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Computation of inception voltage and inception timeof positive impulse corona in rod–plane gaps

M. Abdel-Salam, N.L. Allen and I. Cotton

Abstract: A method is proposed for computing the inception voltage and time of the corona in airin rod–plane gaps under a positive switching impulse and ramp-shaped voltages. The method isbased on the rate of natural production of free electrons in the atmosphere as a result of cosmicray activity, local radioactivity or ultraviolet radiation from the sun. The computed inceptionvoltages agree reasonably with those measured experimentally for different values of the steepnessof the applied switching impulse and ramp-shaped voltages and also with those measured experi-mentally for a rod gap stressed by a switching impulse voltage. The method is applied to differentrod-plane gaps with varying rod radius and gap spacing.

Nomenclature

A constant to determine the crest value of theapplied impulse voltage, kV

c concentration of free electrons in air, cm23

d gap spacing, m

n rate of free electron production in air, cm23 s21

Ne number of free electrons available to start theprimary avalanche

R rod radius m

S steepness of the applied impulse or ramp vol-tages, kV/ms

T0 time on the applied impulse or ramp voltages cor-responding to the onset voltage of steady corona,ms

Ti time on the applied switching impulse or ramp-shaped voltages corresponding to the inceptionvoltage of impulse corona, ms

T time, ms

t0.1Vcr time on the applied impulse voltage correspond-ing to a voltage equal to 0.1 of the crest value, ms

t0.9Vcr time on the applied impulse voltage correspond-ing to a voltage equal to 0.9 of the crest value, ms

v excess volume, m3

vw weighted excess volume, m3

(Voþ)dc onset voltage of steady corona, kV

Vi inception voltage of impulse corona, kV

Vcr crest value of the applied impulse voltage, kV

V(t) voltage of the applied switching impulse or ramp-shaped voltages as function of time t

w weighting function

Zi z-coordinate defining the starting point of theprimary avalanche in steady corona, m

Zt z-coordinate defining the starting point of theprimary avalanche in impulse corona, m

‘1 length of the primary avalanche in steady corona,m

‘10 length of the primary avalanche in impulse

corona, m

a ionisation coefficient, m21

h attachment coefficient, m21

1/b1 front time constant of the applied impulsevoltage, ms

1/b2 tail time constant of the applied impulse voltage,ms

DV overvoltage above the onset value of steadycorona

1 Introduction

Air breakdown in non-uniform field gaps under switchingimpulse is preceded by corona, so it is useful to assess theinception voltage of the corona under such impulses andhow it is influenced by the geometry and the steepness ofthe applied impulse. Mathematical modelling of a coronais complicated. However, an expression has been developed[1] for calculating the inception field (Eoþ)i in kV/cm of theimpulse corona as influenced by the steepness dV/dt of theramp voltage applied to rod–plane gaps in air

(E0þ)i ¼ 22:8 1 þ 1=ffiffiffiR

3p

þ (A=R)ffiffiffiffiffiffiffiffiffiffiffiffidV=dt

ph i(1)

with dV//dt in kV/ms. R is the rod radius in cm. Theconstant A was chosen equal to 0.07 to fit the experimentalresults [1] for 0.1 � R � 2 cm at standard air density.

Expression (1) was developed as an extension to that usedfor calculating the onset field (Eoþ)dc of a steady corona [1]in rod–plane gaps in air

(E0þ)dc ¼ 2:28 1 þ 1=ffiffiffiR

3ph i

(2)

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-smt:20050088

Paper first received 21st November 2005 and in revised form 19th September2006

M. Abdel-Salam is with the Department of Electrical Engineering, AssiutUniversity, Assiut, Egypt

N.L. Allen and I. Cotton are with the School of Electrical Engineering andElectronics, University of Manchester, PO Box 88, Manchester M60 1QD, UK

E-mail: [email protected]

IET Sci. Meas. Technol., 2007, 1, (4), pp. 179–184 179

Similar relationships were developed [2] for the inceptionfields of the corona in rod–plane gaps in air, under bothsteady and impulse voltage conditions. These have beenrelated to the inception voltages and in the impulse caseto the rate of rise or the steepness of the applied voltage.

This paper is a first attempt at computing the inceptionvoltage and time of the impulse corona in atmospheric airin hemispherically capped rod–plane gaps under a positiveswitching impulse or ramp-shaped voltages. The computedinception voltages and times are compared with thosepublished in the literature.

