Journal of Electrostatics, 2 (1976/1977) 223--239 223 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

A METHOD FOR PREDICTION OF GASEOUS DISCHARGE THRESHOLD VOLTAGE IN THE PRESENCE OF A CONDUCTING PARTICLE

MASANORI HARA and MASANORI AKAZAKI

Department of Electrical Engineering, Faculty of Engineering, Kyushu University, Fukuoka (Japan)

(Received March 11, 1975; in revised form March 10, 1976)

Summary

In this paper, calculations are presented of the field strength near a small conducting sphere floating in a parallel plane gap, and also the force acting on the sphere when the sphere is placed at an arbitrary position in a parallel plane gap. A method is proposed for the prediction of the gaseous discharge threshold voltage on the basis of the calculations.

It was found that the breakdown mechanisms in a non-uniform field gap with a free conducting sphere can be classified into four kinds with regard to the particle movement and the occurring position of breakdown. Moreover, the breakdown voltage for each of the four mechanisms has a different pressure dependence. The predicted values were con- firmed by using a concentric hemispherical gap and steel spheres, and are in fairly good agreement with experimental values.

1. Introduct ion

Breakdown phenomena of a gap, when there are conducting particles in a gap, have been studied by many workers [1--9]. However, the analytical discussions of the discharge phenomena in the presence of the particle are not presented in detail. In order to explain the breakdown initiated by the particle, it is necessary to calculate the field strength near the particle placed on or near the electrode and the force acting on the particle.

Lebedev and Skal'skaya [10] studied the potential distribution and the force when the spherical conducting particle is in contact with one of the plates of the parallel plane gap.

In this paper, the field strength near a small conducting sphere floating in a parallel plane gap is calculated, as is the force acting on the sphere. A method for predicting the d.c. breakdown voltage of a uniform and a non-uniform field gap with the particle is proposed. Also presented are measurements on the threshold voltage of a corona discharge and a breakdown. The predicted values for a compressed air and a compressed SF6 gap are in good agreement with experimental results.

224

2. Field strength near a conducting sphere and force acting o n t h e sphere

2.1 Statement and solution of the problem As shown in Fig.l , the gap length is denoted by G, the radius of the parti-

cle by r, the distance between the particle and the earthed cathode of the parallel plane gap by d, the charge on the particle by Q: Under a condit ion of d>>r and G>>r, assuming the external field strength E0 which points in the direction opposite to the z-axis, the potential of the particle becomes:

Q Vo = Eo (d+r) + 4n---~- (1)

and the field strength on the z-axis becomes:

E~ = 1+2 E0 + Q IZ-r-dl/~ - 4~e(Z_r_d) 2 (2)

where the + is for d+r > z and the - for d+r < z. When d=O, the field strength on the z-axis becomes as follows by differen-

tiating the potential obtained by Lebedev and Skal 'skaya[10] with respect to z:

2r r

where ~ is the zeta function. When d is arbitrary and r~ G, the determination of the potential ~ in the

gap reduces to integration of the Laplace equation:

V~¢ = 0 (4)

subject to boundary conditions:

~]z=0 = 0, ~ ) ~ t = v0, ~ j ~ = E ~ (5)

Q = eff ds (6) su

where the subscript "surf" denotes the value on the particle and eqn. (6) is

= E°z fPI,.ANE ELECTRODE

l G

' ~= OtPI.ANE ELECTRODE I "T Fig.1. Conducting sphere in parallel plane gap.

225

necessary to de t e rmine the e lec t ros ta t ic po ten t i a l Vo o f the par t ic le f r o m the given charge Q.

Here, bispherical coord ina tes (7, 0, $ ) shown in F i g . 2 [ l l - - 1 2 ] are intro- duced , and coord ina tes 77 and 0 are re la ted to the cyl indrical coord ina tes z and p as fol lows:

a sinh 7 a sin 0 z = cosh 7 - cos O' P = cosh 77 - cos O' O<O<n, -oo<77<oo (7)

where a=r sinh 7o and

n0 = In {[(d 2 + 2rd) ~ + d]/[(d 2 + 2rd) ~ - d]} being a coordinate of the particle surface.

