IEEE Zkansactions on Electrical Insulation Vol. 28 No. 2, April 1993 281

Streamer Formation and Monte Carlo Space-charge Field Calculation in SFG

Jianfen Liu and G. R. Govinda Raju Department of Electrical Engineering,

University of Windsor, Windsor, Ontario, Canada

ABSTRACT A Monte Carlo simulation is carried out in SFe in uniform electric fields with 7 and 14% overvoltages and at two gas den- sities at each overvoltage. The electron motion and avalanche growth is simulated by tracing individual paths and the effect of space charge is included by solving the Poisson equation. The streamer propagation, electron, positive and negative ion distribution and space charge fields are studied in detail as more time has lapsed after voltage application. The simulated streamer shape explains for the first time the dark space in SF6 streamers observed experimentally. It is found that the mechanism of streamer propagation in an attaching gas is dif- ferent from that in a non-attaching gas. The maximum field enhancement is just behind the streamer or between two suc- cessive streamers in SF6. The anode directed streamer prop- agates with a velocity of lo7 to los cm/s, dependent on the percentage of overvoltage and the gas number density.

1. INTRODUCTION

HE streamer mechanism of electrical breakdown in T gases under high overvoltage and pressure has been studied extensively both experimentally and theoretically in Nz, Hz, He, 0 2 and SFG. Besides the electron mul- tiplication by ionization, a secondary mechanism, pho- toionization, is necessary for electrical breakdown to oc- cur. Also, because the electron and ion densities are high (> lo'* ~ m - ~ ) , space charge field distortion has to be considered [l]. Due to the complexity of photoionization and space charge, many approximations have been as- sumed in the previously published theoretical analyses. Most of the published results are obtained by solving the continuity equation for electron and ion species, and use the Poisson equation for the space charge field. Novak and Bartnikas [2-51 (in He and Hz) considered the two- dimensional continuity equation for electron, positive ion

and excited molecules and included the photoflux, the ion flux and the metastable flux to cathode as cathode emission, rather than photoionization directly in the gap. The continuity equation is then solved by a finite element method. Because of the very steep, shock-like density gra- dients, the solution by ordinary finite difference method is difficult and is limited to the early stage of stream- er formation. Dhali and Pal [6] in SF6 and Dhali and Williams [7] in NZ handled the steep density gradients by using flux-corrected transport techniques which improved the numerical method for the two-dimensional continuity equation. In order that a streamer form immediately, a spheroid or hemispheroid of relatively dense plasma ( 1013 to ~ m - ~ ) is placed either in the gap center or on an electrode as an initial condition. This limits the study to the later stages of streamer propagation. Also, they sim- ulated photoionization which is a key point in streamer formation, by including a tenuous neutral ionization of

0018-9367 $3.00 @ 1993 IEEE

262 Liu et al.: Streamers and Space-charge Calculation in sF6

density IO4 to 10' cmP3 uniformly deposited through- out the gap as a n initial condition. Morrow [8] included photoionization as a secondary source term and observed cathode and anode directed streamers. It is noted that Morrow used a one-dimensional equation. Solution of the one-dimensional Poisson equation is equivalent to a three dimensional problem with the charge homogeneously dis- tributed in the z - y directions [9]. In this context, the terminology of a homogeneous planar wave of ionization may be more appropriate, but to avoid confusion we re- tain the term streamer, consistent with Morrow and oth- er authors. Novak and Bartnikas [lo] also improved their numerical method by flux-corrected transport algorithm to extend the calculations to the later stages of break- down. Yoshida and Tagashira [ll] in Nz related gas pho- toionization with ionization rate directly and used the im- age method to improve the accuracy of one-dimensional space charge field calculation. Because of the large CPU time needed for Monte Carlo simulations, almost all the theoretical studies are performed by solving the conti- nuity equation. Only Kline et al. [12,13] have studied the streamer formation by both methods. In their Monte Carlo simulation, Kline and Siambis [13] followed the elec- tron motion in a modified field which included both the applied field and space charge field. Since the electron multiplication is very fast, scaling is necessary to limit the number of charges to be followed. A new group of 'larger' particles is generated to represent the old, large group of 'smaller' particles by randomly selecting some fraction f of the old group of ,)articles. Because of the unoptimized scaling procedure used in their calculation, the particles were not conserved exactly, and this resulted in some er- ror in the space charge field calculation. Also, their sim- ulation is a t low gas number densities ( N < 3.54 x 10l6 cmP3). Recently a Monte Carlo simulation in N2 , SF6 and gas mixtures containing SF6 has been reported [14] which is a follow up of the investigations of Dincer and Govinda Raju [15] in SF6. Only discharge parameters such as ionization, attachment coefficients and drift ve- locity are reported. N o details about streamer formation and space charge field have been reported.

