Section 1
Why This Topic Matters
Early backflashover calculations consider only one phase. The insulation voltage is simplified to the tower-side voltage minus the coupled phase voltage, using a single coupling factor \(C\):
\[ V_{\text{ins}} = V_{\text{tower}} - V_{\text{phase}} \]
But a real line has three phase conductors, each with a different position on the tower, a different coupling factor, a different tower-side voltage, and a different instantaneous power-frequency voltage. So the backflashover problem is not a single-phase problem, and the key question is:
The question
Which phase flashes over first? The answer depends on both the lightning surge voltage and the instantaneous AC power-frequency voltage at the moment of the stroke.
This is a standalone self-study companion note: it explains how the instantaneous three-phase power-frequency voltage modifies the critical lightning current \(I_c\) used in CIGRE-style backflashover calculations.
What this page teaches
- why a three-phase line cannot be reduced to one fixed phase without care;
- how \(V_{IA}, V_{IB}, V_{IC}\) vary with the power-frequency angle;
- why the flashover phase can change during the AC cycle;
- how \(I_{cL}\) and \(I_{cH}\) define the flashover-probability range;
- how the exact phase-angle method works;
- why the practical CIGRE method uses \(K_{PF}V_{LN}\);
- why \(K_{PF}\) differs for horizontal and vertical configurations.
Section 2
The Single-Phase Approximation
For a single phase the surge insulation voltage is proportional to the lightning current, and flashover occurs when it reaches the non-standard CFO:
\[ V_I = K_I\,I, \qquad V_I = \text{CFO}_{\text{NS}} \;\Rightarrow\; I_c = \frac{\text{CFO}_{\text{NS}}}{K_I} \]
Table 1 — Notation for the single-phase surge insulation-voltage equation.
| Symbol | Meaning |
|---|
| \(V_I\) | Surge voltage across the line insulation |
| \(K_I\) | Insulation-voltage coefficient |
| \(I\) | Lightning stroke current |
Simple — but it ignores the three-phase AC voltage entirely.
Section 3
Three Phases, Three Coefficients
Each phase has its own insulation-voltage coefficient (set by tower geometry, phase position, coupling and the tower voltage at that location), so each phase sees a different surge voltage:
\[ V_{IA} = K_{IA}\,I, \qquad V_{IB} = K_{IB}\,I, \qquad V_{IC} = K_{IC}\,I \]
If the AC voltage is ignored, the worst phase is simply the one with the largest coefficient:
\[ K_m = \max(K_{IA}, K_{IB}, K_{IC}) \;\Rightarrow\; I_c = \frac{\text{CFO}_{\text{NS}}}{K_m} \]
Section 4
Adding the Power-Frequency Voltage
At the instant of the stroke, each phase carries a different AC voltage:
\[ V_A = V_{LN}\sin(\omega t), \quad V_B = V_{LN}\sin(\omega t - 120^\circ), \quad V_C = V_{LN}\sin(\omega t + 120^\circ) \]
Table 2 — Notation for the instantaneous three-phase power-frequency voltages.
| Symbol | Meaning |
|---|
| \(V_{LN}\) | Crest line-to-neutral power-frequency voltage |
| \(\omega t\) | Instantaneous power-frequency angle |
So the total insulation voltage on each phase is the surge component plus the instantaneous AC voltage — the key three-phase result:
\[ V_{IA} = K_{IA}\,I + V_{LN}\sin(\omega t) \]
\[ V_{IB} = K_{IB}\,I + V_{LN}\sin(\omega t - 120^\circ) \]
\[ V_{IC} = K_{IC}\,I + V_{LN}\sin(\omega t + 120^\circ) \]
Table 3 — Notation for the total per-phase insulation voltage combining surge and AC.
| Symbol | Meaning |
|---|
| \(V_{IA}, V_{IB}, V_{IC}\) | Total insulation voltage across phases A, B and C |
| \(K_{IA}, K_{IB}, K_{IC}\) | Per-phase insulation-voltage coefficients |
| \(I\) | Lightning stroke current |
| \(V_{LN}\) | Crest line-to-neutral power-frequency voltage |
| \(\omega t\) | Instantaneous power-frequency angle |
The controlling phase is not fixed: it depends on both the lightning surge coefficient and the instantaneous AC phase angle.
