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Flashover in a vacuum
Flashover
in a v a c u u m
received in final form 10 February 1977 A A A v d i e n k o and M D M a l e v , Institute of Nuclear Physics, Novosibirsk 90, USSR
It is shown that vacuum flashover is due to one or another process alternatively: (a) a thermal breakdown in the thin pre-surface layer of dielectric and (b) a streamer discharge in the desorbed gas cloud. The thermal flashover is observed in dielectrics with pv ~ 1011-1012 #2 cm under dc voltage. The desorption flashover exists under impulse voltage (and under dc voltage in insulators with #v > 1013 .(2 cm). Quantitative models are proposed to describe both the flashover mechanisms. The calculations are in good agreement with experimental data.
I. Introduction As is well known, tile insulation strength of a v a c u u m gap bridged by a solid insulator is less by a factor of 2-3 t h a n it is in an open v a c u u m gap o f the same dimensions. Various models have been p r o p o s e d to a c c o u n t for this p h e n o m e n o n ~-5 a n d the b i b l i o g r a p h y of surface flashover is very large. Nevertheless, it is difficult to answer even the main question: what is the m e c h a n i s m whereby flashover develops ? It is well established that the first stage of flashover is initiated by electron emission at the c a t h o d e - i n s u l a t o r junction.-" The electric field intensity in the contact gap increases by a factor of 2-10 a c c o r d i n g to the dielectric c o n s t a n t of the insulator, a n d the flashover voltage U . is a m a x i n l u m when the electrodes press close to the insulator. 2 Such m a x i m u m b r e a k d o w n data are those most reproducible. (See Fig. I, where there are given data of various a u t h o r s q u o t e d in ref I.)
in the vacuum-dielectric b o u n d a r y leading to b r e a k d o w n remains vague. Neither model can m a k e clear the dependences of the b r e a k d o w n voltage a n d the time lag upon insulator properties, v a c u u m conditions, temperature, magnetic field, surface state, etc. It is u n k n o w n why the flashover strength decreases with increase of the insulator length, a n d why the flashover voltage increases in cone insulators. 3 In this paper, an a t t e m p t is m a d e to answer these questions by m e a n s of the quantitative flashover model. We propose two flashover models according to two groups o f dielectrics with different b r e a k d o w n characteristics. On the surface of dielectrics with relatively high conductivity ( p , , < 1 0 " - 1 0 1 2 D_ cm) u n d e r dc voltage the b r e a k d o w n strength decreases with t e m p e r a t u r e rise, a n d it does not change practically if the insulator shape is modified. T h e flashover current impulse has a distinctive s h a p e with a long rise-time a n d the b r e a k d o w n frequency vs overvoltage d e p e n d e n c e is close to being parabolic (Fig. 2 I). This b r e a k d o w n class can be called "thermal flashover'.
2O
2 i
i
l0
50 1",
E
I00 ns
E
£ &
o
o
2
Me,
o
o
'+-
I
k V
Figure 1. Flashover voltage for cylindrical insulators, d - 1.8-2.2 cm ; Pv ~ 10tt .Qcm; T = 20°C; no = 46.
L
50
Ioo U.
U n f o r t u n a t e l y , the s u b s e q u e n t steps of the flashover f o r m a tion are not so clear. T h e c o m p e t i n g hypotheses are surface c h a r g e a c c u m u l a t i o n 3"4 a n d discharge in the a d s o r b e d gas layer, s But in b o t h cases the c o m p l e t e sequence o f p h e n o m e n a Vacuum/volume 27/number 12.
J
,
150
kV
Figure 2. Characteristics of flashover forms. Shape of current pulse: 1. Conducting glass, pv = 10 t° O. cm; 20°C; d = 1.3 cm. 2. Ultrafarfor pv = l0 t.* ~ cm; 20°C; d = 2 cm. Breakdown frequency vs voltage across insulator: d = 4 cm; D = 20 cm; 20°C. 1. Electrotechnical porcelain. 2. UItrafarfor.
Pergamon Press/Printed in Great Britain
643
A A A v d i e n k o a n d M D Malev: Flashover in a v a c u u m
A second group of dielectrics has Ov ~ 1013 -('2cm (Plexiglas, teflon, alumina, some kinds of glasses, etc.) and it differs from the first one by the higher flashover voltage. In this case, the flashover strength depends strongly upon the angle between the insulator surface and the electric field direction. The time rise of the breakdown current is very small (Fig. 22), and thc breakdowns follow one after the other quite irregularly so that it is possible to give only statistical information about 'the breakdown frequency' (Fig. 22). Tlae failure of the surface insulation in such materials is closely connected with gas desorption.
Tile insulator-enviromnent heat exchange is described by the parameter: d II -
~t (2.3)
221 + ~ f i / ) . t
where d, A and 6, A, are a height (in cm), a heat conductivity (W cm-~ deg-~) of the insulator and electrode respectively and a is the heat transfer coefficient (W cm -z deg- 1). The minimum breakdown voltage (at h,, ~- z ) is U~- = - ~ 1
2
p
<2.4>
2. Thermal flashover The thermal breakdown in a bulk insulator is well known. It develops in dielectrics which have a positive conductivity coefficient. Characteristics of the thermal breakdown are determined by the heating conditions of the insulator with a specific conductivity a (D. cm)- t in the electric field E. 6 In the adiabatic approximation the formative time lag to, is a square-law function of the electric field : const tot - -
E2
(2.1)
•
Equation (2.1) is fulfilled well for the surface breakdown of porcelain ( 2 0 - 2 5 ~ A1203) samples--see Fig. 3. It is possible to assume that the peculiar form of the thermal breakdown is D I ]
"
S"
,
,,
, Ii
I
./ ,/
,
and the formative time lag is cd 2 th, -
tl tTO
U2 q~(y),
(2.5)
where c is the volumetric heat capacity ('J cm-3 deg- ~), 3p 2 2=i1+ ~ a o . o U 2
( ' ~ dz • and 4~(),) = j 0 e'- ----- )'2 .
To a first approximation it is possible to apply the Grinberg theory to the flashover of the insulator (Fig. 3). Then the surface conductance of porcelain can be calculated from the experimental data by means of the equations (2.3)-(2.5). This value is more by a factor 100-1000 than the volume conductance of such ceramic materials 8 (see Fig. 4). To all appearances this fact is associated closely with instability of the pre-breakdown current in the porcelain samples. It is seen from
i0 -e
(o)
IO ~ 4.../
? E
I0
2
/~,/
(b)
./.
L I
8
I ,"I i / t2
v
ill I0
~# ,
i0
~2
I-~ IO-"
b
IO
20
30
4.0
50
u,
kV
60
70
80
90
~00
Figure 3. Breakdown frequency for electrotechnical porcelain. D = 20 cm; d = 4 cm; 8 = 2 cm. I. Ceramic ring. 2. Outer electrodes. 3. Metal top. 4. Inner-electrodes, 5. To vacuum system. • 70°C, x50°C, +30°C.
located in the thin pre-surface layer in the dielectric-vacuum boundary, where heat transfer is worst (so it is subject to the flashover of the other 'conducting' dielectric too). The general solution of the heat-conduction equation for an arbitrary insulator shape is difficult mathematically since the a (T) dependence is nonlinear. The approximate theory of thermal breakdown in a plane capacitor has been proposed by Grinberg,~ who takes into account the heat transfer along the electric field direction only and who supposes that the o (T) dependence is exponential: (2.2)
where To is the initial temperature and ao is the dielectric conductivity at this temperature. 644
il
i" /
i0 -'4 50
a = ao e x p [ a ( T - To)],
,f
I00
T,
150
°C
200
250
I O- 9
50
I00
U,
150
kV
Figure 4. Conductivity of ceramics. (a) Temperature dependence. 1. Electroporcelain. a 2. Electroporcelain from data of Fig. 3.3. Ultrafarfor. (b) Pre-breakdown currents. I. Electroporcelain. 2. Ultrafarfor.
Fig. 4 that the conductance is close to the standard value before the current jump and after this jump it is equal practically to the value calculated from the curves Fig. 3 (see the asterisk on Fig. 4a). In high-alumina ceramic (--~70--75% A]203) the current jumps are absent and the thermal flashover is not observed up to a temperature of 150°C. The mechanism of the current instability is rather unclear. Perhaps it is a result of local heating by the pre-breakdown current. But the experimental fact is that under dc voltage the thermal breakdown develops in the dielectric-vacuum boundary if the insulator conductivity is more than 1 0 - i t - 1 0 -12 (f~ cm)- I. Such a conclusion is confirmed by Gleichauf's data: a the flashover voltage of the glass cylinders (p, ~ 10 ~l f2 cm)
A A Avdienko
and M D Malev:
Flashover in a vacuum
increases by 30-600/; if the glass surface is covered by a silicon oil film. A value of the parameter tt determines the dependence of the thermal breakdown voltage upon the insulator dimensions. Under free convcction ,~ ~=i I and for the short samples (d -< 2A/a)
The relaxation time is of the same order of size as the time of the molecular free path:
Uh, ~ x / d .
where A and i; are the mean free path and the average thermal velocity of molecules respectively. The desorption time is the ratio of the electron free path time to the average number of electrons to one molecule divided to the desorption probability:
This dependence is reduced with increase of the insulator length. ~ It is evident that the division of the dielectric materials into the flashover categories is very provisional. The desorption flashover replaces the thermal one with decrease of the voltage pulse duration, and a heating of an insulator has to lead to the thermal flashover appearance. In fact, on testing of the alumina insulators (94 ~,,. A1203) the typical 'thermal' shape of a current pulse was observed at 250-300"C. The sample cooling led to the back transition to the desorption picture of flashover. If flashover develops by thermal mechanism the breakdown voltage has to be independent of the shape of the insulator surface. Indeed, it was found by Milton 1° that the flashover voltage rises with the cone angle for teflon, alumina, Plexiglas, but it does not do so for a polyurethane foam.
