table 3 — Notation for the power-frequency-factor critical-current approximation.
| Symbol | Meaning |
|---|
| \(K_{PF}\) | Equivalent power-frequency factor |
| \(V_{LN}\) | Crest line-to-neutral voltage, not RMS |
| \(V_{LL}\) | Nominal line-to-line RMS voltage |
\(V_{LN}\) must be the crest value \(\sqrt{2}\,V_{LL}/\sqrt{3}\); using the RMS value would understate the AC voltage contribution.
Do not confuse with the IEEE phase-angle method
This CIGRE treatment folds the AC voltage into a single equivalent factor \(K_{PF}V_{LN}\). IEEE-style methods instead step through the power-frequency angle explicitly — the two approaches are compared in the sensitivity-analysis part.
Table 4 — Recommended power-frequency factor.
| Phase Configuration | \(K_{PF}\) | Range |
|---|
| Vertical (double-circuit) | ≈ 0.40 | 0.25–0.55 (depends on V/CFO) |
| Horizontal (single-circuit) | ≈ 0.70 | 0.65–0.76 |
If unsure, use \(K_{PF} = 0.70\). For example, \(V_{LL} = 230\) kV gives \(V_{LN} = \sqrt{2}\,(230/\sqrt{3}) \approx 188\) kV.
Section 7
The Non-Standard CFO
Standard CFO uses the \(1.2/50\,\mu\text{s}\) impulse, but the backflash waveform is different — power-frequency voltage, a footing-resistance component, a tower surge component, an exponential tail and reflections. So the strength is the non-standard \(\text{CFO}_{\text{NS}}\), and the condition is \(V_{\text{ins}} \ge \text{CFO}_{\text{NS}}\). The actual insulation voltage is built from:
\[ V_{\text{ins}}(t) = V_{PF}(t) + V_{IF}(t) + \Delta V_T(t) \]
Table 5 — Notation for the components of the non-standard insulation voltage.
| Symbol | Meaning |
|---|
| \(V_{PF}\) | Power-frequency component |
| \(V_{IF}\) | Footing-resistance component |
| \(\Delta V_T\) | Tower surge component |
Section 8
The Leader Progression Model
Rather than “flashover when the crest exceeds a fixed value”, the Leader Progression Model (LPM) tracks the physical leader crossing the gap: streamers bridge part of the gap, a leader starts once the gradient exceeds a critical value, it propagates, the remaining distance shrinks, the field rises, the leader accelerates, and flashover occurs when it bridges the gap. The leader velocity has the general form:
\[ v(t) = k\left(\frac{e(t)}{x} - E_0\right) \]
\[ x_{n+1} = x_n - v_n\,\Delta t \]
Table 6 — Notation for the leader-progression-model velocity and gap equations.
| Symbol | Meaning |
|---|
| \(v(t)\) | Leader velocity at time \(t\) |
| \(e(t)\) | Voltage across the gap at time \(t\) |
| \(x\) | Unbridged gap distance |
| \(E_0\) | Critical leader inception gradient |
| \(k\) | Empirical leader-progression constant |
| \(\Delta t\) | Calculation time step |
| \(x_{n+1}\) | Remaining unbridged distance after the time step |
Here \(x\) is the remaining unbridged gap distance (not the leader length); it shrinks as the leader advances, hence \(x_{n+1}=x_n-v_n\Delta t\). If the leader length \(\ell\) is preferred, replace \(x\) with \(g-\ell\), the unbridged distance. Flashover is predicted when the leader bridges the remaining gap, i.e. when \(x \le 0\); as \(x\) shrinks, the same voltage gives a higher gradient \(e/x\), so the leader accelerates, and flashover depends on magnitude and waveform duration, not just crest. This page uses the LPM only to obtain \(\text{CFO}_{\text{NS}}\) — the detailed flashover models (integral method, leader progression, Shindo–Suzuki, predischarge current and EMTP® implementation) are on the air-gap and insulator flashover modelling page.
