Download ELECTRIC FIELD MODELLING FOR POINT

ELECTRIC FIELD MODELLING FOR POINT-PLANE GAP
V. Repän, M. Laana and T. Plank
Institute of Experimental Physics and Technology, University of Tartu, Tähe 4, 51010 Tartu, ESTONIA
Abstract. The electric field distribution for point-plane gap is modelled both for stressed point and
stressed plane electrodes. In simulations, the influence of the discharge chamber walls is taken into
account. The size of an avalanche and the corresponding current pulse are calculated. The results
are compared with those got using other field distribution approximations.
1. INTRODUCTION
For gas discharges, the distribution of the electric field is the most important parameter, which
determines the spatio-temporal development of discharges as well as the waveform of the recorded
current. In the case of the modelling of corona discharges in point-plane gaps, the simplest Laplacian
field approximation is the spherical one and it is widely used [1, 2]. On the basis of numerical
modelling, Abou-Seada [3] presented for the ionisation zone of the gap an analytical expression of the
field distribution E(x) along the gap axis. The approximation does not satisfy the normalization
d
condition U = ∫0 E ( x )dx , where U is the applied voltage and d is the gap length. Coelho and Debeau
[4] derived a formula for field distribution in hyperbolic approximation. The latter approximation
satisfies the normalization condition. It should be pointed out that the approximations mentioned do
not take into account the influence of the grounded walls of the discharge chamber on the field
distribution.
Our experience of the modelling of non-self-sustained negative DC corona pulses confirms that the
using of above-described approximations does not assure a satisfactory coincidence of simulated and
recorded current waveforms. For this reason we undertook the numerical simulation of the field
distribution which considers also the role of the discharge chamber walls. This paper presents the main
results of the modelling and compares waveforms of avalanches calculated using different electric
field E(x) approximations.
2. EXPERIMENTAL
We studied [5] the regularities of negative corona discharge at atmospheric pressure in air initiated by
UV light pulses. UV pulses of 3.5 ms duration were formed using a deuterium lamp and a chopper and
focused to the point tip. The discharge was ignited in a 4-cm point-plane gap. The point electrode was
a 1-mm wire with hemispherical tip and the diameter of the plane electrode was 15 cm. The diameter
of the metal walls surrounding the discharge chamber was 22 cm. Under these conditions, the onset
potentials of the current pulses of negative corona (Trichel pulses) depend on which electrode is
stressed. The onset potential for stressed point is ≈ 8 kV and for stressed plane it is ≈ 13 kV. Below the
onset potential, the discharge was non-self-sustained, i.e. the current arises only due to the action of
UV light. For a certain voltage, the maximum current recorded depends on properties of the point
coating material and the intensity of UV light but in any case the current waveform has a strong
dependence on which electrode is stressed. Figure 1 presents current pulses induced by UV light
pulses at ≈ 20% below the corresponding onset of Trichel pulses (6.5 and 12 kV for stressed point and
a
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plane, respectively). The most striking difference between the current pulses is the duration of the
current decay, which exists after the action of UV light. As in our time-scale the current of positive
ions and electrons stops immediately, the large difference in current tail durations can be caused only
by different drift-times of negative ions. It was the first reason for modelling the electric field
distribution for stressed point and plane separately.
25
Current i , nA
20
15
Stressed point;
6.5 kV
10
Stressed plane;
12 kV
5
0
-1
0
1
2
3
4
5
6
7
8
Time t , ms
FIGURE 1. UV-initiated current pulses. The duration of the light pulse is 3.5 ms.
3. MODELLING
3.1. Procedure and data
The distribution of the Laplacian electric field E(x,r), where r denotes the distance from the gap axis,
is calculated numerically. The rectangular computation domain is divided first into uniform square
grid having 5 nodes per millimetre. The domain is taken a little larger than the discharge gap. On the
gap axis, the condition of symmetry is used. In the vicinity of the point electrode (about 1 mm around
the point electrode tip), additional iterations are made. The domain is divided into a square grid having
100 nodes per millimetre.
The potential of each node is calculated using finite difference method together with the successive
over-relaxation method [6]. For the convergence criterion of the iterations is taken the requirement
that the relative residual in the iterative process should not exceed 10-14 at each interior node.
The accuracy of the computing algorithm is tested by the numerical evaluation of the well-known field
distributions of coaxial cylinders and concentric spheres. The difference between analytical and
numerical results is less than 0.5%.
Finally, the electric field strength is found for each node using the values of potential of that node and
surrounding nodes.
To check the quality of calculation of the electric field strength, we have calculated the normalization
integral. It differs from the applied voltages less than 10 V.
For the calculation of the drift times of charged particles, ionisation and attachment coefficients in air
and nitrogen, the data from [7 - 11] are used. For the calculation of the current in the external circuit,
the gap is split into discs. The discs size in the axial direction coincides with the grid size used in
modelling. According to Ramo-Shockley theorem, the current is recorded as the sum
e
i = ∑ N k v k E k , where k is a disc number, Nk, vk, Ek are charge carrier number, drift velocity and
U k
field strength in the disc, respectively.
3.2. Results
Figure 2 gives the distribution of the equipotential lines for the gap used in experiments (Section 2).
The distributions of equipotential lines for stressed plane and point differ considerably. In the case of
the stressed plate in the remarkable part of the space between the electrodes, the field is almost
homogeneous both in the axial and the radial directions.
120
Electric field E , kV/cm
Electric field E , kV/cm
FIGURE 2. Distribution of equipotential lines at the onset of Trichel pulses: on the left - stressed plate; on the
right - stressed point; the gap between the equipotential lines is 1 kV; the gap axis is presented by
the dashed line; the gap length is 4 cm and the walls are at the distance of 11 cm from the axis.
