Section 1
From Phase-Ground to Phase-Phase Stress
Earlier switching-surge notes treated the insulation stress between a single phase conductor and the grounded tower — phase-ground flashover, controlled by the phase-ground strike distance and the strength-stress ratio \(V_3/E_2\). This note moves the stress across the gap: between phase conductor A and phase conductor B. The design question is the mirror image of the phase-ground one.
Two problems are solved with the same machinery: either calculate the phase-phase SSFOR (switching surge flashover rate) for a given clearance, or, given an acceptable SSFOR, calculate the required phase-phase clearance \(S_p\). The complication — and the whole reason phase-phase needs its own treatment — is that for long air gaps the strength does not depend on the phase-phase voltage alone, but on how that voltage is split between the two conductors.
Section 2
When Phase-Phase Clearance Has to Be Checked
Not every tower needs a separate phase-phase check. In most conventional towers, grounded steel members sit between the phases, so a surge will flash phase-to-ground before phase-to-phase — the phase-ground clearance controls and the phase-phase gap never gets the chance to break down. The phase-phase check only becomes live when only air separates the phase conductors.
Table 1 — Where the phase-phase air gap can control.
| Configuration | Phase-Phase Path | Check Needed? |
|---|
| Conventional tower (grounded steel between phases) | Blocked by earthed steel | No — phase-ground controls |
| Low-profile double-circuit towers | Air only | Yes |
| German “Delta” type towers | Air only | Yes |
| Chainette towers | Air only | Yes |
| Compact / conceptual arrangements | Air only | Yes |
| Substations (bus, apparatus terminals, breakers, bushings, GIS/AIS) | Air / internal | Always |
Substations: always consider phase-phase
Phase-phase insulation exists naturally between busbars, conductors, apparatus terminals, disconnectors, circuit-breaker terminals, transformer bushings and GIS/AIS clearances. This note is framed around transmission lines, but the same concept is central to substation insulation coordination — covered in the next note.
Section 3
The Voltage Split: Vₚ = V⁺ + V⁻
The phase-phase test applies opposite-polarity switching impulses to the two conductors: one receives \(V_+\), the other \(V_-\) (treated as a positive magnitude). The conductors are stressed by the sum:
\[ V_p = V_+ + V_- \qquad\text{e.g.}\quad 900 + 600 = 1500\ \text{kV} \]
- \(V_p\)
- total phase-phase voltage across the gap
- \(V_+,\ V_-\)
- positive and (magnitude of) negative conductor voltages
For internal insulation (e.g. a transformer) only the magnitude \(V_p\) matters. For long external air gaps the split between \(V_+\) and \(V_-\) also matters — two cases with the same \(V_p\) need not have the same flashover probability:
\[ \underbrace{V_+ = 750,\; V_- = 750}_{\text{Case A}} \qquad \underbrace{V_+ = 1200,\; V_- = 300}_{\text{Case B}} \qquad V_p = 1500\ \text{kV (both)} \]
Positive and negative components drive discharge development differently, so for large clearances Case A and Case B can have different strengths. That single fact is why phase-phase air insulation needs more than “line voltage” — and why two test methods exist to describe it.
Section 4
The Alpha Method
In the alpha method the ratio of negative voltage to total phase-phase voltage is fixed, and \(V_p\) is then varied to trace the flashover-probability curve:
\[ V_+ = (1-\alpha)\,V_p \qquad V_- = \alpha\,V_p \qquad 0 \le \alpha \le 1 \]
- \(\alpha = 0.5\)
- equal split, \(V_+ = V_-\)
- \(\alpha = 0.33\)
- \(V_+ \approx 2V_-\) — the two commonly tested values
With \(\alpha\) held constant, the resulting 50% value is the phase-phase critical flashover voltage \(\text{CFO}_p\) (with scatter \(\sigma_{fp}\)), expressed through a phase-phase gap factor \(k_{gp}\) on the familiar Gallet form:
\[ \text{CFO}_p = k_{gp}\,\frac{3400}{1 + 8/S_p} \]
- \(S_p\)
- phase-phase strike distance (m)
- \(k_{gp}\)
- phase-phase gap factor (differs from the phase-ground value)
Plotting \(\text{CFO}_p\) (or \(k_{gp}\)) against \(\alpha\) gives an approximately straight line — the phase-phase strength changes almost linearly with the voltage division. The gap factor itself depends on geometry, conductor length and height, \(\alpha\), and whether the gap is conductor–conductor, ring–ring, bus fittings, etc.
