GAS DISCHARGES

Breakdown in nonuniform fieldsR.T. Waters, B.Sc, Ph.D., C.Eng., M.I.E.E.

Indexing terms: Breakdown and gas discharges, Discharges, electric, lonisation

Abstract: The paper presents a review of recent studies in the field of air breakdown. Coverage is given bothto technological aspects of UHV systems and to the physics of spark breakdown, reflecting current interest in'sparkover voltages, spark leader growth and space-charge effects.

1 Introduction

There is indeed much work to report, particularly from labora-tories in Europe, USSR and Japan, and I shall try to refer bothto some technological aspects of UHV systems and to themarkedly improved understanding of the physics of sparkgrowth. The session on breakdown at this Conference [1] con-tained eight papers concerned with sparkover voltages, fivecontributions on spark leader growth, and four studies ofspace-charge effects. This review reflects these areas of specialinterest.

2 Sparkover characteristics

2.1 Critical voltages

The positive-impulse sparkover voltage of long air gaps exhibitsthe well known U-shaped dependence on time to crest [2].The minimum sparkover voltage arises when the time to break-down TB is equal to the time to crest Tcr. Under these criticalconditions TB is mainly controlled by the leader velocity.Neither gap length nor geometry has any influence upon thisminimum stable velocity; we shall consider this point further.

It is this optimisation of leader propagation that leads tothe much lower strength of air gaps under switching impulses,and to the nonlinear sparkover-voltage/gap-length charac-teristic. The characteristic for rod/plane configurations hasrecently been extended to very large distances at the CESItesting station at Suvereto [3]. The results for gaps in therange 12—32 m appear to answer an important outstandingquestion on the behaviour of long gaps — is there an upperlimit to the critical positive sparkover voltage? The answerappears to be that there is not. Table 1 shows that the Suveretoresults indicate a rate of increase of sparkover voltage with gaplength of 0.055 MV/m. This implies that this is the limitingmean gradient in the positive leader channel. An interestingsituation remains on the question of negative impulse spark-over. Although a weak U-curve behaviour is found with nega-tive impulses, the negative sparkover voltage remains much

higher than the positive. For gaps up to 14m the empiricalformula

Vs(min) = \A8d0AS

gives a good fit. Bus since recent work [4] has shown that thenegative leader channel is physically indistinguishable from thepositive, we would expect that for very long distances thenegative characteristic would approach the same slope of0.555 MV/m as for positive impulses.

2.2 Statistics

Whereas the impulse sparkover voltage which results in a 50%probability of breakdown is largely governed by the physics ofleader inception and propagation, technological UHV appli-cations frequently require an estimate of the voltage corre-sponding to low or high probability of sparkover. The value oftest data is thus considerably enhanced if the standard deviationa is also given, and if a hybrid strategy which combines up-and-down and constant-level tests is used. Then the conven-tional safety factor

V8(min)-3o

V (SI max)

may be employed; alternatively the more economical 'risk offailure' R may be evaluated:

R = \~p(V)P(V)dV

where p(V) = probability density of surge distribution andP(V) = probability of insulation or clearance failure. Thisapproach is particularly interesting since the risk of failure canbe estimated for any chosen value of the 'statistical safetyfactor' SSF, where

SSF =statistical withstand voltage

statistical overvoltage

Table 1: Sparkover-voltage/gap length

Author V50 formula(MVm)

ExtrapolatedV50(MV)d= 50 m d= 100 m

Alexandrov (1969)

Feser (1970)Galletera/. (1975)Waters (1978)Harbecera/. (1978)Kekezera/. (1979)Piginiefa/. (1979)

tanrT1 >/(p= 9 10"3 (100 + 0.3d)0.7 dos - 0 . 2 5

3.4/(1 + 8/d)0.56 c/05

1.35c/°"/(1.31 +2.11/C/)Volterra model1.40 + 0.055 d

—pld) 3.39 4.27

4.702.933.983.633.04.15

V(CF0)-\3aV(mean) + 2.05a

10% probability of failure

6.753.155.634.643.26.90

Paper 1295A, presented in original form at the IEE 6th InternationalConference on Gas Discharges and their Applications, 8th-llthSeptember 1980Dr. Waters is with the Department of Electrical Engineering, Universityof Wales Institute of Science & Technology, Cathays Park, Cardiff,Wales voltage exceeded by 2% of surges

IEEPROC, Vol. 128, Pt. A, No. 4, MAY 1981 0143-702X/81/040319 + 07 $01.50/0 319

It is strongly to be recommended that those workers reportingtest data should establish the statistical normality of their data,and should quote not only the values found for sparkovervoltage and standard deviation but also the 95% confidencelimits on both Vs and a. Because of the nature of these par-ameters there is usually much greater uncertainty in the valueof a. For a 30-impulse test the confidence limits on V areabout ± 2%; on a they are about ± 50%.