Section 5
The Controlling Phase Changes with Time
The surge component may be largest on phase A (\(K_{IA} > K_{IB} > K_{IC}\)), but the AC voltage shifts with time. At one instant phase A is most stressed; at another, phase B or C becomes critical because its AC voltage adds more strongly. So the controlling phase depends on the coefficients, the angle \(\omega t\) and \(V_{LN}\) together:
Not one phase only
The phase with the largest surge coefficient is not always the only phase that flashes over. A line flashover occurs when any phase reaches the CFO:
\[ V_{Ix} \ge \text{CFO}_{\text{NS}} \quad\text{for } x = A,\,B \text{ or } C \;\Rightarrow\; \text{line flashover} \]
Section 6
The Lower and Upper Critical Currents
The first possible flashover happens when the most-stressed phase just reaches the CFO with the AC voltage adding at its crest — this defines a lower critical current \(I_{cL}\):
\[ I_{cL} = \frac{\text{CFO}_{\text{NS}} - V_{LN}}{K_m} \]
Below \(I_{cL}\) no phase can flash over. As the current rises, flashover becomes possible over a wider part of the cycle, until above an upper current \(I_{cH}\) flashover is certain at any angle. Between the two, the probability lies between 0 and 1:
Table 4 — How flashover probability varies across the lower and upper critical currents.
| Current Range | Flashover probability \(P(\text{FO}\,|\,I)\) | Meaning |
|---|
| \(I < I_{cL}\) | 0% | No phase can reach \(\text{CFO}_{\text{NS}}\) |
| \(I_{cL} \le I < I_{cH}\) | between 0% and 100% | Flashover depends on the AC phase angle |
| \(I \ge I_{cH}\) | 100% | At least one phase flashes for any AC angle |
Section 7
The Exact Phase-Angle Method
The exact approach steps through the cycle (e.g. \(10^\circ\) or \(30^\circ\) steps). At each angle it computes \(V_{IA}, V_{IB}, V_{IC}\), checks each against the CFO, and records whether (and on which phase) flashover occurs. The procedure:
Select an AC angle \(\omega t\)
Compute \(V_{IA}, V_{IB}, V_{IC}\)
Any phase \(\ge \text{CFO}_{\text{NS}}\)?
Repeat over the full cycle
Fraction of angles flashing = \(P_L(I)\)
Integrate over \(f(I)\)
The conditional line-flashover probability and the total probability are:
\[ P_L(I) = \frac{\text{steps causing flashover}}{\text{total steps}}, \qquad P(\text{FO}) = \int_0^\infty P_L(I)\,f(I)\,dI \]
Table 5 — Notation for the exact phase-angle flashover-probability calculation.
| Symbol | Meaning |
|---|
| \(P_L(I)\) | Conditional line-flashover probability for current \(I\) |
| \(f(I)\) | Probability density of the lightning current |
Example. With \(K_{IA} = 18\), \(K_{IB} = 17\), \(K_{IC} = 16\), \(V_{LN} = 400\) kV, \(\text{CFO}_{\text{NS}} = 2000\) kV and \(I = 100\) kA, the 12-step phase-angle calculation gives:
Table 6 — Per-phase flashover counts from the worked 12-step phase-angle example.
| Outcome Over 12 Steps | Count |
|---|
| Flashover on phase A | 5 |
| Flashover on phase B | 3 |
| Flashover on phase C | 1 |
| No flashover | 3 |
\[ P_L(100\ \text{kA}) = \frac{9}{12} = 0.75 \]
This shows flashover is a probabilistic function of phase angle — accurate, but tedious by hand.