3. Gas desorption and discharge formation near an insulator surface Tile connection between flashover and the desorption pheno m e n a is known. Intensive gas release begins when the applied voltage is much less than the flashover threshold. The desorption rate decreases with time but appreciable pressure increases a c c o m p a n y each breakdown. The hypothesis about the gas mechanism of flashover was introduced by Bugaev and Mesyatz. 5 These authors had shown that flashover development is accompanied by the appearance of a luminous spot on the cathode end of an insulator. This spot moves towards the anode with a velocity of about 107-108 cm s - ' and its brightness and dimensions grow. One can see on the photographs 5 that at the same time the spot moves away from the insulator surface at a velocity ~ 1 0 6 cm s-~. ~ On the anode end the spot brightness increases sharply, the luminosity get contracted and a current has risen suddenly. This picture looks like the photographs of streamer discharge development,' -" and the gas hypothesis very soon got recognized. Nevertheless, the estimates of the gas amounts in ref 5 are illustrative only and the breakdown mechanism remains unclear. First of all, discharge development in an adsorbed layer is impossible. The adatom density is so large that the stage of gas desorption and expansion has to precede the discharge formation. Thus, the flashover medium is the desorbed gas cloud being expanded away from the insulator surface. Experimental data of the flashover formative time lag correspond to a single electron transit across the discharge gap. Apparently, the mechanism of the desorption flashover is really similar to the spark discharge in molecular gases at a high pressure. 12 Then the luminous spot 5 is an electron avalanche turning into a streamer at the anode end of an insulator. A. Gas desorption and expansion. It is usually considered that the sorption-desorption processes are quasi-static and adatoms leave the surface with thermal velocities. However, in reality this is true only until the time interval between two desorption acts is much more than the thermal relaxation time in a gas.
2 rrc
I =
(3.1)
--
2,,N t'des = - - ,
(3.1a)
t)d~/11 e
where ,/~.and t'a are respectively the mean free path and the drift velocity of electrons, N and n~ are respectively the molecular and electron concentrations and 7 is the desorption efficiency. By substituting in (3.1) and (3.1a) the relations: Mo N
--
1
i '
I)d/1e
--
ytl"
one can reduce the criterion of the unsteady gas movement during a desorption process: rr~l
i7)~ - > > 1, b Mo2,.y,
-
rd~s
(3.2)
where / is the adsorbed layer thickness, Mo is the surface density of adatoms, / is the magnitude of the pre-breakdown current pulse and y, is the width of the desorption channel. To estimate the desorption efficiency let us consider a gas evolving from the insulator under a strong electric field 5 ( ~ 1 0 s V c m - i ) . As was shown by Avdienko and Kiselev ~s the desorption rate is practically independent of the dielectric material (Plexiglas, teflon, ceramics) and it is equal to (2-5) ~.: l0 -7 torr I s -~ at a current about l0 -a A. The desorption efficiency is equal to 100-200 molecules per electron. It seems at the first sight that such a value is too large since under the electron bombardment of metals the desorption efficiency varies from 10 -4 to 10 -2 molecule/electron. 16' iv But this difference is to be expected. Under the bombardment electrons move to the surface perpendicularly, and the desorption probability is equal in the first approximation to the part of the surface occupied by the adatoms. However, when the electron trajectories are directed along a surface, the number the desorption acts are restricted by departure of electrons from the adsorbed layer only. At r o o m temperature the average velocity of hydrogen molecule is equal to 1.7 x 105 cms - ~ and the ratio A]),c ~ 0.7. I f a channel width is about 1 0 - s - 1 0 -2 cm and Mo ~ 10 ~ c m - ' , the unsteady condition (3.2) is fulfilled at a current >_50-100 mA. The experimental value of the pre-breakdown current is approximately equal to 1 A, s and so the supersonic expansion of the desorbed gas cloud is explained. U n d e r the unsteady expansion in a vacuum, the leading edge of the gas cloud moves away from the surface at a velocity 5Co and a rarefaction wave moves in the opposite direction. The duration of this stage is determined by the adsorbed layer thickness I and by the velocity o f sound Co :
l
r o = ~ ~ 10-llS. CO 645
A A Avdienko and M D Malev: Flashover in a vacuum
At t > re a density distribution is a superposition of the reflected wave and the expanding substance. At t ~ ro such a distribution is described by an asymptotic formula.t 3 For the diatomic gases it is: N(,', t ) =
0.375 M l k r - _ - ( ,. --.~212 (2rrr) k ~-~otL1 \5--~0t] J ,
(3.3)
where r is the distance from a surface in cm; M is the amount of the desorbed gas per c m - 2 ; and the parameter k depends on the wave geometry: k = 0 for a plane wave and k = l for a cylindrical one. The width of the desorption channel ),~ is large enough and it is possible to neglect the deviation of tile wave shape from a plane and to consider k = 0. The main components of the desorbed gas are hydrogen. nitrogen, carbon oxide and water. The relative amounts of these substances vary widely, but the anaount of hydrogen always comes to greater than 30-60%. ~6 Let us assume for simplification that the desorbed gas is pure hydrogen. The sound velocity is equal to 1.3 >: 105 cm s -~ at r o o m temperature. Then, as is seen from (3.3), the edge of a gas cloud moves from the dielectric surface at a velocity about 106 cm s - t.
The electron velocity distribution is not changed duc to the gas cloud expanding, but gas density varies during the avalanche drift. Nevertheless, the ionization growth in this case differs from the usual solutions a2-~'* for the unreslricted gas volume. Let us consider the electron movement near the gas-vacuum and gas-dielectric boundaries. The electron concentration in the leading edge of the expanding gas cloud is equal to zero, since electrons coming to this boundary have gone away in a vacuum never to return. The second boundary condition depends upon the value of the surface charge. If it is large, we have the usual boundary condition on the non-conducting wall in a plasma: grad n,. = 0. If it is small, the electron concentration at the dielectric surface is nearly equal to zero. It is known that a difference between these two conditions is equivalent to a doubling of tile dimensions of the diffusion region. Such a deviation is about the same as it is due to the errors in the value of Co. Below we shall assume at both boundaries that lie =
O.
The distribution function for electron moving in a gas under an electric field is described by the Lorentz approximation: t9 .f(v) = fo(v) + .ft ( v)cos fl,
(3.6)
N
where fl is an angle between the electric field and the electron drift directions. The symmetrical part of the distribution function fo describes a thermal movement of electrons and the directed component f~ cos ,8 corresponds to an electron current in a gas (as a rule in the gas dischargeJ', ~-~Jo). It follows from (3.6) that tile ionization growth in tile restricted gas cloud is a maximum if electrons move parallel to an insulator surface.
IO
6 0
I
2
4-
r/Cc, t
E
Figure 5. Density distribution in a desorbed gas cloud.
The size of a gas cloud is equal to the distance where the gas density decreases by a factor of 2.5 (Fig. 5). The transverse size and the average density in the desorbed gas cloud are: R ( t ) = 3Cot;
N(t) -
0.295 M -
Cot
NR
2~ = 3.5 x 10162,o = 0.25 x 10 - 1 6 M )~eo
(3.5)
where the mean free path at 1 torr A% ~ 0.03-0.05 cm 2° (H2; electron energy ,--,1-10 eV). Thus the transverse size of the desorbed gas cloud is large enough for an electron avalanche growth if M _> 10 ~6 c m - 2 . 646
R
(3.4)
B. The growth of an electron avalanche. The temporal growth of ionization in the restricted gas cloud is possible only if the cloud size is not less than the electron mean free path. The same condition is enough to describe the electron diffusion and drift by the usual equations of gas discharge theory ~2-~'*, ~9 (since a steady distribution of the electron velocities has set in after 1.5-2 free paths~9). It is seen from the equation (3.4) that the cloud size to electron free path ratio is time independent and that it depends on the desorbed gas a m o u n t only: R
A
Cothode
Anode
Y
Figure 6. Electron avalanche drift in a desorbed gas cloud.
The ionization growth coefficient in some direction 0A (Fig. 6) is equal: R ~x
= ~
cos(~,
-
fl),
sin fl where ff is an angle between the electric field and the insulator surface and a is the Townsend primary ionization coefficient. As the angle fl decreases from 'b to arctan R i d the value a X increases regularly. F o r a small thickness of the gas cloud (R <~ d), if an avalanche moves along the surface, R ¢ - fl ~ t,b; sin fl ~ ~- ; (ctX)ma, = ctd cos ~b.
(3.7)
A A Avdienko and M
D Malev:
Flashover in a vacuum
For any elemental volume of the gas cloud (Fig. 6) we have: ?n,. " ~ (n,,ra), ~t -- ~.n,a'a cos ip + V'(Dn,.) - ?--~
(3.8)
where n,., ira, D are respectively the conccntration, drift velocity and diffusion coefficient of electrons in a gas. The first term on the right-hand side of the equation (3.8) describes primary ionization by electrons, the second one describes electron diffusion and the third one gives a change of electron concentration due to expansion of the electron avalanche, By neglecting an alteration of the drift velocity in the electron bunch dimension it is possiblc to transform equation (3.8) to Lagrange co-ordinates:
dX = rd COS I[/ dt ~11 t,
- - = ~n,.ea cos Ct
¢
(3.9a) -~
+ V'(Dn,.).
(3.9b)
The initial condition to the equation (3.9b) corresponds to an appearance of an initial electron in the cathode end of the insulator. Mathematically it is a delta-function: 6(1" = I'1)
Ilt,(X,l',y,O)=f(X)
(3.10a)
¢J(1.') ,
where X - 0, r - - r , , y - = 0 are the co-ordinates of the avalanche birthpoint. The gas cloud is infinite in the directions X and y (d ~. A,.; .v, ~>- A,.). The boundary conditions along coordinate r are : n,.(X, 0, y.
t,)
= ,,.(X, R, y, t) = 0 .
(3.10b)
The equation (3.9a) is integrated by the variables dividing.-" The distribution of the electron concentration in an avalanche is found by averaging of the initial condition from r, = 0 to r~ = R:
n,.(X,
r. v, t) =
"
4 I - - - rtR4nDt
;< exp
x sin
(f
exp
(
va cos
-
X2_+)'z) 4Dt ]
¢
dt
o
(2k + I)Trr R
x exp [ -
)
×
Thus, the electron diffusion in the restricted gas cloud decreases the rate of electron generation in comparison with the usual case of unrestricted gas volume. The diffusion coefficient depends upon the average (thermal) energy of electrons in a gas U: D-
2 0L'. 3
E
Dependence of the electron drift parameters on an electric field and a gas density is described usually by a semi-empirical formt, lae. For hydrogen the next expressions are valid over a limited range of the similarity parameter E/N: :~=5Pexp
(;) -
130
E 0 = 0.085 - eV P Vd=2-4X
'E 10~'/~cms-'
R4Dt
-~
n,,(X,
20 <_ E/P <_ 250
-,2 (3.14b)
10
t3` (3.14c)
:~*d =
A - e -~-o
Cz
EdZ+
1~/i2 COS
In
z, (3.15)
where 2 = N/E; A = 1.4 :./ 1016; B = 3.72 :. 1 0 - i s ; C = 6.3 :.: 10ts; F = 1.2 :.: 10 zs. In the first stage of the expansion when a gas density is large, the diffusion losses exceed the rate of the electron generation, (a)
,i
2 3
4
,.
