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A general empirical formula of current–voltage characteristics for point-to-plane geometry

corona discharges

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2008 J. Phys. D: Appl. Phys. 41 065209

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 41 (2008) 065209 (10pp) doi:10.1088/0022-3727/41/6/065209

A general empirical formula of

current–voltage characteristics for
point-to-plane geometry corona
discharges
Xiangbo Meng, Hui Zhang and Jingxu (Jesse) Zhu
Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, ON,
N6A 5B8 Canada

Received 12 January 2008, in final form 8 February 2008

Published 29 February 2008
Online at stacks.iop.org/JPhysD/41/065209

Abstract
With a point-to-plane geometry, the experimental investigation of the current–voltage
characteristics in corona discharges demonstrated that existing empirical formulae met with
some physical difficulties in explaining the results. By mathematically processing the
experimental data and applying the updated knowledge of corona inception, a new general
formula in characterizing the relationship of corona current–voltage was derived and expressed
as I = K(V − V0 )n . It was demonstrated that the exponent n falls into a limited scope of
1.5–2.0, and there always exists an optimal exponent n in the scope, which can be determined
by maximizing the R-square of regression. Of all the potentially influential factors, it was
disclosed that the point radius has the strongest influence on the optimal exponent n, and the
effects of ambient conditions and corona polarities are not noticeable. The optimal exponent n
holds a fixed value of 2.0 for microscopic points and of 1.5 for large points with a radius in
millimetres, but changes decreasingly with the radius for the points of microns. For given
experimental conditions, the optimal exponent n almost does not change with the
inter-electrode distance. Furthermore, it was demonstrated that the formula is applicable not
only for both negative and positive coronas in point-to-plane geometries but also for both
polarities in point-to-ring geometries. With the optimal exponent n, the formula can well
explain the inconsistencies met by other existing formulae and best represent the
characteristics of corona current–voltage with an accuracy of 1 µm.
(Some figures in this article are in colour only in the electronic version)

1. Introduction though the physical mechanism of corona discharge is not

clear even up to the present. In 1914 Townsend derived a
Studies of corona discharges with point-to-plane electrode formula to characterize the dc steady corona current–voltage
geometry started at the end of the 19th century by Röntgen, as relationship for coaxial cylindrical geometry. Later it was
referred to by Ferreira et al [1]. Nowadays corona discharge empirically found that the Townsend relation could also be
is being involved in a number of commercial and industrial used approximately for point-to-plane geometry, as Henson
applications, such as manufacture of ozone, surface treatment, disclosed [8]. This formula is given as
photocopying, electrostatic precipitation, destruction of toxic
I = AV (V − V0 ), (R-I)
compounds and electrostatic powder coating [1–7]. All these
applications, however, require reduction of the production cost where I is the corona discharge current, V the supplied voltage,
and optimization of the configuration and operation conditions. V0 the corona inception voltage and A a dimensional constant
Several empirical formulae were suggested on describing depending on the inter-electrode distance, the needle electrode
the current–voltage characteristics in corona discharges, radius, the charge carrier mobility in the drift region and other

0022-3727/08/065209+10$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

geometrical factors.
With a point-grid electrode geometry, recently Yamada [9]
modified the Townsend relation by considering the influence
of ambient temperature and of inter-electrode distance in his A2 A/D
P
empirical formula, which is expressed as PC
CP
I = C1 (T − 132)S −2.8 V (V − C2 T −1 S 0.39 ), (R-I1 ) A1

where C1,2 are coefficients depending on the electrode V PG

geometry, T (K) the ambient temperature with a Kelvin scale
and S (mm) the inter-electrode distance. Figure 1. Schematic diagram of the measurement system.
Another kind of empirical relation referred to by Ferreira
[1] is as follows:
I = B(V − V0 )2 . (R-II)
Further, in 1980 Henson [8] theoretically developed a
mathematical model for microscopic point-to-plane coronas Steel Wire
in the steady-state regime, which was based on his work in
liquid helium and expressed as

