HIGH VOLTAGE ENGINEERING - UNIT 1

Electrical Breakdown of Gases in Uniform Fields

Introduction

At normal temperature and pressure, gases are excellent insulators but ‘background

currents’ of the order of micro-amps can be measured if an electric field of several

kV/mm is applied. This current results from the electron/ion pairs produced by high-

energy particles, either cosmic rays or derived from natural radioactivity, striking an

air molecule:

high-energy particle + M ==> M+ + e

If the voltage is increased sufficiently, the electron is accelerated by the electric field

towards the positive electrode (or anode) and further ionisation can occur. The electron will collide with gas molecules and most of these will be elastic collisions,

but, if it has gained enough kinetic energy (KE), it will ionise the gas molecule it hits:

e + M ==> M+ + e + e.

‘Enough energy’ means energy greater than the ionisation energy of the molecule.

The process of acceleration until a collision with a molecule occurs, with most of the

collisions elastic, and some inelastic (i.e., ionising) is illustrated in the ‘AVAL-1.exe’

program. The KE gained by the electron is (electric field)*(distance travelled before the next collision) – see box below.

Now there are two electrons and the process can repeat, and repeat, and repeat,

causing an exponential increase in the number of electrons. The situation after 4 such

sets of ionisations by accelerated electrons is illustrated in the AVAL-2.exe program

from which the diagram below is taken. (The original high-energy particle ionisation

occurs at A – subsequent ionisations are caused by the accelerated electrons).

There are 1+1+2+4+8 = 16 positive ions

… and 16 electrons here

Note that while the electron is accelerated towards the anode, the positively-charged

ion is – obviously – accelerated towards the cathode. However the ion, being far

heavier, is accelerated more slowly: the average velocity of the electrons is about ten

times faster than that for the ions. This causes the situation seen above with the

electrons moving swiftly to the right in a group, leaving clumps of 1, 2, 4 and 8 ions

behind. This is seen more clearly in the AVAL-4.exe program in which 10 sets of

ionisations have occurred (so there are 210, or 1024, electrons and the same number of

positive ions).

– 1.1 –

Kinetic energy gained by the electron = work done on the electron

= (force on the electron) * (distance travelled by the electron)

= (e.E)*x – i.e., proportional to both the electric field and the distance gone.

The electron will normally have many elastic (low-energy) collisions before the

ionising collision – hence the crooked paths seen in the animations. Because of the

random nature of the number of collisions before the ionising collision, the distance

between ionisations is also variable – again, as seen in the animations.

Avalanches

In the ‘AVAL-4’ diagram below there are 1024 electrons (and the same number of

positive ions), the electrons and ions being indistinguishable in black and white

reproduction. The distribution graphs for the two charged particles are shown, and

explain the comet-like shape of the ‘avalanche’, as this phenomenon is called.

Note the overlapping of the two graphs: this means that there will be a number of

electron/positive-ion collisions which may result in recombination:

e + M+ ==> M + energy

This recombination energy is usually released as a photon of light energy.

(a) (b)

‘Cloud chamber photography’ of single avalanches (a) in nitrogen (N2) at 0.37 bar

and (b) in carbon dioxide (C02), both in a 36-mm gap.

The voltage was a DC voltage pulse lasting 0.4 ms. [H. Raether, Electron Avalanches and Breakdown in Gases, 1964, p.5]

– 1.2 –

positive

ions

electrons

The ionisation coefficients

The ionisation coefficient, α, is defined as the probability that an electron will make an ionising collision in travelling unit distance in the direction of the anode.

In addition there is a possibility (especially for slower-moving electrons) that an

attaching collision takes place:

e + M ==> M–.

The attachment coefficient, η, is defined as the probability, per unit distance travelled in the direction of the anode, that an electron will attach to a molecule to form a negative ion.

As mentioned earlier, the kinetic energy (KE) gained between collisions needs to

exceed the energy required to ionise the molecule. The distance between collisions is

inversely proportional to the density and hence to the pressure so it should not be

surprising (see box) that

α/p = f(E/p).

For similar reasons it is found that

η/p = g(E/p).

For simplicity, an effective ionisation coefficient, ά, is defined as

ά = α – η

Many text books use the empirical equation

α/p = 1100(exp{-27.4 E/p})

for air but it is very approximate. Better ones are available but are generally more

troublesome to apply (see, for example, MacAlpine & Li, IEEE Trans.D&EI, Vol.7,

pp.752-757, 2000). Here, and in general in this course, the units are assumed to be

mm, kV and bar – or their combinations.

Clearly, if attachment is more likely

than ionisation when a collision

occurs, or η>α, avalanches cannot develop.

In nitrogen the attachment

coefficient, η, is negligible; in

oxygen it is very small. So, in this

graph of ά/p versus E/p for air it is only at low fields, below 2.3 kV/mm

bar, that η>α (i.e., ά is negative). Even then it is only just below zero.