Then, the solution of eqns. (4)--(6) can be expressed by the series:

exp[-(n+~A)~7o] sinh(n+~/~)TPn(cOsO ) + = x /2Vo~/coshv-cosO ~ sinh(n+l/~)7o n=O

+ x/2Eoa x/coshT-cosO ( 2 n + l ) exp[- (n+lh)7]Pn(cOsO ) - n=O

- n~O= sinh(n+l/~)7o(2n+l)exp[-(n+l/2)7°]sinh(n+l'~)~Pn(c°sO) 1 ,

~ t O j / / ' •

x ~ ~ CONST

Fig.2. Bispherical coordinates (~, 0 ) and cylindrical coordinates (o, z).

(8)

226

with 0<,7<,7o and where the Pn(cOsO) terms are Legendre functions of the first kind and Vo remains unknown.

2. 2 Calculation of field strength For electrical breakdown studies, the region of interest is on the z-axis and

the surface of the particle, on which the field component is only the q-one. The ,7-component of the field in the gap is given by

1 ~ Vo sinh,~ ~ 2 ×

E~ - - - (coshv - cos0) L~-~-~-oshv cos0) exp[(2n+l)v0] 1 a ~ _ - -

× sinh(n+lh)V Pn(cOsO) +

+ X/2 V0 x/(coshv-cos0) ~ 2(n+lh) n=0 exp[(2n+l)v0]-I c°sh(n+lh)V Pn(cOsO) +

÷ Eo a sinhv ln~= 0 V~ ,~/(coshv-cos0) (2n+ l )exp[-(n+l/2)v ]Pn(c°sO ) -

- n~O 2(2n+l) sinh(n+V2)~Pn(cOsO)l = exp[(2n+l)~o]-I

- x/2E°a~/(c°shv-c°s0) t ~ 2(n+l/2)2exp[-(n+lh)nlPn(c°sO) + n=0

n = 0 exp [(2n+I)7 0 ] - I c°sh(n+I/~)~Pn(C°S8 ) (9)

Substituting eqn. (9) into eqn. (6), we find:

Vo=t Q +Eoa ~.~j (2n+l) l / n ~ 0 1 (10) 8~ea n=0 exp[(2n+l)vo]-I exp[ (2n+1),~o]-1

in which the first term converges to the second term of eqn. (1) for large d/r.

2. 3 Calculation of the force acting on the sphere The force acting on the sphere can be presented by the expression

(1-cosh~?o cos0) F = ~ea: f En 2 In=no sin0d0

o (coshv o-COS0)S (11)

A direct analytical computation of F from eqns. (9) and (11) is considerably difficult, so F will be obtained numerically later.

227

2.4 Numerical calculation o f the field and the force When the particle is in motion in a parallel plane gap with the direct voltage

below the threshold voltage for discharge, the induced charge on the particle at the momen t the particle touches the cathode is given by [10]:

Q = -2~3eEor=/3 (12)

which is also obtained by substituting zero for V0 and letting ~o approach zero in eqn. (10). As the voltage is increased beyond the onset of the dis- charge, the charge Q on the particle depends on characteristics of the corona discharge on the particle [3].

When the particle has the charge given by eqn. (12), the T-component of the relative field strength in the gap becomes a function of d/r, ~7 and O as illustrated in Figs.3 and 4. In these figures, the computat ion was continued until the last term for each series in eqn. (9) becomes less than 10 -s. Figure 3 shows the field strength on the surface of the particle which is charged with both polarities and is placed near or on the cathode. As the particle leaves the cathode and approaches the anode, the maximum field strength changes from 4.207 to 4.64 (see solid line in Fig.3(a) ) and increases fur thermore (see dot ted line in Fig.3(a) ).

The relative field strength on the z-axis as a function of T/T0 is shown in Fig.4. As the distance d is decreased, the field converges to the value obtained from eqn. (3). In Fig.4(a), which is used in discussion of the discharge thres- hold, curves intersect each other, but if abscissa 77/T0 is transformed to z/r, then, for a given z/r, the higher the field the larger is d/r.