2 Simplifying assumptions

1. At steady corona onset, a primary streamer is followedby successor avalanches growing in the ionisation-zone sur-rounding the stressed rod.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 of boththe applied field and the field due to its own positive spacecharge.4. The successor avalanches grow under the resultant of theapplied field, the field due to the positive space charge of theprimary avalanche and the field due to the positive spacecharge of the successor avalanches.5. The onset criterion of the corona in rod–plane gapsunder positive impulse voltages is assumed to be the sameas that under steady voltage. This is pending the presenceof an initiatory electron in the proper place in the ionisation-zone around the stressed rod and the growth of electrons in atime-increasing field under impulse.

3 Method of analysis

The onset voltage of a steady corona was computed [3]according to an algorithm based on the ionisation and deio-nisation processes in the ionisation zone around the stressedrod of radius R, (see Fig. 1a). Here, in the electric field, theionisation coefficient a everywhere exceeds the attachmentcoefficient h. The first (primary) avalanche starts at theboundary z ¼ zi where a ¼ h. At z ¼ zd, the number ofion pairs at its head (see Fig.1b, is the exponent ofÐ z¼d

z¼zi(a� h) dz, where d is the gap spacing. The avalanche

length ‘1 is equal to d2 zi. Successor avalanches startwhere photoelectrons are produced within the ionisation-zone. At the onset voltage of the steady corona (Voþ)dc,the number of electrons in the primary (critical) avalancheby the end of its growth is assumed to equal the totalnumber of electrons produced by the successor avalanchesfor a self-sustained ionisation [4–6].

With a transient voltage, time is required for an initiatoryelectron to appear in the ionisation zone, so that anincreased voltage, well above the steady value for onset(Voþ)dc, is required for this condition to occur. Thus, therod gap has to be stressed at a higher voltage V with an over-voltage DV above (Voþ)dc, (see Fig. 1c), determined by theapplied positive impulse voltage

V (t) ¼ A[ exp ( � t=t1) � exp ( � t=t2)] (3)

where 1/t1 and 1/t2 are the front and tail time constantsrespectively, of the applied impulse wave. A is a constantdetermining the crest value Vcr of the impulse.

Consequently, the ionisation-zone expands with anexcess volume (see Fig. 1d), bounded by two contours.The inner contour defines the minimum volume of the

ionisation zone around the rod, which is required for theprimary avalanche to yield sustained ionisation. It movestowards the rod with increase of the applied voltage V.The outer boundary extends down to z ¼ zt and enclosesthe volume where a � h. This contour moves away fromthe rod with increase of the applied voltage V, (see Fig. 1d).

Two conditions must simultaneously be satisfied for acorona to occur under impulse voltage of V higher by DVthan (Voþ)dc. First, there must be at least one suitablylocated free electron inside the excess volume. Secondly,the electric field must be sufficiently strong to ensure thatthis electron produces a primary avalanche, followed bysuccessors that yield sustained ionisation. This conditioncan be satisfied since the gap is stressed by a voltage Vhigher than (Voþ)dc.

The free electrons are detached from naturally occurringnegative ions in the electric field near the rod. The rate ofproduction n of the negative ions (or electrons) is quitesmall [7] of the order of 10 cm23 s21 with an equilibriumconcentration c of 100–300 cm23. It has been suggested[4] that the rate n is around 20 electron-ion pairs per centi-metre3 per second at normal atmospheric conditions.

To check the first condition, the number of free electronsNe available to start the primary avalanche with the gapstressed by the impulse voltage at V corresponding to timet, (Fig. 1c), is determined as

Ne ¼ n

ðtT0

ðV0

dVdt (4)

where T0 is the time on the applied impulse correspondingto the steady onset voltage (Voþ)dc, (Fig. 1c). The excessvolume v is calculated at time t greater than T0, (Fig. 1c).