Section 9
A Regression Equation for CFONS
LPM is physical but inconvenient by hand, so CIGRE fitted a regression equation from LPM results, relating \(\text{CFO}_{\text{NS}}\) to the standard CFO, the tail time constant \(\tau\), the tower and footing voltage components, and the front time. It is valid only within:
\[ 10\,\mu\text{s} \le \tau \le 100\,\mu\text{s}, \qquad 0 \le \frac{\Delta V}{V_{IF}} \le 1.0, \qquad 0.5\,\mu\text{s} \le t_f \le 5\,\mu\text{s} \]
The ratio \(\Delta V/V_{IF}\) compares the tower component with the footing component: \(0\) neglects the tower part, \(1\) makes it as large as the footing part. It changes the waveform shape and so the non-standard CFO. The regression should only be used within its intended ranges of tail time constant, tower-to-footing voltage ratio and front time; outside those ranges, a direct LPM calculation or a more detailed EMT / insulation-strength assessment is needed.
Section 10
Front Time tf
The tower surge component depends on current steepness, \(dI/dt \approx I/t_f\), so a shorter front gives a higher tower voltage and the critical current depends on \(t_f\):
\[ t_f \downarrow \Rightarrow \frac{dI}{dt} \uparrow \Rightarrow \Delta V_T \uparrow, \qquad I_c = I_c(t_f), \quad t_f \uparrow \Rightarrow I_c \uparrow \]
The real front is concave-upward, not linear; a minimum equivalent front is conservative — in the example it overestimates the tower-top voltage by about 9%, which is acceptable for a simplified calculation.
Section 11
BFR Conditioned on Front Time
Because \(I_c = I_c(t_f)\), the BFR is strictly conditional on \(t_f\) and should be integrated over its distribution:
\[ \text{BFR}\,|\,t_f = 0.6\,N_L\,P\bigl(I \ge I_c(t_f)\bigr) \]
\[ \text{BFR} = 0.6\,N_L \int_0^\infty \!\!\int_{I_c(t_f)}^\infty f(I\,|\,t_f)\,f(t_f)\,dI\,dt_f \]
Table 7 — Notation for the front-time-conditioned backflashover-rate integral.
| Symbol | Meaning |
|---|
| \(t_f\) | Lightning stroke current time to crest (front time) |
| \(I_c(t_f)\) | Critical current as a function of front time |
| \(f(I\,|\,t_f)\) | Conditional probability density of current given \(t_f\) |
| \(f(t_f)\) | Probability density of front time |
| \(N_L\) | Number of strokes terminating on the line / shield-wire system |
For hand calculation, the full integration is replaced by a single effective front time, selected so that it is consistent with the median front time associated with the calculated critical current: \(t_f = t_{f,\text{median}}(I_c)\). This is iterative — \(I_c\) depends on \(t_f\), and \(t_f\) is taken from \(I_c\).
Section 12
Corona — Not Dominant Here
On the surge front, corona raises the shield-wire capacitance, lowers \(Z_g\) and raises the coupling factor, so \(V_{\text{ins}} \approx V_T - C V_g\) falls — suggesting corona reduces BFR. But corona acts mainly on the front, not the tail, and for leader progression the tail matters. Since the tail is barely changed:
Corona is not predominant for BFR
Corona may cut the crest but leaves the tail almost unchanged, so the part of the waveform that drives the leader is little affected — in the CIGRE simplified BFR method corona is not expected to dominate and may be neglected with small conservatism for many conventional cases (it can become relevant for high towers, as the sensitivity analysis shows). This is not a general rule for detailed EMT surge-propagation studies, where corona can significantly alter the wavefront and crest — see the corona effect page.
Section 13
The CIGRE Iterative Method
The method is iterative because the current needed to cause flashover depends on \(R_i\) and \(t_f\), yet both \(R_i\) and the representative \(t_f\) depend on the current itself. The overall calculation flow is:
Assume \(t_f\)
Assume \(R_i\)
Compute \(\text{CFO}_{\text{NS}}\)
Compute \(I_c\)
Update \(R_i(I)\)
Update \(t_f(I_c)\)
\(\text{BFR} = 0.6\,N_L\,P(I \ge I_c)\)
It resolves into two nested loops:
- Inner loop — impulse resistance: \(R_i = R_i(I)\) but \(I_c = I_c(R_i)\), so assume \(R_i \approx 0.5\,R_0\), compute \(I_c\), find the footing current, update \(R_i\), repeat to convergence. (Here \(R_i\) is just an iteration input; its current-dependence through soil ionisation and the low-current value \(R_0\) are developed on the impulse resistance of ground electrodes page.)
