Onset voltage of positive glow corona in rod-planegaps as influenced by temperature

M. Abdel-Salam and N.L. Allen

Abstract: The postulate put forward by Hermstein to account for the positive glow corona waswidely accepted at first and later criticised. Loeb’s alternative postulate has not been fullysatisfactory. In this paper, a method is described for calculating the onset voltage of positive glowcorona in rod-plane gaps in air at room and high temperatures. The method has a flavour of boththe Hermstein and Loeb postulates, but is based on the physics of air discharges occurring in theionisation zone around the stressed rod. The threshold voltage of onset streamers and theirdevelopment at higher voltages are prerequisites for calculating the onset voltage of positive glow.The threshold and onset voltages increase with the gap spacing at room temperature and decreasewith the increase of air temperature for the same gap geometry in conformity with previousfindings. The calculated threshold voltage and repetition rate of onset streamers, in addition to theonset voltage of the positive glow agree reasonably with those measured experimentally.

List of symbols

De electron diffusion coefficientDz thickness of layers dividing the avalanche heade electron chargef repetition rate of onset-streamersL clearing length of space chargesm number of layers dividing the avalanche headN number of positive charge clouds simulta-

neously in transit process in the gapN1 number of electrons in the head of the primary

avalancheN2 number of electrons produced by the second

generation of avalanchesNc number of electrons in the head of the primary

avalanche at its critical sizeNe number of electrons at the avalanche headne electron density at the avalanche headqj jth point charge simulating the avalanche headR rod radiusr radius of avalanche head at z-coordinate¼ zrj r-coordinate of the jth layer of avalanche head

at z¼ zjS gap spacingT air temperatureTo air room temperatureTt transit time of positive space chargeV applied voltageVe electron drift velocityVo threshold voltage of onset-streamer corona

Vgl onset voltage of positive-glow coronazi z-coordinate defining the boundary of the

ionisation zonezc z-coordinate defining where the avalanche gets

chokeda ionisation coefficientZ attachment coefficientt clearing time of space charges

1 Introduction

Electrical corona is an extremely complex phenomenon,which takes many forms under various conditions andinvolves numerous microscopic mechanisms. When apositive DC voltage is applied to an electrode in anelectronegative gas such as air, positive corona dischargetakes place in the concentrated field at the electrode. Thedischarge is known to take the form of intermittentstreamer discharges extending from some point on theelectrode; or otherwise as a quasi-steady glow coveringsome area of the electrode surface [1].

The positive glow corona is accompanied by power loss,and radio and TV interference; hence, it is objectionable topower transmission and communication engineers. Never-theless, it has several applications [2], e.g. in electrostaticHVDC generators, electrostatic printing, in textile andplastic industry and in electrostatic precipitators [3], forcleaning flue gases at temperatures up to 200o C, and forozone generation [4].

The positive glow has been the subject of intensiveresearch work by both physicists and engineers. This type ofdischarge is not as well understood as one would expectfrom the amount of work and effort spent on it with a lackof theoretical analysis and great number of parameters thatinfluence the observations [5]. The electric field in theneighbourhood of the stressed electrode is distorted as

M. Abdel-Salam is with Electrical Engineering Department, Assiut University,Assiut, Egypt

N.L. Allen is with Department of Electrical Engineering & Electronics,The University of Manchester, Sackville Street, Manchester, UK

E-mail: norman.allen@umist.ac.uk

r IEE, 2005

IEE Proceedings online no. 20045024

doi:10.1049/ip-smt:20045024

Paper first received 24th August 2004 and in final revised form 31st March 2005

IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 5, September 2005 227

ionisation is building up and the situation is getting morecomplicated by the accumulation of space charges of eitheror both polarities. This makes a theoretical description ofthe problem very difficult without a deep look into thephysics of gas discharges.

Two distinct postulates have been put forward byHermstein [6] and Loeb [1] in attempts to explain thephenomena in the positive glow. In his mechanism of glowformation, Hermstein assumed the presence of negative ionsin a sheath close to the anode inhibiting further onsetstreamer development. Although Hermstein’s mechanismhad long been accepted by Loeb [1], he later doubted it[7, 8]. Instead, he attributed the positive glow to burst pulsesmerging in time and space, thus creating the steady glowregime. Meanwhile, some recent experimental evidence hasbeen reported [9, 10], which challenges either or bothpostulates.

This paper is a first attempt at computing the onsetvoltage of positive glow discharge in rod-plane gaps in air asinfluenced by temperature at atmospheric pressure. Thisrequires investigating the preceding onset streamer coronaand the space charge left behind in the gap. The method ofcomputation enjoys a flavour of both the Hermstein andLoeb postulates.