- x k=o2k + 1
0
x
fto(2k+l)2n2D']~T
o
(3.14a)
where P -- N/3.5 -: 10 '~ torr is an equivalent gas pressure. F r o m the equations (3.13) and (3.14) we obtain:
cltJ.
(3.11)
20
The average concentration is equal to
=
200 <_ E/P <_ 80022
cm
40
60
£o
~6o
Z x I013
r, 3', t) d X d y dr. 4
(b)
It is possible to limit the series in the equation (3.11) to the first term only. Then we have by integrating:
2
n~(t)=~exp
[I'(
7z2DX~dt]
o CtVdCOS¢-- R2 ]
].
(3.12)
The ionization growth coefficient in an avalanche moving along an insulator in the desorbed gas cloud is found from the equations (3.9a) and (3.12):
fa(oc_ n2D ) ct*d = o vaR 2 cos ¢
dX+
In
2
7t2RD---~ t.
(3.13)
Z
X
1 0 13
Figure 7. Ionization rate in a desorbed gas cloud. (a) 1. Ionization rate; 2,3,4. Diffusion rate M = 1.5; 2.0; 3.0 × 10 t6 cm -2. (b) Effective ionization rate. 1,2,3. M = 1.4; 1.45; 1.5 × I0 t6 cm -z. 647
A A A v d i e n k o a n d M O M a l e v : Flashover in a vacuum
and the value in brackets (3.15) is negative. That is why the ionization growth in the desorbed gas cloud can begin only after parameter - is less than the critical value Zo (see Fig. 7a): ./'(:o) =
Azoexp(-B-o)
Cz0 M 2 cos ~//
0.
(3.16) I I
Expansion of the gas cloud leads to the subsequent decrease of the z parameter from Zo to zero. At the same time the.f(-) value increases first, passes through a maximum and decreases to zero (Fig. 7b).
I;
I 0 0 -)
~'OI~)C' LI.
300( I I J
V
Figure 8. Secondary electron emission curve for typical dielectrics. 2-~ 4. Mechanism of desorption flashover When an avalanche has come from the cathode to a point A" a gas density in the desorbed gas cloud is equal:
N(X)
0.295 =
X l'----4-
M/co [-.x
Udc s
Jo
(4.1)
dX
4U°c°s¢ Eo sin 2 cot20
6.25 × I0 a8 7i
~ I0 a" i,
(4.3)
.|' I I'des
; V d e , = 4 . 2 5 X 107x./U2;
2\Uo-1
M =
I' d C O S
where Co = 1.3 - I0 "~ cm s - t is the velocity of sound in hydrogen, r is a delay time of the avalanche start and t'dc, is a velocity of the desorption front shift along the insulator. The Vd~ value is determined by the trajectory and energy of the desorbing electrons. Some of emitted electrons get on the insulator surface. If the secondary emission coefficient o > 1, the strip of a surface is charged positively and the energy of the incident electrons increases. A surface charge accumulates until the energy of the primary electrons reaches eU2 value when a = 1 on the descending branch of the secondary yield curve (Fig. 8). As is known this condition is stable if initial electron energy is more than the value of eU, value (50-100 eV), otherwise the local surface potential falls to zero. -~3 Similarly, secondary electrons charge the next strip of the insulator surface. As a result a stable chain of the surface charges is generated on the dielectric, and electrons travel along this chain from cathode to anode. 24 It follows from an analysis of the electron trajectories, that the chain parameters (step AX, average electron velocity va,, and angle ~ between insulator surface and resulting electric field) depend neither on the current value nor on the initial field direction Eo : AX=
insulator with length ~ 0 . 2 cm is about I-2 ns and rd~, 2 l0 s cm s - ~. The desorbed gas amount is proportional to the desorption time and the field-emission ctmrrent i:
,
where it is assumed: 7 ~- 100-200 tool/electron and .r, ~ 10 -3 10 -2 cm. It follows from the equation (3.14) Vd <
<
I'dc, ,,
up to an E/P value of about 10"~ V c m - ~ t o r r - ~. With such conditions, the equation (4.1) is simplified, and the gradient of the gas density along the insulator is equal to: dN
10 . 9 f
dX = -
~
(4.4)
Equation (3.15) determines the electron concentration in an avalanche having travelled along the insulator from cathode to anode. Flashover occurs when this concentration reaches some critical value corresponding to streamer formation. The avalanche-to-streamer transition is observed in the molecular gases at high pressures ( N d > ~ 5 ",: 1018 cm-2). A streamer theory does not give the quantitative criterion of the selfmaintained discharge, but to calculate the flashover voltage it is possible to use the experimental dependence of the critical (breakdown) electron concentration upon the air gap length ~z (see Fig. 9).
(4.2)
where Uo is the average energy of the secondary electrons (--,5 eV). The formation of a surface charge chain was described for the first time by ref 24, but the surface charge values are too large in this work while the authors by mistake believed that the stable point is eU~ on the ascending branch of the secondary emission curve (Fig. 8). F o r the typical dielectrics (glass, Plexiglas, ceramics) U2 = 2-4 kV 23 and
Ns/2
[ M cos ,1, V ' [ E ( X ) ] J
4-c
2
/.3
4- "
,'o 2C
~0
iO0
200
300
U,
kV
d,
cm
4-00
Vaes ~ 109 c m s - 1. O n the other hand, one can estimate roughly the same value f r o m experimental data: s the delay time from a voltage supply to appearing of the luminous spot in the cathode end of an 648
Figure 9. Ionization growth coefficient and criterion of streamer discharge. 1,2,3. M = 1.4; 1.45; 1.5 :<1016 cm-Z; 4. Streamer cri terion (~d)c = f(d). 12
,4 A Avdienko and M D Malev." Flashover in a v a c u u m
Increasing the electric field intensity leads to a rise of the field-emission current and, consequently, the anaount of the desorbed gas increases. As a result, the diffusion rate decreases [see equation (3.15)]. the effective rate of ionization increases and the flashover strength falls. When this last value becomes equal to the applied field intensity, flashover is developed. Let us call "critical" the corresponding value of a current and a gas amount i~ and M,.. If M = M,. and E is constant the ionization growth coefficient ,~*d depends only on the time delay of the avalanche start (or upon the gas density at the cathode end of the insulator): 0.295 M Ark = - C O "r
(4.5)
By differentiation of (3.15) with respect to Nk, one can see that the ~*d(ND dependence has a maximum trader the next condition (the logarithmic term varies very slowly and it can be neglected):
E,,.f(--,) = E~l'tzk), where
(a)
> a: D
ac
I
5,JL
,-,
2UO
32
(Io)
/
2,3
I 4, ;
3 d ,
4cm
5
6
7
O J,
08 d.
12
16
cm
Figure 10. Dependence of flashover characteristic upon the insulator length. (a) Breakdown voltage 1,2,3. M = 1.4; 1.45; 1.5 - 1016 cm -z. Experimental data: 0 , 26 , 2 s ~29 -c..,3o +,31 :<,3z 033 (b) Formative lag time. O, 26, 0 . 27
Cz
.f(z) = A z e - ' :
m " cos i,b,
and the subscripts k and a correspond to the cathode (X and anode (X = d) ends of insulator respectively.
O)
A. Flashover in the uniform field. It follows from equations (4.2) that the surface charge is not so large, and the angle ~b ~ 3-3.5:1. If the external field direction is parallel to the insulator surface, it is possible to neglect such small field distortion. Let us assume the electric field is uniform near the smooth surface of the cylindrical insulator: E ( X ) = Eo; = 0. With these approximations the equations (3.15) and (4.4) are integrated easily: c ~ * d = 2 . 0 4 x 109 M ( 1 . 4 x 1 0 - ' 6 {exp[-~ B z ' ] e x p [ - BZk] -k
- erf\/(Bz,)]
\.'ORB) x [erfx/(Bz k) 2.52 x 1016 M-'
( z , - l/_, _ z~
g,, 3 + In (1.2 x 10 z S - ~ %) U = Ed = 0.68
voltage: the breakdown strength decreases with increasing of the insulator length. Growth of the electric field intensity has to lead to a small increase of Mc in short insulators. Such a tendency is seen well from Fig. 10. Many other experimental facts become clear in the light of the desorplion flashover model.
x
l09 M (2a -3/2 - - Z k - 3 / 2 ) .
- l/_,))
,/ (4.7) (4.8)
Values of z, and zk correspond to maximum a*d and they are found from the equation (4.6) graphically (see Fig. 7b; E, = E~ = Eo). To calculate the flashover voltage it is necessary to know a correlation between the electric field intensity and a field-emission current. This dependence is determined by the random microgeometry of the cathode surface and it is too complicated to compute. Nevertheless, one can estimate the critical gas amount by comparing the calculated values of flashover voltage (at a given M value) with experimental data. The computing procedure is clear from Fig. 9. It is seen from Fig. 10 that the calculated plot Ub(d) coincides with the experimental one quite satisfactorily at M = (1.4-1.5) x 1016 cm -2. So the desorption model of flashover explains naturally the effect of the total
By substituting Mc = (1.4-1.5)., 10 t6 c m - : in equation (4.3), one can find the critical current is equal to 1-2 A. This value is in a good agreement with the pre-breakdown current data. 5 Roughness increases the real adsorption surface by a factor of 5-10 even for polished metals, 25 and Mc = 1.5 × 1016 c m - 2 corresponds to nearly 0.5-1 monolayer. The adsorption energy is greatest for the first monolayer and it decreases sharply with the polylayer adsorption. 25 It is obvious that the desorption efficiency has to decrease with increase of the adatom bound energy. Therefore, if the critical current remains invariable, the Mc value is the smaller, the stronger the desorbed atoms were bound to the insulator surface. It results from this that immediately after a pumping the breakdown voltage is rather small. As outer adsorbed atoms are moved away the flashover voltage rises gradually (it is seen from Fig. 9 that if the desorbed gas amount decreases by 1 - 2 ~ , Ub doubles). Gleichauf 9 has shown experimentally that insulator conditioning is promoted not only by the electrode phenomena, but also by alteration of the surface state. As a result of the insulator conditioning the equilibrium density of adatoms occurs in the breakdown channel. This concentration is supported by readsorption from the gas phase, by surface migration and mainly by gaseous diffusion from the dielectric bulk. 16.t7 A high stability of flashover voltage demonstrates that a small amount of a firmly bounded gas (1-2 monolayer) remains on the well-conditioned surface. To remove this gas is practically impossible. Under such conditions readsorption is small and the residual pressure in a vacuum system has no influence on the flashover strength, t We have found that the flashover voltage is practically independent of the composition of the residual atmosphere. In these investigations a vacuum system was filled by Ne or CC14. After a long-term exposure to such an atmosphere the breakdown strength of the insulator sample decreased (Ne) or increased (CCI4), but for some first breakdowns only. The short conditioning had returned a sample to the previous stable state and the flashover voltage became the same as it was before a gas filling. The bound energy of the first monolayer depends upon the 649
A A Avdienko and M D Malev: Flashover in a vacuum
dielectric properties rather weakly. That is why a correlation between tlte flashover parameters and the insulator characteristics is absent. The long-term exposure of the conditioned insulator sample in a vacuum leads to readsorption of some gas and the voltage of tlte first breakdown decreases, 9 but after additional conditioning the sample returns again to the stable state. 9 Bake-out of an insulator makes easy the desorption of the outer monolayer and insulator conditioning time becomes shorter. Thus, dependence of the flashover voltage upon the vacuum conditions, insulator conditioning, etc. is determined completely by bound energy of the adatoms. Tire influence of the magnetic field on the flashover characteristics was investigated by Avdienko. "-6 Tire typical dependencc of Ua on H is shown in Fig. II. The formative time lag decreases together with decrease of the flashover voltage; a plot of UB against H has the same character up to H = 40 kOe.