Fig. 1 Ionisation zone

a Rod–plane gap stressed by V ¼ (Voþ)dc with the ionisation zoneshown around the rodb Growth of the primary avalanche inside the ionisation zone atV ¼ (Voþ)dc

c Impulse voltage waveform showing the voltage (Voþ)dc and the cor-responding time T0 as well as the overvoltage DVd Development of the excess volume

IET Sci. Meas. Technol., Vol. 1, No. 4, July 2007180

Because the primary critical avalanche length ‘12, being

equal to (d2 zt) under impulse stress is larger than the ‘1

at steady voltage, so thatÐ z¼d

z¼zt(a� h) dz exceedsÐ z¼d

z¼zi(a� h) dz which corresponds to the steady applied

voltage with sustained ionisation. This increase in the inte-gral value has to be expressed by an increase of the excessvolume from which the initiatory electron starts to triggerthe primary avalanche for sustained ionisation under animpulse. This is why the excess volume in (4) is weightedby a weighting function w equal to this percentage increase.

Therefore, (4) takes the form for the impulse voltage

Ne ¼ n

ðtT0

ðvw

0

w dvw dt (5)

where vw is the weighted excess volume, that is, the excessvolume shown in Fig. 1d after being weighted to accommo-date the increased growth of the primary avalanche startedin the excess volume under impulse voltage when comparedwith that started at z ¼ zi under steady voltage.

The inception voltage Vi of the impulse corona is thevoltage value at which the number Ne of (5) reaches 1

Ne � 1 (6)

The inception voltage Vi does not appear explicitly in(6). However, the applied voltage value V on the positiveimpulse wave affects a, h, zi, and zt. The inception voltageVi is the critical value, which fulfils the equality (6).

4 Numerical data and computational steps

The Appendix (Section 9.1) describes how the values of aand h are determined by the applied electric field as wellas the air pressure, which is kept constant at atmosphericvalue.

The applied electric field is computed by the accuratecharge simulation technique [8, 9] as detailed in theAppendix (section 9.2).

The computational steps for computing the inceptionvoltage Vi and inception time Ti are outlined in the flowchart of Fig. 2. The computational time to determine theinception voltage Vi and inception time Ti is in the range40–60 s on a Pentium IV personal computer, dependingon how close the estimated value of V is to the requestedvalue which satisfies (6) as shown in Fig. 2.

5 Results and discussion

Fig. 3 shows how the outer contour of the excess volume, asmeasured radially from the rod surface, changes around therod at the inception voltage Vi of the impulse corona and at avoltage midway between Vi and (Voþ)dc, the onset voltageof a steady corona. It shows how the radius of the contourincreases after weighting. It is quite clear that this radiusincreases significantly with increase of the applied positiveimpulse voltage V above (Voþ)dc as shown in Fig. 2. This issimply explained by the increase of the field in the rodvicinity with a subsequent shift of the outer contour fromthe rod surface.

The weighted and non-weighted excess volumes increasewith the applied positive impulse voltage, V, that is, with theincrease of V and the time t above T0, which is the time onthe impulse wave corresponding to the steady onset voltage(Voþ)dc of corona. The increase of the excess volume afterbeing weighted above that before weighting becomesnoticeable at higher voltages.

The values of the double integration of (4) and (5)increase with applied impulse voltage V, that is, with the

increase of time t above T0 for weighted and non-weightedexcess volumes. Of course, the multiplication of the valueof the double integration by the rate n of free electronproduction gives Ne, the number of initiatory electronsavailable to start the primary avalanche. The value of V whenNe ¼ 1 is the inception voltage Vi of the impulse corona.

Fig. 2 Flow chart describing the steps of computing the coronainception voltage and inception time

Fig. 3 Ratio of the excess voltage after being weighted to thatbefore weighting as a function of the applied impulse voltagereferred to the onset voltage of steady corona

Radius of rod ¼ 5 mm, gap spacing ¼ 10 cm

IET Sci. Meas. Technol., Vol. 1, No. 4, July 2007 181

The corresponding time Ti is the inception time. Thisdepends on the value assigned to the rate n.

Table 1 shows how the computed inception voltage Videcreases slightly with the increase of n in the order of10. It is satisfying that the calculated Vi value of theimpulse corona is close to those measured (45–56 kV)[10]. Not only the computed inception voltage Vi, but alsothe computed inception time Ti values of the impulsecorona in Table 1 agreed reasonably with those measured(66–145 ms) [10]. The values assigned to the rate nin Table 1 follow the numbers reported in the literature[4, 7] for the natural production of free electrons in theatmosphere by cosmic rays, local radioactivity or ultravioletradiation. However, some authors have assumed [11, 12]that the rate n is equal to N2/t, where N2 is the densityof negative ions and t is their mean life time. A value of1.7 � 108 cm23 s21 was assigned for N2/t without justifi-cation in a model aimed at studying the flashover character-istics in needle-plane gaps stressed by positive impulsevoltages. This is based on the availability of negative ionsin a field high enough to shorten their lifetime and dissociatethem for high production of free electrons.

