520 IEEE Transactions on Dielectrics and Electrical Insulation Vol. 1 No. 3, June 1994

Simulation of Corona Discharge Negative Corona in SFG

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

University of Windsor, Windsor, Ontario, Canada

ABSTRACT Negative coronas in SF6 under three voltage levels are inves- tigated by Monte Carlo simulation. For the purpose of sim- ulation, the gap between a hyperboloidal needle and a plane is divided into two regions: region I where the electric field is very steep and most electrons and ions are accumulated, and region I1 the rest of the gap. In region I electron motion is simulated by dividing the region into a number of cells and a small cell size improves the accuracy. The magnitude of current pulse increases with increasing voltage and there appears to be more than one peak in each pulse as observed by experiment. The development of electron avalanches is due to ionization and photoionization in the high field region, while the quenching of the avalanches is due to the space charge field suppression. Al- so the accumulation of positive and negative ions are displayed in detail. The space charge field distortion is studied and found to increase with increasing applied voltage.

1. INTRODUCTION ORONA is a self-sustained electrical discharge in a C gas where the Laplacian electric field confines the

primary ionization process to regions close to high-field electrodes or insulators. With the increasing importance of SF6 as an insulating medium, it is important to under- stand the mechanism of corona discharge. Experimental studies [1,2] of corona activity in SF6 under dc condi- tions have shown that the inception occurs in the form of current pulses. Van Brunt and Leep [2] investigated the corona pulses for point-plane, positive and negative dc coronas in SF6 over the pressure range of 50 to 500 kPa. The corona pulse height distributions, pulse shapes and repetition rates were measured under various volt- ages and pressures. For reviews of experimental results in other gases, such as Nz, 0 2 , air and Ha, the reader is referred to Sigmond [3], Goldman and Goldman [4] and Sigmond and Goldman [5]. The only complete quanti- tative theory of corona was proposed by Morrow [S-91. Morrow [6] described a numerical solution of Poisson’s equation in conjunction with the continuity equations

for electrons, positive ions and negative ions in 0 2 . The calculated shape of current pulse and light pulse agreed with experimental observations. Later, Morrow [7] con- sidered the secondary processes a t cathode by photons and ions to explain the step observed on the leading edge of the current pulse in negative corona in 0 2 . Morrow also studied the positive corona in SF6 under impulse [8] and constant voltages [ 9 ] . In Morrow’s theory, there is an implied equilibrium assumption, i.e. the electron, ion transport coefficients and rate coefficients are only a func- tion of the local reduced electric field E / N where E is the electric field, N the gap number density. As pointed out by Sigmond and Goldman [5], this assumption be- comes questionable in very high or very inhomogeneous fields. The study of the nonequilibrium effects of electron swarms in nonuniform fields in SF6 by Liu and Govinda Raju [lo] shows that the ioniBation and attachment co- efficients are quite different from the equilibrium values based on local field.

The purpose of this paper is to analyze the corona dis- charge in SF6 while avoiding the equilibrium assumption.

1070-9878/94/ $3.00 @ 1994 IEEE

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 1 No. 3, June 1994 521

T h e T= t t d t

M ouision

I I I I & & & l Attachment Ioniuticm Mommt.&Erclte.

no

r - VCS

Choore NENS out orNc

Space charge ncld cakuhtlon

t E=Eo+Es

Figure 1. Flow chart of Monte Carlo simulation of corona discharge. e = electron, N. = number of elec- trons, N+ = number of positive ions, N- = num- ber of negative ions.

This could be accomplished by a Monte Carlo simula- tion of electrons, considering the ionization, attachment and photo-ionization processes and finding the solution of Poisson's equation for the space charge field. The cur- rent of the corona pulse is calculated from the number and drift velocity of charge carriers. The development and quenching of electron avalanches are followed in great detail. The accumulation of positive and negative ions and the development of space charge field are also fol- lowed in time sequences.

