“Paschen's law”的意思、由来-开放百科全书

Paschen studied the breakdown voltage of various gases between parallel metal plates as the gas pressure and gap distance were varied:

For a given gas, the voltage is a function only of the product of the pressure and gap length.^[1]^[2] The curve he found of voltage versus the pressure-gap length product (right) is called Paschen's curve. He found an equation that fit these curves, which is now called Paschen's law.^[2]

At higher pressures and gap lengths, the breakdown voltage is approximately proportional to the product of pressure and gap length, and the term Paschen's law is sometimes used to refer to this simpler relation.^[4] However, this is only roughly true, over a limited range of the 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 minimal distance between the electrodes. For example, in air, at a pressure of one atmosphere, the distance for minimal breakdown voltage is about 7.5 µm. 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 3.5 µm 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 µm distance, but would arc at a 7.5 µm distance.

where

is the breakdown voltage in volts,

is the pressure in pascals,

is the gap distance in meters,

is the secondary-electron-emission coefficient (the number of secondary electrons produced per incident positive ion),

is the saturation ionization in the gas at a particular

(electric field/pressure), and

is related to the excitation and ionization energies.

The constants

and

are determined experimentally and found to be roughly constant over a restricted range of

for any given gas. For example, air with an

in the range of 450 to 7500 V/(kPa·cm),

= 112.50 (kPa·cm)⁻¹ and

= 2737.50 V/(kPa·cm).^[6]

The graph of this equation is the Paschen curve. By differentiating it with respect to

and setting the derivative to zero, the minimal voltage can be found. This yields

and predicts the occurrence of a minimal 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 minimal arc voltage and the distance at which it occurs. For argon, the minimal arc voltage is 137 V at a larger 12 µm. For sulfur dioxide, the minimal arc voltage is 457 V at only 4.4 µm.

Long gaps

For air at standard conditions for temperature and pressure (STP), the voltage needed to arc a 1-meter gap is about 3.4 MV.^[7] The intensity of the electric field for this gap is therefore 3.4 MV/m.

The electric field needed to arc across the minimal-voltage gap is much greater than what is 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^[8]

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.

Physical mechanism

The mean free path of a molecule in a gas is the average distance between its collision with other molecules. This is inversely proportional to the pressure of the gas. In air the mean free path of molecules is about 96 nm. Since electrons are much smaller, their average distance between colliding with molecules is about 5.6 times longer, or about 0.5 µm. This is a substantial fraction of the 7.5 µm spacing between the electrodes for minimal arc voltage. If the electron is in an electric field of 43 MV/m, it will be accelerated and acquire 21.5 eV of energy in 0.5 µm of travel in the direction of the field. The first ionization energy needed to dislodge an electron from nitrogen molecule is about 15.6 eV. The accelerated electron will acquire more than enough energy to ionize a nitrogen molecule. This liberated electron will in turn be accelerated, which will lead to another collision. A chain reaction then leads to avalanche breakdown, and an arc takes place from the cascade of released electrons.^[9]

More collisions will take place in the electron path between the electrodes in a higher-pressure gas. When the pressure–gap product

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 and is decelerated by the field.

Collisions reduce the electron's energy and make it more difficult for it to ionize a molecule. Energy losses from a greater number of collisions require larger voltages for the electrons to accumulate sufficient energy to ionize many gas molecules, which is required to produce an avalanche breakdown.

On the left side of the Paschen minimum, the

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 have fewer ionizing collisions. A greater voltage is therefore required to assure ionization of enough gas molecules to start an avalanche.

Derivation

Basics

To calculate the breakthrough voltage, a homogeneous electrical field is assumed. This is the case in a parallel-plate capacitor setup. The electrodes may have the distance

. The cathode is located at the point

To get impact ionization, the electron energy

must become greater than the ionization energy

of the gas atoms between the plates. Per length of path

a number of

ionizations will occur.

is known as the first Townsend coefficient as it was introduced by Townsend^[10]. The increase of the electron current

, can be described for the assumed setup as

(So the number of free electrons at the anode is equal to the number of free electrons at the cathode that were multiplied by impact ionization. The larger

and/or

, the more free electrons are created.)

Neglecting possible multiple ionizations of the same atom, the number of created ions is the same as the number of created electrons:

is the ion current. To keep the discharge going on, free electrons must be created at the cathode surface. This is possible because the ions hitting the cathode release secondary electrons at the impact. (For very large applied voltages also field electron emission can occur.) Without field emission, we can write

where

is the mean number of generated secondary electrons per ion. This is also known as the second Townsend coefficient. Assuming that

, one gets the relation between the Townsend coefficients by putting (4) into (3) and transforming:

Impact ionization

What is the amount of

? The number of ionization depends upon the probability that an electron hits a gas molecule. This probability

is the relation of the cross-sectional area of a collision between electron and ion

in relation to the overall area

that is available for the electron to fly through:

As expressed by the second part of the equation, it is also possible to express the probability as relation of the path traveled by the electron

to the mean free path

(distance at which another collision occurs).