I ~_~E (( rio, O) 5,0

| - % , ~ )

" ---- E o

d+r d ~ ,

~,5 S

40

30 o

W

- - 20

4,( 1 I l I I . I I I I I 1 2 3 4 5 "10 ]50 50 70 90

a i r

( ~ ..... Q- "Gin, Q,,,O - - ~ Q-Oe, $ - r

• E(I. (3)

W 2

[ ~ o , O )

d+r d ~E(rto,O); Qo ~ ~'

o I l i l I ~I 1 1 [ L 0 ~ 2 3 4 s zo 3o so zo 9o

d / r (b) q : -Qo, e : ~

- - - - - q.O.,, 8-O

Fig. 3. Fie ld s t r e n g t h fo r ~ffino a n d e = 0 o r ~. Qo = 2~SeEor2/3; ~7=no .

228

5 -- _ .11'l'E(r/'O)

1

0 [ I [ I I ,0 0,8 0,6 0,4 0.2 0

( z = + 2 r ) ~/r/° ( Z = CO) #

(a) 8 =0

2[. o r__O ? _~..~ ~l~,~

i ~ d / r = i00

1,0 ( z = d )

0.1

1 I L I I 0,8 0,6 0,4 0.2 0

q/,'qo ( z = 0 )

(b) O=m"

2

d+2r

\ d / r = 0,1 ~' ~,

l lo?- ~--

o| I I i I 1.0 0,8 0.6 0,4 0,2

(Z =d + 2 r ) ~/~o

(c) ~ =o

- - Q=-2/~3~Eor~3 =-Qo

- - - - - - Q= 2 ~ E o r ~ 3= Qo

Fig.4. Field strength on the z-axis.

~Eo 20- -

uJ

. . . . . . . . d / r = 0,1

~ o - -~-

-~z - - IE(~ '~) 4 ~

~%- i~ ..... L .....

l 0I °° 7 - - m - - - - T - - - V J 0 1.0 0,8 0,6 0,4 0,2 0

( z = oo) ( z = d ) r~/rto ( z = 0 )

(d) 8=/~

~ Eo

Figure 5 shows that the relative force obtained by numerical integration of eqn. (11) by dividing the range of 0 into 100 increments of equal length is monotonic in d/r and also converges to Lebedev's value of 0.832 at d/r=O.

229

1.0

0.9 Y

0,8 I I I I I . I I [ I I 0 1 2 3 ~ 5 10 30 50 70 90

d/ r

Q = m 2;Ta¢Eor2/3

- - - - - - Q - 2~3EEor2/3

• Lebedev ~ VCllUe (F= 4~IEE2o r ~ {E(3) + ~})

Fig.5. Force acting on conducting sphere.

3 o

3. Prediction of breakdown voltage in the presence of a conducting sphere

3.1 Parallel p lane gap As Pedersen [14--16] and Takuma [17] stated, Meek's or Raether's crite-

rion for discharge threshold may be written simply as:

z2

/ (z-p)d~ = K (13) zl where ~ and p are the first Townsend ionization coefficient and the electron at tachment coefficient, respectively, z l the place in a gap where the electric field strength is the highest and z2 the place where ~ = p. K is a constant determined from measuring breakdown voltage in a uniform field gap with d.c. voltage [14]. It has a value of about 13--20 [17], and here is taken to be 15.

Assuming that the threshold of self-sustaining discharge in the main gap is independent of microdischarge in the gap between the particle and the elec- trode, the discharge threshold equation (13) will be satisfied in the region shown in Fig.4(a) at first, because the field in this region is a maximum in the gap. The calculated threshold voltage of the discharge by using (c~-p)]p = 1.6X10 -4 ( E / p - 3 1 . 6 ) 2 for air and ( a - p ) / p = 2.77 X 10 -2 ( E / p - l l 7 ) for Sf6, where p is expressed in tor t and E in V/cm, is illustrated in Fig.6 as a func-

230

~33 SF6

~922

~200[_ 7 AIR I 1 I~I I 18 4 6 8 90 iQO

d/r

Fig.6. Calculated value of discharge threshold field strength as a function of d/r.

tion of d/r for a sphere of 0.75-mm radius. From this figure, as d/r is increased after the particle attaches to the electrode, the threshold voltage suddenly decreases and becomes constant at large d/r.

