Paschen's law

Paschen's Law, named after Friedrich Paschen, was first stated in 1889.^[1] He studied the breakdown voltage of parallel plates in a gas as a function of pressure and gap distance. The voltage necessary to arc across the gap decreased up to a point as the pressure was reduced. It then increased, gradually exceeding its original value. He also found that decreasing the gap with normal pressure caused the same behavior in the voltage needed to cause an arc.

Paschen curve

Paschen found that breakdown voltage was described by the equation

.

Where V is the breakdown voltage in volts, p is the pressure, d is the gap distance. The constants a and b depend upon the composition of the gas. For air at standard atmospheric pressure of 101 kPa, a = 43.6×10⁶ V/(atm·m) and b = 12.8 , where p is the pressure in atmospheres and d is the gap distance in meters. ^[2]

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

pd = e^{1 − b}

and predicts the occurrence of a minimum breakdown voltage for pd = 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. With sulfur dioxide, the minimum arc voltage is 457 V at only 4.4 µm.

For air at STP, the intensity of the electric field needed to arc the minimum voltage gap is much greater than that necessary to arc a gap of one meter. For 7.5 µm, the field is 43 MV/m and for one meter it is only 3.4 MV/m. This is about 13 times greater. The phenomenon is well verified experimentally and is referred to as the Paschen minimum. The equation fails for gaps under about ten micrometers in air at one atmosphere ^[3] and incorrectly predicts an infinite arc voltage at a gap of about 2.7 micrometers.

Early vacuum experimenters found a rather surprising behavior. An arc would take place in a long irregular path rather than at the minimum distance between the electrodes. For example, at a pressure of 133 Pa, the distance for minimum breakdown voltage is about 5.7 mm. The voltage required to arc that distance is 327 V and is greater for gaps above and below that point. For a 2.85 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 5.7 mm distance.

This minimum can be understood in terms of the mean free path for electrons in the gas. When the pressure-gap product pd is high, an electron will collide with many different gas molecules as it travels from the cathode to the anode. Each of the collisions randomizes the electron direction, so the electron is not always being accelerated by the electric field -- sometimes it travels back towards the cathode for some time and is decelerated by the field. In this situation, large voltages are required for the electrons to accumulate sufficient energy to ionize gas molecules and produce an avalanche.

On the left side of the Paschen minimum, the pd product is small. The electron mean free path can become long compared to the gap between the electrodes. In this case, the electrons might gain lots of energy, but they often arrive at the anode before getting a chance to bump into a gas molecule and start the avalanche.

Practically speaking, the breakdown voltage can be different from the Paschen curve prediction, for example when field emission from the cathode surface becomes important.

References

External links

• High Voltage Experimenter's Handbook
• Breakdown Voltage vs. Pressure
• Paschen Equation
• Electrical Discharges