The ratio of the excess volume after weighting to thatbefore weighting against the applied impulse voltage Vreferred to the steady corona onset voltage (Voþ)dc increasesnot only with voltage V but also with the steepness of theimpulse wave. The steepness s of the impulse wavedescribed by (3) is expressed as

s ¼ (0:9Vcr � 0:1Vcr)=(t0:9vcr � t0:1vcr) (7)

where t0.9Vcr and t0.1Vcr are the times on the applied impulsevoltage corresponding to voltages equal to 0.9 and 0.1 of thecrest value, respectively.

The increase of the weighted excess volume with thesteepness and the corresponding limited time available on

the impulse front is reflected in the increase of the calcu-lated inception voltage Vi of impulse corona in conformitywith previous findings [1, 2].

Table 2 shows how the computed inception voltage Vi ofimpulse corona increases with the steepness of the appliedpositive impulse for the same rod–plane gap. The computedvalues of Vi agree better with those measured experimen-tally [2] when compared with those estimated before [2]using (1). The Table dictates a decrease of the inceptiontime Ti of the impulse corona with the increase of the steep-ness s as does the time T0 corresponding to the steadycorona onset voltage (Voþ)dc.

Fig. 4 shows how the computed inception voltage Viincreases with the increase of the steepness of the appliedpositive impulse as well as with the increase of the radiusR of the stressed rod for the same gap spacing d. Here,the applied impulse voltage is a ramp function with aslope equal to the steepness s and is expressed as

V (t) ¼ st (8)

The increase of Vi with the increase of R is self-explanatorybecause of the corresponding decrease of the field in the rod

Table 1: Calculated inception voltage and inceptiontime as influenced by the production rate n of freeelectrons in air

n (cm23 s21) 20 30 40 50 60

Vi (kV) 60 59 58.6 58 57.4

Ti (ms) 137 110 100 90 70

R ¼ 5 mm, H ¼ 10 cm, d ¼ 10 cm, 1/t1 ¼ 3155 ms, 1/t2 ¼ 62.5 ms

Table 2: Calculated and measured [2] inceptionvoltages as well as calculated inception time for differentsteepness values of the applied switching impulse

Steepness, kV/ms 9 17 34

inception voltage, (kV) Vi,

measured [2]

102.5–115 104–125 103–130

inception voltage, (kV)Vi,

measured [2]

92 95 99

inception voltage (kV) Vi, present

calculation

100 110 118

inception voltage, (ms) Ti,

present calculation

216 61 25.5

onset voltage kV (Voþ)dc, present

calculation

85 85 85

time T0 corresponding to (V0þ)dc

on impulse wave

100 41 17

R ¼ 1.416 cm, d ¼ 60 cm, 1/t1 ¼ 4545.45 ms, 1/t2 ¼ 3.838 ms

Fig. 4 Calculated and measured [1] inception voltage againstthe steepness of the applied ramp-shaped voltages for differentvalues of the rod radius spacing is constant at 1 m

Fig. 5 Calculated inception time against the steepness of theapplied ramp-shaped voltages for different values of the rod radius

Gap spacing is constant at 1 m


vicinity. The computed values of Vi agree satisfactorily withthose measured experimentally [1, 2] under applied rampvoltages as shown in Fig. 4.

Fig. 5 shows how the computed inception time Tiincreases with the decrease of the steepness of the appliedpositive ramp impulse as well as with the increase of theradius R of the stressed rod for the same gap spacing d. Asthe steepness of the applied ramp voltage decreases, thegreater is the time available to counterbalance the slowrising voltage and the associated decrease of the excessvolume to meet the inception criterion of impulse corona,given in (6). As the radius R of the rod increases, theexcess volume decreases because of the field in the vicinityof the rod. Subsequently, the inception time increases tocounterbalance the decrease of the excess volume to meetthe inception criterion, (6). This is why the inception timeTi increases with the decrease of the steepness s for thesame rod radius R and decreases with the decrease of therod radius for the same steepness s of the applied impulse.