I = (2π Kε/α)[F (δ/α)]−2 (V − V0 )2 , (R-II1 )

where δ is the minimum corona glow radius, α the distance

between the needle tip and the plane, K a dimensional constant
and F (δ/α) a polynomial function of δ/α.
In essence, the above-mentioned relations can be Figure 2. Schematic diagram of the employed needle (corona point).
categorized into two types: (R-I) and (R-II). By comparing
the linear dependences of I /V versus V and I 1/2 versus V , from the inconsistencies met by the formulae (R-I) and (R-II),
Ferreira et al [1] found that for negative corona discharges as discussed by Ferreira et al [1]. Based on the experimental
(R-II) performs better in all the inter-electrode ranges than data of this study for negative corona discharges with a point-
(R-I), and (R-I) may hold only when the inter-electrode to-plane geometry, and the knowledge of corona inception,
distance is greater than 15 mm; in contrast, for positive corona a new general formula is derived in this paper. Further
discharges (R-I) performs better in all the inter-electrode examinations of this proposed formula with available results in
ranges than (R-II), and (R-II) may hold at the inter-electrode earlier studies demonstrated that this formula could provide a
distance greater than 15 mm. However, Kip [10] asserted that better characterization of current–voltage in corona discharges
for positive corona discharges (R-II) fits his results rather than
with a very high accuracy in a wide range of applications.
(R-I). All these indicate the existence of some inconsistencies
Potentially this model will be a very important tool for more
in the applications of the formulae mentioned and that there
theoretical studies of the corona phenomenon.
were still no determinate answers.
Besides the above-mentioned efforts to improve the
effectiveness of current–voltage characteristics, another 2. Experiment
important feature with the above-discussed formulae is that
corona inception voltage is reflected in them. In earlier studies, The experimental setup in this study is illustrated in figure 1.
Ferreira et al [1], Yamada [9] and Giubbilini [11] determined This system consists of a needle P (corona point), a copper
the corona inception voltages V0 by extrapolating the fitting plane CP (corona plane), a negative dc high-voltage supply
straight lines to the voltage axis. And the extrapolated V (Sure Coat high-voltage supply, Nordson Corporation,
voltage physically should always locate on the right side of the USA) connected to the needle, two ammeters A1 and A2, an
ordinate and has the same polarity as the corona. Henson [8] exterior pulse generator PG, an A/D board (Lab-PC-1200) for
discussed that mathematically extrapolating corona data to converting the analogous current signals to digital current data
zero current allows one to empirically determine the corona and a PC to store the digital current data.
inception voltage V0 , though it is not physically meaningful to In earlier studies, Loeb et al [12] revealed that the
extrapolate the corona formulae to zero current and in general materials (brass, copper aluminum, iron, and various alloys,
the current must be greater than some critical value for the etc) of points have no measurable difference in corona
formulae to be valid. inception potential or current. It has been disclosed that
In the applications of the above-mentioned formulae to the the electric field at the tip of a corona point depends on
experimental results of this study, it was found that they met point sharpness, i.e. radius of curvature, but for point radii
with some difficulties in providing valid extrapolated corona less than 50 µm no further current gain occurs as electrode
inception voltages in some cases. And this is one of the strong sharpness is increased, because the corona discharge becomes
motivations for exploring a better relation for the current– so conducting that it shields the electrode tip [13]. In this study,
voltage characteristics in this paper. Another motivation comes the needle (shown in figure 2) is made of steel with a curvature

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J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

(a)
Figure 3. Negative corona current–voltage characteristics.

radius ≈50 µm (determined with a Hitachi Model S-2600N

SEM), and the plane is a blank printed circuit board, which is
covered with a layer of 0.2 mm thick copper and with a diameter
of 300 mm. In this measurement, a pulse generator is employed
to produce two exterior pulses simultaneously: one is sent to
turn on the high-voltage supply V , and another externally
triggers the LabVIEW program to collect the experimental data
through the A/D board. The occurrence of the corona and the
data collection thereby can be realized at the same instant.
In addition, the duration of high-voltage supply is adjustable
from 5 to 100 s.
As for the ammeters (A1 and A2), A1 is a built-in ammeter
with the commercial Nordson high-voltage supply and used to
monitor the total current flowing in the needle through its LCD (b)
monitor; A2 is a Model 6514 Keithley electrometer (Keithley
Instruments Inc., USA) and used to measure the current passing
through the plane. It was found that the currents from Al (I1 )
and A2 (I2 ) become different when the inter-electrode distance
S is higher than 150 mm, and I1 is larger than I2 , resulting
from the plane being not big enough to collect the total corona
current at larger inter-electrode distances. To explore the
relationship between the total corona current and the supplied
high-voltage, the current (I1 ) flowing in the needle is used
in this study. The supplied high-voltage locates between 25
and 90 kV with a 5 kV increment in most cases. The inter-
electrode distance varies from 30 to 600 mm with a 50 mm
increment mostly. All the measurements are made in air at
room temperature (23 ◦ C) (T ), 760 mm Hg of atmospheric
pressure (P ) and at 60% of relative humidity (RH).