Tile equation (4.9) givcs H,. ~ 3-5 kOe; this value coincidcs with experimental data (see Fig. II). The formative time lag decreases since the electron trajectories in the magnetic lield are cycloidal and on drifting to the anode, the electrons pass part of their way in a vacuum. The effective electron free path increases and the drift velocity also. Tlte formative time lag of flashover consists of a statistical time delay of the avalanche start r and the drift time of electrons from cathode to anode (it is possible to neglect the time of streamer travel which is less than 10 -'~ s ~-'- i.,):
f
I.i- ---- r "Jr"
ddY "--F,d" 0
It is easy to show that the ts- value is inversely proportional to the gas density at the anode end of the insulator:
Ir -
0.295 M CoN.
(4.10)
'ooI
X 2C
[
2
5
4 H,
5
E
7
8
kOe
Figure l l . Flashover voltage in magnetic field, d :- 0.2-0.4 cm.
Such effects are observed for any magnetic field direction (along or across an insulator axis). The desorption flashover model explains these phenomena. The electron drift velocity in a magnetic field is determined by a factor [1
+ (o)T) 2] -
1,
ell~me
where T is the time of the electron free path and ,o = is the electron gyrofrequency. In the desorbed gas cloud a , T . ~ 1 up to H ~ 50 kOe and the drift velocity is practically independent of the magnetic field. But the trajectories of each electron are crooked. It is obvious that somewhere on the insulator perimeter the electron velocity has to turn to a surface for any magnetic field direction. In that region the magnetic field gives back a part of the electrons diffusing from a gas cloud, and the ionization growth coefficient increases. This mechanism is corroborated experimentally: in the transverse magnetic field the flashover channels are located just in those sections of the insulator where the electron velocity turns to a surface. 26 A fraction of the returned electrons grows (and the flashover voltage decreases) with increase of the magnetic field as long as the diffusion losses compensate completely. To estimate the critical magnetic field one can equate the electron L a r m o r radius to the gas cloud dimension R: H ~ = 3"4 x/ [7 -~ -
650
x
10ax/(EN).M
(4.9)
The equation (4.10) determines the value of the m a x i m u m formative time when the applied voltage is exactly equal to the breakdown value. Figure 10 shows the calculated plot of ts+ against d. These data are in quite good agreement with the time delay data at a small overvoltage -'7. With rise of overvoltage both components of the formative time lag decrease. To make a qualitative comparison, Fig. I 0b shows the dots corresponding to a large overvoltage -'6. It follows front the calculations that flashover formation begins for a gas density of about (2--4) ..: 1018 cm -3 (the equivalent pressure is equal to 50-100 torr). As an avalanche drifts to the anode, the gas density decreases, and it is equal to 1.5 . l0 Is cm -3 ( ~ 4 0 tort) when the flashover formation is over (for an insulator lengtlt ~ 1 cm). The N d product varies front 2 - 1018 to 5 ~.~ l 0 t s cm -2 for the insulators of length 0.1-10 cm, i.e. flashover forms in the typical range of gas density for the streamer discharge. If the gas density is smaller the self-maintaining discharge cannot form in tlte expanding gas cloud. The logarithmic term in tlte equation (3.15) is related to decrease of the electron concentration in an avalanclte owing to expansion of the bunch. At N = 10 t7 cnt -3 this term gets negative and the ionization growth stops _~ 0). That is why flashover cannot develop by the Townsend exchange mechanism: the gas density falls to 10t6-1017 cm - s during the ion path along the insulator
(a*d
(10-6-10
- 5 S).
The first assumption of the desorption flashover model is a single avalanche approximation, as is made in a streamer theory usually. In reality a criterion of tire avalanche-to-streamer transition is the ion concentration, not the electron, tz-I'* Ions accumulate in a discharge gap : I1 i ~
£
Ile k,
k=l
where s is the number of avalanches. F o r the single avalanche model s = 1 and n~ = n~. In reality the avalanches start continuously from the cathode end of the insulator, and flasbover can form at M < M~ because of the accumulation o f the ions. Therefore the above quoted values of i~ and m~ are too high, apparently. The second rough assumption is an extrapolation of the equation (3.14) far over the experimental limits (flashover develops at = 1000-2000 V c m - t t o r r - t).
E/P
A A Avdienko and M D Malev: Flashover in a v a c u u m
Despite these and some other assumptions the flashover desorption model coincides well with experimental data and the estimation is quite reasonable o f values which are not measured directly (M,., N). The model is rather stable; the final conclusions are the same as if it is assumed the desorbed gas is hydrogen, nitrogen, oxygen, o r air.
5. Conclusions The diversity o f p h e n o m e n a under flashover in a vacuum is described quantitatively by means o f two models: the thermal one and the desorption one. The thermal flashover develops in dielectrics with relatively low specific resistance ( p , . ~ 1 0 " - I 0 ' - " ~--~ cm) if the voltage pulse duration is more than 10 -3 s. In this case, the flashover characteristics are similar to the ordinary thermal b r e a k d o w n of the solid dielectrics. The desorption form o f the flashover is a streamer discharge in the desorbed gas cloud. An analysis o f the electron avalanche drift in the restricted cloud o f e x p a n d i n g gas (with supersonic velocity) makes it possible to calculate the b r e a k d o w n voltage and the formative time lag for the cylinder insulators. The physical properties o f the dielectric and the vacuuna conditions have practically no influence on the characteristics o f desorption flashover, since to form the discharge, it is enough to desorb less than one monolayer o f adatoms. If a flashover develops in the uniform electric field the surface charge takes a slight part, but this part becomes essential if" the electric field is not parallel to the insulator surface. Both p r o p o s e d models make it possible to systematize the n u m e r o u s flashover experimental data and to outline ways of constructing high-voltage vacuum insulators.
References t 1 N Slivkov, Electrohlsuhttion am/ I~bcaum Discharge, Atomizdat, Moscow (1972). z M J Kofoid, AIEE Trans, No 6, 1960, 991,999. 3 A. Watson, J appl Phys. 38, 1967, 2019.
~C H De Tourrcil and K D Srivastava, IEEE Trans Electr hlsal, El-8, 1973, 17. 5 S P Bugaev, A M Iskoldski and G A Mesyatz, Zh Tekh Fiz, 37, 1967, 1855, 1861. " W Franz, Break~hnt,n o f Dielectri¢w, lnostr Liter, Moscow (1961). G A Grinberg, M I Kontorovitch and N N Lebedev, Zh Tekh Fiz, 10, 1940, 199. 8G A Widrik and N S Kostjukov, Prothwthm o f Electroceramh', Energia, Moscow ( 1971 ). " P H G leichauf, J appl Phys, 22, 1951,535,766. ,o Milton, IEEE Trans Electr hlsul, El-7, NI, 1972, 9. ,t V 1 Rakhovsky, Physical Faundathm o f Electron Carrent Consma/ticathm itl Va('tatm, Nauka, Moscow (1970). '-' J Mick and J Craggs, Electrical Breakdown o f Gases, Inostr Liter, Moscow (1960). ,3 H Raether, Electron Aralanches and Breakdown in Gases, Mir, Moscow (1968). '" E D l.ozansky and O B Firsov, Spark Theoo,, Atomizdat, Moscow (1975). ,5 A A Avdienko and A V Kiselev, Zh Tekh Fiz, 37, 1967, 533. t6 N V Tscherepnin, Vacattm Properties o f MateriaL~ fin" Electron Tubes, Sov Radio, Moscow (1966). ,7 M D Malev, I/aeuttm, 23, 1972, 43. t, K L Stanjukovitsch, Unsteady 1%4ol'ementafSalid Medium, Nauka, Moscow ( 1971 ). t,, V L Granovsky, Eh, ctrieal Current hi Gas, Vol I, Gittl, Moscow (1961). 20 S S Brown, Elenwntao, Processes hi Gas Discharge Plasma, Atomizdat, Moscow (I 961 ). -'~ H S Carslaw and J. C. Jaeger, Conduction o f Heat h/ Solids', Nauka, Moscow (1961). .,z A Engel, hmized Gases, Fizmatgiz, Moscow (1959). ,-.3 1 M Bronshtein and B S Fraiman, Secondary Electron Emission, Nauka, Moscow (1969). z.* H Boersch, H Hamisch and W Ehrlich, Z angew Phys, 15, 1963, 61. "~ S Dushman, Scientific Foundation o f Vacaant Technique, Mir, Moscow (1964). .,6 A A Avdienko, Zh Tekh Fiz, 47, 8, 1977 (to be published.) _,7 S P Bugaev and G A Mesjatz, Zh Tekh Fiz, 35, 1965, 1202. 28 E S Borovik and B M Batrakov, Zh Tekh Fiz, 28, 1958, 1971. z,, K D Srivastava and C H De Tourreil, quoted by Slivkov.' 30 K D Srivastava, Proc 2rid hit S),mp htsulation hi Vacaunl, Boston, MA (1966). (p.229) 31 m K Walter, Electrostatie Accelerator o f Charged Particles, Atomizdat, Moscow (1963). 3-, R C Finke, Proc 2rid hit Syrup htsa/ation ht Vaeatun, Boston, MA (1966). (p.217) 33 N Hamidov, Proc Akad Nauk Uz SSR, No I1, 1968, 16; No 1, 1969, 25.