Although it is useful that breakdown probability oftenclosely resembles the statistical 'normal distribution', it isimportant to be alert to the occurrence and causes of seriousdepartures from this behaviour. The breakdown probabilitycurve often shows a binormal form [5], indicating the coexis-tence of two breakdown modes. Use of the calculated a thenleads to an over-pessimistic value for the safety factor. Anexample of this binormal distribution is given for insulatingstructures in the paper by Volkova and Sloutskin [6]. Fortu-nately, where the distribution is more normal, the estimate ofwithstand voltage usually errs on the safe side by about onestandard deviation.

2.3 Gap geometry

Many physical studies are carried out in rod/plane gaps, whichfor positive unidirectional impulses exhibit the minimumsparkover voltage. For engineering purposes, the effect of gapgeometry is naturally of considerable interest. In this case, theproblem begins with an adequate definition of the dimensionsof the test configuration.

A very useful contribution was made by Paris [7] over tenyears ago when he defined a gap factor k for any configuration:

k =V^configuration)

Vs (rod/plane)

The gap factor, which ususally lies in the range 1—2.2, wouldbe an especially valuable tool if

(a) k remained constant with varying gap length and im-pulse shape, and

(b) k could be evaluated from electric field or potentialcomputations.The effect of gap length can be illustrated by means of a logar-ithmic plot of Vs and d, since

In Ve{configuration) = \nk 4- In Vs(rod/plane)

would imply a constant vertical displacement between thebreakdown characteristics. In fact, a constant horizontal dis-placement is indicated. Jones and Waters [8] thus suggestedthat a spacing index S could be defined by the ratio of theconfiguration gap length to the rod/place gap length having thesame sparkover gradient. Then the sparkover voltage Vs(d) forany configuration of spacing index S can be obtained from thebasic rod/plane characteristic using the equation

Vs(d) = S VRP ( I I

In view of the considerable progress made in recent years inthe digital computation of electric field distributions bycharge-simulation, finite-element and Monte Carlo methods, itshould now be the aim to examine the possibility of predictinggap factor, spacing index or similar aids to design.

Certainly some useful progress has already been made toclassify air-gap behaviour in terms of the degree of nonunifor-mity of the field distribution. It is well known that the spark-over voltage of a sphere/plane gap approaches that of therod/plane gap if the ratio gap-length/sphere-radius is suf-ficiently large. Carrara and Thione [9] introduced the critical-radius model, based upon the fact that the voltage Kx required

for leader inception is approximately independent of sphereradius below a critical radius Rc and is equal to the corona-inception voltage (which is easily calculable) for radii greaterthan Rc. The sparkover voltage is obtained by adding to Vx afurther increment VL necessary to maintain propagation ofthe leader. The concept of critical radius is a useful one inseveral respects. Eriksson [10] uses it to examine the initiationof an upward lightning leader from a tall mast; in anotherpaper Bazelyan [11] et al. point out the critical range of fieldnonuniformity mcr

= E max l& mean (which, we may note, isequivalent to d/Rc) which will cause low sparkover voltageson sign-reversing voltages.

3 Leader channel and space charges

Much of the macroscopic behaviour of air-gap breakdown canbe explained in terms of the behaviour of the leader channel.An interesting example is in the case of an oscillatory impulseshape. Many surges occurring on power systems have a largeoscillatory component, and sparkover often takes place on alater voltage crest even when significant damping is present.This can arise because of the resumption of leader propagationnear the voltage crests following partial breakdown of the gapduring the previous oscillations, as shown by the image-converter records of Pesavento [12]. Similar effects can alsooccur with partly chopped impulses.

It is in the study of the parameters and the physics of theleader channel that the most remarkable progress has beenmade in the last five years. This is in part due to the concen-tration of research in the breakdown of very long gaps. Atlarge spacings not only is the longer formative timelag helpfulin experimental studies, but processes involving significantlylong time constants have the opportujity to evolve [8].

The following account is based mainly upon the work ofmembers of the Les Renardieres Group [13—15] and a recentreview by Jones and Waters [8].

