Transients and insulation
Uniform-field air breakdown (Paschen, Kuffel, 3 kV/mm)
- where
- pressure reduced to 20 °C (kPa)
- atmospheric pressure (kPa)
- air temperature (°C)
- breakdown voltage, Kuffel empirical (kV), pd in bar·cm
- gap length (mm)
- breakdown voltage, Paschen (kV)
- 365 V/(cm·Torr) for air
- pressure × gap at 20 °C (Torr·cm) in the Paschen equation
- 15 /(cm·Torr) for air
- secondary emission coefficient
- 3 kV/mm rule of thumb (kV)
Explain this equation
Gives the breakdown voltage of a uniform-field air gap three ways: Kuffel’s empirical fit, Paschen’s law, and the 3 kV/mm rule. Use it to understand gap strength for plates, large spheres at small spacing, and test set-ups.
p₂₀ is the pressure corrected to 20 °C, so it represents gas density. d is the gap. In the empirical form pd is in bar·cm. In Paschen’s law pd is in Torr·cm, A and B describe how readily electrons ionise air, and γ is the secondary emission of electrons from the cathode.
Breakdown depends on the number of ionising collisions across the gap, which depends on gas density times distance, hence pd. Paschen’s form comes from the Townsend criterion; the 6.72√(pd) + 24.36pd fit tracks measured data much better around atmospheric pressure. The ÷1000 gives kV.
Sense check: a 10 mm gap at 1 bar and 20 °C breaks down at about 31 kV. Paschen’s constant-coefficient form gives about 35 kV, and warns because its A and B are fitted at higher E/p.
Watch out: real equipment gaps are non-uniform and can break down at a fraction of these values.
Breakdown voltage of a uniform-field air gap (parallel plates or large spheres at small spacing) from Paschen’s law and from Kuffel’s empirical fit to measured data, with the 3 kV/mm rule of thumb.
Enter your values to see the result.
Good to know
Pressure is reduced to 20 °C (density correction) before use. The empirical form 6.72√(pd) + 24.36 pd kV (pd in bar·cm at 20 °C) fits measurements closely from about 0.01 to several bar·cm and is the better estimate at atmospheric pressure. Paschen’s law with constant A = 15 /(cm·Torr), B = 365 V/(cm·Torr) and γ ≈ 0.01 over-estimates at around 1 bar·cm because A and B are only fitted for E/p 100–800 V/(cm·Torr). 3 kV/mm is a 1 cm-gap rule; the strength per mm rises for smaller gaps and falls towards about 2.4 kV/mm for long uniform gaps.
Common mistake
Applying uniform-field values or 3 kV/mm to rod, point or conductor-to-structure gaps, whose non-uniform field breakdown can be several times lower; use IEC 60071-2 gap factors and standard withstand data instead.
Limits
Uniform field, dry air, DC or power-frequency peak, no humidity correction. Not valid for non-uniform fields, very small pd near the Paschen minimum (about 330 V) or for altitude-corrected withstand of real insulation (see IEC 60071-2 and IEC 60060-1 for atmospheric correction).
Sources
- Kuffel, E., Zaengl, W.S. & Kuffel, J. High Voltage Engineering Fundamentals, 2nd ed. (Newnes, 2000)
- Naidu, M.S. & Kamaraju, V. High Voltage Engineering (McGraw-Hill Education)
- Paschen, F. Ueber die zum Funkenübergang in Luft, Wasserstoff und Kohlensäure bei verschiedenen Drucken erforderliche Potentialdifferenz. Annalen der Physik 273 (1889) 69–96
For competent high-voltage engineers. Results are calculations from the published methods shown and depend entirely on the data you enter. They are not a substitute for a system study, manufacturer data, the applicable standards and Engineering Recommendations, or your organisation’s safety rules. Never use a result to justify working on or near live HV equipment.