A
80
Stressed point; 8.3 kV
&
Stressed plane; 13 kV
40
5
B
4
3
Stressed plane; 13 kV
2
Stressed point; 8.3 kV
1
0
0
0
0.1
0.2
0.3
Distance x , cm
0.4
0.5
0.5
1
1.5
2
2.5
3
3.5
4
Distance x , cm
FIGURE 3. Field distributions along the gap axis at the onset of Trichel pulses: A - ionisation zone; B - drift
zone.
In the ionisation zone at the onset potential of Trichel pulses, the field distributions along the gap axis
demonstrate a perfect coincidence (Figure 3A). In the drift zone, axial fields have a large difference
(Figure 3B): in the case of the stressed plane electrode, the field is much higher than it is for stressed
point electrode. First of all, the difference in field distributions appears in drift times of negative ions.
As a sample, the drift time of negative ions is calculated at voltages where curves of Figure 1 are
recorded. Calculated drift times (0.9 and 3.6 ms for stressed plane and stressed point, respectively) are
very close to the durations of current tails in Figure 1. Here it should be pointed out that in spite of the
movement of the negative ions in low electric fields and consequent small value of the field factor in
Ramo-Shockley formula, the contribution of negative ions to the total current is large. This is caused
by the high concentration of negative ions in the drift zone.
3.3. Comparison with other field approximations
Figure 4 shows that Abou-Seada formula [3] gives in the ionisation zone the field distribution, which
is closest to that simulated by us. There is an obvious trend: the larger is the diameter of the
surrounding walls the better Abou-Seada’s approximation works. Two other approximations give
considerably larger deviations from our results.
(OHFWULFILHOG( N9FP
Sp
M4
M15
AS
C-D
'LVWDQFH[ FP
FIGURE 4. Field distribution for different approximations; applied voltage U = 9 kV. Sp - spherical field
distribution; M4 - our field modelling for stressed point (4 cm walls radius); M15 - our field
modelling for stressed point (15 cm walls radius); AS - field distribution according to [3]; C-D field distribution according to [4].
The calculation of the avalanche size gives even more drastic difference (Figure 5): the spherical
approximation overestimates the avalanche by five orders of magnitude while according to Coelho and
Debeau formula the ionisation is practically missing.
1E+10
Sp
Avalanche size, Ne
1E+8
M4
M15
1E+6
1E+4
AS
1E+2
C-D
1E+0
0
0.02
0.04
0.06
0.08
0.1
Distance x , cm
FIGURE 5. Number of electrons in an avalanche as a function of the distance from the point calculated for the
stressed point in nitrogen at U = 9 kV. Explanation of the abbreviations is at the capture of
Figure 3. NB! Note the logarithmic scale on the ordinate.
In spite of a huge discrepancy in avalanche sizes, the spherical and Abou-Seada approximations
describe the temporal changes of the electron component of the current in nitrogen correctly. Choosing
the proper fitting coefficients (inset in Figure 6), the waveforms of the current of different
approximations practically coincide. In the case of Coelho and Debeau formula, the voltage as well as
the effective radius of the point are varied but the current waveform does not coincide with those
presented in Figure 6.
Electron component i e , nA
200
160
M15: U = 1 x U0
AS: U = 1.07 x U0
Sp:
U = 2/3 x U0
120
80
40
0
0
5
10
15
20
Time t , ns
25
30
35
40
.
FIGURE 6. Fitting of electron components of the current for U0 = 9 kV in nitrogen.
4. CONCLUSIONS
If we are interested only in the temporal evolution of the phenomena in the ionisation zone even the
spherical approximation gives a good presentation of the reality.
Abou-Seada approximation gives quantitatively correct results for ionization zone if the ratio of the
walls distance to the gap length is larger than three.
In the case of discharges in electronegative gases where the contribution of the negative ion
component to the total current is large, a correctly modelled field distribution must be used.
REFERENCES
[1] Morrow R., Theory of negative corona in oxygen., Phys. Rev. A , 32, 1799-1809 (1985)
[2] Sigmond R. S., The oscillations of the positive glow corona., In: Proc. XXIII Int. Conf. Phenom. Ionized
Gases, Inv. Papers, C4-383, Toulouse, 1997
[3] Abou-Seada M. S., Calculating of high-frequency breakdown voltages of point-to-plane air gaps. In: IEEE
IAS 15th Annual Meeting, 1980, Vol. 2, pp. 1118-22
[4] Coelho R., Debeau J., Properties of the tip-plane configuration., J. Phys. D: Appl. Phys., 4, 1266-80 (1971)
[5] Laan M., Repän V., Paris P., Waveforms of UV-induced current pulses of negative corona, In: 11thSymp. El.
Processes & Chem. Reactions, Contr. Papers 2, Low Tatras, 1998, pp 271-273
[6] Zhou P., Numerical Analysis of Electromagnetic Fields, Berlin: Springer, 1993
[7] Tammet H., Reduction of air ion mobility to standard conditions., J. Geophys. Res., 103, 13 933-7 (1998)
[8] Badaloni S., Gallimberti I., Basic data of air discharges, Upee-72/05, 1972
[9] Wen C., Wetzer J. M., Determination of swarm parameters in dry air with a fast timeresolvet swarm
technique. In: Proc. XIX Int. Conf. Phenom. Ionized Gases, Vol. 3, 1989, p. 592
[10] Kennedy J. T., Study of the avalanche to streamer transition in insulating gases, Ph. D Thesis, Eindhoven
University of Technology, 1995
[11] Wen C., Time resolved swarm studies in gases with emphasis on electron detachment and ion conversion,
Ph. D Thesis, Eindhoven University of Technology, 1989