Paschen's Law - Paschen Curve

Paschen Curve

Early vacuum experimenters found a rather surprising behavior. An arc would sometimes take place in a long irregular path rather than at the minimum distance between the electrodes. For example, at a pressure of 10-3 atmospheres, the distance for minimum breakdown voltage is about 7.5 mm. The voltage required to arc this distance is 327 V which is insufficient to ignite the arcs for gaps that are either wider or narrower. For a 3.75 mm gap, the required voltage is 533 V, nearly twice as much. If 500 V were applied, it would not be sufficient to arc at the 2.85 mm distance, but would arc at a 7.5 mm distance.

It was found that breakdown voltage was described by the equation:

Where is the breakdown voltage in Volts, is the pressure in Atmospheres or Bar, and is the gap distance in meters. The constants and depend upon the composition of the gas. For air at standard atmospheric pressure of 101 kPa, = 4.36×107 V/(atm·m) and = 12.8.

The graph of this equation is the Paschen curve. By differentiating it with respect to and setting the derivative to zero, the minimum voltage can be found. This yields

and predicts the occurrence of a minimum breakdown voltage for = 7.5×10-6 m·atm. This is 327 V in air at standard atmospheric pressure at a distance of 7.5 µm. The composition of the gas determines both the minimum arc voltage and the distance at which it occurs. For argon, the minimum arc voltage is 137 V at a larger 12 µm. For sulfur dioxide, the minimum arc voltage is 457 V at only 4.4 µm.

For air at STP, the voltage needed to arc a 1 meter gap is about 3.4 MV. The intensity of the electric field for this gap is therefore 3.4 MV/m. The electric field needed to arc across the minimum voltage gap is much greater than that necessary to arc a gap of one meter. For a 7.5 µm gap the arc voltage is 327 V which is 43 MV/m. This is about 13 times greater than the field strength for the 1 meter gap. The phenomenon is well verified experimentally and is referred to as the Paschen minimum. The equation loses accuracy for gaps under about 10 µm in air at one atmosphere and incorrectly predicts an infinite arc voltage at a gap of about 2.7 micrometers. Breakdown voltage can also differ from the Paschen curve prediction for very small electrode gaps when field emission from the cathode surface becomes important.