3.1 First corona

For positive impulses, streamer growth develops at a velocityof about lm/jus. Each streamer channel can be regarded ascomposed of a passive region (the filament) behind an activeregion (the tip). The passive region is quasineutral, and con-tains electrons, negative ions and positive ions. The number ofcharges is about 7 x 10n — 6 x 1012 cm"1, with an excess ofpositive ions of 6x 108 — 3x 109cm"1. Measurements offilament radius (10—30/um) and current, suggest electrondensities in the range 5 x 1013 — 10ls cm"3. The gas in thefilament remains cold, with a rotational temperature notgreater than 330 K.

In the active region, Stark broadening measurements indicatean electron density of 1015cm"3; since UV spectroscopyshows many molecular transitions of N2 with energies abovethe ionisation threshold of O2 , photoionisation seems entirelypossible as a secondary mechanism.

The stability or guiding field needed for streamer propaga-tion is known to increase with gas density and humidity,whereas the injected charge decreases with increasing humidity.These effects can be explained by replacing the usual spherical-tip model by a linear charge, whose length would be stronglyaffected by variations in attachment rate.

3.2 Leader velocity

Depending on the anode curvature and the form of the appliedvoltage, an incipient positive leader is formed as a small highlyionised filament following the first or subsequent corona. Alarge leader corona is usually associated with the formation ofthe leader; in gaps of up to 2 m there is little elongation of theleader filament before the leader corona bridges the gap. A

320 IEE PROC, Vol. 128, Pt. A, No. 4, MA Y 1981

rapid rise in current and acceleration of the leader tip thenfollows. When little or no leader development occurs beforethis phase, the discharge is usually termed as a direct break-down. In longer gaps, appreciable leader growth is observedbefore the acceleration phase, the latter then being known asthe final jump.

If an impulse of the optimum or critical time to crest isapplied to the gap (for example, T^n = 500/xs at 1.8 MV fora 10 m gap), the velocity or rate of elongation of the narrow,tortuous core of the leader is quite constant and reproducible.Measurements of the 'real' velocity vL, which involves3-dimensional analysis, shows it to be 18mm/jus with a stan-dard deviation of only 3%. It has not proved possible topredict from such velocity measurements which leaders willculminate in the final jump, or those which will terminatewithout resulting in sparkover. The stable velocity is even lessaffected by

(a) application of overvoltage: a doubling of the appliedvoltage increases VL by less than 5%

(b) anode geometry(c) gap length.

At high humidity (greater than 12g/m3), however, re-strikes — sudden elongation and brightening of the leader — canoccur.

The variability of time to breakdown TB arises mainlybecause of tortuosity and off-axis growth of the leader. Theaxial velocity vz is consequently lower (about 0.75 vL) andmore variable {a(yz) ~ 20%). The proportion of the gap lengthtraversed during the leader propagation phase before onset ofthe final jump depends on gap length and the applied voltage.In a 10 m gap, the final jump commences for axial-leaderlengths in the range 6—8 m, although leaders rarely exceed4—5 m axial length in cases of eventual withstand. The final-jump onset occurs earlier in the leader development when anovervoltage is applied; this earlier acceleration of the leaderfrom its stable velocity vL is the main reason for the observedvoltage/time characteristic of air gaps.

3.3 Impulse shape

There seems little question that the optimum conditions forleader propagation at 18 mm/jus arise when the rate of appliedvoltage matches the rate of leader elongation. For double-exponential impulses, the critical time to crest results in themost effective leader propagation and lowest sparkover voltage.A faster application of the applied voltage, as in lightning-impulse tests, merely results in a faster-developing but short-lived leader channel. Sparkover is then achieved mainly byincreasing the applied voltage so as to bridge the gap by theleader corona and to initiate the final-jump phase. On theother hand, because 18mm/jus appears to be the minimumstable leader velocity, too slow an application of the appliedvoltage causes interruptions to leader extention and a lesseffective, discontinuous discharge growth.

3.4 Leader velocity

During the stable regime of leader propagation with theminimum applied voltage needed for leader growth, the dis-charge current also remains approximately constant (0.6 A at5 m, 0.8 A at 10, gap length). In these circumstances it ispossible to visualise an injection of charge into the gap corre-sponding to about 45/iC/m of leader length. However, thecharge injection can be much greater than this, since applicationof 100% overvoltage can increase the current to about eightfoldcompared with only a 5% increase in vL. In this case it seemsreasonable to suppose that most of this increased charge residesin the extended leader-corona region ahead of the leader.