The adjoining sketch illustrates that

. As the radius of an electron can be neglected compared to the radius of an ion

it simplifies to

. Using this relation, putting (7) into (6) and transforming to

one gets

The alteration of the current of not yet collided electrons at every point in the path

can be expressed as

According to its definition

is the number of ionizations per length of path and thus the relation of the probability that there was no collision in the mean free path of the ions, and the mean free path of the electrons:

It was hereby considered that the energy

that a charged particle can get between a collision depends on the electric field strength

and the charge

Breakdown voltage

For the parallel-plate capacitor we have

, where

is the applied voltage. As a single ionization was assumed

is the elementary charge

. We can now put (13) and (8) into (12) and get

Putting this into (5) and transforming to

we get the Paschen law for the breakdown voltage

that was first investigated by Paschen in^[11] and whose formula was first derived by Townsend in,^[12] section 227:

Plasma ignition

Plasma ignition in definition of Townsend (Townsend discharge) is a self-sustaining discharge, independent of an external source of free electrons. This means that electrons from the cathode can reach the anode in the distance

and ionize at least one atom on their way. So according to the definition of

this relation must be fulfilled:

Conclusions, validity

Effects with different gases

Different gases will have different mean free paths for molecules and electrons. This is because different molecules have different diameters. Noble gases like helium and argon are monatomic and tend to have smaller diameters. This gives them greater mean free paths.

Ionization potentials differ between molecules, as well as the speed that they recapture electrons after they have been knocked out of orbit. All three effects change the number of collisions needed to cause an exponential growth in free electrons. These free electrons are necessary to cause an arc.

See also

References

1. ^¹{{cite web | title = Paschen's Law | website = Merriam-Webster Online Dictionary | publisher = Merriam-Webster, Inc. | date = 2013 | url = http://www.merriam-webster.com/dictionary/paschen's%20law | doi = | accessdate =June 9, 2017}}
2. ^¹²{{cite book | last = Wadhwa | first = C.L. | title = High Voltage Engineering | edition = 2nd | publisher = New Age International | date = 2007 | location = | pages = 10–12 | url = https://books.google.com/?id=4rQu1M0sjRAC&pg=PA10&dq=paschen's+law#v=onepage&q=paschen's%20law&f=false | doi = | id = | isbn = 978-8122418590}}
3. ^{{cite journal | title = Ueber die zum Funkenübergang in Luft, Wasserstoff und Kohlensäure bei verschiedenen Drucken erforderliche Potentialdifferenz (On the potential difference required for spark initiation in air, hydrogen, and carbon dioxide at different pressures) | author = Friedrich Paschen | journal = Annalen der Physik | volume = 273 | issue = 5 | pages = 69–75 | date = 1889 | url = | doi = 10.1002/andp.18892730505|bibcode = 1889AnP...273...69P | hdl = 2027/uc1.$b624756 }}
4. ^{{cite book | last = Graf | first = Rudolf F. | title = Modern Dictionary of Electronics | edition = 7th | publisher = Newnes | date = 1999 | location = | page = 542 | url = https://books.google.com/books?id=uah1PkxWeKYC&pg=PA542&dq=%22paschen's+law | doi = | id = | isbn = 978-0750698665}}
5. ^¹{{cite book|title=Principles of plasma discharges and materials processing|last1=Lieberman|first1=Michael A.|last2=Lichtenberg|first2=Allan J.|publisher=Wiley-Interscience| location=Hoboken, N.J.|at=546|isbn=978-0471005773|oclc=59760348|edition=2nd|date=2005|url=http://www.slideshare.net/elquseooso/principles-of-plasma-discharges-and-materials-processing-2nd-edition}}
6. ^{{cite journal|last1=Husain|first1=E.|last2=Nema|first2=R.|title=Analysis of Paschen Curves for air, N2 and SF6 Using the Townsend Breakdown Equation|journal=IEEE Transactions on Electrical Insulation|date=August 1982|volume=EI-17|issue=4|pages=350–353|doi=10.1109/TEI.1982.298506}}
7. ^{{cite book | last = Tipler | first = Paul | title = College physics | publisher = Worth Publishers | location = New York, NY | year = 1987 | isbn = 978-0879012687 |p=467 }}
8. ^{{cite journal | title = Submicron gap capacitor for measurement of breakdown voltage in air | author = Emmanouel Hourdakis | author2 = Brian J. Simonds | author3 = Neil M. Zimmerman | last-author-amp = yes | journal = Rev. Sci. Instrum. | volume = 77 | issue = 3 | pages = 034702–034702–4 | date = 2006 | doi = 10.1063/1.2185149 |bibcode = 2006RScI...77c4702H | url = https://zenodo.org/record/1231939 }}
9. ^Electrical Discharges-How the spark, glow and arc work.
10. ^J. Townsend, [The Theory of Ionization of Gases by Collision http://www.worldcat.org/wcpa/oclc/8460026]. Constable, 1910. Section 17.
11. ^{{cite journal | last1 = Paschen | first1 = F. | year = 1889 | title = Ueber die zum Funkenübergang in Luft, Wasserstoff und Kohlensäure bei verschiedenen Drucken erforderliche Potentialdifferenz | url = | journal = Annalen der Physik | volume = 273 | issue = 5| pages = 69–96 | doi = 10.1002/andp.18892730505 | bibcode=1889AnP...273...69P| hdl = 2027/uc1.$b624756 }}
12. ^J. Townsend, Electricity in Gases. Clarendon Press, 1915. Online: http://www.worldcat.org/wcpa/oclc/4294747