651
in a v a c u u m
received in final form 10 February 1977 A A A v d i e n k o and M D M a l e v , Institute of Nuclear Physics, Novosibirsk 90, USSR
It is shown that vacuum flashover is due to one or another process alternatively: (a) a thermal breakdown in the thin pre-surface layer of dielectric and (b) a streamer discharge in the desorbed gas cloud. The thermal flashover is observed in dielectrics with pv ~ 1011-1012 #2 cm under dc voltage. The desorption flashover exists under impulse voltage (and under dc voltage in insulators with #v > 1013 .(2 cm). Quantitative models are proposed to describe both the flashover mechanisms. The calculations are in good agreement with experimental data.
I. Introduction As is well known, tile insulation strength of a v a c u u m gap bridged by a solid insulator is less by a factor of 2-3 t h a n it is in an open v a c u u m gap o f the same dimensions. Various models have been p r o p o s e d to a c c o u n t for this p h e n o m e n o n ~-5 a n d the b i b l i o g r a p h y of surface flashover is very large. Nevertheless, it is difficult to answer even the main question: what is the m e c h a n i s m whereby flashover develops ? It is well established that the first stage of flashover is initiated by electron emission at the c a t h o d e - i n s u l a t o r junction.-" The electric field intensity in the contact gap increases by a factor of 2-10 a c c o r d i n g to the dielectric c o n s t a n t of the insulator, a n d the flashover voltage U . is a m a x i n l u m when the electrodes press close to the insulator. 2 Such m a x i m u m b r e a k d o w n data are those most reproducible. (See Fig. I, where there are given data of various a u t h o r s q u o t e d in ref I.)
in the vacuum-dielectric b o u n d a r y leading to b r e a k d o w n remains vague. Neither model can m a k e clear the dependences of the b r e a k d o w n voltage a n d the time lag upon insulator properties, v a c u u m conditions, temperature, magnetic field, surface state, etc. It is u n k n o w n why the flashover strength decreases with increase of the insulator length, a n d why the flashover voltage increases in cone insulators. 3 In this paper, an a t t e m p t is m a d e to answer these questions by m e a n s of the quantitative flashover model. We propose two flashover models according to two groups o f dielectrics with different b r e a k d o w n characteristics. On the surface of dielectrics with relatively high conductivity ( p , , < 1 0 " - 1 0 1 2 D_ cm) u n d e r dc voltage the b r e a k d o w n strength decreases with t e m p e r a t u r e rise, a n d it does not change practically if the insulator shape is modified. T h e flashover current impulse has a distinctive s h a p e with a long rise-time a n d the b r e a k d o w n frequency vs overvoltage d e p e n d e n c e is close to being parabolic (Fig. 2 I). This b r e a k d o w n class can be called "thermal flashover'.
2O
2 i
i
l0
50 1",
E
I00 ns
E
£ &
o
o
2
Me,
o
o
'+-
I
k V
Figure 1. Flashover voltage for cylindrical insulators, d - 1.8-2.2 cm ; Pv ~ 10tt .Qcm; T = 20°C; no = 46.
L
50
Ioo U.
U n f o r t u n a t e l y , the s u b s e q u e n t steps of the flashover f o r m a tion are not so clear. T h e c o m p e t i n g hypotheses are surface c h a r g e a c c u m u l a t i o n 3"4 a n d discharge in the a d s o r b e d gas layer, s But in b o t h cases the c o m p l e t e sequence o f p h e n o m e n a Vacuum/volume 27/number 12.
J
,
150
kV
Figure 2. Characteristics of flashover forms. Shape of current pulse: 1. Conducting glass, pv = 10 t° O. cm; 20°C; d = 1.3 cm. 2. Ultrafarfor pv = l0 t.* ~ cm; 20°C; d = 2 cm. Breakdown frequency vs voltage across insulator: d = 4 cm; D = 20 cm; 20°C. 1. Electrotechnical porcelain. 2. UItrafarfor.
Pergamon Press/Printed in Great Britain
643
A A A v d i e n k o a n d M D Malev: Flashover in a v a c u u m
A second group of dielectrics has Ov ~ 1013 -('2cm (Plexiglas, teflon, alumina, some kinds of glasses, etc.) and it differs from the first one by the higher flashover voltage. In this case, the flashover strength depends strongly upon the angle between the insulator surface and the electric field direction. The time rise of the breakdown current is very small (Fig. 22), and thc breakdowns follow one after the other quite irregularly so that it is possible to give only statistical information about 'the breakdown frequency' (Fig. 22). Tlae failure of the surface insulation in such materials is closely connected with gas desorption.
Tile insulator-enviromnent heat exchange is described by the parameter: d II -
~t (2.3)
221 + ~ f i / ) . t
where d, A and 6, A, are a height (in cm), a heat conductivity (W cm-~ deg-~) of the insulator and electrode respectively and a is the heat transfer coefficient (W cm -z deg- 1). The minimum breakdown voltage (at h,, ~- z ) is U~- = - ~ 1
2
p
<2.4>
2. Thermal flashover The thermal breakdown in a bulk insulator is well known. It develops in dielectrics which have a positive conductivity coefficient. Characteristics of the thermal breakdown are determined by the heating conditions of the insulator with a specific conductivity a (D. cm)- t in the electric field E. 6 In the adiabatic approximation the formative time lag to, is a square-law function of the electric field : const tot - -
E2
(2.1)
•
Equation (2.1) is fulfilled well for the surface breakdown of porcelain ( 2 0 - 2 5 ~ A1203) samples--see Fig. 3. It is possible to assume that the peculiar form of the thermal breakdown is D I ]
"
S"
,
,,
, Ii
I
./ ,/
,
and the formative time lag is cd 2 th, -
tl tTO
U2 q~(y),
(2.5)
where c is the volumetric heat capacity ('J cm-3 deg- ~), 3p 2 2=i1+ ~ a o . o U 2
( ' ~ dz • and 4~(),) = j 0 e'- ----- )'2 .
To a first approximation it is possible to apply the Grinberg theory to the flashover of the insulator (Fig. 3). Then the surface conductance of porcelain can be calculated from the experimental data by means of the equations (2.3)-(2.5). This value is more by a factor 100-1000 than the volume conductance of such ceramic materials 8 (see Fig. 4). To all appearances this fact is associated closely with instability of the pre-breakdown current in the porcelain samples. It is seen from
i0 -e
(o)
IO ~ 4.../
? E
I0
2
/~,/
(b)
./.
L I
8
I ,"I i / t2
v
ill I0
~# ,
i0
~2
I-~ IO-"
b
IO
20
30
4.0
50
u,
kV
60
70
80
90
~00
Figure 3. Breakdown frequency for electrotechnical porcelain. D = 20 cm; d = 4 cm; 8 = 2 cm. I. Ceramic ring. 2. Outer electrodes. 3. Metal top. 4. Inner-electrodes, 5. To vacuum system. • 70°C, x50°C, +30°C.
located in the thin pre-surface layer in the dielectric-vacuum boundary, where heat transfer is worst (so it is subject to the flashover of the other 'conducting' dielectric too). The general solution of the heat-conduction equation for an arbitrary insulator shape is difficult mathematically since the a (T) dependence is nonlinear. The approximate theory of thermal breakdown in a plane capacitor has been proposed by Grinberg,~ who takes into account the heat transfer along the electric field direction only and who supposes that the o (T) dependence is exponential: (2.2)
where To is the initial temperature and ao is the dielectric conductivity at this temperature. 644
il
i" /
i0 -'4 50
a = ao e x p [ a ( T - To)],
,f
I00
T,
150
°C
200
250
I O- 9
50
I00
U,
150
kV
Figure 4. Conductivity of ceramics. (a) Temperature dependence. 1. Electroporcelain. a 2. Electroporcelain from data of Fig. 3.3. Ultrafarfor. (b) Pre-breakdown currents. I. Electroporcelain. 2. Ultrafarfor.
Fig. 4 that the conductance is close to the standard value before the current jump and after this jump it is equal practically to the value calculated from the curves Fig. 3 (see the asterisk on Fig. 4a). In high-alumina ceramic (--~70--75% A]203) the current jumps are absent and the thermal flashover is not observed up to a temperature of 150°C. The mechanism of the current instability is rather unclear. Perhaps it is a result of local heating by the pre-breakdown current. But the experimental fact is that under dc voltage the thermal breakdown develops in the dielectric-vacuum boundary if the insulator conductivity is more than 1 0 - i t - 1 0 -12 (f~ cm)- I. Such a conclusion is confirmed by Gleichauf's data: a the flashover voltage of the glass cylinders (p, ~ 10 ~l f2 cm)
A A Avdienko
and M D Malev:
Flashover in a vacuum
increases by 30-600/; if the glass surface is covered by a silicon oil film. A value of the parameter tt determines the dependence of the thermal breakdown voltage upon the insulator dimensions. Under free convcction ,~ ~=i I and for the short samples (d -< 2A/a)
The relaxation time is of the same order of size as the time of the molecular free path:
Uh, ~ x / d .
where A and i; are the mean free path and the average thermal velocity of molecules respectively. The desorption time is the ratio of the electron free path time to the average number of electrons to one molecule divided to the desorption probability:
This dependence is reduced with increase of the insulator length. ~ It is evident that the division of the dielectric materials into the flashover categories is very provisional. The desorption flashover replaces the thermal one with decrease of the voltage pulse duration, and a heating of an insulator has to lead to the thermal flashover appearance. In fact, on testing of the alumina insulators (94 ~,,. A1203) the typical 'thermal' shape of a current pulse was observed at 250-300"C. The sample cooling led to the back transition to the desorption picture of flashover. If flashover develops by thermal mechanism the breakdown voltage has to be independent of the shape of the insulator surface. Indeed, it was found by Milton 1° that the flashover voltage rises with the cone angle for teflon, alumina, Plexiglas, but it does not do so for a polyurethane foam.