IEEPROC, Vol. 128, Pt. A, No. 4, MAY 1981

3.5 Energy dissipation

The observed currents mean that even at critical breakdownthe mean power input to the leader growth is about 1 MW.Some of the energy supplied, amounting under these conditionsto about 40 J/m extension of the leader, is stored in the redis-tributed electrostatic field, and most of the remainder is utilisedin discharge processes. This division of energy was studied [13]by examination of the space charge in the gap using a rotatingfluxmeter, which indicated an approximatly equal sharing.Much of the utilised energy causes heating and expansion ofthe gas; calibrated-photomultiplier/slit-measurements showthat very little energy is lost by radiation.

3.6 Final jump

The space charge associated with leader growth increases thefield in the unbridged gap between the leader corona and theearthed plane. Many investigations have shown that a fieldstrength of 0.45—5 MV/m is necessary for the propagation ofpositive streamers. When the leader corona reaches the plane,cathode secondary processes ensure an increase in leadervelocity, and in this final-jump phase the velocity can increaseby two orders of magnitude. This phase has practical impli-cations since it determines the steepness of voltage collapse.In the absence of a negative leader, as for plane cathodes orbecause of negative space-charge effects, the final-jump processarises from an acceleration of the positive leader when theleader corona bridges the leader-to-cathode space. Reductionof local capacitance will suppress this process, and it has beenshown that there is an upper limit to the charge flow in thegap even for very large capacitance. This limitation is almostcertainly imposed by the rate of electron generation in theleader-to-cathode corona. The main electron generation musttake place very near to the cathode surface: it is known fromsmall-gap experiments that a cathode fall exists at the streamer/cathode interface. There has been no direct observation ofthese effects in long-gap breakdown. It may be possible todetermine a limiting current density which a plane cathode iscapable of providing. It would then be possible to model moreprecisely the final-jump process. Suitable probes reveal largefields at the plane electrode (up to 1.5 MV/m just beforebreakdown) which indicates the large effect of positive spacecharge just in front of the cathode.

3.7 Field probes

Since the discharge growth takes place in a strongly modifiedspace-charge field, the use of field probes to examine fieldchanges during and after the breakdown or withstand is invalu-able. Simple small-area capacitive probes may be used in regionswhere negligible conduction occurs, and we have examples ofits use in the present papers by Isa et at [16] in air gaps, andby Volkova, Sloutskin and Zargarian [16, 17] in the break-down of insulating structures. In regions where both displace-ment and conduction currents are present, other types ofprobe are available, such as the rotating fluxmeter, the fieldfilter and the biased-current probe. The availability of fieldmeasurements is all the more useful when combined with therecent capability of computer space-charge simulation by ring,line or point charges.

3.8 Leader gradient

Perhaps the most important single parameter governing thesparkover characteristic in highly nonuniform fields is the elec-trical gradient within the leader channel, and its influence onthe stability of leader propagation. Field probes have played auseful part in obtaining measurements of this gradient. Oneexample of this is in the use of fluxmeters in long gaps, where

321

the location of space charge is shown to be closely related tothe position of the leader tip. The measurement of chargedensity then allows the mean gradient in the leader to beinduced. An example is given in Table 2.

Table 2: Mean leader gradient g in 10 m rod/plane gap at Vso (criticalsparkover)

Lz(m) 2 3 4 5ff(MV/m) 0.5 0.35 0.25 0.18

This characteristic means that the potential at the tip of thedischarge varies only slowly during its development, which isconsistent with the uniform velocity and current flow ob-served. It is also, of course, the underlying cause of the fallingbreakdown-gradient/distance characteristic observed for longgaps, which has been referred to earlier.

3.9 Leader diameter

All the recent theories of the leader channel attempt, in variousways, to account for this negative-gradient characteristic.These theories have one important need in common, which is aquantitative knowledge of the leader diameter as a function ofcurrent and time. Measurement of this diameter by directphotography is difficult because of the limited resolving powerof this method. It is pleasing to report, therefore, that thisproblem has largely been solved by the use of strioscopy(Schlieren photography), combined with high-speed imagerecording. We may note an interesting paper by Kekez andSavic [18] also using this technique. In experiments in a 1.5 mrod/plane gap, with positive impulse of 30/us time to crest,Ross [13] found that the leader-channel thermal boundarywas so clearly defined that the diameter could be measuredwith a precision of ± 0.1 mm. The diameter of a given sectionof the channel increased with time, giving a tapered appearanceto the leader at any instant. No strong shock wave was observedunder these conditions, and the rate of radial expansion wasless than 100 m/s in all cases. This indicated that the gaspressure within the leader channel remained constant at theambient value. The diameter in this case increased from aninitial value of about 0.3 mm to about 1.8 mm just beforesparkover.