Comparing the two methods of analysis, the continu- ity equation method is concise and consumes less CPU time, but it requires far too many assumptions and is not as straightforward as a Monte Carlo simulation. Another disadvantage of the continuity equation method is that all the transport and rate coefficients have to be well known prior to the analysis. Usually these coefficients are as- sumed to be in equilibrium with the field and expressed as a function of E I N only where E I N is the reduced field units of T d (1 T d = V cm2; the conversion factor from E / N to E l p , p the gas pressure, is 1 T d =

Table 1. List of symbols.

time (9)

distance from cathode (mm) gas number density ( ~ m - ~ ) electric field (V/cm) reduced electric field Td scaling factor of electron numbers a t time t k

solid angle (sr) number of ion pairs per second formed by photoionization number of ion pairs per second formed by ionization probability of photoionization per excitation probability to decide the photoionization position electron density ( ~ m - ~ ) positive ion density ( ~ m - ~ ) negative ion density (cmW3) applied field (V/cm) total field = applied field plus space charge field

269 V cm-' bar-' = 2 . 6 9 ~ 1 0 - ~ V cm-' Pa-'). Since in the streamer mechanism the space charge field strongly distorts the uniform field, the maximum total field could be as large as 5 x the minimum field and the field slope could be very steep, the equilibrium assumption (for rate coefficients a t least) is erroneous according to the non- equilibrium study of Boeuf and Marode [16] in Nz and Liu and Govinda Raju [17] in SF6.

In the Monte Carlo method, which takes a longer CPU time and starts from the collision cross sections, one need not know the swarm parameters prior to the study. Fur- thermore, the nonequilibrium behavior of electron motion in nonuniform fields is accounted for as the simulation proceeds. For a full understanding of various discharge simulation methods, the reader is referred to a review paper by Davies [18].

In their experimental study of mechanism of spark breakdown in N 2 , 0 2 and SF6, Chalmers et al. [19] found that the propagation of streamers in SF6 is different from that in N2 and 0 2 : there is a broad dark space (Fig- ure 14(e) in [19]) in the center of the gap persisting for - 0.3 ns before it brightens to form a diffuse discharge bounded on anode and cathode (Figure 14(f) in [19]). Similar observations of dark space are also found in Nz by Koppitz [20]. No satisfactory explanation has been provided for this observed phenomenon so far. There is no such phenomenon reported in the theoretical study in SF6 by Dhali and Pal [6], and Pfeiffer and Welke [14].

IEEE Transactions on Electrical Insulation Vol. 28 NO. 2, April 1993 263

In this paper, a Monte Carlo simulation is carried out in SF6 under 7 and 14% overvoltages and a t two gas number densities for each overvoltage. When the num- ber of electrons, positive ions and negative ions exceeds a certain criterion, the space charge field is included by a one-dimensional Poisson equation. The streamer prop- agation, electron, positive and negative ion distributions and space charge fields are studied in detail as time in- creases. The streamer shape and propagation velocity in SFs agree with the earlier experimental observations of Chalmers et al. [19].

2. SIMULATION METHOD

ONSIDERING the CPU time of the Monte Carlo sim- C ulation of a large group of electrons for - 0.3 ns, we assume one dimension in position space, and three dimensions in velocity space. According to Kline and Siambis [13] this assumption is valid for gas number den- sities N < 1 . 4 ~ lo1' ~ m - ~ .