6 Conclusions

On the basis of the present analysis, the following con-clusions may be drawn: first, a method is proposed for com-puting the inception voltage and the inception time of animpulse corona in air in rod–plane gaps stressed by positiveswitching impulse and ramp-shaped voltages. The methodis based on the rate of natural production of free electronsin the atmosphere as a result of the arrival of cosmic raysand the presence of local radioactive materials or thepenetration of ultraviolet radiation from the sun.

Secondly, the computed inception voltages of the impulsecorona agree reasonably with those measured experimen-tally for different steepness of the applied switchingimpulse and ramp-shaped voltages. Thirdly, the computedinception time of the impulse corona agree reasonablywith those measured experimentally for a rod gap stressedby a switching impulse. Fourthly, the weighted excessvolume increases with both the voltage value and the steep-ness of the applied positive switching impulse. Fifthly, thecomputed inception voltage increases with the increase ofthe steepness of the applied positive ramp-shaped impulsesand the increase of the radius of the rod for the same gapspacing. And finally, the computed inception time decreaseswith the increase of the steepness of the applied positiveramp-shaped impulses and the decrease of the radius ofthe rod for the same gap spacing.

7 Acknowledgment

One of the authors (M. A.-S.) wishes to acknowledge theEngineering Physical Sciences Research Council for thesupport he received in 2005 while carrying out this studyat the University of Manchester, Manchester, UK.

8 References

1 Boehm, A.: ‘Der Entladungseinsatz einer Stab-Platte- Funkenstreckeals Funktion des Elektodenradius und der Spannungssteilheit’, Arch.Elektrotehnik, 1976, 58, pp. 225–231

2 Abdel-Salam, M., and Allen, N.L.: ‘Inception of corona and rate ofrise of voltage on diverging electric field’, IEE Proc., pt. A, 1990,137, pp. 217–220

3 Abdel-Salam, M., and Allen, N.L.: ‘Current-voltage characteristics ofcorona in rod-plane gaps as influenced by temperature’, IEE Proc.,Sci. Meas. Technol., 2003, 150, pp. 135–139

4 Loeb, L.B.: ‘Electrical coronas: their basic physical mechanisms’(California, University Press, Berkeley, CA, USA, 1965)

5 Nasser, E.: ‘Fundamentals of gaseous ionization and plasmaelectronics’ (Wiley, New York, USA, 1971)

6 Khalifa, M., and Abdel-Salam, M.: ‘Corona Discharges’ inAbdel-Salam, M. et al. (Eds.): ‘High Voltage Engineering Theoryand Practice’ (Marcel Dekker, New York, USA, 2000)

7 Morgan, C.G.: ‘Irradiation and Time Lags’ in Meek, J.M., and Craggs,J.D. (Eds.): ‘Electrical Breakdown of Gases’ (John Wiley & Sons,New York, USA, 1978)

8 Abou-Seada, M., and Nasser, E.: ‘Digital computer calculation of theelectric potential and field of a rod gap’, Proc. IEEE, 1968, 56,pp. 813–820

9 Singer, H., Steinbigler, H., and Weiss, P.: ‘A charge simulationmethod for the calculation of high voltage fields’, IEEE Trans.Power Appar. Sys., 1974, 93, pp. 1660–1668

10 Kong, J.: ‘Corona and breakdown characteristics in air at elevatedtemperatures’, Ph.D. thesis, University of Manchester Institute ofScience and Technology, 2003

11 Arima, I., and Watanabe, T.: ‘Study of predischarge phenomena inneedle-to-plane electrode geometry’. Proc. XIIIth Int. Conf. OnPhenomena in Ionised Gases, 1977, pt. 1, pp. 441–442

12 Arima, I., and Watanabe, T.: ‘Study of V-shaped flashover voltagecharacteristics in needle-to-plane electrode geometry by a dischargemodel’. Proc. Gas Discharge Conf, 1980, pt. 2, pp. 198–201

9 Appendices

9.1 Field calculation in hemispherically cappedrod-to-plane gaps

The charge on the stressed rod is simulated by a set ofcharges consisting of point, ring and finite-line chargesextending along the axis of the rod. The hemisphericalcap of the rod is simulated by a point charge at the hemi-sphere centre and nine ring charges equally distributedalong the z-axis as shown in Fig. 6. The radius of eachring charge is 0.5 times the radius of the rod at the samez-level. The shank of the rod is simulated by 20 finite linecharges extending along the axis of the rod. These linecharges are increasing in length in the direction awayfrom the cap. Thus, the total number of simulationcharges is 30. This set of charges must produce an equipo-tential surface, whose potential value is equal to the appliedimpulse voltage, coinciding with the rod boundary.