2 Hermstein’s postulate

Finding that the glow could be helped by the presence ofnegative ions, Hermstein proposed that there must be a roleplayed by them [6]. According to his assumption, photo-electrons, created outside the ionisation zone around theanode, attach to gas molecules to form negative ions. Thesewould advance towards and accumulate in the region nearthe anode, thus sharply enhancing the field between it andthe anode surface.

An observation supporting this assumption is that innon-attaching gases, such as nitrogen, where negative ionsdo not form, the individual streamers starting at the coronathreshold level propagate outward into the gap with voltagerise, until they lead to a complete breakdown. Such anobservation indicates some truth in Hermstein’s postulate[9]. A trace of 0.1% of an electronegative gas in purenitrogen could furnish the negative ions claimed byHermstein as necessary for the positive glow.

The current-voltage relation developed empirically byTownsend [9, 11] and theoretically by the authors [12] forthe steady corona discharge is incompatible with Herm-stein’s assumption. The negative ions playing a role in thepositive glow cannot be asumed to originate in bulk outsidethe ionisation zone as claimed by Hermstein. We assumethat only positive ions would carry the current outside theionisation zone around the anode.

3 Loeb’s postulate

The onset of the steady positive glow following the pre-onset region of onset streamers and burst pulses (corruptstreamers) was explained by Loeb [7, 8] as an increase of theduration and spread of burst pulses to steady glow. Theresulting space charge reduces the field at the anode surfaceand prevents streamer formation. At some fairly sharplydefined current density and potential, the burst pulses mergein time and space and create the steady-current glow regime.

Merging in time is understandable and agrees with theobserved current fluctuations with a glow [9, 13, 14].Merging in space, however, is contradicted by theappearance of the glow mode as basically distinct from

the onset streamers [5, 9, 15] and would not be expected tocause current fluctuations.

4 Method of analysis

4.1 Simplifying assumptions

(1) The onset streamer is composed of a primary avalanchefollowed by successor avalanches of the succeedinggenerations.

(2) Photo-ionisation of the air is the main ionising processfor creating the initiatory electrons of the successoravalanches.

(3) The primary avalanche grows under the resultant ofboth the applied field and the field owing to the positivespace charge of the avalanche.

(4) The successor avalanches grow under the resultant ofthe applied field, the field owing to the positive spacecharge of the primary avalanche and the field owing tothe positive space charge of the successor avalanches.

(5) Based on the availability of the initiatory electrons, thestreamer repetition rate is the reciprocal of the clearingtime, which is the time required to clear the ionisationzone around the rod from the existing ions until theapplied field at the outer boundary of the zone isrecovered.

(6) Each streamer emits in space a positive ion cloud andthe number of positive clouds in transit process is theratio of the ion transit time along the gap length and theclearing time. These ion clouds contribute to the fieldinside the ionisation zone where there are growingavalanches.

(7) At glow formation, the avalanche grows under theresultant of the applied field, the field of positive cloudsin transit process across the gap, the field owing todiscrete charges simulating the avalanche head and thepositive space charge of the avalanche.

4.2 Threshold voltage of onset-streamercoronaThe threshold voltage of onset streamer corona has beencomputed [12] according to an algorithm based on theionisation and deionisation processes [16] acting inthe ionisation zone around the stressed rod of radius R.The rod is spaced a distance S from the ground plane,Fig. 1. In this zone, the electric field is sufficiently high thata, the ionisation coefficient, exceeds Z, the attachmentcoefficient. The dependency of a and Z on the applied fieldand air pressure and temperature has been reportedelsewhere [12, 16, 17]. For the case of positive polarity,the primary avalanche (first generation) proceeds to growtowards the rod and ends at its surface. Such growth takesplace under the resultant of both the applied field owing tothe applied voltage V and the field owing to the positivespace charge of the avalanche. The avalanche develops acloud of electrons at its head and a cloud of positive ions inits wake. This is in addition to the photons that are emittedfrom the avalanche through the ionisation zone, Fig. 1.Subsequently, the photons generate photoelectrons to startthe successor avalanches of the second generation. With thesuccession of avalanches and accumulation of spacecharges, the ionisation zone expands and its contourchanges, Fig. 1. The applied field is calculated using theaccurate charge simulation technique [1, 16, 18]. Thecharges simulating the rod are changing during the growth

228 IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 5, September 2005

of the generation avalanches to keep the potential of the rodsurface constant at the applied voltage V.