3. The empirical formula (c)

3.1. The difficulties of existing formulae Figure 4. Testing of three empirical formulae with the experimental
data: (a) I /V as a function of the voltage V by applying (R-I) to the
The experimental data of this study are summarized in figure 3. experimental data; (b) I /((T − 132)S −2.8 V) as a function of the
Among the mentioned formulae, three ((R-I), (R-I1 ) and (R-II)) voltage V by applying (R-I1 ) to the experimental data; (c) I 1/2 as a
function of the voltage V by applying (R-II) to the experimental
are examined with the data and illustrated in figures 4(a)–(c), data. The solid lines are obtained by least squares fittings.
respectively. ((R-II1 ) is not tested particularly due to our

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J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

Figure 5. The corona current (I : µA) as a function of the applied

voltage (V : kV) on a log scale for several inter-electrode distances. Figure 6. The dependence of m on the inter-electrode distance
The solid lines are least squares fitting lines. (S: mm). The solid line is obtained by least squares fitting.

inability to fit the data with Henson’s detailed expression. As

a matter of fact, however, Henson [8] validated (R-II1 ) by
directly observing the linear regressions of the dependence
of I 1/2 versus V . Hence, the examination of (R-II1 ) with
the data of this study has been covered by figure 4(c). A
common difficulty met by the tested formulae is that, in
some cases (S < 150 mm), no physically valid inception
voltages can be obtained by extrapolating the straight line to
the V -axis. In those cases, the extrapolated inception voltages
are ‘negative’ in the V -axis, and these formulae lose their
physical meaning. Another obvious phenomenon is that the
dependence of I /V versus V in figures 4(a) and (b) presents
no good linear behaviours but downward curvatures while (R-I)
and (R-I1 ) are examined. By comparison, the dependence
of I 1/2 versus V in figure 4(c) shows a better linear fitting
behaviour while (R-II) is examined. As discussed in section I,
Ferreira et al [1] disclosed the inconsistencies between (R-I) Figure 7. The dependence of the dimensional constant K1 on the
and (R-II) in negative and positive coronas. Upward and inter-electrode distance (S: mm).
downward tendencies of current–voltage characteristics were
also observed by Ferreira et al while the formulae could not unknown. Theoretically and physically, the meaningful scopes
behave well in fitting the experimental results. Obviously, in of the current and voltage fall into the ranges of [I0 , +∞) and
the application of the mentioned formulae to the experimental [V0 , +∞), respectively. Through least squares fittings, on a log
data, they all met with some difficulties. scale a linear behaviour for the dependence of I versus V is
obtained in the entire voltage range and shown in figure 5. As
a result, the relationship of current–voltage can be generally
3.2. Derivation of the new empirical formula
expressed as
The new empirical formula derived in this paper is based on
log10 (I ) = m log10 (V ) + a (1)
the experimental data of current–voltage and the knowledge
of corona inception. All the measurements were carried out in or I = 10 V .
a m
(2)
the regular Trichel pulse regime [14] of a negative corona in a
Replacing 10 with K1 ,
which the current is becoming stable.
I = K1 V m . (3)
3.2.1. The relationship between the current (I) and the applied The parameter m is determined by the slopes of the fitting
voltage(V). As the starting point of the deduction of the new lines and presents a linear dependence on the inter-electrode
formula, logarithm functions of the voltage (log10 (I )) and the distance S (illustrated in figure 6). As a dimensional constant,
current (log10 (V )) are set up respectively, due to logarithms K1 presents a non-linear descending tendency with the distance
being useful in solving equations in which exponents are (illustrated in figure 7).