2. SIMULATION METHOD A flow chart for the simulation is shown in Figure 1.

Anode Figure 2.

A schematic diagram of the electrode geometry.

simulation of streamer formation and propagation in uni- form fields by Liu and Govinda Raju [ll]. The initial electrons are emitted from the cathode according to a cosine distribution. The energy gain of the electrons in a small time interval At is governed by the equation of motion. Whether a collision between an electron and a gas molecule occurs or not is decided by generating a random number R uniformly distributed between 0 and -

This model is an extension of the earlier Monte Carlo 1.0. A collision is assumed to have occurred a t the end

522 Liu et al.: Simulation of Corona Discharge

, I 1 I I I l I I 1 XI

[I 2 4 6 8 10 1 2 14

t(ns)

Figure 4. Current density of a corona pulse at N = 2 . 1 2 ~ 10" cm-'3 and V = 3.0 kV. The current in the external circuit (mA) is obtained by multiplying the ordinates by 3 . 1 4 ~ 1 0 - ~ .

of a time step if the condition P > R is satisfied, where P is the probability of collision. In the event of an elas- tic collision the fractional energy loss is 2m/M' where m and M' are the masses of the electron and SFe molecule. The direction of electron motion after a collision is de- termined according to equations of motion. To reduce the CPU time of the simulation of the motion of a large group of electrons during - 20 ns, we assume that the electrons move in one dimension in space with a velocity which has three components (Vz, V,, Vz). In the simula- tion of a corona discharge, the time step and the cell size should be paid much attention, since the field is steep in region I which is close to the smaller electrode and less steep in region 11. Also the accumulation of space charge causes the field in region I to change abruptly, and there- fore a smaller cell size Az1 is used while in region 11, a larger cell size Azz is adequate (Figure 2) . The length of region I may vary with various voltages and gap sep- arations, and should be tested in the calculation. The electric field close to the smaller electrode is so high that the electron mean collision time T,, which is defined as

1 NQTV

T, = -

t(nd

Figure 5. Current density of a corona pulse at N = 2 . 1 2 ~ 10" cm-' and V = 3.5 kV. The current in the external circuit (mA) is obtained by multiplying the ordinates by 3.1&10-'.

is very small. Here NQ, is the total collision cross sec- tion and v the electron velocity. The time step At used to calculate the motion of electrons and space charge field is chosen to be very small (At << T,), and At may differ for different applied voltages. As shown later, in negative coronas under 2.4, 3.0 and 3.5 kV, At is in the range of 0.5 to 1 ps.

At each time step, the new position and energy is cal- culated according to the equation of motion. New elec- trons, positive and negative ions may be produced by ionization, photoionization and attachment collisions. At the end of each time step, the space charge field is calcu- lated from the Poisson's equation as a function of charge distribution and is stored for use in the next time step. Kline and Siambis [12] presented the calculation of space charge field from Poisson's equation in a uniform cell. In subuniform cells, it is more complicated. Let

IEEE Dansactions on Dielectrics and Electrical Insulation Vol. 1 No. 3, June lQQ4 523

O I I

0.2 0 . 1 0 . 6 0 . 8 1.0 1 . 2 1 . 4 1 . 5

z(") Figure 6.

Total field distribution in the gap at N = 2.12~ lo1* cm-' and V = 2.4 kV.

( I l l 1 1 1 1 I l l 1 [ I l l 1 / 1 1 I , [ , ( 1 1 1 I , [

dl Mi

A z l = -

M = M I + M2

where d is the gap length, dl the length of region I; M I the number of cells in region I, M2 the number of cells in region I1 and M the total number of cells in the gap (Figure 2). The change in the electric field in the k-th cell AEk depends on the charge in that cell

1

EO AEk = - ( - Q e k + Q p k - Qnk)

(3) k = 1,2, . . .MI k = M1 + 1,. . . M

in region I in region I1

where E, is the permittivity of free space. QLk, QPk and Qnk are electron, positive and negative ion charges in cell k. The space charge field in adjacent cells are related by