Section 8
The Power-Frequency Factor KPF
For hand calculation, CIGRE replaces the phase-angle stepping with an equivalent power-frequency factor \(K_{PF}\) that subtracts a representative AC voltage from the CFO. The practical critical-current equation becomes:
KPF is an equivalent factor, not a voltage
\(K_{PF}\) is not the instantaneous voltage of one phase. It is an equivalent factor used to represent the statistical effect of the three-phase power-frequency voltage in a simplified critical-current calculation.
\[ V_{PF} = K_{PF}\,V_{LN}, \qquad I_c = \frac{\text{CFO}_{\text{NS}} - K_{PF}\,V_{LN}}{(K_{TA} - C_A K_{TT})\,K_{sp}} \]
Table 7 — Notation for the CIGRE equivalent power-frequency critical-current equation.
| Symbol | Meaning |
|---|
| \(K_{PF}\) | Equivalent power-frequency factor (statistical, not a per-phase multiplier) |
| \(V_{LN}\) | Crest line-to-neutral power-frequency voltage |
| \(K_{TA}\) | Tower-voltage coefficient at the selected phase |
| \(C_A\) | Coupling factor for the selected phase |
| \(K_{TT}\) | Tower-top / ground-wire coefficient |
| \(K_{sp}\) | Span / adjacent-tower correction factor |
Section 9
The Line-to-Neutral Crest Voltage
\(V_{LN}\) is the crest line-to-neutral voltage — not the RMS value:
\[ V_{LN} = \sqrt{2}\,\frac{V_{LL}}{\sqrt{3}} \qquad\Longrightarrow\qquad V_{LL} = 230\ \text{kV} \;\Rightarrow\; V_{LN} \approx 188\ \text{kV} \]
Table 8 — Notation relating line-to-line RMS voltage to the line-to-neutral crest.
| Symbol | Meaning |
|---|
| \(V_{LL}\) | Nominal line-to-line RMS voltage |
| \(V_{LN}\) | Crest line-to-neutral voltage |
Section 10
Why KPF Depends on the Phase Arrangement
\(K_{PF}\) depends on how the conductors are arranged, because the relative values of \(K_{IA}, K_{IB}, K_{IC}\) differ with geometry. For a horizontal line the two outside phases have roughly equal coupling and the centre phase slightly more — the AC contribution is relatively strong. For a vertical line the coefficients are more unequal and one phase tends to dominate, so the equivalent AC contribution is lower:
Table 9 — Recommended power-frequency factor for horizontal, vertical and exact methods.
| Phase Configuration | Recommended \(K_{PF}\) | Interpretation |
|---|
| Horizontal configuration | 0.70 | AC voltage contribution is relatively strong |
| Vertical configuration | 0.40 | One phase tends to dominate more strongly |
| Exact phase-angle method | Not needed | AC phase angle is calculated directly |
\[ \text{horizontal: } I_c = \frac{\text{CFO}_{\text{NS}} - 0.70\,V_{LN}}{(K_{TA} - C_A K_{TT})\,K_{sp}} \qquad \text{vertical: } I_c = \frac{\text{CFO}_{\text{NS}} - 0.40\,V_{LN}}{(K_{TA} - C_A K_{TT})\,K_{sp}} \]
These are engineering approximations, not universal constants — they exist to avoid detailed phase-angle integration in hand calculations. The two arrangements compared:
Table 10 — Comparing horizontal and vertical phase arrangements and their equivalent factors.
| Item | Horizontal Configuration | Vertical Configuration |
|---|
| Phase stress distribution | More balanced between phases | One phase may dominate more strongly |
| Role of AC angle | Stronger statistical contribution | Smaller equivalent contribution |
| Typical \(K_{PF}\) | 0.70 | 0.40 |
| Practical conclusion | AC voltage has a stronger equivalent effect | AC voltage still matters, but with a lower equivalent factor |
Section 11
A 115 kV Worked Example
A 115 kV single-circuit horizontal line, with the power-frequency voltage handled by computing the critical current and BFR for each phase over many AC angles:
Table 11 — Input parameters for the 115 kV horizontal-line backflashover worked example.