When the free conducting sphere is placed in the gaseous gap with d.c. voltage, the particle is lifted by the external field:

Eo = 5.19V~pr (kV/cm) (14)

where pp is the specific gravity of the particle and is derived from the balance between the gravitational force and the electric force at d/r=O in Fig. 5.

In Fig.7, the external field strength for the discharge threshold calculated from eqn. (13) for two cases of d/r ( = 0 and 100) and lifting field strength are shown as a function of the radius of a steel sphere, in air and SF6, and are compared with the experimental values in the 2.48-cm parallel plane gap. The experimental values for the lifting field is in good agreement with the theore- tical value. It is also clear from Fig.7 that when the free particle is placed, if its radius is small, the particle is lifted before the discharge begins and the breakdown is developed with no corona. In cases of large particles which are unavoidably fixed or at rest due to their weights, the corona precedes the breakdown, and the corona threshold curve follows the same trend as the breakdown voltage for the small free particle, bu t is slightly higher than that for the free particle. The breakdown threshold gradient for the small free particle is very little influenced by the polarity, bu t that for the fixed particle on the anode is higher than on the cathode.

For the free particle, the external threshold field strength for the break- down is considerably lower than the calculated value of d/r=O, and, contrary to Fig.6, the breakdown takes place on the top of the particle when the particle is not in the center of the gap bu t very near the electrode. This was confirmed by still photographs. Also, according to the experiment with a simulated particle suspended by a nylon string in an air gap, the breakdown voltage Vb is minimal and coincides with the corona starting voltage when the particle is placed very near the anode. If the particle is in contact with an

231

1o 2

z

F

J

~io

~I~CRITICAL RADIUS FOR REDUCING ~ { ~ THE FRASHOVER VOLTAGE

, ",'.% \ . d / r = 0 ICALCUL. VALUES -- \ ~ x ~ ' d / r = IOOIFROM Ed.(13) . . . . . . /

r / 7,38P( 1 + 0,952-I,587 )

S \ I IFTING FIELD STRENGTH:

1 / I 14'5~ I I

10 -3 10 -2 10 -I 1 10 10 2 PARTICLE RADIUS r (MM)

(a ) Air

0 B r e a k d o w n threshold field for f ree par t ic le

<> Corona threshold field for still or f ixed part ic le

r l B reakdown threshold field for still or f ixed per t ic le when the partlcle is in contact with negetive electrode

• BreQkdown threshold fleld for still or flxed perticAe when the particle is ,n contact with posit{re electrode

• Lifting field strength

1031~CRITICAL~" RADIUS FOR REDUCING

IV\ THE FRASHOVER VOLTAGE

b ' ~ c \ , 21,19~( i + o,3z9 -o.63:, ~

- k - - ~ c ~ , " - , " $ . . ,,~Ir. : 0 ~CALoUL.VALOES / \',.~'~,"-.."-~ _/z,,r : IOOSERO. Eo.<13~

/ \ ,,\" , . / \ \" "',-',- / - - "-.~'-RG~,_ /

q [ r . ~ l t _ f 2i,i9.

~ TH : 1 4 . 5 V ' F "

10-5 i0 -2 i0 - I i i0 10 2 PARTICLE RADIUS r (MM)

(b) SF 6

Fig.7. Comparison of calculated and experimental values of discharge threshold field ~trength.

electrode, Vb increases by the effect o f the stable corona. The cause of such characteristics is presumed to be due to the fol lowing reasons.

(1) As the particle approaches the electrode, eqn. (13) is satisfied, and the gap between the electrode and the particle is bridged by a microdischarge with high conductivity. This is the same effect as a protrusion which is longer than the particle diameter, and generates a highly non-uniform field and, thus, is considered to lower the breakdown voltage. For example, when a particle having a 0.075-cm radius is charged at the cathode and approaches the anode of a parallel plane gap with a 5.6 kV/cm external field, eqn. (13) is satisfied at G-d=O.O075 cm and the microdischarge will occur.