Ea = E k - l + $ ( A E k - l + A E k ) 16 k 6 M (4)

The field in the first cell is given by

(5) 1 2

El = E , + - A E l

where E , is the field a t the cathode. The integration

charge produces no external voltage on the gap).

of space charge field over all cells should be iero (space 0

d 1 Ed1

524

: \ e o ns

Liu et al.: Simulation of Corona Discharge

I I I I I ! # I I I I ~ I I I O , I t , i l l

0 1 2 3

zo" Figure 9

Total field distribution in the gap at N = 2 . 1 2 ~ 10" cm-3 V = 3.5 kV, t from 4.5 to 10.85 ns.

or Mi M

Z E k d z l 4- E k d Z 2 = 0 (7) k = l k=M1+1

Substituting Equations (4) and (5) to Equation (7), E , is derived as

Mi

Ec = - M l d r l : M a ~ z s A21 + ( M I - k + 0 5 ) + 12 1 d ~ 2 ~ 2 ) A E k + ( A z ~ ( z - k + 0.5)dEk)

k=zi+l

(8) The space charge field a t other cells are derived from

Equation (4). The final field is the sum of the applied Laplacian field and the space charge field.

The photoionization process is simulated in the same way as in [ll], where a probability of 5 ~ 1 0 - ~ per exci- tation collision is assumed and the onset energy of pho- toionization is assumed to be the same as the onset ener- gy of ionization. The lifetime of the excited state is not known and following Morrow [9], the radiation is assumed to be emitted instantly. Collision cross sections in SF6 are taken from Itoh [13]. The calculations are performed

4 U1 T

2.5

2.0

r E p 1.5 5

0.5

, . . . 0.0 13 1 0 . 1 0 . 2 0.3 0.4 0.5 0.6 0.7 0 - 8 0.0 1.')

z ( m d Figure 10.

Electron density distributions at N = 2.12 x lo1* cm-3 V = 3.5 kV, 1 = (0.5,l.O) ns.

corresponding to a gap number density N = 2.12 x lo1* ~ m - ~ . It is noted that the principal advantage of the Monte Carlo method lies in the fact that swarm param- eters are not required for the simulation.

3. RESULTS AND DISCUSSION The electric field in the gap is [14]

( 9 ) 2Vd

ln(4d/re)[d(2z + re) - z 2 ] E L ( Z ) =

where EL is the Laplacian Field, V the applied voltage and re the tip radius. The gap parameters a t three volt- ages are d = 0.5m, r , = 0.05 cm, dl = 0.1 cm, M I = 50, M2 = 100, b z 1 = 2 x cm. The time steps for three voltages 2.4, 3.0 and 3.5 kV are 1, 0.8 and 0.5 ps respectively. The current density J in the external circuit due to the motion of electrons and ions between the electrodes is calculated using Sato's equation [15]

cm and A22 = 4 x

J = - [NpWp - NnWn - N e W e ] E ~ d 2 (10) V e./

where e is the electronic charge, EL the Laplacian field, W,, W,, and We are the drift velocities of positive ions, negative ions and electrons respectively, N p , N,, and Ne

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 1 No. 3, June 1084 525

E increasing

E(O) 6 E' E ( 0 ) 2 E * , E ( d ; ) 6 E* E ( d ; ) 2 E' (just exceeded) E ( & ) > E'

1 .o

0.0 2 . ~ 1 . 0

initial secondary current pulse avalanche avalanche attached no no attached no no

developed no one peak avalanche type

developed developed multiple peaks streamer type

i

zmm) Figure 11.

Electron density distributions at N = 2.12 x 1 O I 8 cmV3 V = 3.5 kV, t = (1.2,1.6) ns.

the corresponding densities. The current is the product of J and streamer channel area which is nr2. According to Morrow [9], the channel radius is T x 100 pm. The drift velocities of electron, positive and negative ions are taken from Morrow's review paper of transport coeffi- cients in SFB [l6] and reproduced here for convenience.