- Outer loop — front time: \(I_c = I_c(t_f)\) but the representative \(t_f\) is picked from \(I_c\), so assume \(t_f\), solve the inner loop, compute the median front time for that \(I_c\), and iterate until consistent.
Typical starting front times: \(t_f \approx 2.5\,\mu\text{s}\) for 115–230 kV lines and \(t_f \approx 4.0\,\mu\text{s}\) for 345 kV and above. It can be done by hand in principle, but is normally a computer program.
CIGRE workflow
- Select an initial front time (\(2.5\,\mu\text{s}\) for 115–230 kV; \(4.0\,\mu\text{s}\) for 345 kV+).
- Assume \(R_i \approx 0.5\,R_0\).
- Compute \(\text{CFO}_{\text{NS}}\) from the waveform parameters.
- Compute \(I_c\) (tower + footing voltage, coupling, \(K_{PF}\), \(K_{sp}\), \(\text{CFO}_{\text{NS}}\)).
- Update \(R_i(I)\) and iterate the inner loop to convergence.
- Update \(t_f\) to the median for \(I_c\); iterate the outer loop until consistent.
- Compute \(\text{BFR} = 0.6\,N_L\,P(I \ge I_c)\).
Section 14
The Simplified Method — and an Example
If the tower component of voltage is neglected, \(t_f\) is no longer a major parameter and the outer loop disappears — a practical hand method. The stress is then footing-dominated:
\[ V_{\text{ins}} \approx (1 - C)\,I_i R_i + K_{PF}\,V_{LN} \qquad I_c \approx \frac{\text{CFO}_{\text{NS}} - K_{PF}\,V_{LN}}{(1 - C)\,R_i} \]
Table 8 — Notation for the simplified footing-dominated critical-current equation.
| Symbol | Meaning |
|---|
| \(R_i\) | Impulse footing resistance, not automatically the measured \(R_0\) |
| \(C\) | Coupling factor between shield wire and phase conductor |
| \(K_{PF}V_{LN}\) | Equivalent power-frequency contribution |
| \(\text{CFO}_{\text{NS}}\) | Non-standard critical flashover voltage |
| \(I_i\) | Current flowing through the tower footing path |
A hand approximation with limits
This is a hand-calculation approximation valid only when the tower component of voltage is neglected — do not use it blindly for tall towers or where the tower surge is significant. The sensitivity analysis shows it is mainly suitable for towers below about 50 m. Note also that \(R_i\) here is the impulse resistance, not automatically the measured low-current \(R_0\) (see Part Two).
with \(\text{CFO}_{\text{NS}}\uparrow \Rightarrow I_c\uparrow\), \(V_{LN}\uparrow \Rightarrow I_c\downarrow\), \(C\uparrow \Rightarrow I_c\uparrow\), \(R_i\uparrow \Rightarrow I_c\downarrow\). Example — a 230 kV single-circuit line:
Table 9 — Example 230 kV line inputs.
| Quantity | Value |
|---|
| \(Z_g\) / CFO | 400 Ω / 960 kV |
| \(C\) / \(R_0\) / \(\rho\) | 0.30 / 50 Ω / 1000 Ω·m |
| \(V_{LN}\) / \(K_{PF}V_{LN}\) | 188 kV / 0.70×188 = 131 kV |
| Ground-wire / phase height | 30 m / 24 m |
| Span / \(N_g\) / \(S_g\) | 300 m / 4 / 5 m |
\[ N_L = 88.2, \quad P(I > I_c) = 0.235 \;\Rightarrow\; \text{BFR} = 0.6 \times 88.2 \times 0.235 \approx 12.4 \]
The full computer CIGRE method gives \(\approx 13.2\) flashovers per 100 km-year — the source reports about a 17% tolerance for the simplified estimate. Note \(\text{BFR}/N_L = 0.14\): about 14% of strokes to the line cause a backflashover, neatly separating exposure (\(N_L\)) from severity probability (\(P(I \ge I_c)\)).