This work has been extended by Gibert et al. [4] by com-bining strioscopy with image convertor recording. It is thenpossible to detect not only the density change within theleader core but also the generation of a sonic wave associatedwith the leader. The increase of the positive leader cross-section with time shows a similar increase to that previouslyobserved. Perhaps an even more striking method of employingthe strioscopic technique is the slit method recently employedby Gibert et al. A field of view 0.66 mm by 35 mm at a distanceof lmm below the HV rod in a 2.18 m gap was examinedstrioscopically. An image convertor in a sweep mode timeresolved the slit image at right angles to its length, i.e. parallelto the gap axis. The 130/3000/us positive impulse had a crestvoltage of 885 kV. With this method, the birth of the leaderchannel with an associated pressure wave, the growth of theleader channel diameter, the instant of breakdown and theresulting channel expansion and shock wave are all clearly seen.In the case quoted the boundary-expansion velocity of theleader was 18 m/s.

3.10 Gas-density reduction

The expansion of the leader channel and the decreasing axialgradient characteristic are clearly related parameters. Theabsence of strong shock waves and the nature of the strioscopicrecords of the channel show that the neutral-particle densitywithin the channel is a decreasing function of time. This

reduced density is the basic explanation of how the leader-channel conductivity can be sustained even at the low overallelectric fields present in switching impulses. This process is asfundamental in understanding the long-spark breakdown pro-cess, as are the Townsend and streamer mechanisms to thebehaviour of corona phenomena.

Whereas, at atmospheric density, a field of about 3 MV/m isnecessary to maintain direct-impact ionisation, an expansionof the leader radius by a factor of three will reduce the requiredfield to little over 0.3 MV/m, which approaches the averagefields available in long-gap sparkover. Indeed, recent work byKline and Denes [19] considered the effect of gas densityupon the necessary value of E/n at which ionisation andattachment rates become equal.

It was found that this value decreased from about 94 Td at760torr to about 78 Td at lOOtorr. This would seem to haveimportant implications for the leader channel. If one considersa leader at increasing temperature T expanding at constantpressure, then at a temperature of 2700K these data suggestthat E/n = 78 Td = ET = 5.7 108 V K/m is sufficient, ratherthan E/n = 100 Td = ET = 7.3 108 VK/m. No data were givenfor lower gas density but an extrapolation to 5400 K gives55 7tf = 4 108 VK/m.

3.11 Physical theories of leader

Recent theories have indicated that the value of ET in theleader remains approximately constant. In the approach ofRoss et al. [13], it is noted that the constant-current con-dition observed in the leader channel at critical breakdownimplies that

ne

— ve — constantn

where ne,n are the electron and neutral-particle volume den-sititesat any instant during leader growth, and ve is the electrondrift velocity. This equation supposes that the mass of gaswithin the expanding leader channel remains constant.

Now the equilibrium electron density in the leader, whenthe positive-ion density of nh is given by the condition

Kxnne—K2nen2 — K3nent = 0

where Kx, K2, K3 are, respectively, the rate coefficients forionisation, free-body attachment and recombination. Since theleader channel can certainly be regarded as a quasineutralplasma with ne — nh we have

ne_ _ Kj -K2n

n K3

If, as n decreases because of channel expansion, we supposethat the parameter E/n increases, then the ratio ne/n wouldalso increase strongly; this is because Kt is much increased, K2

and K$ are decreased. Furthermore, the electron drift velocityve increases approximately linearly with E/n the range ofinterest. Thus the increased value of nevejn would imply anincreasing leader current with time, contrary to observation.The constant-current condition indicates a constancy of E/nand ET. The value of ET in the leader can be obtained byassuming the value of E/n in the channel to be 70 Td. Then wehave

ET = K (constant) = 5.1 108 VK/m

If a constant specific heat Cp is now assumed, then the rate ofrise of temperature in any element of the leader is

dt

322 IEEPROC, Vol. 128, Pt. A, No. 4, MAY 1981

where B is constant. We may combine these equations toobtain

JL dJLBE3 dt

Under critical breakdown conditions, where leader current andvelocity (dL/dt) are constant, we then have

dL_

dt

KBqE3 dt

where q is the injected change for unit leader length. Then theelectric field strength at an element of the leader at distance xfrom the electrode is

-v-l/2

2Bq _E~ =

where Eo is the initial field when the channel is formed. Themean electric gradient within the leader channel of length L is

8L = T \ Exdx

K

BqL

If Eo is taken as 0.5 MV/m (corresponding to the transitionfrom the positive streamer filament) and the initial leaderradius of 1 mm, Table 3 shows the variation in gL and leaderradius a with leader length.

