Paschen law breakdown voltage calculator

The Paschen law calculator determines the breakdown voltage V_b for electrical discharge in gas between two parallel plate electrodes per the law formulated by German physicist Friedrich Paschen in 1889 at University of Münster. The classic formula: V_b = (B · p · d) / [ln(A · p · d) − ln(ln(1 + 1/γ))], where p is gas pressure in pascal (or Torr), d electrode distance in meters (or cm), A and B gas-specific constants (air: A = 11.28 (cm·Torr)⁻¹, B = 273.8 V/(cm·Torr)), and γ Townsend secondary emission coefficient (typically 10⁻²-10⁻¹). The critical variable is pressure-distance product pd in pascal·meter or Torr·cm: V_b vs pd curve has a universal minimum ("Paschen minimum") around pd ≈ 0.75 Pa·m (= 0.7 Torr·cm) for air at 327 V. Below this minimum (high vacuum MEMS) voltage rises rapidly due to scarcity of ionizable molecules ("left branch" regime); above it (atmospheric and high pressures) it rises due to inelastic collision saturation ("right branch" regime). Typical Paschen values for various gases at 1 atm and d = 1 cm: air 30 kV/cm; argon 4 kV/cm; helium 2 kV/cm; nitrogen 33 kV/cm;

SF₆ sulfur hexafluoride 89 kV/cm (industry max insulator MT/HV breakers, GWP 23,500 greenhouse gas phasing-out F-gas Reg. EU 517/2014); hydrogen 20 kV/cm; mercury vapor 4 kV/cm; neon Ne 1.4 kV/cm (noble gas minimum, neon signs). Italian industrial applications:

• engine spark plugs (Magneti Marelli Corbetta Milan 1919, NGK-NTK Italy, gap 0.7-1.1 mm × 80-120 kPa Otto cylinder = pd 0.5-1.3 Pa·m, V_b 5-30 kV)
• neon signs (Lampade Neon Italia 1932, p ≈ 0.5-3 Torr Ne, V_b 100-400 V)
• high-voltage SF₆ breakers (ABB SACE Bergamo 1948, Schneider Electric Crema 1885, Hitachi Energy Genoa ex-ASGEN 1925)
• X-ray generators (Esaote Genoa 1980, Saes Pure Gas Lainate Milan 1940)
• plasma fusion reactors (Consortium RFX Padua 1996 reverse field experiment, ITER Cadarache Italian contribution ENEA Frascati 1960, DTT Divertor Tokamak Test Frascati 2024)
• MEMS (STMicroelectronics Agrate Brianza 1987 - micron gaps, anomalous sub-Paschen regime). Italian high-voltage labs: CESI Centro Elettrotecnico Sperimentale Italiano Milan 1956 (800 kV AC + 1,200 kV DC test facility), Polytechnic of Turin 1859 energy department, Polytechnic of Milan 1863, ENEA Frascati Rome 1960 FTU tokamak 1990-2020 + DTT 2028, INFN Frascati National Labs 1957 (DAΦNE φ-factory accelerator), Consortium RFX Padua 1996 + SPIDER ITER injector + MITICA NBI 1 MV, CNR-IFP Plasma Physics Institute Milan 1971

Italian Paschen gas constant tables

(Source: Friedrich Paschen 1889 Annalen der Physik + Lieberman & Lichtenberg "Principles of Plasma Discharges" Wiley 2005 + CEI EN 60060-1:2010 + IEEE Std 4-2013 + ENEA Frascati 1960).