The field at which α=η is called the Critical Field.

– 1.3 –

-5

0

5

10

15

20

25

0 1 2 3 4 5 6

E/p (kV/mm.bar)

alpha/p (1/mm.bar)

Li-MacA

Geballe & H

Prasad

Morruzzi & P

The energy gained between collisions = e.E.λ = eE/p

because λ (= mean free path between collisions) is proportional to 1/p (p = pressure)

The probability of a collision resulting in ionisation is a function of the energy gained between

collisions, that is, from the above, a function of E/p, say F(E/p).

The number of collisions (any kind) per unit distance = 1/λ = Ap (A is a constant)

∴ the number of ionising collisions, α, is F(E/p)*Ap,

or, α/p = f(E/p)

Avalanche calculations

The average size of an avalanche may be

calculated for uniform-field conditions by

considering the number of electrons n passing through a plane at a distance x from the cathode

in the direction of the electric field (towards the

anode) in a time ∆t. Simple integration (see

box) gives

n(x) = eάx

How big is an avalanche? Consider an electron produced at x = 0 in a 10-mm gap (e.g. as an

electron-ion pair due to a cosmic ray, or by

emission from the electrode) in air at

atmospheric pressure.

If the applied voltage is 25 kV, E = 2.5 kV/mm,

and ά is found (from the graph on the previous

page) to be close to zero.

Now try 30 kV: ά = 1.3 mm-1

∴the number of electrons in the head of the

avalanche when it strikes the anode will be

n(10mm) = exp(13) = 4.4x105

Now try 35 kV n(10mm) = exp(30) = 1.1x1013 …. This is a huge increase!

Clearly if the current (the sum of all the electrons in all the avalanches which occur in

a second) increases at this rate, breakdown MUST occur near this voltage.

The reverse calculation: what is the voltage which gives, say, 108 electrons in the

head of the avalanche when it strikes the anode?

n(10mm) = exp(10.ά) = 108

Therefore ά = 1.84 mm-1,

So, V = 32 kV.

Avalanches in non-uniform fields

In a non-uniform field, the same approach as used above gives

n(x) = exp( ∫ ά(E).dx ) This is useful in for example, coaxial cable or busbar systems – see next lecure.

Sulphur hexafluoride, SF6

This is a very important insulating gas and is widely used in equipment for electrical

power transmission and distribution. It is colourless, odourless and heavy.

For SF6 the ‘critical field’ is at 8.85 kV/mm – as shown in the graph overleaf – and

the attachment coefficient is much larger than that for oxygen.

From the right-hand graph, an empirical expression for the ‘effective ionisation

coefficient’, ά or (α-η), may be obtained as

ά/p = 26E - 230.

– 1.4 –

The average size of an avalanche

Consider the number of electrons n passing through a plane at a distance x from the cathode in the direction of the

electric field (towards the anode) in a

time ∆t.

The number of electrons passing through

a plane at a distance x + dx may be

written as n + dn (again in a time ∆t) where,

dn = n.αdx, using the definition of α,

so that, integrating between limits of x = 0 and x, and remembering that each

avalanche is started by a single electron, n(x) x

∫ dn/n = ∫ α.dx, 1 0

or, n(x) = eαx

Unless the field is non-uniform, varying

with x, which means α varies with x,

so n(x) = exp(∫ α.dx).

60

70

80

90

100

110

120

130

140

8 9 10 11

E/p, kV/mm.bar

Ionisation coefficients, 1/mm.bar

-50

0

50

100

150

200

8 10 12 14 16

E/p kV/mm.bar

Effective alpha / p, 1/(mm.bar)

This formula gives the critical field as Ec = 230/26 = 8.85 kV/mm.

How big is an SF6 avalanche? Consider an electron produced at x=0 in a 10-mm gap,

as previously, but in SF6 at atmospheric pressure. If the applied voltage is 88.5, 89.0

and 90.0 kV, then, from the formula, ά = 0, 1.4 and 4.0, respectively, and the number

of electrons in the head of the avalanche is n(10mm) = 1, 1.2 x 106 and 2.35 x 1017…!

Clearly the size of an SF6 avalanche increases hugely within 1 to 2% above the

critical field. But what is the criterion for breakdown?

The Streamer Theory of electrical breakdown in gases

The electrons in the head of the avalanche increase the field ahead of it. Similarly, the

positive ions, particularly those in the high concentration zone just behind the head,

increase the field in the ‘tail’. When the avalanche is small, this is an insignificant

effect, but as the avalanche grows there must be a critical size when it is the

concentration of charge is great enough for the field to be increased to twice the

original gap field.