3. Gas desorption and discharge formation near an insulator surface Tile connection between flashover and the desorption pheno m e n a is known. Intensive gas release begins when the applied voltage is much less than the flashover threshold. The desorption rate decreases with time but appreciable pressure increases a c c o m p a n y each breakdown. The hypothesis about the gas mechanism of flashover was introduced by Bugaev and Mesyatz. 5 These authors had shown that flashover development is accompanied by the appearance of a luminous spot on the cathode end of an insulator. This spot moves towards the anode with a velocity of about 107-108 cm s - ' and its brightness and dimensions grow. One can see on the photographs 5 that at the same time the spot moves away from the insulator surface at a velocity ~ 1 0 6 cm s-~. ~ On the anode end the spot brightness increases sharply, the luminosity get contracted and a current has risen suddenly. This picture looks like the photographs of streamer discharge development,' -" and the gas hypothesis very soon got recognized. Nevertheless, the estimates of the gas amounts in ref 5 are illustrative only and the breakdown mechanism remains unclear. First of all, discharge development in an adsorbed layer is impossible. The adatom density is so large that the stage of gas desorption and expansion has to precede the discharge formation. Thus, the flashover medium is the desorbed gas cloud being expanded away from the insulator surface. Experimental data of the flashover formative time lag correspond to a single electron transit across the discharge gap. Apparently, the mechanism of the desorption flashover is really similar to the spark discharge in molecular gases at a high pressure. 12 Then the luminous spot 5 is an electron avalanche turning into a streamer at the anode end of an insulator. A. Gas desorption and expansion. It is usually considered that the sorption-desorption processes are quasi-static and adatoms leave the surface with thermal velocities. However, in reality this is true only until the time interval between two desorption acts is much more than the thermal relaxation time in a gas.
2 rrc
I =
(3.1)
--
2,,N t'des = - - ,
(3.1a)
t)d~/11 e
where ,/~.and t'a are respectively the mean free path and the drift velocity of electrons, N and n~ are respectively the molecular and electron concentrations and 7 is the desorption efficiency. By substituting in (3.1) and (3.1a) the relations: Mo N
--
1
i '
I)d/1e
--
ytl"
one can reduce the criterion of the unsteady gas movement during a desorption process: rr~l
i7)~ - > > 1, b Mo2,.y,
-
rd~s
(3.2)
where / is the adsorbed layer thickness, Mo is the surface density of adatoms, / is the magnitude of the pre-breakdown current pulse and y, is the width of the desorption channel. To estimate the desorption efficiency let us consider a gas evolving from the insulator under a strong electric field 5 ( ~ 1 0 s V c m - i ) . As was shown by Avdienko and Kiselev ~s the desorption rate is practically independent of the dielectric material (Plexiglas, teflon, ceramics) and it is equal to (2-5) ~.: l0 -7 torr I s -~ at a current about l0 -a A. The desorption efficiency is equal to 100-200 molecules per electron. It seems at the first sight that such a value is too large since under the electron bombardment of metals the desorption efficiency varies from 10 -4 to 10 -2 molecule/electron. 16' iv But this difference is to be expected. Under the bombardment electrons move to the surface perpendicularly, and the desorption probability is equal in the first approximation to the part of the surface occupied by the adatoms. However, when the electron trajectories are directed along a surface, the number the desorption acts are restricted by departure of electrons from the adsorbed layer only. At r o o m temperature the average velocity of hydrogen molecule is equal to 1.7 x 105 cms - ~ and the ratio A]),c ~ 0.7. I f a channel width is about 1 0 - s - 1 0 -2 cm and Mo ~ 10 ~ c m - ' , the unsteady condition (3.2) is fulfilled at a current >_50-100 mA. The experimental value of the pre-breakdown current is approximately equal to 1 A, s and so the supersonic expansion of the desorbed gas cloud is explained. U n d e r the unsteady expansion in a vacuum, the leading edge of the gas cloud moves away from the surface at a velocity 5Co and a rarefaction wave moves in the opposite direction. The duration of this stage is determined by the adsorbed layer thickness I and by the velocity o f sound Co :
l
r o = ~ ~ 10-llS. CO 645
A A Avdienko and M D Malev: Flashover in a vacuum
At t > re a density distribution is a superposition of the reflected wave and the expanding substance. At t ~ ro such a distribution is described by an asymptotic formula.t 3 For the diatomic gases it is: N(,', t ) =
0.375 M l k r - _ - ( ,. --.~212 (2rrr) k ~-~otL1 \5--~0t] J ,
(3.3)
where r is the distance from a surface in cm; M is the amount of the desorbed gas per c m - 2 ; and the parameter k depends on the wave geometry: k = 0 for a plane wave and k = l for a cylindrical one. The width of the desorption channel ),~ is large enough and it is possible to neglect the deviation of tile wave shape from a plane and to consider k = 0. The main components of the desorbed gas are hydrogen. nitrogen, carbon oxide and water. The relative amounts of these substances vary widely, but the anaount of hydrogen always comes to greater than 30-60%. ~6 Let us assume for simplification that the desorbed gas is pure hydrogen. The sound velocity is equal to 1.3 >: 105 cm s -~ at r o o m temperature. Then, as is seen from (3.3), the edge of a gas cloud moves from the dielectric surface at a velocity about 106 cm s - t.
The electron velocity distribution is not changed duc to the gas cloud expanding, but gas density varies during the avalanche drift. Nevertheless, the ionization growth in this case differs from the usual solutions a2-~'* for the unreslricted gas volume. Let us consider the electron movement near the gas-vacuum and gas-dielectric boundaries. The electron concentration in the leading edge of the expanding gas cloud is equal to zero, since electrons coming to this boundary have gone away in a vacuum never to return. The second boundary condition depends upon the value of the surface charge. If it is large, we have the usual boundary condition on the non-conducting wall in a plasma: grad n,. = 0. If it is small, the electron concentration at the dielectric surface is nearly equal to zero. It is known that a difference between these two conditions is equivalent to a doubling of tile dimensions of the diffusion region. Such a deviation is about the same as it is due to the errors in the value of Co. Below we shall assume at both boundaries that lie =
O.
The distribution function for electron moving in a gas under an electric field is described by the Lorentz approximation: t9 .f(v) = fo(v) + .ft ( v)cos fl,
(3.6)
N
where fl is an angle between the electric field and the electron drift directions. The symmetrical part of the distribution function fo describes a thermal movement of electrons and the directed component f~ cos ,8 corresponds to an electron current in a gas (as a rule in the gas dischargeJ', ~-~Jo). It follows from (3.6) that tile ionization growth in tile restricted gas cloud is a maximum if electrons move parallel to an insulator surface.
IO
6 0
I
2
4-
r/Cc, t
E
Figure 5. Density distribution in a desorbed gas cloud.
The size of a gas cloud is equal to the distance where the gas density decreases by a factor of 2.5 (Fig. 5). The transverse size and the average density in the desorbed gas cloud are: R ( t ) = 3Cot;
N(t) -
0.295 M -
Cot
NR
2~ = 3.5 x 10162,o = 0.25 x 10 - 1 6 M )~eo
(3.5)
where the mean free path at 1 torr A% ~ 0.03-0.05 cm 2° (H2; electron energy ,--,1-10 eV). Thus the transverse size of the desorbed gas cloud is large enough for an electron avalanche growth if M _> 10 ~6 c m - 2 . 646
R
(3.4)
B. The growth of an electron avalanche. The temporal growth of ionization in the restricted gas cloud is possible only if the cloud size is not less than the electron mean free path. The same condition is enough to describe the electron diffusion and drift by the usual equations of gas discharge theory ~2-~'*, ~9 (since a steady distribution of the electron velocities has set in after 1.5-2 free paths~9). It is seen from the equation (3.4) that the cloud size to electron free path ratio is time independent and that it depends on the desorbed gas a m o u n t only: R
A
Cothode
Anode
Y
Figure 6. Electron avalanche drift in a desorbed gas cloud.
The ionization growth coefficient in some direction 0A (Fig. 6) is equal: R ~x
= ~
cos(~,
-
fl),
sin fl where ff is an angle between the electric field and the insulator surface and a is the Townsend primary ionization coefficient. As the angle fl decreases from 'b to arctan R i d the value a X increases regularly. F o r a small thickness of the gas cloud (R <~ d), if an avalanche moves along the surface, R ¢ - fl ~ t,b; sin fl ~ ~- ; (ctX)ma, = ctd cos ~b.
(3.7)
A A Avdienko and M
D Malev:
Flashover in a vacuum
For any elemental volume of the gas cloud (Fig. 6) we have: ?n,. " ~ (n,,ra), ~t -- ~.n,a'a cos ip + V'(Dn,.) - ?--~
(3.8)
where n,., ira, D are respectively the conccntration, drift velocity and diffusion coefficient of electrons in a gas. The first term on the right-hand side of the equation (3.8) describes primary ionization by electrons, the second one describes electron diffusion and the third one gives a change of electron concentration due to expansion of the electron avalanche, By neglecting an alteration of the drift velocity in the electron bunch dimension it is possiblc to transform equation (3.8) to Lagrange co-ordinates:
dX = rd COS I[/ dt ~11 t,
- - = ~n,.ea cos Ct
¢
(3.9a) -~
+ V'(Dn,.).
(3.9b)
The initial condition to the equation (3.9b) corresponds to an appearance of an initial electron in the cathode end of the insulator. Mathematically it is a delta-function: 6(1" = I'1)
Ilt,(X,l',y,O)=f(X)
(3.10a)
¢J(1.') ,
where X - 0, r - - r , , y - = 0 are the co-ordinates of the avalanche birthpoint. The gas cloud is infinite in the directions X and y (d ~. A,.; .v, ~>- A,.). The boundary conditions along coordinate r are : n,.(X, 0, y.
t,)
= ,,.(X, R, y, t) = 0 .
(3.10b)
The equation (3.9a) is integrated by the variables dividing.-" The distribution of the electron concentration in an avalanche is found by averaging of the initial condition from r, = 0 to r~ = R:
n,.(X,
r. v, t) =
"
4 I - - - rtR4nDt
;< exp
x sin
(f
exp
(
va cos
-
X2_+)'z) 4Dt ]
¢
dt
o
(2k + I)Trr R
x exp [ -
)
×
Thus, the electron diffusion in the restricted gas cloud decreases the rate of electron generation in comparison with the usual case of unrestricted gas volume. The diffusion coefficient depends upon the average (thermal) energy of electrons in a gas U: D-
2 0L'. 3
E
Dependence of the electron drift parameters on an electric field and a gas density is described usually by a semi-empirical formt, lae. For hydrogen the next expressions are valid over a limited range of the similarity parameter E/N: :~=5Pexp
(;) -
130
E 0 = 0.085 - eV P Vd=2-4X
'E 10~'/~cms-'
R4Dt
-~
n,,(X,
20 <_ E/P <_ 250
-,2 (3.14b)
10
t3` (3.14c)
:~*d =
A - e -~-o
Cz
EdZ+
1~/i2 COS
In
z, (3.15)
where 2 = N/E; A = 1.4 :./ 1016; B = 3.72 :. 1 0 - i s ; C = 6.3 :.: 10ts; F = 1.2 :.: 10 zs. In the first stage of the expansion when a gas density is large, the diffusion losses exceed the rate of the electron generation, (a)
,i
2 3
4
,.