Section 5
The V⁺/V⁻ Method
The second method holds \(V_-\) constant and varies \(V_+\) until flashover. For each fixed negative voltage it finds the positive voltage required for a 50% phase-phase flashover, \(\text{CFO}_+\), which falls along an approximately straight line:
\[ \text{CFO}_+ = \text{CFO}_0 - K_L\,V_- \]
- \(\text{CFO}_+\)
- positive voltage for 50% phase-phase flashover
- \(\text{CFO}_0\)
- positive CFO when \(V_- = 0\) (the other phase grounded)
- \(K_L\)
- slope — how strongly \(V_-\) contributes
The physical reading is direct: if the other conductor already carries a negative voltage, less positive voltage is needed to flash over. As \(V_-\) rises, \(\text{CFO}_+\) falls — but only one-for-one if \(K_L = 1\). For long air gaps \(K_L < 1\), so the negative conductor contributes, but not fully.
When \(V_- = 0\) the second conductor is effectively grounded, so \(\text{CFO}_0\) is really a phase-ground-type CFO whose “ground” is another conductor:
\[ \text{CFO}_0 = K_{Gp}\,\frac{3400}{1 + 8/S_p} \]
- \(K_{Gp}\)
- gap factor with the second phase grounded
- \(S_p\)
- phase-phase strike distance (m)
Why the V⁺/V⁻ form wins
Many older tests used the alpha method, and its results convert into \(K_L\) and \(K_{Gp}\). Both methods describe the same physical insulation. But the \(\text{CFO}_+ = \text{CFO}_0 - K_L V_-\) form is far more convenient for combined phase-phase / phase-ground probabilistic SSFOR work — it is the form used for the rest of this note.
Section 6
What Kₗ Means — and the Transmission-Line Values
\(K_L\) measures how much the negative-polarity conductor reduces the positive CFO. It separates two regimes:
- \(K_L \approx 1\) — only the total \(V_p\) matters. Typical of internal insulation, small air gaps below about 2–3 m, and roughly systems below 500 kV.
- \(K_L < 1\) — the split between \(V_+\) and \(V_-\) matters. Typical of long external air clearances, EHV lines, and large conductor–conductor gaps.
Real transmission-line phase-phase insulation is a long conductor–conductor air gap, not a short laboratory gap. For spans around 300–400 m the recommended, test-based values are:
Table 2 — Recommended phase-phase parameters, transmission-line 300–400 m spans.
| Parameter | Value | Role |
|---|
| Slope factor \(K_L\) | \(\approx 0.68\) | Weight of \(V_-\) in the equivalent stress |
| Gap factor \(K_{Gp}\) | \(\approx 1.26\) | Sets \(\text{CFO}_0\) from \(S_p\) |
| Strength scatter \(\sigma_{pp}/\text{CFO}_0\) | \(\approx 0.02\) | Low scatter for long spans |
Long spans show notably lower scatter than short gaps: where simple gaps may scatter 4–9% (≈6% average), conductor–conductor spans of 300–400 m sit near 2–3%. The low \(\sigma_{pp}/\text{CFO}_0 \approx 0.02\) is important for the probabilistic calculation later.
Section 7
Practical Effects: Rain, Wavefront, Third Phase, Timing
Four real-world effects modify the idealised picture — most can be dismissed for the cases that drive SSFOR, but each should be checked:
- Rain / wet: for a pure air phase-phase gap, wet conditions do not significantly reduce strength. If vertical or I-string insulators form part of the flashover path, strength may fall by about 12%.