Both Raether (in Germany) and Meek (in England) came to the conclusion that when

there were about 108 electrons in the avalanche head, the field due to the avalanche

itself could equal the main field, and that this would lead to the development of a

channel of ionised conducting gas. Meek called this a ‘streamer’ (Raether called it a

kanal, the German word for ‘channel’).

The mechanism they proposed was that in the region where the electrons and positive

ions overlap (see diagram on page 1.2), recombination would take place:

e + M+ ==> M + photon

This region will therefore be a source of photons which would speed off (at the

velocity of light) in all directions. When they strike molecules, photo-ionisation - the

ionisation of molecules by photon impact – can occur:

photon + M ==> M+ + e

These ‘photo-electrons’ initiate new avalanches: those in the higher-field zones ahead

and behind the ‘mother avalanche’, will grow even faster than the mother avalanche

did because the value of ά increases very quickly above the critical field. Between 4

and 8 kV/mm ά in air increases by a factor of 10; in SF6 the situation is even more

spectacular as breakdown tends to be close (within 1 or 2% - see above) to the critical

field.

– 1.5 –

η

α α- η

They would produce further avalanches ahead and behind, thus producing a column

of conducting ionisation. Consider electrons produced by photo-ionisation at A B C,

D and E in the figure below. At A the field is perhaps 5 times in the gap remote from

the avalanche – a ‘baby’ avalanche will develop far faster than the original or

‘mother’ avalanche owards the anode. Even at B the field is perhaps twice the gap

field so the ‘baby’ avalanche will still develop much faster than the ‘mother’

avalanche. At C the field is zero – nothing will happen. At D the field is again about

twice the gap field so the ‘baby’ avalanche will still develop much faster than the

‘mother’ avalanche, and again in the anode direction (or, more precisely, towards the

maximum density of positive ions). At E the field is again about twice the gap field,

but in the opposite direction, so the ‘baby’ avalanche will still develop much faster

than the ‘mother’ avalanche, but in the cathode direction.

All this development of avalanches of similar size in electron numbers, though

occupying a smaller space, occurs immediately ahead of the ‘head’ of the avalanche,

behind it in the ‘tail’ region, and close besides the dipole area (in the mother

avalanche near E) – thus extremely quickly, almost explosively, forming a column of

avalanches bridging of the gap between the electrodes (see the STREAMER

program). This column of free charges (positive ions and electrons) is conductive so

current flows. Positive feedback occurs as, while the field is maintained, more and

more avalanches occur, the number of charged particles increases, the current

increases – until the source impedance limits the current to the short-circuit current,

thus dropping the voltage across the gap to near zero. This defines electrical

breakdown.

The electric field

around a large

avalanche

The critical size for an avalanche to transform into a streamer for air is usually taken

as 108, which is therefore the breakdown criterion for uniform-filed gaps. It is easy to

demonstrate that taking the critical size as 108 or, say, 2x10

8 makes a change to the

calculated breakdown voltage which is usually negligible in practical terms.

For SF6, many workers have suggested that 107 gives better agreement with

experiment.

Taking N = 108 = eάx = eK, gives K = 18.4, an alternative criterion.

For N = 107, K = 16.1.

– 1.6 –

B A

C

D

E

Paschen's Law – Vbdn is a function of ‘pd’

This was discovered experimentally by a German scientist, F. Paschen, who published

it in 1889. It may be derived as follows: breakdown occurs when

exp(αd) = 108, or αd = K

∴ α/p = f(E/p) = K/pd

Hence, Vbdn = F(pd) … or Paschen's Law.

A log/log graph of the

breakdown voltage of

nitrogen against pd.

Gallagher & Piermain, page 49

Note: 1 atm.cm = 1.013 bar.cm = 10.13 bar.mm = 1013 kPa.mm

The graph shown is for nitrogen, but similar shapes are found for all gases.

Note that for pd > 100 bar.mm (= 10 atm.cm in the figure) Paschen’s Law fails. This

is because of the field distortions around small imperfections, and dust particles, on

the electrode surface. This will be covered in Unit 2 on breakdown in compressed

gases.

Similarly, for a ‘near vacuum’, pd < 10-3 bar.mm (= 10

-4 atm.cm in the figure),

Paschen’s Law also fails. This is because the number of mean free paths in the

distance from cathode to anode becomes too low for it to be possible for avalanches to

develop. This will also be covered in Unit 2.

The so-called Paschen minimum can be of importance for electrical circuits on, for

example, printed circuit boards in space or near-space conditions. For power

engineering it is seldom of direct interest. Vacuum circuit-breakers operate on the

left-hand part of the curve. Atmospheric-pressure air as insulation involves

dimensions of the order of metres (e.g., overhead lines). SF6 is normally used at a

pressure of 4 to 5 bar and dimensions in the region of 10’s of mm. Do the

calculations yourself.

The breakdown voltages at the

‘Paschen minima’ for various gases

Gallagher & Piermain, page 49

– 1.7 –