Table 3: Calculated leader parameters (at x = 1 m)

Mm) 1 2 3 4 5 6 7SL(MV/m) 0.39 0.28 0.23 0.10 0.18 0.17 0.15a (mm) 1.0 3.8 4.6 5.0 5.4 5.7 6.0

Gallimbert [13] has described a detailed hydrodynamic/thermodynamic analysis of the leader channel, again basedupon the expansion of a constant mass of gas within thechannel. From such considerations as the conservation ofneutral particles, heat transport at the boundary, energy trans-fer from charge carriers to neutrals, ambipolar diffusion and aradial distribution of gas temperature, it is possible to obtain,as a function of time, the rate of expansion, increase in tem-perature and the weak shock generation at the leader boun-dary. The value of ET within the channel again remainsconstant, and the same value of 5.1 108 VK/m emerges asbefore.

There would appear to be some difficulty with both analysesin explaining the extent of the leader expansion observed inthe strioscopic studies. The slit method clearly shows that theincrease of leader diameters before breakdown can be aboutthirty-fold. The Ross model predicts about a three-foldincrease; the Gallimberti equation relating increase of cross-sectional area with power input to the leader is

dt IP

which from the measured values oidS/dt (about 0.1 m2/s) and/ give ET=\ .84107 VK/m, a very low value corresponding toE/n = 2.5 Td. It may therefore be necessary to re-examine thebasic assumptions of these methods, such as the constancy ofthe mass of gas within the leader.

There is much theoretical interest also in how the energyacquired by electrons from the applied field maintains ioni-sation within the leader. The above analyses assume directionising collisions as a result of electron acceleration in thefield. An alternative model is that of thermal ionisation, to

which Saha's equation could be applied. Plausible values of gastemperature and axial gradient can be obtained in this way.However, the electron density of 1017 irf3 is much lower thanthe 1023 m"3 normally associated with thermal equilibrium. Aninteresting paper by Brambilla [20], for a low-current (1A)leader, suggests that a stable electron temperature is reachedwithin a few microseconds, the ionisation being caused bythose electrons on the high-energy tail of the distribution.

It is useful to recall a recent suggestion by Darveniza andHolcombe [21] that, at the gas temperatures found in theleader, both thermal ionisation and direct-implact ionisationmay coexist. This could come about if there are two distinctelectron-energy distributions; the authors indicated that thelow-temperature thermal-equilibrium population would berelatively stable, since a time of about 2 ms would be necessaryto raise the mean electron temperature from 5000 K to20 000 K. This is an interesting proposal which spectroscopicstudies might verify.

3.12 Leader tip processes

It seems generally agreed that the charge separation processesoccurring in the leader corona, itself maintained by the highfield present ahead of the leader, provide the means by whichthe energy input required for the elongation of the leader canbe sustained. It is not yet clear which mechanisms are dominantat the discontinuous boundary which is present at the leadertip.

Alexandrov [22] envisaged that the energy input would bemainly utilised in heating the gas in a leader tip of about 1 mmdiameter to a temperature of the order of 500 K. It is not,however, certain that conditions of local thermal equilibriumcould be established within the time available.

Kekez and Savic [23,24] considered a parabolic profilenear the leader tip, but of very small radius of curvature(about 30/im). A very rapid thermalisation of energetic freeelectrons from the leader-corona streamers would raise the gastemperature in the small-volume tip to extremely high values,of the order of 106 K. The very strong hypersonic shock wavevelocity would then constitute the leader tip velocity. Theyshow in their present paper [18] experimentally observedshock waves which might demonstrate this electrically sup-ported detonation process; as these authors state, it is impor-tant to demonstrate whether this shock wave exists all aroundthe leader tip.

An alternative process, suggested by Gallimberti [14], isthat the leader tip represents the boundary at which thenegative ions present in the passive region of the leader coronabecome subject to thermal detachment. A gas temperature inthe range 1000-2000 K would be required for this process,such localised heating being provided by random field intensi-fications causing an increased power input per unit volume.The thermal-detachment process would be accelerated bylocally increased E/n. Gallimerti also visualised a narrowparabolic profile with a tip radius of about 25 //m, but causedby a weak point explosion whose source is translated at theleader velocity.