In a Monte Carlo simulation of the Townsend type of discharge, the motion of a single electron is followed until a termination time or position and the procedure is re- peated for the next electron. In a simulation of streamer discharge based on the Monte Carlo method, the electrons and ions, if exceeding a certain number, distort the elec- tric field and all the electrons are followed a t the same time intervals. The time interval should be small, less than a mean flight time. Accordingly from the calcula- tion of mean flight times, d t = 0.35 x lo-" s is chosen in this study. At the beginning of each time step, the new position and energy are calculated according to the equa- tion of motion. New electrons, positive ions and negative ions may be produced by ionization, photoionization and attachment collisions. At the end of each time step, the space charge field is calculated from the Poisson equation as a function of the new charge distribution. The gap is divided into equal cells with a cell length of 5 ~ 1 0 - ~ cm. The total field distribution a t the end of the step t k is stored for the use of next time step t k + l . Since the electron multiplication is fast under overvoltage, the to- tal electron number may exceed the maximum allowable number of simulation particles, or require excessive CPU time to follow all of them. At the end of time step t k ,

if the total number exceeds a limit N,,,,,, a statistical subroutine is introduced to choose a new group of larg- er particles to represent the old larger group of smaller particles. The subroutine contains a weighing of velocity distribution of the old group, so that the new group is equivalent in phase space to the old group. Each of the selected particles then has l / f k x as much charge and mass as each of the old particles, where f k is the scaling factor a t t k .

In the experimental study of photoionization in air, 0 2

and N2. Penney and Hummer [21] related the photoion- ization rate with ionization rate through a coefficient $J.

where N p is the number of ion pairs per second formed by photoionization, N D the number of ion pairs per second by ionization collision, w the solid angle extended from the ionization point to the plane contained photoioniza- tion point, N d the product of gas number density and depth. ?c, is a function of N d for a fixed gas, usually it is inversely proportional to N d . A typical value of 4 in 0 2

is Z . ~ Z X I O - ~ ~ to 2 . 8 2 ~ 1 0 - ~ ~ cm2 s r - l . Since very little in- formation is available on photoionization in SFe, in this study a threshold for photoionization of 15.8 eV and a probability of P1 = 5 x l o p 4 per excitation collision is as- sumed. In nitrogen a probability < l o p 3 is measured [21, 131. Here PI is related to excitation directly, rather than to ionization [ 2 1 ] since photons are produced when excit- ed molecules return to lower levels. If PI > R I , where R1 is a uniformly distributed random number between 0 and 1, one ion-electron pair is produced due to pho- toionization. The photoionization position Z, is decided by the probability P2 which is proportional to $JnN&j from Equation ( 1 ) . Assuming $J - l / N Z i , [21] we can approximate

1

Zi, is the distance between excitation point Z, and the photoionization point Z,, R is the discharge radius. We assume tha t the photoionization which occurs outside a cylindrical region with r > 0.1R is negligible. According to [22], R = 4 x lo-' mm is assumed.

N

p 2 , = 1

(3) 3 =1

P21 6 P 2 2 6 . . . 6 P2, 6 . . . P z N

P 2 1 + P22 + . . . P2, -1 < R2 < P21 + Pa2 + . . . Pa,

determines the photoionization position 2,. RZ is a sec- ond random number generated similar to R I . Then a t t k

and Z,, a electron-ion pair is stored with 0.1 eV energy for the electron generated by photoionization.

In the present simulation the electron molecule collision cross sections in SF6 are taken from Itoh et al. [23]. The calculations are performed corresponding to 0°C.

264 Liu et al.: Streamers and Space-charge Calculation in SFe

z ("1 Figure 1.

Electron density distributions at various times at 410 Td and N = 2.12 x 10" cm-3 in SFs. N e should be multiplied by 0.02 to convert to density ~ m - ~ .

3. RESULTS AND DISCUSSION

NE thousand initial electrons are released from the 0 cathode with 0.1 eV energy. The applied field is 385 and 410 T d corresponding to - 7 and 14% overvoltage respectively. The gas number densities investigated are 1 . 4 2 ~ 1 0 ' ~ ~ m - ~ = 5 . 2 6 kPa and 2 . 1 2 ~ 1 0 ~ ' cmP3. During the first 400 time steps (< 1.4 ns), the space charge field is neglected. If the total electron number N > lo4 , the scaling subroutine chooses l o 4 out of the total number. Since the electrons start with 0.1 eV energy, attachment is large during the first several steps, the total number of electrons increases slowly; e.g., a t 410 Td , N = 2 . 1 2 ~ 1 0 ' ~ cm-3 (< 4.0 ns), and the total number of electrons is still

Figure 2. Positive ion distributions at various times under same condition as Figure 1, N + should be multi- plied by 0.02 to convert to density cm-'. Broken lines are the peaks of the electron density in Fig- ure 1.