| Quantity | Value |
|---|
| Ground flash density \(N_g\) | 6.0 flashes/km²-year |
| Nominal voltage | 115 kV |
| Ground-wire / tower surge impedance | 339 Ω / 170 Ω |
| Coupling factors A / B / C | 0.331 / 0.386 / 0.331 |
| Ground-wire / phase height | 57 ft / 46 ft |
| Shield-wire separation / span | 12.5 ft / 750 ft |
| CFO | 1067 kV |
| Footing resistance \(R_0\) / \(\rho\) | 20 Ω / 400 Ω·m |
\[ \text{BFR} = 0.191 \ \text{flashovers per 100 km-year} \]
\[ \text{phase A: } 43.06\%, \quad \text{phase C: } 43.06\%, \quad \text{phase B: } 13.89\% \]
The lessons: the outside phases (lower coupling) dominate; the better-coupled centre phase is not zero — it still flashes at some AC angles; and the exact phase-angle calculation is accurate but tedious, which is why \(K_{PF}\) is used.
Better coupling reduces, but does not eliminate
Better coupling lowers a phase's flashover probability — it does not eliminate it. The centre phase of a horizontal line still flashes over for certain AC phase angles.
Section 12
Coupling, KI, and the Lowest-Coupling Phase
The insulation-voltage coefficient can be written as the tower-side voltage minus the coupled phase voltage:
\[ K_I = K_T - C K_{TT} \;\Rightarrow\; C \uparrow \Rightarrow K_I \downarrow \Rightarrow I_c = \frac{\text{CFO}_{\text{NS}} - K_{PF}V_{LN}}{K_I} \uparrow \Rightarrow \text{BFR} \downarrow \]
So higher coupling is beneficial for backflashover. Conversely, the CIGRE conclusion is to base the critical current on the phase with the lowest coupling factor, because it gives the highest net insulation stress (the phase is lifted least by the shield wire):
\[ V_{\text{phase}} = C V_g, \qquad C \downarrow \Rightarrow V_{\text{phase}} \downarrow \Rightarrow V_{\text{ins}} = V_{\text{tower}} - V_{\text{phase}} \uparrow \]
So the lowest-coupled phase is normally the controlling phase for the simplified calculation.
Section 13
The Final CIGRE Practical Equation
Taking \(C_A\) as the lowest coupling factor and \(K_{TA}\) as the corresponding tower coefficient, the practical critical current and BFR are:
\[ I_c = \frac{\text{CFO}_{\text{NS}} - K_{PF}\,V_{LN}}{(K_{TA} - C_A K_{TT})\,K_{sp}} \]
\[ \text{BFR} = 0.6\,N_L\,P(I \ge I_c) \]
where the \(0.6\) factor accounts for stroke-location effects along the span.
Section 14
Exact Method versus the KPF Method
Table 12 — Comparing the exact phase-angle, factor, and CIGRE practical methods.
| Method | Description | Use |
|---|
| Exact phase-angle | Computes \(I_c\) and BFR over many AC angles | Computer calculation |
| \(K_{PF}\) method | Uses the equivalent \(K_{PF}V_{LN}\) contribution | Hand / simplified calculation |
| CIGRE practical | \(K_{PF} = 0.70\) (horizontal) or \(0.40\) (vertical) | Engineering BFR estimation |
The \(K_{PF}\) method agrees better with the exact method when the system voltage is low relative to CFO, all three phases contribute evenly, and the \(I_{cL}\)–\(I_{cH}\) range is small. It agrees worse when the voltage is high relative to CFO, one phase dominates, and the \(I_{cL}\)–\(I_{cH}\) range is large.
An approximation, not a replacement
\(K_{PF}\) is an engineering approximation, not an exact replacement for phase-angle analysis. For high-importance studies, computer-based phase-angle analysis is preferable.
