ISBN 978-0-620-44584-9
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
A COMPUTATIONAL METHOD FOR POSITIVE CORONA INCEPTION
IN THE COAXIAL CYLINDRICAL ELECTRODE ARRANGEMENT IN
AIR UNDER VARIABLE ATMOSPHERIC CONDITIONS
P.N. Mikropoulos* and V.N. Zagkanas
High Voltage Laboratory, School of Electrical & Computer Engineering, Faculty of Engineering,
Aristotle University of Thessaloniki, Building D, Egnatia Str., 54124 Thessaloniki, Greece
*Email: pnm@eng.auth.gr
Abstract: Corona discharge has many practical applications, thus it has been studied extensively
experimentally as well through modelling in many electrode arrangements. In the present study a
computational method for the estimation of the positive corona inception field strength in the
coaxial cylindrical electrode arrangement in air is presented. It is based on streamer theory and
involves Hartmann’s expression for the field dependent effective ionization coefficient and the
known distribution of the geometric electric field. A very good agreement with literature
experimental data referring to wire-cylinder air gaps has been observed for an avalanche number
of 104 (ionization integral ~9.2) in a wide range of wire radii and under variable atmospheric
conditions. A simple absolute humidity correction factor has been introduced in Peek’s formula,
allowing for an accurate estimation of the corona inception field strength under variable humidity.
1.
experimental data referring to wire-cylinder air gaps in a
wide range of wire radii and under variable atmospheric
conditions. Peek’s formula is modified to consider,
besides air density, absolute humidity variation.
INTRODUCTION
Corona discharge is of paramount importance in high
voltage technology as it has many practical applications
including electrostatic precipitators and printing, ozone
production, spray coating, biological and chemical
surface treatments and the treatment of flue gases. In
power transmission systems corona activity results in
power losses and interference in communication
systems. Moreover, surge voltages propagating on
transmission lines are attenuated and distorted by losses
due to corona discharges; these corona effects are
important in overvoltage protection and insulation
coordination of power systems. Therefore, the better
understanding of the physical processes involved in
corona discharges is important, besides fundamental,
from an engineering point of view.
2.
LITERATURE EXPERIMENTAL RESULTS
A great amount of experimental results on corona
inception voltage obtained in the coaxial cylindrical
electrode arrangement of wire-cylinder in air has been
reported in literature [1-12]. Most commonly,
estimation of the corona inception field strength at the
wire surface is achieved by using the Peek’s empirical
formulation [1], which can be written as:
⎛
B ⎞
Ei = A ⋅ ⎜1 + C ⎟
⎝ r0 ⎠
Because of its important effects, corona discharge has
been studied extensively experimentally as well through
modelling in many electrode arrangements. The coaxial
cylindrical geometry has drawn a great attention as,
besides simulating many practical applications, corona
develops symmetrically around the inner electrode and
the accurate knowledge of the geometric electric field
distribution between the electrodes allows for
computations for the estimation of the basic corona
characteristics. Thus, corona discharge in a coaxial
cylindrical arrangement has been systematically studied
since the beginnings of last century [1-18]. The Peek’s
formula [1], experimentally derived, is still commonly
used for the estimation of the corona inception voltage
in many electrode configurations. It is well established
that corona characteristics depend on the electric field
distribution between the electrodes and upon
atmospheric conditions.
(1)
where r0 is the wire radius and constants A, B and C are
given in Table 1, as experimentally derived [1-4, 6, 7, 9,
10] or through a theoretical approach [14, 18].
Table 1: Constants A, B and C to be used in (1).
Author
In the present study a computational method for the
estimation of the positive corona inception field strength
in the coaxial cylindrical electrode arrangement in air is
presented. A comparison is made with literature
Pg. 1
A
Constants
B
C
Peek [1] from [17]
31.53
0.305
0.5
Farwell [2]
35.00
0.241
0.5
Whitehead and Brown [3]
33.70
0.283
0.5
Whitehead and Lee [4]
39.80
0.212
0.5
Stockmeyer [6]
31.00
0.379
0.5
Zalesski [7]
24.50
0.613
0.4
Waters and Stark [9]
23.80
0.670
0.4
Hilgarth [10]
30.00
0.330
0.5
Hartmann [14]
25.94
0.127
0.4346
Lowke and D’Alessandro [18]
25.00
0.400
0.5
Paper B-10
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
Figure 1 shows the variation of the corona inception
field strength with wire radius at standard atmospheric
conditions, by including experimental data obtained
from literature and an application of (1) with constants
A, B and C according to Peek [1] as adapted in [17]
(Table 1). It is important to note that Peek’s formula
shows a very good agreement with experimental data in
a wide range of wire radii, even for radii smaller than
those employed in his investigations.