(2) According to experiment [3] , the streamer and Trichel corona develop on both ends of the particle floating in the gap and the two coronas greatly interfere via the particle when the particle is very close to the electrode. For example, if a spheroidal particle, 9.6- and 1.l-ram dia., is placed in a 4-cm air gap with a 5 kV/cm external field, the field strength near the particle is in- creased by several tens of a percent due to the electron in a streamer pulse

232

flowing into the particle. The oscillation of this field by the corona results in a disturbance in the space charge in the corona and then a reduction of the breakdown voltage.

(3) When the particle is placed in the center of the gap, if corona discharge occurs on either side of the particle, corona does not always sustain due to neutralization by the charge in the corona. So the probability of a spark occurring on the particle near the electrode is much greater than on a particle far removed from the electrode.

As shown in Fig.7, calculated and experimental external field for break- down with a small free particle and for corona starting with a fixed particle can be expressed by the simple equation for air:

t 0"952~1"587 f (kV/cm) (15, E = 7.38p 1+

and for SF6:

l ' 0.379~0.632 . (kV/cm) (16) E = 21.19p 1+ x / ~

where p is the pressure in atm. The critical particle size r c for reducing the breakdown voltage is obtained

from the intersection of the breakdown curve for a clean gap and the exten- sion of the curve (eqns. (15) and (16) ) for the free particle, and is expressed by the following equations:

r c = (0.0885 ~ 0.246)/p, (ram) in air (17)

rc = {0.014 ~ 0.039)/p, (mm) in SF6 (18)

since, in the case of a parallel plane of a few centimeters, the breakdown field strengths of a clean gap are about 31p (kV/cm) for air and 89p (kV/cm) for SF6.

3.2 Non- un i f o rm field gap

3.2.1 M e t h o d for predic t ion o f the threshold voltage Referring to Fig. 8 and assuming that the particle is placed on the electrode

at the lower potential, the field on the trajectory of the particle mot ion is mono- tonic in z-direction and in the same direction as the gravitational force, and the external field strengths are En(V ) and E m ( V } on the bot tom and the top elec- trode when the applied voltage is V. If there is no effect of the non-uniformity of the main-gap field on the discharge starting condition in the presence of the particle, a method for prediction of the discharge threshold voltage can be writ ten as shown in Fig.9 by using eqns. (14), (15) and (16). The flowchart outlines the main computational steps, which are as follows.

Step (1): Comparison with the lifting voltage and the discharge threshold voltage for the top of the particle when the particle is placed on the bot tom electrode.

E M ~

og

CL~ER ~ AgTI c~

ELECTRODE

G,I,R~r

EN,EM:EXTERNAL FIELD STRENGTH

Fig.8. Conducting sphere in non-uniform field gap.

233

~ N0

LCULATIOL OF Y2- I (YEs L~RoM EN(V 2) . c~(1+-v~> I

1

..,0,952~-1,587 FOR AIR THRESHOLDBREAKDONN MEcHANISMvoLT, I, " '0,379--0,632 FOR SF 6

VTI r-,7,38 FOR AIR ~-t21,1g FOR SF 6 r: (MM), I : (MM)

Fig.9. Flowchart for the prediction of the discharge threshold voltage.

Step (2): Comparison with the lifting voltage and the discharge threshold voltage for the electrode at higher potential with no particle. As mentioned in Section 3.1, since much breakdown in a parallel plane gap occurs when the particle is near an electrode, the value 4,2 (4.207} for d/r=O shown in Fig.3(a} or Fig.4(a) was used as the maximum value of field enhancement produced by the particle to estimate the discharge threshold voltage in the absence of the particle. That is, the threshold field strength for no particle becomes 4.2 times that of eqns. (15) and (16}.

Step (3): Comparison with the lifting voltage and the discharge threshold voltage for the top of the particle floating near the electrode at the higher potential.

Step (4): Comparison with the discharge threshold voltage for the top of the particle floating near the bottom electrode and that for the electrode at the higher potential with no particle,

234

Each type of breakdown is referred to as Breakdown Mechanism 1, 2, 3, 4 or 5 as shown in Fig.9 and these are described as follows.

Mechanism 1 : After the particle is lifted and as the applied voltage is in- creased, the breakdown occurs on the particle in the Em region.