4 Models of breakdown

4.1 Jones's model

For engineering purposes, the objective is to incorporate whatis known of the physical characteristics of leader breakdowninto predictive models of sparkover and its statistical variabi-lity. I will mention briefly only two such models, i.e. that dueto Jones [25] and its subsequent development by Butzler[26]. In Jones's model the developing discharge is representedas a cylindrical column carrying a uniform charge of q Cm"1 so

IEEPROC, Vol. 128, Pt. A, No. 4, MA Y1981 323

that the surface field strength is just too low to supportfurther ionisation. If yd is the column length at any instant(0 <y < 1), and the anode potential is V, then the potential Pat the tip of the advancing column is

P = V-gyd

The column propagation velocity is then obtained from

_ d(yd) _ dV/dt -yd • dg/dtdt ~ g + (l/d)dP/dy

This expression contains the necessary conditions for bothwithstand and sparkover: if the rate of rise in voltage is toolow or the conductivity of the column is insufficient, then thenumerator becomes zero and the discharge is interrupted; asthe column propagation reduces the unbridged gap, so themagnitude of the negative quantity dP/dy increases, and asufficiently low value of g makes the denominator zero andsparkover follows.

The model is quantified by considering the column to havean inverse voltage/current characteristic, so that the limitingcolumn gradient is

go =_ ,—0.4 kV/cm

This gradient is reached only with a time constant r (with anempirical value of 50JUS), so that after a time t of uniformgrowth the mean gradient along the column is

Sit) = go+f (gs -go) {1 - exp ( - t/r)}

gs is the initial column gradient, made equal to the streamergradient of 0.5 MV/m.

4.2 Hutzler model

In the Hutzler model two important further features areadded. First, any interruption to the leader current whichoccurs because of a fall in tip potential P is considered to causea deionisation of the leader core, so that the leader gradientincreases according to the equation

g(t') = *(r ,)exp(r7r ')

where t' = t — tt is the time elapsed since interruption of thedischarge at / = tt. The time constant T is taken as 500jus. Thedischarge restarts if dV/dt is sufficiently high to compensatefor this increasing leader gradient; if g(t')>gs, the leader isassumed to be permanently stopped. Secondly, the effect ofgap geometry and the presence of statistical variations in dis-charge behaviour are taken into account in a way which alsosimplifies computational problems. The leader channel is con-sidered to consist of successive straight segments of length0.2—0.45 m, which make some angle 0 to the gap axis. Thegradient in each segment is considered uniform, and the angle0 for each new element is defined as having a probability densitywhich is a function of both the applied-voltage shape and gapgeometry. The leader will stop if, at the termination of asegment, the angle 0 is so great that the field falls below acritical value Ec — 3.1 MV/m necessary to maintain the coronacurrent into the leader. On the other hand, a value of tipfield E>EC allows the growth of a further segment and deter-mines the new value of propagation velocity and current.

The minimum possible sparkover voltage Vs = K(0%) isobtained by taking 0 = 0 for all segments. The simulation ofcurrent and leader growth are then similar to those obtained inthe calculations of Jones and Waters, and show a good repre-sentation of the constant leader propagation and dischargecurrent at critical breakdown, and the discontinuous nature of

growth under slowly rising impulses. It also simulates the dif-ference between maximum axial leader length at withstandand the leader length at onset of the final jump.

When the tortuosity of the channel is taken into accountby these means, the statistical fluctuations in current andleader growth closely resemble those found in actual dis-chearges. The macroscopic data of time to breakdown andbreakdown probability obtained from tests on a 10 m rod-plane gap can also be simulated, both from constant-voltage-level tests and up-and-down procedures.

5 Negative breakdown

In most practical configurations, such as phase-to-phase orlongitudinal insulation, at least part of the discharge growthwill originate at the negative electrode. Indeed, most of thedangerous surges originating in power networks arise fromnegative lightning strikes. Two major studies [4,27] of negativebreakdown in long gaps have shown that the complexity of theprebreakdown events is greater than for positive discharges.

5.1 Leader structure

Compared with the spatially continuous leader which isobserved to develop in a positive rod/plane gap, the leaderchannels in negative-impulse breakdown may be classified intothree components:

(i) a negative leader, which develops continuously at criti-cal breakdown, but superimposed on the growth of which aresudden elongations each with accompanying reillumination ofthe whole channel, a superimposed current pulse of 10—100A,and a large negative corona burst from the tip of the elongatedchannel. Except for fast-rising impulses, little negative-leadergrowth may occur before the first elongation. Between reil-luminations the propagation velocity is about 10mm//is andthe current is 0.5 —1 A.