350
250
200
E (r ) =
100
50
0.01
0.1
1
10
Figure 1: Variation of corona inception field strength,
Ei, with wire radius, r0, in coaxial wire-cylinder air
gaps; standard atmospheric conditions.
In the coaxial cylindrical electrode arrangement of wirecylinder the spatial distribution of the electric field is
given as:
V
⎛R⎞
r ⋅ ln ⎜ ⎟
⎝ r0 ⎠
for r0 ≤ r ≤ R
(4).
(5).
The critical avalanche length can be derived from the
avalanche number, that is, the number of electrons in
the head of the critical avalanche, as:
THE COMPUTATIONAL METHOD
E (r ) =
Ei ⋅ r0
r
⎛ L ⎞
Ei = E0 ⋅ ⎜1 + a ⎟
r0 ⎠
⎝
r0 (cm)
3.
(3)
According to streamer theory, at threshold corona
inception conditions, near to the wire surface a suitably
positioned initiatory electron starts an avalanche, able to
attain a critical size (critical avalanche) and thus to
develop into streamer. The critical avalanche length, La,
is assumed to be the distance from the wire surface at
which the electric field strength falls to a critical value
E0, (critical field strength) such that the ionization
coefficient becomes zero (Figure 2). Hence, using (4)
the corona inception field strength at r = r0 + La is:
150
0
0.001
Vi
⎛R⎞
r0 ⋅ ln ⎜ ⎟
⎝ r0 ⎠
Thus, by combining (2) and (3), the spatial distribution
of the electric field at corona inception becomes:
Peek [1]
Farwell [2]
Whitehead and Brown [3]
Schumann [5]
Robinson [8]
Waters and Stark [9]
Yehia [11]
Peek's formula [1]
300
Ei (kV/cm)
Ei =
⎛ r0 + La
⎞
exp ⎜ ∫ λ1 (r )dr ⎟ = Q
⎜ r
⎟
⎝ 0
⎠
(2)
(6)
where λ1 (λ1 = α – η + ξ with α, η and ξ the first
ionization, attachment and detachment coefficients
respectively), is the field dependent effective ionization
coefficient according to Hartmann [14], given by the
following expression which takes into account, besides
air density, absolute humidity variation:
where V is the voltage stressing the wire and r0 and R
are the radii of the wire and external cylinder,
respectively (Figure 2).
⎧ ⎡
⎫
⎤
⎪ ⎢
⎪ [14]
⎥ ⎛⎜ − B⋅P0 ⎞⎟
C
⎪ ⎢
⎥ ⋅ e⎝ E ⎠ − O ⋅ Ψ ⎪⎬
= M ⋅ ⎨A⋅ 1+
3
P0
⎪ ⎢ N ⋅⎛ E ⎞ ⎥
⎪
⎜
⎟ ⎥
⎢
⎝ P0 ⎠ ⎦
⎩⎪ ⎣
⎭⎪
λ1
(7)
where:
3
4
12
A = 1.75 × 10 , B = 4 × 10 , C = 1.15 × 10 ,
M = 1 + 10−2 × H , N = 1 + 3.2 ⋅10−2 × H ,
O = 1 + 1.15 ⋅10−1 × H 0.1 , Ψ =
Figure 2: Critical avalanche formation in the coaxial
cylindrical electrode arrangement; positively stressed
inner electrode.
0.9
⎛ − P0 ⎞
⎜
⎟
1.49 + e⎝ 587 ⎠
o
P0 is the pressure in torrs at 0 C and H is the absolute
humidity in g/m3. It must be noted that according to IEC
60060-1:1989 [19] the standard atmospheric conditions
o
are: pressure 760 torrs, temperature 20 C and absolute
3
humidity 11 g/m .
At corona inception voltage, Vi, the electric field
strength at the wire surface, that is the corona inception
field strength, Ei, is given as:
Pg. 2
Paper B-10
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
By solving the system of equations (5)-(7) for a given
avalanche number, the corona inception field in wirecylinder air gaps can be computed as a function, besides
wire radius, of atmospheric conditions, as the latter are
taken into account in (7).
Computational method
350
300
Ei (kV/cm)
4.
Peek's formula [1]
400
COMPUTATIONAL RESULTS
250
200
150
Figure 3 shows the calculated corona inception field
strength at standard atmospheric conditions as a
function of wire radius; experimental data obtained
from literature are also shown. It is obvious that a very
good agreement with data is obtained for an avalanche
number of 104, as the corresponding calculated curve
fits accurately the experimental data in a wide range of
wire radii. Computed curves for avalanche numbers of
103 and 105 are also shown in this Figure; the effect on
corona inception field strength of the avalanche number
becomes less evident with increasing wire radius. It
must be noted that the avalanche number of 104 is used
hereafter in computations, where the effects on the
corona inception field strength of the atmospheric
conditions are demonstrated.