4
J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

3.2.2. The estimation of the corona inception voltage V0 . The

corona inception voltage V0 represents the starting point of
corona discharges, at which a corona discharge happens when
the supplied voltage exceeds a threshold. It is believed [8]
that V0 is essentially independent of the current and applied
voltage, and that V0 has a strong dependence on the tip (corona
point) radius r and the inter-electrode distance S. In the
literature [15, 16], experimental and theoretical efforts were
made to predict V0 , but these are of little help in obtaining its
practical value under different conditions.
However, earlier studies [17–19] disclosed that, for a
point-to-plane geometry corona and relatively small points,
pre-onset currents remain at about 10−7 µA and the currents
change abruptly to the order of 0.1 µA with the appearance of
Trichel pulses after the voltage exceeds a threshold. Thereafter,
the current increases as the supplied voltage to the point
increases [14]. It is further explained that the inception currents
(no matter positive or negative points) from Trichel pulse
inception up to the steady corona at about 0.1 µA are only Figure 8. The influence of the preset corona inception current (I0 :
mean values [18], for the current is continually fluctuating, µA) on the estimated corona inception voltage (V0 : kV). The solid
lines are least squares fitting lines.
owing to the pulsed and intermittent nature of the discharge.
English [20] also pointed out that, due to the more or less abrupt
negative corona inception, it is very difficult to determine the V − V0 are set up. Through least squares fittings, a linear
exact inception voltage. In a systematic study of the electrical dependence of I versus V − V0 on a log scale is observed, no
characteristics of Trichel pulses, Lama and Gallo [15] referred matter what preset corona inception current is adopted. In the
the current at corona threshold on the order of a microampere cases of I0 = 0.1, 0.5 and 1.0 µA, the results are exemplified
(1 µA). In addition, Akishev et al [21] set the corona inception in figures 9(a)–(c), respectively. Generally, the dependence of
current at 1 µA to separate various discharge modes and Cross I versus V − V0 can be expressed as
[22] also referred the corona inception current as about 1 µA.
log10 (I ) = n log10 (V − V0 ) + b (4)
In spite of these discrepancies in the corona inception currents,
it can still be concluded that the corona inception current falls or I = 10b (V − V0 )n . (5)
into the range of 0.1–1 µA.
Based on the above discussion, it is reasonable to Replacing 10b with K2 ,
recognize the voltage corresponding to the initial current of
I = K2 (V − V0 )n . (6)
the steady corona as the corona inception voltage V0 . With
the scope of the corona inception current determined, the For a given inter-electrode distance, the parameter n can be
potential scope of the corona inception voltage could be determined by the slope of the fitting line. The influence of
determined by using equation (3). In this study, ten preset the preset corona inception current I0 on the values of the
corona inception currents (from 0.1 to 1 µA with a 0.1 µA exponent n along the distance S is shown in figure 10. The
increment) are chosen to estimate their corona inception value of the exponent n holds only one digit after the decimal
voltages and to examine their influence on the estimated to facilitate the analysis. Obviously, for a higher preset corona
inception voltages. Substituting ten preset corona inception inception current (e.g. I0 = 1 µA), the exponent n holds as a
currents into equation (3) respectively, the corresponding fixed value 1.5; but for a lower preset corona inception current
estimated corona inception voltages are determined in the (e.g. I0 = 0.1 µA), the exponent n changes in shorter distances
whole range of distances. Figure 8 exemplified the influence but remains at 2.0 since 200 mm. In conclusion, it can be
of the preset corona inception current I0 on the estimated summarized that the potential scope for the exponent n falls
corona inception voltage V0 . The higher the preset current into a range of 1.5–2.0.
is, the higher the resulting inception voltage is. For a given To discriminate the influence of the exponent n on the
preset current, however, the resulting corona inception voltage linear regressions, R-square is introduced into this study.
follows a non-linear trend with the inter-electrode distance. R-square (also called as the coefficient of determination
(COD)) approaches unity as a regression presents a perfect
3.2.3. The relationship between the current (I ) and the voltage- fitting. As discussed above, in the deduction of the relationship
difference (V −V0 ). Based on the preceding work, it becomes of log10 (I )–log10 (V −V0 ), ten preset corona inception currents
possible to explore the relationship of the corona current I with have been applied, then ten resulting exponents n can be
the voltage-difference V − V0 . Physically, the meaningful obtained. In figure 11, it is shown that there exists an
scope of the current and voltage-difference fall into the ranges optimal value n for each inter-electrode distance, at which
of (I0 , +∞) and (0, +∞), respectively. Again, the logarithm the regression lines reach the best fitting. Any discrepancies
functions of the corona current I and the voltage-difference (larger or less) from the optimal value of n result in the

5
J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

(a)
Figure 10. The influence of preset corona inception currents (I0 :
µA) on the values of the exponent n with the inter-electrode
distance (S: mm).