W, = 6 x lO-'E E - < 1 . 2 ~ 1 0 - l ~ N

E N W, = (1.216 x In - + 5.89 x 10-4)E

E 1.2 x 1 0 - l ~ < - < 3.5 x 1o19

N

1 E W, = -1.897 x In - - 7.346 x E [ N E - > 3 . 5 ~ 1 0 - l ~ N

(11)

E N 1.69 x lo3'( -)' + 5.3 x

E - < 5~ lo-" N

L(") Figure 12.

Electron density distributions at N = 2 . 1 2 ~ 10'' cm-3 V = 3.5 kV, t = (1.7,2.2) ns.

E N 1 x < - < 2 x lo-'*

(13)

2000 initial electrons are released from the cathode with 0.1 eV energy at t = 0. While the electrons move toward the anode, some of them may be lost due to attachment, or new electrons may be produced by ionization or pho- toionization if the field is higher than the critical field E' which is defined as the field at which ionization coef- ficient is equal to the attachment coefficient. E* = 7.66 kV/cm a t N = 2.12 x 10" cmh3. The development of a current pulse as the applied voltage increases is shown in Table 1 where d, is the minimum distance required for

0.0 4

2.8

2.7

2.6 ' 2.5

2.4

k2.3ns

1 ' 1 I . I I i ' i 3 i i i * i > T .. ..-

0 . * 2 I > . : . > . a 0.6 0 . g 1.0 1.2 1.4 1.6 1 . 9 I

z@" Figure 13.

Electron density distributions at N = 2.12 x lo1' cm-' V = 3.5 kV, t = (2 .3 ,2 .8 ) ns.

electrons to gain the ionization onset energy Q, i.e.

] E ( z ) dz = E, (14) 0

When the field is low (first two rows in Table 1) the initial electrons are attached and no current pulse will be observed in the external circuit within the ionization time. When the field E ( d i ) just exceeds E* the initial electrons will grow first, then are quenched due to the attachment in the low field region. There is a pulse of current due to the initial avalanche, but its value is small. When the voltage is high enough, the initial electrons de- velop to an avalanche and a streamer formed ( N e 2 10l1 ~ m - ~ ) in which space charge field distortion becomes sig- nificant. In the meantime, successive avalanches are pro- duced by photoionization. The positive ions (caused by ionization and photo-ionization), and the negative ions (caused by attachment) form a reversed space charge field which quenches all the avalanches or streamers. The current appears to be a pulse or several successive puls- es. The magnitude of the pulse current increases with increasing voltage.

Figures 3 to 5 show the pulse current densities in the external circuit as a function of time at various volt-

Liu et al.: Simulat ion of Corona Discharge

2.0

5 0 P 8 p

-

6

iii 1.0

. . , 0.0

9.0 0 . 2 0 . 4 0 . 6 0 . a 1 .0 1.2 1 . 4 1 . 5 1.8 2 ,

W m )

Figure 14. Electron density distributions at N = 2.12~10'~ cm-' V = 3.5 kV, t = (2.9,5.0) ns.

Table 2. Pulse currents for negative corona.

3.5

ages. This current is calculated by numerically evalu- ating Equation (10) with the appropriate value of Ai, and Azz. The results are summarized in Table 2. At V = 2.4 kV (Figure 3), the peak of the current density is 4 mA/m2 at t = 10 ns. The current is of avalanche type with a single pulse. At V = 3.0 kV, the current density splits to two peaks (explained later): the first peak is 36.6 mA/m2 a t t = 3.2 ns and the second one 14.7 mA/mz a t t = 4.9 ns. At V = 3.5 kV, the magni- tude of the current density increases and the second peak exceeds the first. The magnitude of the current density increases with increasing voltage. The current is of the streamer type with more than one pulse and the time at which the peak occurs becomes shorter. This is due to the faster development of the electron avalanche in a higher field.