δ =1
50
0
0.001
0.01
δ = 0.7
0.1
1
10
r0 (cm)
Figure 4: Variation of corona inception field strength,
Ei, with wire radius, r0, in coaxial wire-cylinder air
gaps; relative air density, δ, as parameter.
(8) by using for constants A, B and C the values
suggested by Peek [1] (Table 1) are also included in this
Figure for comparison purposes. The computed corona
inception field strength compares favourably with that
derived from Peek’s formula, yielding a greater air
density effect with increasing wire radius.
Peek [1]
400
Figure 5 shows a comparison between calculated values
of Ei/δ and experimental data obtained from literature
both plotted as a function of the product δr0; an
excellent agreement is observed in a wide range of δr0.
Farwell [2]
Whitehead and Brown [3]
350
Schumann [5]
Robinson [8]
300
Waters and Stark [9]
Ei (kV/cm)
δ = 1.3
100
Yehia [11]
250
Q=10^5
200
Q=10^4
350
Q=10^3
Peek [1]
150
Farwell [2]
Whitehead and Brown [3]
300
Schumann [5]
100
0.01
0.1
1
Ei/δ (kV/cm)
50
0
0.001
Robinson [8]
250
10
r0 (cm)
Figure 3: Variation of corona inception field strength,
Ei, with wire radius, r0, in coaxial wire-cylinder air
gaps; standard atmospheric conditions.
⎞
⎟
⎟
⎠
P
293
⋅
760 273 + T
Computational method
150
50
0
0.001
0.01
0.1
1
10
δr0 (cm)
Figure 5: Variation of corona inception field strength,
Ei/δ, with δr0, in coaxial wire-cylinder air gaps.
It is well known that absolute humidity has only a minor
effect on corona inception field in the coaxial
cylindrical electrode arrangement. This was first
reported by Peek [1], therefore also no humidity
correction was included in his empirical formula. Figure
6 shows a comparison between the calculated corona
inception field and that measured in [12] as influenced
by absolute humidity; surface factors around 0.94 had to
be included in the computations to match the measured
corona inception field strength at standard atmospheric
conditions. The computed curves in Figure 6
satisfactorily fit data, although some discrepancy is
observed for the smallest wire radius of 0.005 cm.
(8)
where δ is the relative air density given as:
δ=
Yehia [11]
100
It is well known that the corona inception field increases
with air density. Actually, (1) is usually used in the
following modified form, taking into account the
variation of atmospheric pressure and temperature:
⎛
B
Ei = δ ⋅ A ⋅ ⎜1 +
⎜ ( r ⋅ δ )C
0
⎝
Waters and Stark [9]
200
(9)
where P is the pressure in torrs and T is the temperature
o
in C.
Figure 4 shows the calculated corona inception field
strength as a function of wire radius with relative air
density as parameter. Computed curves obtained from
The variation of corona inception field with absolute
humidity can be accurately estimated through a
Pg. 3
Paper B-10
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
160
Ei (kV/cm)
140
ro = 0.005 cm
ro = 0.0155 cm
ro = 0.04 cm
Hartmann’s expression for the effective ionization
coefficient [14], an excellent agreement with
experimental data from literature referring to wirecylinder air gaps has been observed for an avalanche
number of 104 in a wide range of wire radii (Figure 3)
and under variable air density (Figures 4 and 5). For the
same avalanche number, a very good agreement is also
observed with data referring to variable absolute
humidity [12], although some discrepancy has been
noted for the smallest wire radius (Figure 6). The
avalanche number of 104 (ionization integral ~9.2),
found from the present computations by assuming only
ionization by electron collision, negligible distortion of
the electric field by space charge and also neglecting
photoionization, is close to that quoted before in several
investigations on corona inception and breakdown in
quasi-uniform and non-uniform fields [18, 22-36].
However, it is generally accepted that for streamer
formation the critical avalanche number is about 108;
such a large difference in avalanche number could be
ascribed to streamer formation, occurring rather through
a multi-avalanche process, assisted by subsidiary
avalanches starting from electrons produced by
photoionization [15, 22, 23, 27, 29, 37, 38].
ro = 0.01 cm
ro = 0.02 cm
Computational method
120
100
80
60
0
2
4
6
8
10
12
14
16
18
20
22
24
26
3
H (g/m )
Figure 6: Variation of corona inception field with
humidity in coaxial wire-cylinder air gaps; δ = 1,
experimental data obtained from [12].
modified Peek’s formula (10), by using the simple
humidity correction factor as given in (11). This can be
deduced from Figure 7, where the calculated corona
inception field strength, Eic, according to the set of (10)
and (11) and by using surface factors around 0.9, is
plotted against the corresponding measured values, Eit,
obtained from [12], shown also in Figure 6; in Figure 7
En designates the corona inception field strength at
standard atmospheric conditions, derived from the
experimental data in [12].