Mechanism 2: Jus t after the particle is lifted, the breakdown occurs on the particle on its way to the electrode at high potential.

Mechanisms 3 and 5: Breakdown occurs on the electrode at high potential before the particle is lifted.

Mechanism 4: Breakdown occurs on the particle in the En region. A typical breakdown process with change of gas pressure is shown in

Fig.10. It would be expected that the discharge threshold voltage in Mechan- ism 2 is not affected by the polarity and the gas pressure. In other mechanisms, the threshold voltages have different pressure-dependences for each mechanism. All these processes do not always appear under a given condition of the parti- cle size and the gap geometry.

3.2.2 Experimental results Experimental set-up. The gap used comprises concentric spherical electrodes

which lead only to spark breakdown with no corona in the absence of a particle, and is mounted inside a steel test cell of 80-cm length and 50-cm inner

p (.Q)

(b)

l / p (c)

Fig.lO. Discharge process wi th eharlge o f pressure. VT: Discharge threshold voltage; p: Pressure; 1 :Breakdown Mechanism 1 ;2: Breakdown Mechanism 2; 3: Breakdown Mechanism 3; 4: Breakdown Mechanism 4;5:Breakdown Mechanism 5.

235

diameter. The central brass sphere the diameter of which could be 1, 1.5, 2 and 3 cm is supported by a brass rod which is glazed to prevent the occurrence of discharge from it and d.c. voltage was applied. The outer hemisphere of 4.24-cm inner diameter is made of copper and earthed.

Steel particles of radii 0.5, 0.75 and 1 mm were used, but only experi- mental results for the 0.75-ram-radius particle and the 2-cm-diameter inner sphere will be presented because the breakdown characteristics for these particles were very similar. The applied voltage was raised slowly {about 0.3 kV/s) until the breakdown threshold occurred. For the experiment using SF~, the test cell was evacuated to 1 mmHg and filled with pure SF6.

Corona threshold. At first, the current is observed in order to investigate the occurrence of self-sustaining discharge. Figures 11 (a) and (c) show that when the particle moves in the gap with a low voltage applied, current pulses of small and large amplitude appear alternately and approximately periodi- cally. The amplitude and its variation increase linearly with increasing voltage

(a) air, positive, p=7 atm., V=44.2 kV, I ~s/div, 125 uA/div.

(b) air, positive, p=7 atm., 17--45.8 kV, 1 us/div, 125 uA/div. CURRENT BY PARTICLE

MOT I ON (d)

(c) air, negative, p=9 atm., 17--38.0 kV, I ~s/div, 50 uA/div.

(d) air, negative, p=9 atm., V=57.0 kV, l~s/div, 125 ~A/div.

(e) SF6, positive, pffi2.5 atm., 11--43.7 kV, 1 us/div, 500 uA/div. Fig.11. Current pulses. (f) SF~, negative, p=2.b atm., V=45.0 kV, 5 us/div,

50 uA/div.

236

and is little affected by the gas pressure, the kind of gas and the voltage polarity. The ratio of the large amplitude to the small amplitude is about Ern/En = (R/I) ~, where R and I are shown in Fig.8. No pulse is detected prior to the lift of the particle and corona occurrence. As the applied voltage is increased, pulses with larger amplitude and higher frequency than the pulses described above occur randomly in a group and are unstable (see Figs. l l ( b ) , (d), (e) and ( f ) ) . The corona threshold voltage is defined as the voltage which initiated the above unstable current pulse. The pulse height of the current by the particle mot ion is far larger than the displacement current estimated from the motion of the charge on the particle. A possible mechanism should be discussed in a further study.

Breakdown voltage. Figure 12 shows the mean value of the breakdown voltage, its variation and the corona threshold voltage versus the gas pressure in the presence or absence of the free steel particle. The predicted values from Fig.9 are also indicated. As can be seen from this figure, the experi- mental result for the discharge threshold voltage shows the same characteristic as that of Fig.10(c), and considerably coincides with the predicted value for both polarities which is that of the inner spherical electrode. In Breakdown Mechanism 2, the variation of the breakdown voltage is very small (less than 2%), because the field strength near the particle approaching the inner spherical

1OO --

90 --

80--

70--

~60 ~

no

3O

2O

10

(a) Air

POS Neg.

o [3

A v

CALCULATED VALUE "~l~z?