< lo4 . For larger times, the scaling subroutine takes over. At 2.8 ns, the total density of electrons > 2 x 10" cmP3 (so does the density of positive ions), space charge field distortion begins. Figures 1 to 3 show the electron, posi- tive ion and negative ion density distribution in the gap a t 410 T d and N = 2 . 1 2 ~ 10" cmP3. Since the electron density is low a t the beginning of the avalanche growth, only the distributions after 2.45 ns are shown. Figures 4 to 6 show the total field distributions a t various times under the same condition. The field behind and ahead of the avalanche is enhanced, while in the bulk of the av- alanche it is weakened by the space charge field. In an electronegative gas, because of the attachment, the elec- tron number is less than that of the positive ion, and en- hances the field behind the avalanche. On the other hand,

IEEE Transactions on Electrical Insulation

ns c _ _ _ _ _ _ --- 3.85 3.68

3.50

3.32

3.15

2.95

2.80

2.62

2.45

J

Negative ion distributions at various times under same condition as Figure 1, N- should be multi- plied by 0.02 to convert to density cm-’.

in a non-attaching gas, the maximum field enhancement is a t the leading edge of the avalanche. At t = 2.8 ns, the maximum field distortion is about 0.8% in the bulk of avalanche (Figure 4). Also the secondary avalanche caused by photoionization, behind the primary avalanche becomes noticeable (Figure 1). The number of electrons is about two orders less than that in the primary. As time increases, the enhancement becomes greater. The weakened field position moves toward the anode as the electrons drift. The secondary avalanche is located in the maximum field enhancement region (between the cath- ode and the leading edge of the primary avalanche), it grows faster than the primary avalanche (Figure I ) , e.g. a t t = 3.32 ns, i t is only one order of magnitude less than the number of electrons in the primary avalanche.

Vol. 28 No. 2, April 1993

1.00;

1.1oc

2 . ? ? E J - U

0 . ? ? 6

0 . $04

0 . O O f 1 2

Dlstar7ce ‘rom cathode(”! ’3

Figure 4. Normalized total field distribution (space charge field plus applied field) under same condition as Figure 1, t = 2.45, 2.62 and 2.8 ns.

i n

265

0 . 0 o.33 0 1 2 3

Distance from cathode(”)

Figure 5. Same as Figure 4, t = 1.98, 3.15, 3.32 and 3.5 ns. Secondary streamer begins to distort the applied field at t = 3.5 ns.

The secondary avalanche begins to distort the field sig-

Liu et al.: Streamers and Space-charge Calculation in SFG

ization. They simulated photoionization by including a tenuous neutral ionization of density lo4 to lo8 cm-3 uni- formly deposited throughout the gap. Since photoioniza- tion is also a function of time and position (dependent on the field), we believe tha t our procedure is more realistic.

-rf LUMINOUS REGION

266

1

2

s J -

1

0 2 3 0 1

Distance from cathode(min)

Figure 6. Same as Figure 4, t = 3.68 and 3.85 ns.

nificantly a t t = 3.5 ns (Figure 5). Since the leading edge of the secondary avalanche and the trailing edge of the primary avalanche are located in this maximum enhanced field (e.g., t = 3.5 ns, E,,, = 1.25E0 (Fig- ure 5); a t t = 3.68 ns, E,,, = 1.7E0, Figure 6), the secondary avalanche is even more accelerated (Figure 1) . At t = 3.68 ns, it has approximately the same number of electrons (within a factor of two) as the primary ava- lanche. Also the trailing edge of the primary avalanche grows faster than its leading edge. At t = 3.85 ns, E,,, is almost 3 x E, (Figure 6). The secondary avalanche now exceeds the primary avalanche (Figure 1 uppermost curve). The positive ion distribution is similar to that of the electron, except that its peak position is trailing behind that of the electron, (Figure 2). At t = 3.85 ns, the fact that the primary electron peak is behind that of the ion indicates cathode-directed streamers. The num- ber of negative ions (Figure 3) is one order less than that of positive ions. Although the number of electrons in the secondary avalanche exceeds that in the primary, the number of negative ions in the secondary avalanche is still less than in the primary because in the secondary ava- lanche, the field is high, and results in fewer attachment collisions.