⎛
0.305 ⎞
Εi = 31.53 ⋅ δ ⋅ kh ⋅ ⎜1 +
⎟
⎜
δ kh r0 ⎟⎠
⎝
kh = 1 +
where
Concerning the humidity effect on the corona inception
field, calculations according to (7) yield with increasing
humidity a slight increase of the critical field strength
but a nonmonotonical variation in the ionization
coefficient as the effect of humidity on the latter
depends upon the electric field strength (Figure 8). It
must be mentioned, as recognised by Hartmann [14],
that (7) has not been verified in a wide range of
humidities especially for field values greater than ~30
kV/cm. These might explain the discrepancy seen for
the smallest wire radius, associated with the highest
electric fields among radii shown in Figure 6.
(10)
0.11
⋅ ( H − 11)
100
(11)
1.015
y=x
27.0
ro = 0.005 cm
ro = 0.02 cm
ro = 0.04 cm
1.000
26.5
26.0
25.5
25.0
24.5
24.0
2
4
6
8
10
12
14
3
Absolute humidity (g/m )
-3
0 gm
1400
1200
1000
800
600
400
16
18
20
22
200
20
30
40
50
60
70
80
90
100
110
120
130
140
E (kV/cm)
Figure 8: Variation of critical field strength with
humidity and effective ionization coefficient with
humidity as parameter.
0.990
0.985
0.985
22 gm
1600
0
0
0.995
0.990
0.995
1.000
1.005
1.010
Peek’s empirical formula [1] satisfactorily yields the
corona inception field strength in the coaxial wirecylinder electrode arrangement in air under variable air
density and in a wide range of wire radii, even for radii
much smaller than those employed in his investigations
(Figure 1). Absolute humidity affects only slightly the
corona inception field in this arrangement; this has been
well established since Peek’s investigations [1] and an
attempt to predict this effect has been made in [12].
Nevertheless, the modified Peek’s formula (10),
involving the simple absolute humidity correction factor
(11), accurately predicts the variation of the corona
inception field strength with humidity in wire-cylinder
air gaps, observed in [12] and shown in Figure 7.
1.015
Eit/En (p.u)
Figure 7: Comparison between calculated from (10)
and measured [12] normalised corona inception fields
considering the effect of humidity.
5.
Effective ionization coefficient (cm-1)
Eic/En (p.u)
1.005
1800
-3
ro = 0.01 cm
ro = 0.0155 cm
Critical field strength (kV/cm).
1.010
DISCUSSION
The presented computational method for the estimation
of the corona inception field strength in the coaxial
cylindrical electrode arrangement in air assumes, based
on streamer theory [20, 21], that corona inception
occurs when an avalanche attaining a critical size
develops into a streamer. By employing in computations
Pg. 4
Paper B-10
ISBN 978-0-620-44584-9
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
[12] Y. Zebboudj and R. Ikene, “Positive corona
inception in HVDC configurations under variable
air density and humidity conditions”, The
European Physical Journal, Applied Physics, vol.
10, pp. 211-218, 2000.
[13] C. H. Gary, B. Hutzler and J. P. Schmitt, “Peek’s
law generalization. Application to various field
configurations”, IEEE Summer Meeting, Paper C
72-549-4, 1972.
[14] G. Hartmann, “Theoretical evaluation of Peek’s
Law”,
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[19] IEC 60060-1: “High-voltage test techniques, Part
1: General definitions and requirements”, 1989
[20] J. M. Meek, “A theory of spark discharge”,
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[21] H. Raether, Z. Phys. 117, p. 375, 1941.
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Finally, similar computations can be performed also for
other electrode arrangements simulating practical
applications, as for the estimation of the corona
inception field strength on stranded or coated
conductors [39-42]; work on this subject is in progress.
6.
CONCLUSIONS
A computational method for the estimation of the
positive corona inception field strength in the coaxial
cylindrical electrode arrangement in air, involving
Hartmann’s expression for the field dependent
ionization coefficient and the known distribution of the
geometric electric field, has been presented. A very
good agreement with experimental data from literature
referring to wire-cylinder air gaps has been observed for
an avalanche number of 104 in a wide range of wire
radii and under variable atmospheric conditions.
Peek’s empirical formula may satisfactorily yield the
corona inception field strength in coaxial wire-cylinder
air gaps under variable air density and in a wide range
of wire radii. A modified Peek’s formula, incorporating
a simple absolute humidity correction factor, has been
introduced, accurately predicting the effect of absolute
humidity on the corona inception field strength.
7.
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Paper B-10
ISBN 978-0-620-44584-9
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
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