FROM EQ, (13) I,~CALCULATED VALUE FROM

/~ /" E M = 31P(1 * ~ ) ,

!

~ ACCORDING TO THE

~ ULATED VALUE ACCORDING TO THE PROCEDURE OF FIG,9 :VT2

I I I I I I I I I 1 3 4 5 6 7 8 9 i0

PRESSURE (ATM)

Corone threshold vol t in the presence of the particle Breakdown threshold volt in the presence of the I:x;Ir ticle Breokdown threshold volt in the absence of the par t ic le

110 CALCULATED VALUE FROM, EQ,(13)

/"

: r / .Y , ~ ' ~ CALC°LATE~ VALUE I - - /~ ~J/~ ACCORDING TO THE

50F . . . . . . . . . . . . F,G.g

F / ~ - " ~ 1 ~ ~ A L CT : . . . . . . . . . E ACCORD]NG

lOI';~ I I I I I I I

PRESSURE (ATM) (b) SF 6

Fig.12. Comparison o f calculated and experimental values o f corona threshold and break- down voltage in concentric hemispherical gap with free steel sphere against pressure.

237

electrode is always greater than that predicted by eqn. (15) or (16). If the electrode system was moved slightly to lift the particle, then the breakdown voltage was lowered by aboy t 15% in comparison with the still electrode system. This is the reason why the lifting voltage falls when d/r increases (see Fig. 6).

In air, for a negative polarity, the breakdown is preceded by the corona and is always higher than for a positive polarity in the pressure region con- sidered in this study. For a positive polarity, the corona discharge was ob- served only at a pressure of 1.5--8.5 arm., but the breakdown voltage has no maximum, unlike the characteristic of the needle to plane gap in the absence of a particle.

In SF6, the characteristics follow the same trend as in air, but a slight maxi- mum occurs at abou t 5 atm. for positive polarity. Boltnik and Cook [9] re- ported that the d.c. breakdown in the coaxial cylindrical gap, having a 7.6-cm outer dia. and a 25-cm inner dia., with a steel particle 0.8 mm in diameter occurred at about 580 kV at 4.4 atm. pressure, and the corona threshold voltage was lower than both this and the particle lifting voltage. The predicted value from Fig.9 is 507--564 kV and Breakdown Mechanism 3 is involved. This shows that the f lowchart of Fig.9 may be of use in estimating the breakdown voltage for a large gap. In the present experiment, only the breakdown process of Fig.10 (c) was developed; an example of the occurrence of the other type is shown in Table 1. The process changes from (a) to (c) in Fig.10 as the non-uniformity of the external field becomes pronounced.

4. Conclusions

The field and the force acting on a conducting sphere are derived and numerically calculated when the particle is placed on or near an electrode, and a method for the prediction of the discharge threshold voltage in the gap with the sphere is proposed on the basis of the calculation and is confirmed by the short gap.

T A B L E 1

Discharge process under various gap c o n d i t i o n s in air

R (cm) I ( cm) r ( r am) B r e a k d o w n Mech. E m / E n

25 10 1 5 - . 3 - -2 - -1 6 .25 25 20 1 4--~2-~ 1 1 .56 50 10 5 5-~3 25 .0 50 20 5 5 ~ 3 - - 2 ~ 1 6 .25 50 28 5 5 ~ 4 - ~ 2 ~ 1 3.19 50 35 5 4--~2~ 1 2 .04

Arrow d e n o t e s change o f B r e a k d o w n M e c h a n i s m w i t h increase in pressure f r o m 0.5 to 10 a tm.

238

The results can be summarized as follows: (1) When a conducting spherical particle on a plate of a parallel plane gap

is lifting in the gap, the T-component of the field strength is given as a func- tion of d/r and the bispherical coordinates (~ and 0).

(2) Under the same condition as (1), the force acting on the sphere be- comes a function of d/r only.

(3) The solutions obtained in this paper for the field strength and the force includes those for d/r=O and d/r ~ 1 obtained by other workers.