- Critical wavefront: the most severe front time scales with clearance, found between \(25S_p\) and \(30S_p\) µs (\(S_p\) in m) — e.g. \(S_p = 5\) m ⇒ ~125–150 µs.
- Third phase: usually a minor effect — for the highest SOVs its voltage is much smaller than the negative second phase, so a two-phase model is acceptable.
- Crest-time delay \(\Delta T\): if the negative impulse leads the positive by several ms, CFO may fall 10–15%; if positive leads, no reduction. For the highest SOVs (which dominate SSFOR) the delay is small, so it is normally ignored.
Do not apply the wet correction
Because phase-phase line insulation is treated as pure air, the 0.96 (4%) wet-condition reduction used for phase-ground tower insulation is not applied: \(\text{CFO}_{wet} \approx \text{CFO}_{dry}\). Altitude correction still applies.
Section 8
Phase-Phase and Phase-Ground Coexist
In a real tower both insulations exist at once: a conductor may flash to another phase or to grounded steel. On a \(V_+\)–\(V_-\) diagram this is bounded by three strength lines — positive phase-ground CFO, the phase-phase CFO line, and negative phase-ground CFO. For low and high \(V_-\) the flashover tends to go to ground; for intermediate \(V_-\) it tends to go phase-to-phase.
The controlling mode is a design choice
Phase-phase flashover is essentially eliminated if the phase-ground strike distance is made small enough that phase-ground flashes first; conversely phase-ground flashover is eliminated by increasing its strike distance. The relative clearances decide which mode controls — a key lever in compact-line design.
Combining the two modes uses the same independence logic as multiple towers. With \(p_p\), \(p_g\) the phase-phase and phase-ground flashover probabilities:
\[ P(\text{either}) = 1 - (1 - p_p)(1 - p_g) \]
In practice the two can simply be added. With typical numbers the combined and summed results agree within a few percent:
\[ \text{SSFOR}_{pp} + \text{SSFOR}_{pg} = \tfrac{1.05}{100} + \tfrac{1.14}{100} = \tfrac{2.19}{100} \;\approx\; \text{SSFOR}_{total} = \tfrac{2.06}{100} \]
Two further simplifications hold for transmission lines: the negative phase-ground strength is much higher than positive, so its flashover probability is negligible; and the theoretically correct reversed-parameter pass (swapping the roles of the positive and negative SOVs) adds only a small contribution and is usually neglected.
Section 9
The Key Simplification: One Equivalent Stress Vₘ
Phase-phase stress has two random variables, \(V_+\) and \(V_-\) — awkward for a probabilistic calculation. The breakthrough is to collapse them into one. Flashover occurs when \(V_+ > \text{CFO}_0 - K_L V_-\), i.e. \(V_+ + K_L V_- > \text{CFO}_0\). Define the equivalent stress:
\[ V_z = V_+ + K_L\,V_- \qquad\Longrightarrow\qquad \text{flashover when } V_z > \text{CFO}_0 \]
Since \(V_- = V_p - V_+\), the same quantity can be written from the two values an EMT study most naturally yields — \(V_+\) and \(V_p\):
\[ V_z = (1 - K_L)\,V_+ + K_L\,V_p \]
- \(V_z\)
- equivalent phase-phase switching stress
- \(K_L\)
- 0.68 for transmission-line conductor–conductor gaps
The whole problem in one line
The phase-phase problem becomes a one-variable comparison: stress \(V_z\) versus strength \(\text{CFO}_0\) — exactly the shape of the phase-ground problem, so Brown’s method can be adapted directly.