The theoretical study of Dhali and Pal [6] showed tha.t the maximum field enhancement is in front of the stream- er. This is possibly due to their assumption of photoion-

- E E - N

C

Figure 7. Calculated luminosity 410 T d and N = 2 . 1 2 x 10" cm-'.

vs. position and time at

The light output from the developing avalanches and streamers is due to the photons emitted by excited mol- ecules when they return to a lower state. Assuming the lifetime of the excited molecules is of the order of - 4 ns, the light output which is proportional to the excitation collisions is calculated in a time step of At , = 50 x At = 0.175 ns. The boundaries of the luminous region for sim- ulated streamers are assumed to be the points where the excited molecule density falls by two orders of magnitude below its peak value a t the same time. The results are shown in Figure 7 for 410 Td , N = 2.12 x 10l8 cmU3 and Figure 8 for 410 Td, N = 1.42 x 10" ~ m - ~ ,

Referring to Figure 7, a t first the avalanche moves to- ward the anode and its size grows. The leading edge of the streamer propagates a t a speed of 6 . 5 ~ 1 0 ~ cm/s. The

IEEE Transactions on Electrical Insulation Vol. 28 No. 2, April lQQ3 267

t ( n s )

Figure 8. Same as Figure 8, but at 410 Td, N = 1.41 x lo1* ~ m - ~ .

trailing edge travels a t a slower speed, - 2.9 x l o 7 cm/s. The velocity of the streamer center is - 4.9 x l o 7 cm/s. At t = 1.4 ns, the primary streamer slows down its prop- agation speed (at A) by shielding itself from the a.pplied field. The velocity of the leading edge decreases to 3 . 9 ~ 1 0 ~ cm/s. The trailing edge propagates faster than before, a t 3 . 8 ~ 1 0 ~ cm/s. We believe that the enhanced field region between cathode and the primary streamer is responsi- ble for the increase in velocity. The secondary streamer caused by photoionization begins a t t = 2 ns and propa- gates very fast in the maximum enhanced field between the two streamers. There is a dark space, shown hatched, between two streamers. At t = 3.85 ns, the field between two streamers is so high (E,,, 3E,), that the trail- ing edge of the primary streamer propagates backward to cathode. This also could be seen from Figure 2 a t 3.85 ns where the peak in the number of electrons (the leading edge of the streamer) is backwards. Also the secondary streamer grows and moves very fast and within - 0.2 ns, the two streamers connect. The dark space between the cathode and the anode exists for - 2 ns, for 410 T d and N = 2 . 1 2 ~ 10" cmp3.

formation depends on the percentage of overvoltage and pressure, the existing time of dark space a t high over- voltage (2 ns a t 14% overvoltage, N = 2.12 x 10" ~ m - ~ ) is shorter than that of a lower value of overvoltage (10 ns, 3% overvoltage). We believe that this is the first time that a theoretical simulation has provided an explanation for the experimentally observed dark space in SFs. The experiments of Chalmers [19] were conducted a t 3 and 8% overvoltages and N = 3.54 x cm-3 (13 kPa) and a simulation a t this pressure and voltage cannot be carried out due to computational limitations and excessive CPU (> 100 h). However, a comparison with present result is still qualitatively valid.

A a2

Figure 9. The schematic streak picture defined by Koppitz [20]

In the experimental study of streamer formation and propagation a t high overvoltage in a homogeneous field in Nz, Koppitz [20] observed the same shape of the streak photographs. The schematic streak picture as defined by Koppitz [20] is shown in Figure 9. The discharge was divided by Koppitz [20] into three stages, a1 and az being the slow and fast stages of the anode directed luminous front and the final stage being the development of the anode and cathode directed luminous layers (a3 and K1). The present calculated light output is very similar to that of Koppitz [20].

The leading edge of the primary streamer a t late stages moves faster again ( 7 . 9 ~ 1 0 ~ cm/s) since the field in front of it increases again (comparing the field curve a t 3.32 ns, 1.05E0 and a t 3.85 ns, 1.14E0). The result by Dhali and Pal [6] shows that the maximum field enhancement is a t the t ip of the streamer leads to the conclusion that a streamer will only propagate in one direction, e.g., Fig- ure 2 in [SI, if the initial charge is a t cathode, only an- ode directed streamer will be observed. Obviously their theoretical results are a t variance with the experimental results.