(ii) one or more space leaders, which originate well aheadof the negative leader. These leaders have the same tortuousnarrow channel as normal leaders; they are elongated in bothdirections by tip velocities which are typical of normal leaders(30mm/jus towards the cathode, 10mm//zs towards the anode).In a 7m gap an average of three or four space leaders arefound; their number is an increasing function of gap length.Adjacent space leaders may make a junction before meetingthe already established negative leader especially for short-fronted impulses. It is when this latter meeting occurs that thewhole channel undergoes the reillumination already mentioned.Space leaders can be up to 1.5 m in length at this point

(iii)when the corona streamers from the leader systemreach the plane, one or more upward-developing positiveleaders are established and the discharge enters the final-jumpstage. In rod/rod gaps such a positive leader inception canoccur as a result of field enhancement at the earthed anodebefore traverse of the gap by corona.

5.2 Leader-corona structure

As with the positive discharge, the negative-leader/space-leadersystem is both preceded by and accompanied with coronadischarges. The first corona event tends to occur with lowerscatter in inception time than for positive corona. It is esti-mated that the mean field strength required to maintainpropagation of negative streamers is about 1.8 MV/m, comparedwith about 0.45 MV/m for positive streamers. The propagationvelocity is about 5 m/fis.

The leader-corona system ahead of the leader tip also has acomplex structure, which between reilluminations consistsagain of three regions:

(i) A space stem, which is a small periodically-luminous

324 IEEPROC, Vol. 128, Pt. A, No. 4, MAY 1981

zone at 0.5—1 m ahead of the negative leader, which progressesin steps towards the anode at an effective velocity of aboutlOOmm/jus. The space stem appears to be the point of originfor the possible launching of a space leader. It is also the originfor the remaining two components of the corona system,which are the negative and positive streamers.

(ii) The negative streamers in front of the space stem arebranched filaments of high velocity (several m/fis) and of lessthan 1 m length. They appear in successive bursts coincidentwith the space-stem steps.

(iii) The positive streamers are also periodic, and are estab-lished in the space between the space stem and the negativeleader. They progress in the retrograde sense from the spacestem with a velocity of about 200mm/^s. The filamentarystructure of the positive streamers is less apparent than that ofthe negative streamers. If a space leader is formed, a furtherpositive streamer zone is present between the space leader andthe negative leader (or the adjacent space leader).The regular periodic illumination of the space stem/negativestreamer/positive streamer system is accompanied by small(1-5 A) current pulses.

5.3 Long gapsThe complex phenomenological developments described aboveindicate successive elongations of the leader in long gaps,arising from space leader inception. The longer the gap, thegreater the relative length of the leader channel. Since the elec-tric gradient within the leader is significantly lower than in thecorona filaments, this leads to the observed nonlinear sparkover-voltage/gap-length characteristic. Also contributing to thisnonlinearity is the increasing leader conductivity as a functionof time, due to channel expansion; recent strioscopic studieshave shown this expansion to occur in the negative leader as inthe positive leader, but with discontinuities associated with thereillumination-current pulses.

The stepped leader observed in lightning discharges mightwell be a larger scale manifestation of the space-leader/negative-leader/reillumination sequence recorded in these long labora-tory gaps. It is thus of considerable interest to gain a quantita-tive insight into the conditions for formation of the spaceleader, and the nature and mechanism of the stepwise spacestem from which it appears to initiate. Pigini et al. [4] haverecently shown that inception of the space leader may occurwhen the potential of the space stem is at a critical value Vx,which is the same as that required to launch the original nega-tive leader from the cathode. The critical impulse shape wouldthen be that for which the increase of applied voltage betweenconsecutive space leaders is sufficient to compensate for thepotential fall along the leader. As in positive discharges, theoptimum condition would be for an approximately constantpotential at the leader tip. Conditions within the space stemitself are at present unknown; Hutzler and Kleimeier haveproposed a model based upon a plasma dipole having an excessnegative charge. Propagation of the stem would occur as aresult of successive negative and positive corona from theanode-and-cathode-directed extremities of the stem. Spaceleader initiation would occur when the critical field strengthwas exceeeded at both extremities, when the current densityin the stem could cause significant gaseous heating.

6 Acknowledgment

In presenting this short review of recent progress in the studyof air breakdown, which represents work done by manycolleagues, I wish to pay special tribute to the late Brian Jones(CEGB) and Alfred Fischer (TU Stuttgart). Their co-workers

deeply regret the premature loss of these friends who havecontributed greatly to the science and technology of UHVbreakdown.

References

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