Section 15
Common Misunderstandings
Table 13 — Common misconceptions about phase flashover and the power-frequency factor corrected.
| Misunderstanding | Correct Interpretation |
|---|
| “The same phase always flashes over” | The most-stressed phase usually dominates, but others can flash depending on the instantaneous AC voltage |
| “Power-frequency voltage can be ignored” | It can significantly reduce \(I_c\), especially where the system voltage is high relative to CFO |
| “\(K_{PF}\) is a physical multiplier on one phase” | It is an equivalent statistical factor for the three-phase AC effect |
| “The centre phase of a horizontal line never flashes” | It has better coupling but still flashes over for certain AC phase angles |
Do not make these mistakes
• Do not use the RMS line-to-line voltage directly as \(V_{LN}\) — use the crest \(\sqrt{2}\,V_{LL}/\sqrt{3}\).
• Do not assume the same phase always flashes over.
• Do not treat \(K_{PF}\) as an exact physical constant.
• Do not ignore power-frequency voltage for high-voltage lines.
• Do not also apply \(K_{PF}\) when the exact phase-angle method is already being used.
Section 16
Summary and Memory Map
Equation Summary
Phase A total voltage
\(\displaystyle V_{IA} = K_{IA}I + V_{LN}\sin\omega t\)
Line-to-neutral crest
\(\displaystyle V_{LN} = \sqrt{2}\,\frac{V_{LL}}{\sqrt{3}}\)
Equivalent PF contribution
\(\displaystyle V_{PF} = K_{PF}V_{LN}\)
Lower critical current
\(\displaystyle I_{cL} = \frac{\text{CFO}_{\text{NS}} - V_{LN}}{K_m}\)
Conditional probability
\(\displaystyle P_L(I) = \frac{\text{FO steps}}{\text{total steps}}\)
Insulation coefficient
\(\displaystyle K_I = K_T - C K_{TT}\)
Practical critical current
\(\displaystyle I_c = \frac{\text{CFO}_{\text{NS}} - K_{PF}V_{LN}}{(K_{TA} - C_A K_{TT})K_{sp}}\)
Backflashover rate
\(\displaystyle \text{BFR} = 0.6\,N_L\,P(I \ge I_c)\)
Memory map. Lightning surge creates the insulation voltage → each phase has a different \(K_I\) and a different instantaneous AC voltage → \(V_{IA}, V_{IB}, V_{IC}\) change with \(\omega t\) → flashover may occur on A, B or C → the exact method steps angle by angle → the practical method uses \(K_{PF}V_{LN}\) → \(I_c = (\text{CFO}_{\text{NS}} - K_{PF}V_{LN})/[(K_{TA} - C_A K_{TT})K_{sp}]\) → \(\text{BFR} = 0.6\,N_L\,P(I \ge I_c)\).
Why power-frequency voltage increases BFR
The power-frequency voltage is not a fixed add-on — it changes with the AC cycle and differs per phase. Its net effect runs one way:
\[ K_{PF}V_{LN}\uparrow \;\Rightarrow\; (\text{CFO}_{\text{NS}} - K_{PF}V_{LN})\downarrow \;\Rightarrow\; I_c\downarrow \;\Rightarrow\; P(I \ge I_c)\uparrow \;\Rightarrow\; \text{BFR}\uparrow \]
So the power-frequency voltage reduces the critical lightning current and therefore can increase the calculated backflashover rate.
Key messages
- Each phase has its own surge coefficient and its own instantaneous AC voltage, so \(V_{IA}, V_{IB}, V_{IC}\) all change with \(\omega t\).
- A line flashover counts if any phase reaches the CFO; the controlling phase is not fixed in time.
- Between \(I_{cL}\) and \(I_{cH}\) the flashover probability runs from 0 to 1; the exact method integrates \(P_L(I)\) over the current distribution.
- The hand method uses \(K_{PF}V_{LN}\): \(K_{PF} \approx 0.70\) (horizontal), \(0.40\) (vertical) — an engineering approximation, not an exact replacement.
- Higher coupling raises \(I_c\) and lowers BFR (\(K_I = K_T - C K_{TT}\)); the lowest-coupled phase controls the simplified calculation.
- Power-frequency voltage reduces \(I_c\) and can raise BFR — the objective stays \(I_c\uparrow \Rightarrow P(I \ge I_c)\downarrow \Rightarrow \text{BFR}\downarrow\).