The experimental results by Chalmers [19] verified the existence of this kind of dark space in the center of the gap a t 4% overvoltage N = 3 . 5 4 ~ 1 0 ' ~ cm-3 (Figure 14(e) in [19]). They reported that the dark space persisted for - 0.3 ns before i t brightens to form a diffuse discharge bounded by the anode and cathode. Since the streamer

The experimental results [19] of the velocity of anode- directed streamer in SFG a t 8% overvoltage, is - l o 7 cm s- ' . Our calculations show that it is between 4 to 8 ~ 1 0 ~ cm/s a t 14% overvoltage. The time between the avalanche start and the time interval for the whole gap to brighten is about 3.6 ns under these conditions. Of

course, the higher the overvoltage and gas number den- sity, the faster the streamer propagates. Dhali and Pal [6] reported a velocity of 10’ cm/s a t 410 T d which is in agreement with the present results.

-1

5.42

5.6

r-1 L

0.96

0.94 4, 1 I I I

0 1 2 3 4

Dis!ance from cathode(”)

Figure 10. Normalized total field distribution (space charge field plus applied field) at 410 Td a.nd N = 1.41~ lo1* cm-’, t = 5.25, 5.42, 5.6, 5.78 and 5.95 ns.

Calculations are also performed a t different overvolt- ages and gas pressures. Figure 8 shows the streamer propagation a t 410 Td , N = 1 . 4 2 ~ lo1’ ~ m - ~ . At a low- er pressure, it is predicted that the streamer will prop- agate slower and the breakdown time should be longer since the net ionization is smaller than a t higher pres- sure ( a / N = f ( E / N ) ) . The primary streamer propa- gates with a velocity between 3.9 to 5 . 7 ~ 1 0 ~ cm/s. There are two dark spaces between cathode and anode before it brightens the whole gap. The first dark space exists for - 3.5 ns and the second one 1.75 ns. The time interval between the initiation of the primary avalanche and the time that the whole gap is brightened is 7.7 ns. The space charge field distribution is shown in Figures 10 to 13 a t various times. Figure 10 shows tha t a t t < 5.95 ns, the

1.2 1 n

268 Liu et al.: Streamers and Space-charge Calculation in SFe

field distortion is < 15%. Figure 11 shows that the field - Figure 12 shows the appearance of the secondary stream-

! 0.9

- -

J . 0 . 6 - W -

0 .3 1

V 6.65ns

0 . 0 1 0 1 2 3 4

Distance from cathode(”)

Figure 11. Same as Figure 10, t = 6.12, 6.3, 6.48, 6.65 ne. Secondary streamer begins to distort the applied field at t = 6.48 ns.

2 3 4 0 1

Distance from cathode(”)

Figure 12. Same as Figure 10, t = 6.82 and 7 ns.

distortion is due to the primary avalanche a t t < 6.48 ns.

IEEE Transactions on Electrical Insulation

0 . 0 I I I I

7 4 0 1 2

Olstance from cathode(")

Figure 13.

Same as Figure 10 , 1 = 7.18 ns. Third streamer begins to distort the applied field.

er and the maximum field enhancement is 2E, between the primary and second streamers. Figure 13 shows the existence of a third streamer.

Vol. 28 No. 2, April 1993

Figure 14 shows the streamer propagation in SF6 a t 385 Td (7% overvoltage) and N = 1.42 x lo1' cmP3. The leading edge of the primary streamer propagates with a velocity of 3 . 8 ~ 1 0 ~ cm/s, while the trailing edge is slower a t 2 . 7 ~ 1 0 ~ cm/s. After 7.3 ns from the release of ini- tial electrons from the cathode, there is a luminous re- gion behind the primary avalanche. At 9.5 ns, the whole gap is brightened. The dark space between the cathode and the anode exists for - 2.5 ns. The streamer velocity and breakdown time a t three sets of overvoltages and gas number densities are summarized in Table 2. At the same overvoltage, the higher the density, the faster the stream- er, and shorter the breakdown time. At the same N , the higher the overvoltage, the faster the streamer and the shorter the breakdown time, as is observed experimental- ly.

4. CONCLUSIONS

HE streamer formation and propagation in SF6 are T studied in great detail by a Monte Carlo simulation. The space charge field is calculated from the solution of

LUMINOUS REGION -1 40 r

Figure 14. Calculated luminosity vs. position and time at 3.85 Td and N = 1.41~10'~ ~ m - ~ .

