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you didnt specify whether you meant high or low voltage. additionally you didnt specify ac or dc. but....

In 1889, F. Pashchen published a paper which set out what has become known as Paschen's Law. The law essentially states that the breakdown characteristics of a gap are a function (generally not linear) of the product of the gas pressure and the gap length, usually written as V= f( pd ), where p is the pressure and d is the gap distance. In actuality, the pressure should be replaced by the gas density.

For air, and gaps on the order of a millimeter, the breakdown is roughly a linear function of the gap length: V = 30pd + 1.35 kV, where d is in centimeters, and p is in atmospheres.

Much research has been done since then to provide a theoretical basis for the law and to develop a greater understanding of the mechanisms of breakdown. It should be realized that there are many, many factors which have an effect on the breakdown of a gap, such as radiation, dust, surface irregularities. Excessive theoretical analysis might help understanding why a gap breaks down, but won't necessarily provide a more accurate value for the breakdown voltage in any given situation.

Paschen's Law reflects the Townsend breakdown mechanism in gases, that is, a cascading of secondary electrons emitted by collisions in the gap. The significant parameter is pd, the product of the gap distance and the pressure. Typically, the Townsend mechanism (and by extension Paschen's law) apply at pd products less than 1000 torr cm, or gaps around a centimeter at one atmosphere. Furthermore, some modifications are necessary for highly electronegative gases because they recombine the secondary electrons very quickly.

In general, an equation for breakdown is derived, and suitable parameters chosen by fitting to empirical data.

Here are three equations:

Breakdown voltage:

Vbreakdown = B * p * d / (C + ln( p * d))

Breakdown field strength:

Ebreakdown = p * ( B / ( C + ln ( p * d)))

where:

C =ln( A / ln ( 1 + 1 / gamma))

where:

gamma is the (poorly known) secondary ionization coefficient.

For air:

A = 15 cm-1Torr -1

B = 365 Vcm-1 Torr-1

and gamma = 10-2

so

C = 1.18

Minimum sparking potential for various gases

Gas Vs min

(V) pd at Vs min

(torr cm)

Air 327 0.567

Ar 137 0.9

H2 273 1.15

He 156 4.0

CO2 420 0.51

N2 251 0.67

N2O 418 0.5

O2 450 0.7

SO2 457 0.33

H2S 414 0.6

Note that the sparking voltage is affected by the electrode material, with cathodes of Barium and Magnesium having higher voltages than Aluminum, for example.

Temperature dependence

Paschen's law ( V = f(pd)) should really be stated as V = f( Nd) where N is the density of gas molecules, which is, of course, affected by the temperature as well as the pressure of the gas ( n/V = p/RT). An empirical formula for air (considering it as an ideal gas) is:

x = 293 * p * d / (760 * T)

Vbreakdown = 24.22 * x + 6.08 * SQRT(x)

p = pressure in Torr (mm Hg),

d = distance in cm,

T = Temperature in Kelvins

Vbreakdown in kV

Humidity dependence

In air, increasing humidity increases the breakdown voltage. The effect is most noticeable in uniform field, and less important in nonuniform gaps (such as sphere gaps where the gap is a large fraction of the sphere diameter, or in rod or needle gaps). An emprical expression for this is:

(not here yet)

Gamma - Townsend's secondary ionization coefficient

Gamma is the net (!) number of secondary electrons produced per incident positive ion, photon, excited or metastable particle. It is a function of gas pressure and E/p. Electronegative gases (SF6, Freon, oxygen, CO2) reattach the electrons very quickly, so they have low gammas.

For nitrogen, gamma ranges between 10-3 and 10-2 for E/p of 100-700 V cm-1 torr-1. Insulating gases like SF6 or Freon have gammas of 10-4 or even less.

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