(b)

Figure 11. The influence of the exponent n on the values of COD

under different inter-electrode distances (S: mm).

deterioration of linear regressions. Furthermore, it is found

that the optimal n holds almost as a fixed value (illustrated in
figure 12) with the inter-electrode distance, which is around
1.6. That may indicate that equation (6) can best describe the
relationship of I versus (V − V0 ) while the exponent n takes a
value of 1.6 in this study. However, the optimal exponent n may
change with experimental conditions, such as relative humidity
(RH), air pressure (P ), temperature (T ) and corona point radius
(r), due to the dependence of V0 on these parameters.

(c) 3.2.4. The proposed empirical formula. In the above

discussion, the relationship of the corona current with the
Figure 9. The dependence of the corona current (I : µA) on the
voltage-difference is deduced, as well as the relationship of
voltage-difference (V − V0 : kV) on a log scale: (a) the preset corona
inception current I0 = 0.1 µA; (b) the preset corona inception current–voltage. As an inherent characteristic for the corona
current I0 = 0.5 µA; (c) the preset corona inception current discharge, the corona inception voltage should always be
I0 = 1.0 µA. The solid lines are least squares fitting lines. reflected in a successful formula. In this case, equation (6)
is directly taken as the choice in developing the empirical
formula. Rewriting equation (6) and replacing K2 with K,

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J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

Figure 12. The change of the optimal exponent n along the Figure 14. The dependence of the corona inception voltage
inter-electrode distance (S: mm). The solid line is obtained by least (V0 : kV) on the inter-electrode distance (S: mm): V0 ∝ S 0.51 . The
squares fitting. solid line is obtained by least squares fitting.

where C1,2 are coefficients depending on the electrode

geometry and ambient conditions, which are equal to
0.525 µA mm0.41 kV−1.6 and 0.759 kV mm−0.51 in this study,
respectively.

4. Discussion

4.1. The effect of the exponent n on the characteristics of

current–voltage
As demonstrated in figure 11, there exists an optimal value in
the range 1.5–2.0, which presents the best linear dependence
of I versus V − V0 on a log scale. For this study, the optimal
value of n is around 1.6. To further examine the effect of
n on the linear dependence of I 1/n versus V , figures 15(a)
and (b) exemplified the cases at inter-electrode distances 200
and 400 mm. It is found that a higher value of n (e.g. 2.0)
Figure 13. The dependence of the dimensional constant K on the results in a downward curvature for the dependence of I 1/n
inter-electrode distance (S: mm): K ∝ S −0.41 . The solid line is versus V ; in contrast, a lower value of n (e.g. 1.5) leads to
obtained by least squares fitting. an upward curvature (see figures 15(a) and (b)). In addition,
the effect of the exponent n on the inception voltage V0 is
the proposed formula is generally stated as that an over-estimated exponent n (e.g. 2.0) leads to a smaller
extrapolated V0 and an under-estimated one (e.g. 1.5) produces
I = K(V − V0 )n , (F-I) a larger extrapolated V0 . As a result, in some cases an over-
estimated exponent n may lead to a physically invalid V0 by
where n falls into the range 1.5–2.0 and takes a value of 1.6 for extrapolation. This well explains the reason why no valid V0
this study, V0 the corona inception voltage and K a dimensional were obtained while (R-I) and (R-II) were applied to the data of
constant, which both depend on the inter-electrode distance, this study for the inter-electrode distance of less than 150 mm
the needle electrode radius and ambient conditions (including (see figures 4(a)–(c)).
relative humidity (RH), temperature (T ) and air pressure (P )). With the optimal exponent n (1.6) of the proposed formula
While n is equal to 1.6, on a log scale K presents a (F-I) determined for this study, the experimental results are
linear behaviour with S and is proportional to S −0.41 , which tested and shown in figure 16. Obviously, the dependence
is illustrated in figure 13. Similarly, V0 is proportional to S 0.51 I 1/1.6 versus V presents good linear behaviours and physically
and shown in figure 14. Finally, a more specific form for this valid inception voltages V0 can be obtained by extrapolation.
study is expressed as follows: Further, by comparing the corona currents measured with the
ones calculated by (F-I) (n = 1.6) in this study, it is found that
I = C1 S −0.41 (V − C2 S 0.51 )1.6 , (F-I1 ) the accuracies are more than 99%. However, if (R-I) and (R-II)