At threshold voltage Vo of onset streamers, the numberof electrons N1 in the primary avalanche by the end of itsgrowth is equal to that of N2 produced by the secondgeneration of avalanches and the primary streamer becomesunstable [12]. This instability results in the initiation ofluminous filamentary onset streamers [1, 5]. Thus, it isimplied that the level of ionisation in the successoravalanches is such that sufficient photons and hencephotoelectrons will be generated to make the ionisationself-sustained when

N2 � N1 ð1ÞIt has been reported in the literature that corona occurswhen the primary avalanche reaches a critical size Nc. It hasbeen generally believed that Nc should be approximately 108

[19]. However, later studies [20–22] showed that the size ofthe critical avalanche depends on the gap geometry.

4.3 Growth of onset streamersThe onset streamer is built up by a number of successivegenerations of electron avalanches [23], which take place inthe ionisation zone around the anode. Thus, at a voltagehigher than the threshold value Vo, succeeding generationsof avalanches develop in the ionisation zone. Photons areemitted by the generating avalanches in various directions.Photoelectrons are produced in various locations fromwhich they start the new generation of avalanches. Most ofthe new avalanches get started almost at the instant oftermination of the previous avalanche when its populationof electrons is highest. The electrons at the avalanche’s headof a given generation are swept into the positive spacecharge left by the preceding generation, thus transform intonegative ions. This results in a plasma streamer (highdensity positive and negative ions along its core) withpositive ions accumulating only at the streamer tip, Fig. 1.Of course, recombination takes place between positive andnegative ions along the streamer core. However, it is notsignificant because the duration of the streamers is so short.

The motion of electrons and ions of the succeedinggenerations in the ionisation zone results in a corona currentthrough the high-voltage circuit. This motion takes placeunder the resultant of the applied field, the field due topositive charge at the streamer tip and the field due to thepositive self-space charge of the growing avalanches of thesucceeding generations. The computations of the coronacurrent are continued until the magnitude of the generationcurrent falls below 0.1% of the maximum value reached forany generation. Thus, the streamer growth is terminatedand the corresponding current pulse is considered to haveended [23].

For calculating the corona-current pulse repetition rate(frequency) f, one has to define the conditions to be fulfilledafter the termination of a current pulse in order that thesubsequent pulse can be initiated. The favourable conditionis taken as the negative ion space charges are swept into theanode and the positive ion space charges are swept, as acloud, far from the anode so that their residual field strengthat the boundary of the ionisation zone becomes negligiblewhen compared with the field produced by the appliedvoltage V. This defines the clearing time t whose reciprocalgives the pulse repetition rate, assuming the availability ofan initiatory electron for triggering the next current pulse.The corresponding distance L moved by the positive spacecharges to clear the ionisation zone is named the ‘clearinglength’ [1]. The clearing time would be a small fraction ofthe positive ion transit time Tt, the time required for positiveions to cross the whole gap length. The number N ofpositive charge clouds simultaneously in transit processacross the gap is not given by the ratio of the gap spacing Sto the clearing length L, i.e. NaS/L. This inequality is aresult of the non-uniformity of the electric field across thegap. The maximum separation of positive charge clouds(near the rod tip) is equal to the clearing length L. Furtherfrom the tip, the separation between charge clouds will besubstantially less than L because of the weaker field. Sincethe separation is always less than or equal to L, thenN4S/L.Therefore, the number of positive charge clouds N isexpressed as the ratio of the transit time Tt and the clearing

ZZ Z

z= S

firstavalanche

α =2 α =2

positive ionsleft by firstavalanche

segment ofionisationzone

successoravalanches

R R R

plasma positive ions electrons photons photoelectron

a b

Fig. 1 Schematic diagram showing the threshold and the growth of onset streamersa Threshold of onset streamersb Growth of onset streamers

IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 5, September 2005 229

time t, i.e.

N ¼ Tt=t ð2Þ

4.4 Onset of glow coronaDuring the growth of a given generation of avalanches, theionisation coefficient a, the attachment coefficient Z, andthe ionisation zone boundary around the stressed rod aredetermined by the resultant field of the applied field and thefield of the N positive clouds departing outside theionisation zone.

Figure 2 shows one of the growing avalanches that startsits growth at the ionisation zone boundary (z¼ zi) andreaches a distance z along the gap axis with a head of radiusr and population number of Ne electrons.

The radius of the growing avalanche is

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6

Z z

zi

De

Ve

sdz ð3Þ

where De is the electron diffusion coefficient and Ve is theelectron drift velocity under the prevailing field strength.