Table 2. Streamer Velocity and Breakdown Time at Vari- ous E I N and N. VSL Propagation velocity of the leading edge of the streamer. V.T Propagation velocity of the trailing edge of the streamer. t g Time that the whole gap is brightened.

VST

T d 1 0 ' ~ cmp3

kV/cm io7 cm/s i o 7 cm/s

ns

410 2.12 8.72 4.8

3.6 2.9 - 3.8

9.5

269

Poisson equation. The simulation results explain the ob- served experimental phenomena in SF6, we believe for the first time. The conclusions are:

1. The maximum field enhancement in attaching gas (SF6) is between the cathode and streamer or between the two streamers.

2. The streamer propagates with a velocity between 3x107 to 10' cm/s under 385 to 410 Td.

3. At the same overvoltage for a higher N , the streamer propagates faster and the breakdown time is shorter. At

2 70 Liu et al.: Streamers and Space-charge Calculation in SF6

the same N , the higher the overvoltage, the faster the [12] L. E. Kline, “Calculations of Discharge Initiation streamer, and the shorter the breakdown time.

4. One or two dark spaces exist between cathode and anode, due to photoionization and space charge field en- hancement, before the whole gap is brightened in SF6. The dark space exists for - 3 ns.

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[2] J . P. Novak and R. Bartnikas, “Breakdown Model of a Short Plane- Parallel Gap”, J . Appl. Phys. Vol. 62, pp. 3605-3615, 1987.

[3] J . P. Novak and R. Bartnikas, “Density Profiles, Electric Field and Energy Dissipation in a Short Gap Breakdown: A Two-dimensional Model”, J . Phys. D: Appl. Phys., Vol. 21, pp. 896-903, 1988.

[4] J . P. Novak and R. Bartnikas, “Early Stages of Cha.n- ne1 Formation in a Short-gap Breakdown”, J . Appl. Phys. Vol. 64, pp. 1767-1774, 1988.

[5] J. P. Novak a.nd R. Bartnikas, “Collisional Phenom- ena and Current Build-up in the Electrical Break- down of Hydrogen”, IEEE Trans. Plasma Sci., Vol. 18, pp. 775-783, 1990.

[6] S . K. Dhali and A. K. Pal, “Numerical Simulation of Streamers in SFs”, J . Appl. Phys., Vol. 63, pp. 1355-1362, 1988.

[7] S. K. Dhali and P. F. Williams, “Two-dimensional Studies of Streamers in Gases”, J . Appl. Phys., Vol. 62, pp. 4696-4707, 1987.

[8] R. Morrow, “Properties of Streamers and Streamer Channels in Sf6”, Phys. Rev. A., vol. 35, pp. 1778- 1785. 1987.

[9] We thank the referee for elucidating this point.

[ lo] J . P. Novak and R. Bartnikas, “Theoretical and Experimental Investigations on the Process of High- pressure Discharge Development” , IEEE Trans. Plasma Sci., Vol. 19, pp. 95-101, 1991.

[ll] K. Yoshida and H. Tagashira, ‘Computer Simulation of a Nitrogen Discharge a t High Overvoltages”, J. Phys. D: Appl. Phys., Vol. 9, pp. 491-505, 1976.

in Overvolted Parallel-plane Gaps”, J . Appl. Phys.,

131 L. E. Kline and J . G. Siambis, “Computer Simulation of Electrical Breakdown in Gases: Avalanche and Streamer Formation”, Phys. Rev. A., Vol. 5, pp. 794- 805, 1972.

Vol. 45, pp. 2046-2054, 1974.

141 W. Pfeiffer and H. Welke, “Simulation of Discharge Development in N z , SF6 and Gas Mixtures Contain- ing sf,”, Conf. on Electrical Insulation and Dielectric Phenomena, Oct. 20-21, 1991, Knoxville, Tennessee.

1151 M. S. Dincer and G. R. Govinda Raju, “Monte Car- lo Simulation of the Motion of Electrons in SF6 in Uniform Electric Fields”, J . Appl. Phys., Vol. 54, pp. 6311-6316. 1983.

[16] J . P. Boeuf and E. Marode, “A Monte Carlo Analysis of an Electron Swarm in a Non-uniform Field: The Cathode Region of a Glow Discharge in Helium”, J. Phys. D: Appl. Phys., Vol. 15, pp. 2169-2187, 1982.

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Manuscript was received on 3 June 1992, in revised form 26 October 1992.