- x k=o2k + 1
0
x
fto(2k+l)2n2D']~T
o
(3.14a)
where P -- N/3.5 -: 10 '~ torr is an equivalent gas pressure. F r o m the equations (3.13) and (3.14) we obtain:
cltJ.
(3.11)
20
The average concentration is equal to
=
200 <_ E/P <_ 80022
cm
40
60
£o
~6o
Z x I013
r, 3', t) d X d y dr. 4
(b)
It is possible to limit the series in the equation (3.11) to the first term only. Then we have by integrating:
2
n~(t)=~exp
[I'(
7z2DX~dt]
o CtVdCOS¢-- R2 ]
].
(3.12)
The ionization growth coefficient in an avalanche moving along an insulator in the desorbed gas cloud is found from the equations (3.9a) and (3.12):
fa(oc_ n2D ) ct*d = o vaR 2 cos ¢
dX+
In
2
7t2RD---~ t.
(3.13)
Z
X
1 0 13
Figure 7. Ionization rate in a desorbed gas cloud. (a) 1. Ionization rate; 2,3,4. Diffusion rate M = 1.5; 2.0; 3.0 × 10 t6 cm -2. (b) Effective ionization rate. 1,2,3. M = 1.4; 1.45; 1.5 × I0 t6 cm -z. 647
A A A v d i e n k o a n d M O M a l e v : Flashover in a vacuum
and the value in brackets (3.15) is negative. That is why the ionization growth in the desorbed gas cloud can begin only after parameter - is less than the critical value Zo (see Fig. 7a): ./'(:o) =
Azoexp(-B-o)
Cz0 M 2 cos ~//
0.
(3.16) I I
Expansion of the gas cloud leads to the subsequent decrease of the z parameter from Zo to zero. At the same time the.f(-) value increases first, passes through a maximum and decreases to zero (Fig. 7b).
I;
I 0 0 -)
~'OI~)C' LI.
300( I I J
V
Figure 8. Secondary electron emission curve for typical dielectrics. 2-~ 4. Mechanism of desorption flashover When an avalanche has come from the cathode to a point A" a gas density in the desorbed gas cloud is equal:
N(X)
0.295 =
X l'----4-
M/co [-.x
Udc s
Jo
(4.1)
dX
4U°c°s¢ Eo sin 2 cot20
6.25 × I0 a8 7i
~ I0 a" i,
(4.3)
.|' I I'des
; V d e , = 4 . 2 5 X 107x./U2;
2\Uo-1
M =
I' d C O S
where Co = 1.3 - I0 "~ cm s - t is the velocity of sound in hydrogen, r is a delay time of the avalanche start and t'dc, is a velocity of the desorption front shift along the insulator. The Vd~ value is determined by the trajectory and energy of the desorbing electrons. Some of emitted electrons get on the insulator surface. If the secondary emission coefficient o > 1, the strip of a surface is charged positively and the energy of the incident electrons increases. A surface charge accumulates until the energy of the primary electrons reaches eU2 value when a = 1 on the descending branch of the secondary yield curve (Fig. 8). As is known this condition is stable if initial electron energy is more than the value of eU, value (50-100 eV), otherwise the local surface potential falls to zero. -~3 Similarly, secondary electrons charge the next strip of the insulator surface. As a result a stable chain of the surface charges is generated on the dielectric, and electrons travel along this chain from cathode to anode. 24 It follows from an analysis of the electron trajectories, that the chain parameters (step AX, average electron velocity va,, and angle ~ between insulator surface and resulting electric field) depend neither on the current value nor on the initial field direction Eo : AX=
insulator with length ~ 0 . 2 cm is about I-2 ns and rd~, 2 l0 s cm s - ~. The desorbed gas amount is proportional to the desorption time and the field-emission ctmrrent i:
,
where it is assumed: 7 ~- 100-200 tool/electron and .r, ~ 10 -3 10 -2 cm. It follows from the equation (3.14) Vd <
<
I'dc, ,,
up to an E/P value of about 10"~ V c m - ~ t o r r - ~. With such conditions, the equation (4.1) is simplified, and the gradient of the gas density along the insulator is equal to: dN
10 . 9 f
dX = -
~
(4.4)
Equation (3.15) determines the electron concentration in an avalanche having travelled along the insulator from cathode to anode. Flashover occurs when this concentration reaches some critical value corresponding to streamer formation. The avalanche-to-streamer transition is observed in the molecular gases at high pressures ( N d > ~ 5 ",: 1018 cm-2). A streamer theory does not give the quantitative criterion of the selfmaintained discharge, but to calculate the flashover voltage it is possible to use the experimental dependence of the critical (breakdown) electron concentration upon the air gap length ~z (see Fig. 9).
(4.2)
where Uo is the average energy of the secondary electrons (--,5 eV). The formation of a surface charge chain was described for the first time by ref 24, but the surface charge values are too large in this work while the authors by mistake believed that the stable point is eU~ on the ascending branch of the secondary emission curve (Fig. 8). F o r the typical dielectrics (glass, Plexiglas, ceramics) U2 = 2-4 kV 23 and
Ns/2
[ M cos ,1, V ' [ E ( X ) ] J
4-c
2
/.3
4- "
,'o 2C
~0
iO0
200
300
U,
kV
d,
cm
4-00
Vaes ~ 109 c m s - 1. O n the other hand, one can estimate roughly the same value f r o m experimental data: s the delay time from a voltage supply to appearing of the luminous spot in the cathode end of an 648
Figure 9. Ionization growth coefficient and criterion of streamer discharge. 1,2,3. M = 1.4; 1.45; 1.5 :<1016 cm-Z; 4. Streamer cri terion (~d)c = f(d). 12
,4 A Avdienko and M D Malev." Flashover in a v a c u u m
Increasing the electric field intensity leads to a rise of the field-emission current and, consequently, the anaount of the desorbed gas increases. As a result, the diffusion rate decreases [see equation (3.15)]. the effective rate of ionization increases and the flashover strength falls. When this last value becomes equal to the applied field intensity, flashover is developed. Let us call "critical" the corresponding value of a current and a gas amount i~ and M,.. If M = M,. and E is constant the ionization growth coefficient ,~*d depends only on the time delay of the avalanche start (or upon the gas density at the cathode end of the insulator): 0.295 M Ark = - C O "r
(4.5)
By differentiation of (3.15) with respect to Nk, one can see that the ~*d(ND dependence has a maximum trader the next condition (the logarithmic term varies very slowly and it can be neglected):
E,,.f(--,) = E~l'tzk), where
(a)
> a: D
ac
I
5,JL
,-,
2UO
32
(Io)
/
2,3
I 4, ;
3 d ,
4cm
5
6
7
O J,
08 d.
12
16
cm
Figure 10. Dependence of flashover characteristic upon the insulator length. (a) Breakdown voltage 1,2,3. M = 1.4; 1.45; 1.5 - 1016 cm -z. Experimental data: 0 , 26 , 2 s ~29 -c..,3o +,31 :<,3z 033 (b) Formative lag time. O, 26, 0 . 27
Cz
.f(z) = A z e - ' :
m " cos i,b,
and the subscripts k and a correspond to the cathode (X and anode (X = d) ends of insulator respectively.
O)
A. Flashover in the uniform field. It follows from equations (4.2) that the surface charge is not so large, and the angle ~b ~ 3-3.5:1. If the external field direction is parallel to the insulator surface, it is possible to neglect such small field distortion. Let us assume the electric field is uniform near the smooth surface of the cylindrical insulator: E ( X ) = Eo; = 0. With these approximations the equations (3.15) and (4.4) are integrated easily: c ~ * d = 2 . 0 4 x 109 M ( 1 . 4 x 1 0 - ' 6 {exp[-~ B z ' ] e x p [ - BZk] -k
- erf\/(Bz,)]
\.'ORB) x [erfx/(Bz k) 2.52 x 1016 M-'
( z , - l/_, _ z~
g,, 3 + In (1.2 x 10 z S - ~ %) U = Ed = 0.68
voltage: the breakdown strength decreases with increasing of the insulator length. Growth of the electric field intensity has to lead to a small increase of Mc in short insulators. Such a tendency is seen well from Fig. 10. Many other experimental facts become clear in the light of the desorplion flashover model.
x
l09 M (2a -3/2 - - Z k - 3 / 2 ) .
- l/_,))
,/ (4.7) (4.8)
Values of z, and zk correspond to maximum a*d and they are found from the equation (4.6) graphically (see Fig. 7b; E, = E~ = Eo). To calculate the flashover voltage it is necessary to know a correlation between the electric field intensity and a field-emission current. This dependence is determined by the random microgeometry of the cathode surface and it is too complicated to compute. Nevertheless, one can estimate the critical gas amount by comparing the calculated values of flashover voltage (at a given M value) with experimental data. The computing procedure is clear from Fig. 9. It is seen from Fig. 10 that the calculated plot Ub(d) coincides with the experimental one quite satisfactorily at M = (1.4-1.5) x 1016 cm -2. So the desorption model of flashover explains naturally the effect of the total
By substituting Mc = (1.4-1.5)., 10 t6 c m - : in equation (4.3), one can find the critical current is equal to 1-2 A. This value is in a good agreement with the pre-breakdown current data. 5 Roughness increases the real adsorption surface by a factor of 5-10 even for polished metals, 25 and Mc = 1.5 × 1016 c m - 2 corresponds to nearly 0.5-1 monolayer. The adsorption energy is greatest for the first monolayer and it decreases sharply with the polylayer adsorption. 25 It is obvious that the desorption efficiency has to decrease with increase of the adatom bound energy. Therefore, if the critical current remains invariable, the Mc value is the smaller, the stronger the desorbed atoms were bound to the insulator surface. It results from this that immediately after a pumping the breakdown voltage is rather small. As outer adsorbed atoms are moved away the flashover voltage rises gradually (it is seen from Fig. 9 that if the desorbed gas amount decreases by 1 - 2 ~ , Ub doubles). Gleichauf 9 has shown experimentally that insulator conditioning is promoted not only by the electrode phenomena, but also by alteration of the surface state. As a result of the insulator conditioning the equilibrium density of adatoms occurs in the breakdown channel. This concentration is supported by readsorption from the gas phase, by surface migration and mainly by gaseous diffusion from the dielectric bulk. 16.t7 A high stability of flashover voltage demonstrates that a small amount of a firmly bounded gas (1-2 monolayer) remains on the well-conditioned surface. To remove this gas is practically impossible. Under such conditions readsorption is small and the residual pressure in a vacuum system has no influence on the flashover strength, t We have found that the flashover voltage is practically independent of the composition of the residual atmosphere. In these investigations a vacuum system was filled by Ne or CC14. After a long-term exposure to such an atmosphere the breakdown strength of the insulator sample decreased (Ne) or increased (CCI4), but for some first breakdowns only. The short conditioning had returned a sample to the previous stable state and the flashover voltage became the same as it was before a gas filling. The bound energy of the first monolayer depends upon the 649
A A Avdienko and M D Malev: Flashover in a vacuum
dielectric properties rather weakly. That is why a correlation between tlte flashover parameters and the insulator characteristics is absent. The long-term exposure of the conditioned insulator sample in a vacuum leads to readsorption of some gas and the voltage of tlte first breakdown decreases, 9 but after additional conditioning the sample returns again to the stable state. 9 Bake-out of an insulator makes easy the desorption of the outer monolayer and insulator conditioning time becomes shorter. Thus, dependence of the flashover voltage upon the vacuum conditions, insulator conditioning, etc. is determined completely by bound energy of the adatoms. Tire influence of the magnetic field on the flashover characteristics was investigated by Avdienko. "-6 Tire typical dependencc of Ua on H is shown in Fig. II. The formative time lag decreases together with decrease of the flashover voltage; a plot of UB against H has the same character up to H = 40 kOe.