Section 15
What Each Factor Does
Table 10 — The correction factors and their meaning.
| Factor | Accounts for |
|---|
| 0.6 (span location) | Not all shield-wire strokes are as severe as direct tower strokes |
| \(K_{PF}\) (power-frequency) | AC voltage can add to the surge on one phase — effectively lowers the required surge |
| \(C\) (coupling) | Phase conductor lifted by the shield wire — reduces stress |
| \(R_i\) (impulse resistance) | Soil ionisation; usually below the measured \(R_0\) |
| \(\text{CFO}_{\text{NS}}\) | The backflash waveform is not a standard \(1.2/50\,\mu\text{s}\) impulse |
BFR rises when \(N_L\uparrow\) (higher \(N_g\), taller towers, wider geometry) or \(I_c\downarrow\) (high footing resistance, low coupling, low insulation, high power-frequency contribution, steep front, poor geometry). BFR falls when \(N_L\downarrow\) or \(I_c\uparrow\) — the most common practical lever being lower footing resistance (\(R_i\downarrow\)).
Section 16
Summary and Memory Map
Equation Summary
Phase insulation voltage
\(\displaystyle V_{IA} = (K_{TA} - C_A K_{TT})K_{sp}I + V_{LN}\sin\omega t\)
Critical current (phase)
\(\displaystyle I_{cA} = \frac{\text{CFO}_{\text{NS}} - V_{LN}\sin\omega t}{(K_{TA} - C_A K_{TT})K_{sp}}\)
Controlling phase
\(\displaystyle I_c = \min(I_{cA}, I_{cB}, I_{cC})\)
Approx. with KPF
\(\displaystyle I_c \approx \frac{\text{CFO}_{\text{NS}} - K_{PF}V_{LN}}{(1-C)R_i}\)
Line-to-neutral crest
\(\displaystyle V_{LN} = \sqrt{2}\,\frac{V_{LL}}{\sqrt{3}}\)
Leader velocity
\(\displaystyle v(t) = k\left(\frac{e(t)}{x} - E_0\right)\)
Conditional BFR
\(\displaystyle \text{BFR}\,|\,t_f = 0.6\,N_L\,P(I \ge I_c(t_f))\)
Backflashover rate
\(\displaystyle \text{BFR} = 0.6\,N_L\,P(I \ge I_c)\)
Key messages
- Computing \(I_c\) is the whole calculation chain: the three-phase power-frequency voltage, per-phase coupling, tower geometry, \(R_i\), \(\text{CFO}_{\text{NS}}\), front time, corona and adjacent-tower reflections all feed it.
- A line flashover counts if any phase flashes, so \(I_c = \min(I_{cA}, I_{cB}, I_{cC})\); coupling alone does not pick the phase — the AC angle matters.
- The AC effect is captured by \(K_{PF}\) (vertical ≈ 0.40, horizontal ≈ 0.70); the non-standard waveform by \(\text{CFO}_{\text{NS}}\) via the Leader Progression Model / CIGRE regression.
- The CIGRE method iterates an inner \(R_i\) loop and an outer \(t_f\) loop; corona is not dominant and may be neglected with small conservatism.
- Neglecting the tower component gives a simple hand method; the 230 kV example yields BFR ≈ 12.4 vs 13.2 (computer) — \(\text{BFR}/N_L = 0.14\), i.e. ~14% of strokes flash over.
- The objective is unchanged: \(I_c \uparrow \Rightarrow P(I \ge I_c) \downarrow \Rightarrow \text{BFR}\downarrow\).
References
References
The standards, technical brochures, key papers and reference works behind this page.
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.
IEEE Std 1243-1997, IEEE Guide for Improving the Lightning Performance of Transmission Lines. New York, NY, USA: IEEE, 1997.
CIGRE Working Group C4.407, Lightning Parameters for Engineering Applications, Technical Brochure 549. Paris, France: CIGRE, 2013.
IEC 60071-2:2023, Insulation Co-ordination – Part 2: Application Guidelines. Geneva, Switzerland: International Electrotechnical Commission, 2023.
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.