To assess the enhancement of the electric field ahead ofthe avalanche while growing, the head of the avalanche isdivided into m layers, each of thickness dz. The radius of thejth layer rj is related to the z-coordinate of the layer zj,j¼ 1, 2,y,m as

rj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðz� zjÞð2rþ zj � zÞ

qð4Þ

where zj¼ z�(j�0.5). dz, j¼ 1, 2,y,mThe layer thickness dz is related to the radius of the

avalanche head as

dz ¼ 2r=m ð5ÞAs assumed before [5], the electron density in the head ofthe growing avalanche is uniform, so the density ne is

ne ¼ Ne=ð4pr3=3Þ ð6ÞTherefore, the number of electrons in the jth layer is equalto 2p rj dz ne. Subsequently, each layer is represented by apoint charge qj, j¼ 1, 2,y,m

qj ¼ 2 p e rj dz ne ð7Þwhere e is the electron charge.

The positive ions left behind the avalanche head wereassumed to be concentrated at a distance 1/a from theavalanche head [24].

The avalanche shown in Fig. 2 is growing under theresultant of the applied field, the field of the N positiveclouds in transit process across the gap, the field due to them discrete charges simulating the head and the positivespace charge of the avalanche. With the higher appliedvoltage V, the avalanche chokes off further growth and theavalanche electrons are attached forming negative ions. Thiswould be the source of negative ions needed for theHermstein postulate. Attachment of photoelectrons in theweak field outside the ionisation zone (as claimed byHermstein) is possible from the theoretical point of view.But their number is very small and their penetrationthrough the dense plasma in the ionisation zone would notbe efficient as proposed by Hermstein. In case the aboveresultant field exceeds the limit 70 kV/cm at roomtemperature (To¼ 293K) [1, 13], which is necessary forelectron detachment from negative ions, electrons are freedand start burst avalanches forming burst pulses. These burstpulses merge in time to form the steady glow in conformitywith Loeb’s postulate. The repetition rate of these pulsesdetermines the high frequency relaxation oscillationsobserved with positive glow [9, 13]. The many growingavalanches as shown in Fig. 2 are to pump positive ions inthe gap to carry the steady component of the coronacurrent. Not only is the avalanche choked and the fieldahead of its tip reaches the 70kV/cm limit at To, but it isalso important where choking occurs, the burst avalanchesstart at sufficient distance from the anode surface (z¼ zc,Fig. 2), to allow growth following (1). Therefore, the onsetvoltage of positive glow Vgl is the critical value of theapplied voltage V which results in (i) an avalanche chokingfor negative ion formation, (ii) a field at the avalanche tipexceeding the 70kV/cm limit for negative ion detachment,and (iii) a sustenance of the growing burst avalanches. Atother temperatures T, the limit is assumed equal to 70 timesTo/TkV/cm.

The division of the avalanche head into layers is anattempt to assess accurately the electric field enhancementahead of the growing avalanche better than the use of singlepoint charge simulating the head.

Z

r

ZR

z = Sj =1

j = m qm

qj

rj

q1

jth layer

z =zc

z =zi

z =zj

dzz =z

avalanche head enlarged

sustainedburstavalanches

growingavalanche

1

2

N

onset of positive glow

α =2

Fig. 2 Schematic diagram showing the streamer-to-glow transition

230 IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 5, September 2005

5 Results and discussion

To test the proposed method of analysis for calculating theonset voltage of glow corona, the threshold voltage Vo andthe repetition rate f (i.e. the reciprocal of the clearing time t)of onset streamers are determined. As stated, the thresholdvoltage is a pre-requisite to follow-up the growth of onsetstreamers for determining their repetition rates. The clearingtime defines the number N of positive charge cloudssimultaneously in the transit process across the gap. Theseclouds contribute to the prevailing field where avalanchesare growing in the glow mode.

For calculating the threshold voltage of onset streamercorona, two rod-plane gaps are investigated. The first gaphad rod radius R¼ 10mm with variable gap spacing S(0.05mrSr0.35m). The gap was tested [5] at roomtemperature (T¼ 293K). The calculated threshold voltageVo increases with the gap spacing S, Fig. 3, and agreedreasonably with the values measured experimentally [5]. Thedeviation between the measured and calculated thresholdvalues did not exceed 6%.