To assess the values of the simulation charges (Qi, I ¼ 1,2, . . . , 30), a set of 30 boundary points is chosen on the rodsurface, where the calculated potential is equated to theapplied voltage.

The boundary points are chosen as one boundary point atthe rod tip, nine boundary points on the rod cap at the samez-level as the simulation ring charges and 20 boundarypoints on the rod shank corresponding to the simulationfinite line charges. Each boundary point on the shank islocated at the same z-level as the mid-point of the corre-sponding line charge as shown in Fig. 6.

To account for the ground plane, images of the simulationcharges are considered.

Fig. 6 Simulation charges for field computation in rod–planegaps and boundary points on rod surface

IET Sci. Meas. Technol., Vol. 1, No. 4, July 2007 183

The computed potentialfi due to all the simulation chargesand their images at the ith point in space is expressed as

fi ¼X30

i¼1

QiPi, j (9)

where Pi,j is the potential coefficient at the ith space point dueto the jth simulation charge and its image.

For a point charge:

Pi,j ¼1

4p10

1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(ri � rj)

2 þ (zi � zj)2

q�

�1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(ri � rj)

2 þ (zi � zj)2

q � (10)

Where (ri, zi) and (rj, zj) are the r- and z- coordinates of theith space point due to the jth simulation charge.

For a ring charge:

Pi, j ¼1

4p10

2

p

K(k1)

b1

�K(k2)

b2

� �(11)

Where K(k) is the complete elliptic integral of the firstkind.

b1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(ri þ rj)

2 þ (zi � zj)2

qb2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(ri þ rj)

2 þ (zi þ zj)2

qk1 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffirjri=b1

qk2 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffirjri=b2

qHere, rj is the radius of the jth ring charge and zj is thez-coordinate of the ring charge.

For a finite-line charge

Pi; j ¼1

4p10(zj2 � zj1)ln

(zj2 � zi þ g1)(zj1 � zi þ g2

(zj1 � zi þ d1)(zj2 � zi þ d2)

" #

(12)

where zj1 and zj2 are the z-coordinates of the start and end ofthe jth finite line charge.

g1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2i þ (zi � zj2)2

qg2 ¼


qd1 ¼


qd2 ¼


qSatisfaction of the equality of the computed potential fi

at the 30-chosen boundary points to the voltage applied tothe rod results in a set of 30 simultaneous equations

whose solution determines the 30 unknown simulationcharges. Once these charges are determined, the electricfield at any point within the volume of the ionisation-zone,(Fig. 1a), and the excess volume, (Fig. 1d), is computed asits r- and z- components expressed as

Eri ¼X30

i¼1

Qi fri, j (13)

Ezi ¼X30

i¼1

Qi fzi, j (14)

Where fr and fz are the field coefficients along the r- and z-directions calculated at the ith space point due to the jthsimulation charge and its image. The expressions for thesecoefficients for point, ring and finite-line charges arereported elsewhere [8, 9].

9.2 Ionisation and attachment coefficients in air

The natural logarithm of the experimental values of a/P(cm21 torr21) in dry air was plotted against E/P (Vcm21

torr21). The resulting curve was very closely approximatedby five straight line portions, making it possible to accu-rately express a/P as an exponential function of E/P overeach of the five ranges of E/P. The resulting formulas areas follows [3]

(a) 30 ,E/P , 32.5

a=P ¼ 9:36 � 10�6 exp (0:805E=P� 20) (15a)

(b) 32.5 , E/P , 42.5

a=P ¼ 6:09 � 10�6 exp (0:2E=P) (15b)

(c) 42.5 , E/P , 64

a=P ¼ 1:59 � 10�3 exp (0:07E=P) (15c)

(d) 64 , E/P , 100

a=P ¼ 1:283 � 10�3 exp (0:039E=P) (15d)

(e) 100 �E/P

a=P ¼ 9:682 exp ( � 264:2P=E) (15e)

The experimental value of h/P (cm21torr21) in dry air isrelated to E/P in the mathematical form [3]

h=P ¼ 0:012983 � 0:00054E=Pþ 0:87 � 10�5(E=P)2

(16)


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