Tile equation (4.9) givcs H,. ~ 3-5 kOe; this value coincidcs with experimental data (see Fig. II). The formative time lag decreases since the electron trajectories in the magnetic lield are cycloidal and on drifting to the anode, the electrons pass part of their way in a vacuum. The effective electron free path increases and the drift velocity also. Tlte formative time lag of flashover consists of a statistical time delay of the avalanche start r and the drift time of electrons from cathode to anode (it is possible to neglect the time of streamer travel which is less than 10 -'~ s ~-'- i.,):
f
I.i- ---- r "Jr"
ddY "--F,d" 0
It is easy to show that the ts- value is inversely proportional to the gas density at the anode end of the insulator:
Ir -
0.295 M CoN.
(4.10)
'ooI
X 2C
[
2
5
4 H,
5
E
7
8
kOe
Figure l l . Flashover voltage in magnetic field, d :- 0.2-0.4 cm.
Such effects are observed for any magnetic field direction (along or across an insulator axis). The desorption flashover model explains these phenomena. The electron drift velocity in a magnetic field is determined by a factor [1
+ (o)T) 2] -
1,
ell~me
where T is the time of the electron free path and ,o = is the electron gyrofrequency. In the desorbed gas cloud a , T . ~ 1 up to H ~ 50 kOe and the drift velocity is practically independent of the magnetic field. But the trajectories of each electron are crooked. It is obvious that somewhere on the insulator perimeter the electron velocity has to turn to a surface for any magnetic field direction. In that region the magnetic field gives back a part of the electrons diffusing from a gas cloud, and the ionization growth coefficient increases. This mechanism is corroborated experimentally: in the transverse magnetic field the flashover channels are located just in those sections of the insulator where the electron velocity turns to a surface. 26 A fraction of the returned electrons grows (and the flashover voltage decreases) with increase of the magnetic field as long as the diffusion losses compensate completely. To estimate the critical magnetic field one can equate the electron L a r m o r radius to the gas cloud dimension R: H ~ = 3"4 x/ [7 -~ -
650
x
10ax/(EN).M
(4.9)
The equation (4.10) determines the value of the m a x i m u m formative time when the applied voltage is exactly equal to the breakdown value. Figure 10 shows the calculated plot of ts+ against d. These data are in quite good agreement with the time delay data at a small overvoltage -'7. With rise of overvoltage both components of the formative time lag decrease. To make a qualitative comparison, Fig. I 0b shows the dots corresponding to a large overvoltage -'6. It follows front the calculations that flashover formation begins for a gas density of about (2--4) ..: 1018 cm -3 (the equivalent pressure is equal to 50-100 torr). As an avalanche drifts to the anode, the gas density decreases, and it is equal to 1.5 . l0 Is cm -3 ( ~ 4 0 tort) when the flashover formation is over (for an insulator lengtlt ~ 1 cm). The N d product varies front 2 - 1018 to 5 ~.~ l 0 t s cm -2 for the insulators of length 0.1-10 cm, i.e. flashover forms in the typical range of gas density for the streamer discharge. If the gas density is smaller the self-maintaining discharge cannot form in tlte expanding gas cloud. The logarithmic term in tlte equation (3.15) is related to decrease of the electron concentration in an avalanclte owing to expansion of the bunch. At N = 10 t7 cnt -3 this term gets negative and the ionization growth stops _~ 0). That is why flashover cannot develop by the Townsend exchange mechanism: the gas density falls to 10t6-1017 cm - s during the ion path along the insulator
(a*d
(10-6-10
- 5 S).
The first assumption of the desorption flashover model is a single avalanche approximation, as is made in a streamer theory usually. In reality a criterion of tire avalanche-to-streamer transition is the ion concentration, not the electron, tz-I'* Ions accumulate in a discharge gap : I1 i ~
£
Ile k,
k=l
where s is the number of avalanches. F o r the single avalanche model s = 1 and n~ = n~. In reality the avalanches start continuously from the cathode end of the insulator, and flasbover can form at M < M~ because of the accumulation o f the ions. Therefore the above quoted values of i~ and m~ are too high, apparently. The second rough assumption is an extrapolation of the equation (3.14) far over the experimental limits (flashover develops at = 1000-2000 V c m - t t o r r - t).
E/P
A A Avdienko and M D Malev: Flashover in a v a c u u m
Despite these and some other assumptions the flashover desorption model coincides well with experimental data and the estimation is quite reasonable o f values which are not measured directly (M,., N). The model is rather stable; the final conclusions are the same as if it is assumed the desorbed gas is hydrogen, nitrogen, oxygen, o r air.
5. Conclusions The diversity o f p h e n o m e n a under flashover in a vacuum is described quantitatively by means o f two models: the thermal one and the desorption one. The thermal flashover develops in dielectrics with relatively low specific resistance ( p , . ~ 1 0 " - I 0 ' - " ~--~ cm) if the voltage pulse duration is more than 10 -3 s. In this case, the flashover characteristics are similar to the ordinary thermal b r e a k d o w n of the solid dielectrics. The desorption form o f the flashover is a streamer discharge in the desorbed gas cloud. An analysis o f the electron avalanche drift in the restricted cloud o f e x p a n d i n g gas (with supersonic velocity) makes it possible to calculate the b r e a k d o w n voltage and the formative time lag for the cylinder insulators. The physical properties o f the dielectric and the vacuuna conditions have practically no influence on the characteristics o f desorption flashover, since to form the discharge, it is enough to desorb less than one monolayer o f adatoms. If a flashover develops in the uniform electric field the surface charge takes a slight part, but this part becomes essential if" the electric field is not parallel to the insulator surface. Both p r o p o s e d models make it possible to systematize the n u m e r o u s flashover experimental data and to outline ways of constructing high-voltage vacuum insulators.
References t 1 N Slivkov, Electrohlsuhttion am/ I~bcaum Discharge, Atomizdat, Moscow (1972). z M J Kofoid, AIEE Trans, No 6, 1960, 991,999. 3 A. Watson, J appl Phys. 38, 1967, 2019.
~C H De Tourrcil and K D Srivastava, IEEE Trans Electr hlsal, El-8, 1973, 17. 5 S P Bugaev, A M Iskoldski and G A Mesyatz, Zh Tekh Fiz, 37, 1967, 1855, 1861. " W Franz, Break~hnt,n o f Dielectri¢w, lnostr Liter, Moscow (1961). G A Grinberg, M I Kontorovitch and N N Lebedev, Zh Tekh Fiz, 10, 1940, 199. 8G A Widrik and N S Kostjukov, Prothwthm o f Electroceramh', Energia, Moscow ( 1971 ). " P H G leichauf, J appl Phys, 22, 1951,535,766. ,o Milton, IEEE Trans Electr hlsul, El-7, NI, 1972, 9. ,t V 1 Rakhovsky, Physical Faundathm o f Electron Carrent Consma/ticathm itl Va('tatm, Nauka, Moscow (1970). '-' J Mick and J Craggs, Electrical Breakdown o f Gases, Inostr Liter, Moscow (1960). ,3 H Raether, Electron Aralanches and Breakdown in Gases, Mir, Moscow (1968). '" E D l.ozansky and O B Firsov, Spark Theoo,, Atomizdat, Moscow (1975). ,5 A A Avdienko and A V Kiselev, Zh Tekh Fiz, 37, 1967, 533. t6 N V Tscherepnin, Vacattm Properties o f MateriaL~ fin" Electron Tubes, Sov Radio, Moscow (1966). ,7 M D Malev, I/aeuttm, 23, 1972, 43. t, K L Stanjukovitsch, Unsteady 1%4ol'ementafSalid Medium, Nauka, Moscow ( 1971 ). t,, V L Granovsky, Eh, ctrieal Current hi Gas, Vol I, Gittl, Moscow (1961). 20 S S Brown, Elenwntao, Processes hi Gas Discharge Plasma, Atomizdat, Moscow (I 961 ). -'~ H S Carslaw and J. C. Jaeger, Conduction o f Heat h/ Solids', Nauka, Moscow (1961). .,z A Engel, hmized Gases, Fizmatgiz, Moscow (1959). ,-.3 1 M Bronshtein and B S Fraiman, Secondary Electron Emission, Nauka, Moscow (1969). z.* H Boersch, H Hamisch and W Ehrlich, Z angew Phys, 15, 1963, 61. "~ S Dushman, Scientific Foundation o f Vacaant Technique, Mir, Moscow (1964). .,6 A A Avdienko, Zh Tekh Fiz, 47, 8, 1977 (to be published.) _,7 S P Bugaev and G A Mesjatz, Zh Tekh Fiz, 35, 1965, 1202. 28 E S Borovik and B M Batrakov, Zh Tekh Fiz, 28, 1958, 1971. z,, K D Srivastava and C H De Tourreil, quoted by Slivkov.' 30 K D Srivastava, Proc 2rid hit S),mp htsulation hi Vacaunl, Boston, MA (1966). (p.229) 31 m K Walter, Electrostatie Accelerator o f Charged Particles, Atomizdat, Moscow (1963). 3-, R C Finke, Proc 2rid hit Syrup htsa/ation ht Vaeatun, Boston, MA (1966). (p.217) 33 N Hamidov, Proc Akad Nauk Uz SSR, No I1, 1968, 16; No 1, 1969, 25.
651