Section 10
Collecting the Phase-Phase Stress
Three voltages are relevant — \(V_+\), \(V_-\) and \(V_p = V_+ + V_-\) — and any two give the third. The current practice is to collect \(V_+\) (already needed for phase-ground SSFOR) and \(V_p\), then take \(V_- = V_p - V_+\). Treating them as Gaussian, \(V_z\) is also Gaussian with:
\[ \mu_z = (1 - K_L)\,\mu_+ + K_L\,\mu_p \]
\[ \sigma_z = \sqrt{(1-K_L)^2\sigma_+^2 + K_L^2\,\sigma_p^2 + 2K_L(1-K_L)\,\rho_{p+}\,\sigma_p\,\sigma_+} \]
- \(\rho_{p+}\)
- correlation between \(V_p\) and \(V_+\) (they are not independent)
EMTP® warning — maxima at different instants
A transient program readily reports the maximum phase-ground voltage and the maximum phase-phase voltage, but these generally occur at different time instants. Combining \(V_+\) from one instant with \(V_p\) from another is technically inconsistent — though conservative, since it pairs the worst of each. For rigour, collect both at the same instant; for a quick bound, evaluate SSFOR at the time of maximum positive SOV \(T_+\) and at the time of maximum phase-phase SOV \(T_p\) and take the larger.
Which instant is worse depends on the gap: for large air clearances (\(S > 3\) m, ≈500 kV and above) the maximum positive SOV usually governs; for internal insulation and small clearances the maximum phase-phase SOV usually governs. Where studies are thin, the 2% phase-phase statistical overvoltage \(E_{2p}\) can be estimated from its phase-ground counterpart \(E_2\) using the IEC ratio:
\[ \frac{E_2}{E_{2p}} \approx 1.55 \;\Rightarrow\; E_{2p} \approx 1.55\,E_2 \qquad \frac{\sigma_p}{E_{2p}} \approx \frac{\sigma_0}{E_2} \]
For example \(E_2 = 1.8\) pu gives \(E_{2p} \approx 2.8\) pu; \(E_2 = 2.8\) pu gives \(E_{2p} \approx 4.3\) pu.
Section 11
Phase-Phase SSFOR by Brown’s Method
With strength as \(\text{CFO}_0\) and stress as \(V_z\), the phase-phase SSFOR takes the familiar multi-span form — just substitute \(V_z\) for \(V\), \(\text{CFO}_0\) for the phase-ground CFO, \(\sigma_{pp}/\text{CFO}_0 \approx 0.02\) for the strength scatter, and the \(V_z\) profile \(\gamma_z\):
\[ \text{SSFOR}_p = \tfrac{1}{2}\int f(V_z)\,\bigl[\,1 - q^{\,n}(V_z)\,\bigr]\,dV_z \]
Worked example. Gaussian \(V_+\) and \(V_p\) with \(\gamma = \gamma_+ = 0.90\), \(\rho_{p+} = 0.80\), \(n = 625\), \(1\,\text{pu} = 449\) kV. The equivalent stress works out to:
\[ \mu_z = 2.238\ \text{pu} = 1004.8\ \text{kV} \qquad \sigma_z = 0.272\ \text{pu} = 121.9\ \text{kV} \]
\[ \gamma_z = 0.9,\quad n_e = 50,\quad Z_f = -2.204 \;\Rightarrow\; \text{CFO}_n = 1221.4\ \text{kV} \]
\[ \text{SSFOR}_p = \tfrac{1}{2}P(V_z > 1221.4) = \frac{1.89}{100} \quad(\text{computer: } 1.94/100) \]
The simplified method lands within a few percent of the full computation. Assuming perfect correlation \(\rho_{p+} = 1.0\) is a convenient conservative choice: it raises \(\sigma_z\) to 0.279 pu (126.5 kV) and SSFOR to 2.17/100 (computer 2.22/100).
Correlation is not very sensitive
Moving \(\rho_{p+}\) from 0.7 to 1.0 changes SSFOR by only 7–16% and the required strike distance by ~0.04 m (≈1%). Taking \(\rho_{p+} = 1.0\) is both safe and practical. The reversed-parameter contribution is likewise negligible for line work.
