Engineering & Science
Paschen's Law: Gas Breakdown Voltage (Vb) & Minimum Paschen (p*d) Curve
Calculate Townsend DC gas breakdown voltage Vb = (B * p * d) / (LN(A * p * d) - LN(LN(1 + 1 / gamma_se))) and the minimum breakdown product (p * d)_min = (EXP(1) / A) * LN(1 + 1 / gamma_se).
Prevent high-voltage spacecraft, vacuum-chamber, and switchgear arcing!
High-voltage engineers designing semiconductor plasma-etch chambers, satellite power buses, and SF6/air switchgear quickly discover Friedrich Paschen's counterintuitive 1889 law: breakdown voltage Vb depends ONLY on the product of gas pressure p and electrode gap d (p * d, in Torr*cm)—and Vb shoots toward infinity BOTH at high pressure-distance (frequent collisions prevent electrons from gaining ionization energy) AND at ultra-low vacuum p * d (too few gas molecules to form a Townsend electron avalanche)!
Combining Townsend's first ionization coefficient alpha/p = A * EXP(-B / (E/p)) with cathode secondary-electron yield gamma_se gives Vb = (B * pd) / (LN(A * pd) - LN(LN(1 + 1 / gamma_se)))! Differentiating with respect to pd locates the dangerous Paschen minimum at (pd)_min = (EXP(1) / A) * LN(1 + 1 / gamma_se), where Vb_min = (EXP(1) * B / A) * LN(1 + 1 / gamma_se)! For Argon (A = 11.5 Torr^-1*cm^-1, B = 176 V/(Torr*cm), gamma_se = 0.07), minimum breakdown occurs at (pd)_min = 0.645 Torr*cm, rising to Vb = 143.12 V at pd = 1.5 Torr*cm and 470.37 V at pd = 10 Torr*cm!
| Argon Gap Regime (A=11.5, B=176, gamma=0.07) | Pressure * Gap (Torr*cm) | Breakdown Voltage Vb | Townsend Physics |
|---|---|---|---|
| Below Minimum (Vacuum Left Branch) | pd <= 0.237 Torr*cm | No Townsend Breakdown! | Denominators <= 0 -> Vacuum Holdoff |
| Paschen Curve Minimum | pd_min = 0.645 Torr*cm | Vb_min = 113.52 V | Easiest Glow-Discharge Ignition |
| Near-Minimum Plasma Chamber | pd = 1.500 Torr*cm | 143.12 V | Right-branch glow regime |
| Higher Pressure Switch | pd = 10.00 Torr*cm | 470.37 V | Linear high-pd climb |
Paschen DC Gas Breakdown Voltage Vb (Volts)
Guards against the left-hand vacuum asymptote where LN(A * pd) <= LN(LN(1 + 1 / gamma_se)), then evaluates Vb.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, LN(B2 * B5) <= LN(LN(1 + (1 / B4)))), "Below Vacuum Breakdown Asymptote", ROUND((B3 * B5) / (LN(B2 * B5) - LN(LN(1 + (1 / B4)))), 2))Paschen Minimum Pressure-Distance Product (p * d)_min (Torr * cm)
Computes (EXP(1) / A) * LN(1 + 1 / gamma_se) where Vb reaches its absolute minimum.
=IF(OR(B2 <= 0, B4 <= 0), "Invalid Gas Constants", ROUND((EXP(1) / B2) * LN(1 + (1 / B4)), 3))Try the example
Use the original values or change them to test your understanding.