As already mentioned, Van Brunt and Leep [2] mea- sured the positive and negative coronas in SFe as a func- tion of applied voltage and pressure in the range 50 to 500

IEEE l'kansactions on Dielectrics and Electrical Insulation Vol. 1 No. 3, June 1994 52 7

0.0 1).2 0 . 4 0 . 6 0.8 1.0 1.2 1.4 1.5 1 . 8 2 . p t(")

Figure 15. Electron density distributions at N = 2.12 x 10" cm-' V = 3.5 kV, t = (6.0,S.O) ns.

ymm) Figure 16.

Positive ion density distribution at N = 2 . 1 2 ~ 10" cm-' V = 3.5 kV.

kPa. The pulse shapes, pulse repetition rates and pulse- height distributions are measured for positive coronas. But for negative coronas, because of their irregular oc-

Z@" Figure 17.

Negative ion density distribution at N = 2 . 1 2 ~ 10" cm-' V = 3.5 kV.

currence, the pulse shapes are not shown in their paper. The observed quantities are dependent on gas pressure, electrode diameter and gap spacing. The larger pulses are usually followed by a long tail, or a burst of lower level pulses as in positive corona, or both [2]. Morrow's calculation [6] shows that the negative corona current is a single pulse with very sharp front and long tail in 0 2

in agreement with that of Van Brunt and Leep [2].

The total field distribution in the gap under three volt- ages are shown in Figures 6 to 9. At V = 2.4 kV (Fig- ure 6), before 8 ns, there is no significant field distortion. Later, the space charge enhances the field at both bides and weakens the field in between. At t = 12 ns, the smallest total field is Emin = 4.7 kV/cm a t 0.7 mm. Af- ter 12 ns, the total field remains almost unchanged in a time scale of - 200 ns, but changes in a scale of ps as the positive ions and negative ions drift towards the electrodes. At voltage V = 3.0 kV, the space charge field distortion begins a t 2.8 ns (Figure 7). The field distor- tion increases with increasing time up to 5.95 ns, then remains unchanged for - 5 ns because no more new ions are produced after this time. The smallest total field Emjn is only 400 V/cm a t Zmin = 0.36 mm a t 5.95 ns. At V = 3.5 kV (Figures 8 and 9), the space charge field distortion begins a t t = 1.3 ns. The smallest total field is -1.0 kV/cm a t 0.3 mm (Figure 8). The negative sign means the space charge field at 0.3 mm at t = 7.5 ns exceeds the applied field a t the same position, but with a reversed direction. This is due to the fact that ioniza-

. . ; /

, . , , , ( / , I , , . . I I 1 " 0 . 2 0 . 4 0 . 6 0 . R 1.0 1.2 1.4 1 . 5 1 . 8 2 .

Z(") Figure 18.

Net charge density distributions at N = 2 . 1 2 x 10'' cm-' V = 3.5 kV, to 2.5 ns.

, V (kV) t~ (ns) I Emin (kV/cm) I Zmjn (mm) 2.4 8 1 4.7 0.7

2.0 4 i j!

3.0 3.5

Figure 19 Net charge density distributions at N = 2 . 1 2 ~ 10" cm-' V = 3.5 kV, t = (3.5,10.85) ns.

tion processes in the high field region at a higher voltage cause more positive and negative ions in a small region, the space charge field due to ions is very high, even higher than the applied field. The results are summarized in Ta-

2.5 0.4 0.36 1.3 -1.0 0.30

Liu et al.: Simulation of Corona Discharge

Table 3. Total field distribution in the gap. t ~ : the time at which space charge field distortion begins, Emin: the smallest total field, ,%"in: the position of E-;.. .

ble 3. With increasing voltage from 2.4 to 3.5 kV, space charge field distorts the Laplacian field earlier in time, more in magnitude and positions it closer to the cath- ode. This could easily be explained on the basis that the electron avalanche could reach a density of lo1' cm-3 in a shorter time a t higher fields.