Section 12
The Design Rule and Phase-Phase Strike Distance
The phase-phase strength is characterised by its lower tail \(V_{30}\), and the stress by the 2% value of \(V_z\), called \(E_{2z}\). For transmission-line scatter \(\sigma_{pp}/\text{CFO}_0 = 0.02\):
\[ V_{30} = \text{CFO}_0\Bigl(1 - 3\,\tfrac{\sigma_{pp}}{\text{CFO}_0}\Bigr) = 0.94\,\text{CFO}_0 \]
The phase-phase design rule
For a target of about 1 flashover per 100 operations, set \(V_{30} \approx E_{2z}\) — the phase-phase twin of the phase-ground rule \(V_3 \approx E_2\).
To find the required clearance from a target SSFOR, run the sequence:
Table 3 — From target SSFOR to phase-phase strike distance \(S_p\).
| Step | Compute |
|---|
| 1 | Required \(V_{30}/E_{2z}\) for the target SSFOR (\(\approx 1.0\) at 1/100) |
| 2 | \(V_{30} = (V_{30}/E_{2z})\,E_{2z}\) |
| 3 | \(\text{CFO}_0 = V_{30}/0.94\) |
| 4 | Solve \(\text{CFO}_0 = K_{Gp}\,\dfrac{3400}{1 + 8/S_p}\) for \(S_p\), with \(K_{Gp} = 1.26\) |
| 5 | Apply altitude correction; do not apply wet correction (pure air) |
Section 13
Internal Insulation and Small Air Gaps
For internal insulation — transformers, cables, GIS — strength depends only on the magnitude \(V_p\), not on how it splits. Effectively \(K_L = 1\) and \(\text{CFO}_p = \text{CFO}_0\). The strength is usually a conventional BSL rather than a statistical CFO, so a step model applies (0% below, 100% at/above), and there is no \(\tfrac{1}{2}\) factor because original and reversed polarity give the same \(V_p\):
\[ K_L = 1 \;\Rightarrow\; V_z = V_p \qquad \text{SSFOR} = P(V_p > \text{BSL}) \]
For small external air gaps (below ~2–3 m, roughly <500 kV) \(K_L \approx 1\) again, so \(V_z = V_p\) and \(\text{CFO}_0 = \text{CFO}_p\) — but the external insulation may still be statistical. When calculating a strike distance in this case, enter the calculation with half the desired SSFOR (e.g. use 0.5/100 for a 1.0/100 target) to account for the original and reversed-parameter contributions.
Section 14
EMTP® Workflow and Summary
Putting it together for a modern phase-phase switching-surge design:
EMTP® post-processing workflow
- For each switching case, collect \(V_+\) and \(V_p\) at the same instant.
- Form the equivalent stress \(V_z = (1-K_L)V_+ + K_L V_p\), with \(K_L = 0.68\) for line conductor–conductor gaps.
- Build the distribution of \(V_z\) and read off the 2% value \(E_{2z}\).
- Apply the rule \(V_{30} \approx E_{2z}\) for SSFOR ≈ 1/100, then \(\text{CFO}_0 = V_{30}/0.94\).
- Solve \(\text{CFO}_0 = K_{Gp}\,\tfrac{3400}{1+8/S_p}\) with \(K_{Gp} = 1.26\) for the clearance \(S_p\); altitude-correct, no wet correction.
Key messages
- Phase-phase strength of a long air gap depends on the split of \(V_p\) into \(V_+\) and \(V_-\), not the magnitude alone — captured by \(\text{CFO}_+ = \text{CFO}_0 - K_L V_-\).
- The equivalent stress \(V_z = V_+ + K_L V_-\) reduces a two-variable problem to one variable, so Brown’s method carries straight over.
- The design rule is \(V_{30} \approx E_{2z}\) — the phase-phase twin of \(V_3 \approx E_2\).
- Phase-phase and phase-ground SSFOR can be computed separately and added (within a few percent); relative clearances decide which mode controls.
- For transmission-line air gaps use \(K_L \approx 0.68\), \(K_{Gp} \approx 1.26\), \(\sigma_{pp}/\text{CFO}_0 \approx 0.02\) (so \(V_{30} \approx 0.94\,\text{CFO}_0\)); do not apply the wet correction to pure air.