Italian Paschen calculation examples

• Fiat 500 1957 spark plug Magneti Marelli Corbetta Milan: electrode gap 0.7 mm = 0.07 cm; cylinder end-compression pressure 1:8.5 ratio → 850 kPa = 6,375 Torr. pd = 6,375 × 0.07 = 446 Torr·cm (right branch saturation). V_b = (273.8 × 446) / [ln(11.28 × 446) − ln(ln(1 + 1/0.01))] = 122,139 / [8.52 − 1.53] = 17.5 kV required (Magneti Marelli Champion 1947 transformer ignition system supplies 25-35 kV margin 1.5×)
• "TRATTORIA NONNA MARIA" neon sign Bologna Emilia-Romagna 1,200 mm Ne tube: neon pressure 1.5 Torr; electrode distance 1,200 mm = 120 cm. pd = 1.5 × 120 = 180 Torr·cm. V_b = 1,700-2,500 V (saturated right branch). 9-12 kV start neon transformer, 5 kV maintenance (Lampade Neon Italia Salerno 1932)
• 145 kV SF₆ breaker ABB SACE Bergamo 1948 Terna Rome substation: SF₆ pressure 6 bar = 4,500 Torr; nominal open contact distance d = 5 cm. pd = 22,500 Torr·cm. SF₆ exceeds air 100×, V_b operative 145 kV margin 4×
• STMicroelectronics Agrate Brianza Milan 1987 MEMS gyroscope iPhone 15: clean air gap p = 760 Torr × d = 5 µm = 5×10⁻⁴ cm. pd = 0.38 Torr·cm (sub-Paschen-minimum left regime). Reduced V_b ~150 V "modified Townsend" field emission
• RFX Padua 1996 plasma fusion experiment chamber: hydrogen-deuterium p = 0.1 Pa = 7.5×10⁻⁴ Torr; machine radius d = 0.46 m = 46 cm. pd = 0.034 Torr·cm (extreme left branch). RF 7.2 MHz + 2 MW NBI required
• Esaote Genoa 1980 dental X-ray tube: residual vacuum 10⁻⁷ Torr × d 5 cm = 5×10⁻⁷ Torr·cm. Pure vacuum regime: tungsten W cathode 2,500°C thermionic emission, V_acc 60-90 kV independent of Paschen

Italian high-voltage and insulating gas norms

Italian high-voltage and insulating gas norms:

• CEI EN 60060-1:2010 "High-voltage test techniques - General definitions and test requirements" + CEI EN 60060-2:2014 measurements + CEI EN 60060-3:2008 site testing
• IEC 60270:2015 partial discharge measurement techniques + IEEE Std 4-2013 American standard
• CEI 0-21:2024 + CEI 0-16:2022 LV/MV grid distributor connection rules (ENEL Italy 1962, Terna SpA Rome 1999 national dispatcher, Areti Rome, Unareti Milan, A2A Reti Brescia)
• Reg. EU 517/2014 F-gas phasing-out fluorinated greenhouse gases: SF₆ GWP 23,500 reduced 79% by 2030 (replaced with C₄F₇N Novec 4710 3M, CO₂ technology, clean air Hitachi Energy "EconiQ" Genoa ex-ASGEN 1925)
• D.Lgs. 81/2008 art. 80-87 electrical risk + PPE installer insulating gloves CEI EN 60903 class 3 (26.5 kV) + insulating rubber mats CEI EN 61111
• DPR 462/2001 grounding periodic verifications: industrial 2 years high fire risk MARCI. Italian high-voltage research: CESI Italian Experimental Electrotechnical Center Milan 1956 (Maggiore SpA, 800 kV AC, 1,200 kV DC, 4 MV impulse facility); Polytechnic of Turin 1859 Galileo Ferraris energy lab; Polytechnic of Milan Bovisa 1863 plasma energy; ENEA Frascati Rome 1960 FTU tokamak 1990-2020 + DTT Divertor Tokamak Test 2028 (€600M fusion research); INFN Frascati National Laboratories 1957 DAΦNE accelerator + Frascati electron-positron; Consortium RFX Padua 1996 Reverse Field eXperiment + SPIDER ITER injector + MITICA NBI 1 MV; CNR-IFP Plasma Physics Institute Milan 1971; Galileo Avionica Turin 1986 (Leonardo group). Italian gas insulator industry: ABB SACE Bergamo 1948 SF₆ MV breakers 24-36 kV, HV 145-550 kV; Hitachi Energy Genoa ex-ASGEN 1925 EconiQ clean air HV breakers; Schneider Electric ex-Cabur Crema 1885; Magneti Marelli Corbetta Milan 1919 ignition spark plugs (Marelli Holdings group 2019); STMicroelectronics Agrate Brianza Milan 1987 MEMS sub-Paschen; Esaote Genoa 1980 X-ray, MRI; SOL Industrial Gases Monza Brianza 1927 nitrogen, argon, helium; Saes Pure Gas Lainate Milan 1940 high-purity vacuum gases. Italian R&D tax bonuses 2025: R&D Tax Credit art. 1 c. 200 L. 160/2019 up to 25% innovative spend, max €4M/year; Patent Box L. 178/2020 + Industry 4.0 + Industry 5.0 lab renovation; Decreto Sostegni-bis DL 73/2021 university and CNR research

Updated July 2026