(4) When the free particle is present in a uniform gap, the experimental value of the external field for the self-sustaining discharge are in reasonable agreement with the calculated value for d/r=O and are included in the simple expression:

t 0.952~1.587 f E = 7.38p 14 ~ (kV/cm) for air

and

l 0"378~0"632 f (kV/cm) forSF6 E = 21.19p 14 vrP ~

(5) It is supposed from the intersection of the breakdown curve for a clean gap and the extension of the curve for the free particle that the effect of the particle on the breakdown voltage is eliminated in the region less than the following particle radius rc:

r c = (0.0885 ~ 0.246)/p (mm) for air

r c = (0.014 ~ 0.039)/p (ram) for SF6

where p is the pressure in atm. (6) When the particle is present in a non-uniform field gap, the breakdown

mechanism can be classified into five mechanisms (essentially four mechanisms) with regard to the particle movement and the starting position of the spark. Each breakdown voltage for four mechanisms has a different pressure-depen- dence.

Acknowledgements

The authors wish to thank Dr S. Masuda of the University of Tokyo, Dr T. Miyazoe of Kyushu University and Dr T. Oshige of Kyushu Institute of Technology for useful discussions.

R e f e r e n c e s

1 M. Akazaki and M. Hara, Photographic studies of breakdown processes under impulse voltage at the existence of simulated floating particles in the air, J. Inst. Electr. Eng. Jap., 89(5) (1969) 972.

239

2 M. Akazaki and M. Hara, Influence of floating particles in air on impulse breakdown voltage, J. Inst. Electr. Eng. Jap., 89(10) (1969) 1988.

3 M. Akazaki and M. Hara, The mechanism and characteristics of d.c. corona from floating particles, J. Inst. Electr. Eng. Jap., 90 (1970) 1611.

4 A.H. Cookson, Electrical breakdown for uniform fields in compressed gases, Proc. Inst. Electr. Eng., 117 (1970) 288.

5 A. Diessner and J.D. Trump, Free conducting particles in a compressed gas insulated system, IEEE Trans. Power Appar. Syst., PAS-89 (1969) 1970.

6 M. Hara and M. Akazaki, Influence of floating particles on switching surge flashover characteristics of long air gaps, J. Inst. Electr. Eng. Jap., 91(1971) 557.

7 A.H. Cookson and 0. Farish, Particle-initiated breakdown between coaxial electrodes in compressed SF,, IEEE Trans. Power Appar. Syst., PAS-92 (1972) 871.

8 A.H. Cookson, 0. Farish and G.M.L. Sommerman, Effect of conducting particles on a.c. corona and breakdown in compressed SF,, IEEE Trans. Power Appar. Syst., PAS-91 (1972) 1329.

9 I.M. Boltnik and C.M. Cook, SuIfer Hexafluoride discharge at superhigh voltages, Sov. Phys. Tech. Phys., 17 (1973) 1850.

10 N.N. Lebedev and I.P. Skal’skaya, Force acting on a conducting sphere in the field of a parallel plane condenser, Sov. Phys. Tech. Phys., 7 (1962) 268.

11 P. Moon and D.E. Spencer, Field Theory for Engineers, Van Nostrand, Princeton, N.J., 1961, p. 361.

12 P. Moon and D.E. Spencer, Field Theory Handbook, 2nd Edn., Berlin, Springer-Verlag, 1971, p. 110.

13 P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, McGraw-Hill, 1957, p. 1297.

14 A, Pedersen, Calculation of spark breakdown or corona starting voltage in non-uniform fields, IEEE Trans. Power Appar. Syst., PAS-86 (1967) 200.

15 A. Pedersen, J. Lebeda and S. Vibholm, Analysis of spark breakdown characteristics for sphere gaps, IEEE Trans. Power Appar. Syst., PAS-86 (1967) 975.

16 A. Pedersen, Criterion for spark breakdown in Sulfer Hexafluoride, IEEE Trans. Power Appar. Syst., PAS-89 (1970) 2408.

17 T. Takuma, Discharge characteristics of gases, Part 1: Corona inception voltage, Tech. Rep. of CRIEPI, No. 69015,1969.