Figures 10 to 15 show the development of the initial avalanche and the successive avalanches at a voltage 3.5 kV. When the initial electrons are released from cathode, the primary avalanche (#1 in Figure 10) drifts toward anode and multiplies fast to 0.8 ns. At t = 0.9 ns, a second avalanche (#2 in Figure 10) is initiated by pho- toionization and grows faster than the primary avalanche because the former (secondary avalanche) is in the high field region close to cathode. At t = 1.1 ns (Figure l l ) , avalanche #2 exceeds #l. From 1.2 to 1.6 ns, avalanche #1 grows slowly, while #2 reaches a density lo1' cmw3 a t t = 1.3 ns. Space charge field distortion is now signif- icant (as shown in Table 3 and Figure 8). At t = 1.6 ns, a third avalanche is initiated (in Figure 11) in the high field region. It should be noted that photoionization may occur in a low field region, but the avalanche will extin- guish soon due to attachment. Only those avalanches started in a high field region will develop.

Avalanche #2 decays as time increases. Avalanche #3 grows from 1.6 to 1.9 ns and starts to decrease (Figure 12) later because the total field in its region (- 0.4 mm) de- creases to less than E* (Figure 8). The maximum elec- tron density of 1 . 4 ~ 1 0 ~ ~ cm-3 a t t = 1.9 ns causes the first peak of current density (Figure 5 and Table 2). The avalanches #3 and #2 continue to decline between 2.3 to 2.8 ns (Figure 13), then a t t = 4.5 ns, there appears a fourth avalanche close to the cathode (Figure 14) which increases until 4.3 ns, then decreases. The second max- imum of electron density 2.7~10" cm-3 at t = 4.3 ns corresponds to the second peak of current density in Fig- ure 5 and Table 2. Now all the electrons are in the field region where E 6 E' (Figure 9, t 2 4.5 ns) and they all decrease with time and finally attach to molecules (Fig- ure 15). After 4.5 ns, there is only a narrow high field region close to cathode (- 0 to 0.16 mm) usually called the cathode fall region.

IEEE Bansactions on Dielectrics and Electrical Insulation Vol. I No. 3, June 1QO4 520

The positive ion distributions in the gap at V = 3.5 kV are shown in Figure 16. The peak of the positive ion moves toward cathode because at a later stages ioniza- tion could only happen in the high field region. At t = 10.85 ns, this peak is 3.3~10‘’ cm-3 at 0.22 mm. Af- ter that, no more new positive ions are produced. The negative ion distributions are shown in Figure 17. They are similar to those of positive ions with a slightly small- er peak a t 3 . 1 ~ 1 0 ~ ’ cm-3 a t 0.24 mm at t = 10.85 ns. The net charge distribution a t V = 3.5 kV is shown in Figure 18 up to t = 2.5 ns and in Figure 19 from 3.5 to 10.85 ns. Because the net charge also depends on electron density which changes fast with time, the net charge dis- tributions do not stabilize. A large positive peak always seems to be followed by a negative peak. This general net charge distribution leads to field enhancement on either side of the electron avalanche and suppression of field in between. In negative corona, the number of negative ions has the same order of magnitude as that of positive ions and the summation of net charge along the whole gap is almost aero at the declining stage of the current pulse.

4. CONCLUSIONS Monte Carlo simulation of negative corona discharges A is carried out in SF6. We believe this to be the first

such study. The current pulses a t various voltages are simulated and are found to have more than one peak. This agrees with the experimental observation. The cur- rent lasts - 5 ns depending on the voltage. The develop- ment of electron avalanches due to ionization and pho- toionization are displayed in great detail. The quenching of the electrons a t later stages is due to the space charge field suppression. The positive and negative ions, net charge distributions and space charge field distortions are also studied in detail.

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Manuscript was received on 22 December 1992, in revised form 1 ley, New York, 1978. December 1993.