The other investigated rod-plane gap had rod radiusR¼ 5 mm and gap spacing S¼ 0.01m. The gap was tested[25] at varying temperature in the range 288–494K. Thecalculated threshold voltage Vo decreases with the increaseof air temperature T, Fig. 4, and agreed reasonably with thevalues measured experimentally [25]. The deviation betweenthe measured and calculated threshold values did not exceed5%.

Table 1 shows the calculated repetition rate f of onsetstreamers for different rod radii and gap spacings atdifferent air temperatures. The repetition rate increases withthe increase of the applied voltage V for the same airtemperature and gap geometry. The agreement of thecalculated repetition rates with those measured experimen-tally [5, 25] is satisfactory. The maximum deviation betweenmeasured and calculated repetition rate values did notexceed 5%.

The onset voltage Vgl of positive glow corona wasmeasured [5, 25] for the same two investigated rod-planegaps. Figure 5 shows the increase of the calculated onsetvoltage Vgl at room temperature with the gap spacing. Thecalculated onset voltage values agreed with those measuredexperimentally [5] with a deviation not exceeding 5%.Figure 6 shows the decrease of the calculated onset voltageVgl with the increase of air temperature T. The calculatedonset voltage values agreed with those measured experi-mentally [25] with a deviation not exceeding 4%.

For the investigated gaps, it is worth mentioning that theavalanches at the onset of glow-corona get choked at about70–80% of the ionisation zone thickness in order to yieldsustained burst avalanches. This depends on the geometricfield (determined by the applied voltage V, the rod radius Rand the gap spacing S), the air pressure and temperature inaddition to the space charges in the ionisation zone.

It is satisfying that the deviation between the calculatedthreshold voltage Vo, onset voltage Vgl and repetition rate fvalues from the corresponding measured values lies within

00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

1020304050607080

gap spacing, m

thre

shol

d vo

ltage

, kV

measured computed

Fig. 3 Calculated and measured threshold voltage of onsetstreamer corona for different gap spacings at room temperature

25

27

29

31

33

35

37

39

41

200 250 300 350 400 450 500 550

temperature, K

thre

shol

d vo

ltage

, kV

measured [25]calculated

Fig. 4 Calculated and measured threshold voltage of onsetstreamer corona as influenced by air temperature for the same rodgap

Table 1: Measured and calculated repetition-rate of onset streamers in rod-plane gaps at different temperatures

Reference [5] [5] [5] [25] [25] [25] [25]

Radius R, (mm) 0.5 0.5 0.5 5 5 5 5

Spacing S, (m) 0.08 0.08 0.08 0.1 0.1 0.1 0.1

Temperature T, (K) 293 293 293 319 319 288 288

Voltage V, (kV) 8.0 8.1 8.7 38.5 40 42 45

Repetition rate f measured, (Hz) 300 500 6500 170 340 240 630

Repetition rate f calculated, (Hz) 330 526 6050 163 345 228 598

00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

20

40

60

80

100

120

gap spacing, mon

set v

olta

ge, k

V

measured [5]calculated

Fig. 5 Calculated and measured onset voltage of positive glowcorona for different gap spacings at room temperature

IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 5, September 2005 231

the experimental scatter usually experienced in the measure-ments.

6 Conclusions

From the present analysis of corona in rod-plane gaps, onecan conclude that:

1. The onset voltage of positive glow corona can becalculated by a purely theoretical method that avoids thecontroversy between Hermstein and Loeb postulates.

2. The calculated onset voltage of positive glow coronaincreases with the gap spacing at room temperature anddecreases with the increase of air temperature for thesame gap geometry in agreement with the experiment.

3. Not only the calculated onset voltage of positive glowbut also the calculated threshold voltage of onsetstreamer corona increases with the gap spacing at roomtemperature and decreases with the increase of airtemperature for the same gap geometry in agreementwith the experiment.

4. The calculated repetition rate of onset streamersincreases with the applied voltage for the same airtemperature in agreement with the experiment. Itdepends also on the gap geometry.

7 Acknowledgment

One of the authors (M. A.-S.) wishes to acknowledge theEngineering and Physical Sciences Research Council for thesupport he received in 2004 while carrying out this study atUMIST, Manchester, UK.

8 References

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22 Khaled, M.: ‘New method for computing the inception voltage of apositive rod-plane gap in atmospheric air’, ETZ Arch., 1974, 95,pp. 369–373

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00 100 200 300 400 500 600

5

101520253035404550

temperature, K

onse

t vol

tage

, kV

measure [25]calculated

Fig. 6 Calculated and measured onset voltage of positive glowcorona as influenced by air temperature for the same rod gap

232 IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 5, September 2005