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J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

(a)
Figure 16. The dependence of I 1/1.6 versus V with the experimental
data of this study. The solid lines are least squares fitting lines.

to coronas from larger points, no further evidence provides

support. Later, Ferreira et al [1] discussed the inconsistencies
of (R-I) and (R-II) and suggested (R-II) for negative coronas
and (R-I) for positive coronas. However, Kip [10] claimed
that for positive corona discharges it is (R-II) not (R-I) that
fits his results. Furthermore, Giubbilini [11] disclosed that
there is an analogy between point-to-ring and point-to-plane
coronas by applying (R-I). More recently, Allibone et al [23]
investigated positive coronas from large tips and suggested
(R-I) for the current–voltage characteristics. It might be
neglected in earlier studies that behind the inconsistencies are
different experimental conditions.
With the results in earlier studies [1, 8, 11, 23], the
(b) effectiveness of the formula (F-I) is further examined in order
to clarify the effect of experimental conditions. The optimal
Figure 15. The influence of the exponents n of the formula (F-I) on
the dependence of I 1/n versus V , and the extrapolated inception exponents n were determined by maximizing the COD value
voltages V0 : (a) S = 200 mm; (b) S = 400 mm. The solid lines are of I 1/n versus V and summarized in table 1. To exemplify
least squares fitting lines. the determination of the optimal exponent n, figures 17 and 18
illustrate the processes for the results of Ferreira et al [1] and
are applied to the same data, regardless of the invalid corona Allibone et al [23], respectively. Obviously, there always
inception voltages, the accuracies decrease to around 96%. exists an optimal n in the range 1.5–2.0, with which the COD
value for the dependence of I 1/n versus V is maximized and
4.2. The effects of experimental conditions on the exponent n mostly higher than the one for the correlation of I /V versus
of the formula (F-I) V (as shown by the dashed lines).
From table 1, it can be learned that, of all the potentially
For the new proposed formula (F-I), the exponent n is crucial influential factors, the influence of the point radius on the
for describing the actually accurate characteristics of current– optimal exponent n is especially evident. For microscopic
voltage in coronas and it is essential to further explore point-to-plane coronas, the point radius is smaller than 0.2 µm
the influence of experimental conditions on the exponent and usually the experiments run in liquid nitrogen and its
n. In earlier studies, in spite of extensive investigation on vapour, or in helium vapour to protect the tip from erosion [8].
coronas, there have been inconsistencies in describing the As a result, the formula (F-I) has the highest exponent 2.0,
current–voltage characteristics. Henson [8] developed the which holds as a fixed value for microscopic points. On the
mathematical model (R-II1 ) for microscopic point-to-plane other side, for coronas from large rods whose radii are at an
coronas and successfully validated it with the experimental order of magnitude of millimetres, as did by Allibone et al [23],
results obtained in helium vapour by observing the correlations the formula (F-I) receives the lowest exponent 1.5, which
of I 1/2 versus V better than 99.9%. As a matter of fact, Henson seems not to change any more with the increasing point radius.
described the case of (F-I) with an exponent 2.0. Whereas For the point with a radius of microns, as used in some earlier
Henson simply postulated that (R-II1 ) should also be applicable studies [1, 11] and this study, however, the optimal exponent n

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J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

Table 1. Effect of experimental conditions on the optimal exponent n of the proposed formula (F-I).

Inter-electrode The optimal

Geometry Coronas Ambient conditions distance Point radius exponent n
Negative Helium vapour Microscopic points: 2.0
Henson [8] Point-to-plane T = -239.1 ºC 4 – 10 mm 200 – 2000 Å (or
Positive P = 760 mm Hg 0.02 – 0.2 µm) 2.0

Negative Air 1.9

Ferreira et al [1] Point-to-plane T = 26 ºC 4 – 30 mm 10 and 15 µm
P = 760 mm Hg
Positive 1.8
RH = 60%
Air
T = 21.5 ºC
Negative 1.6
P = 1016 mbar
Giubbilini [11] Point-to-ring RH = 61% 0 – 12 mm 45 µm
Air
T = 19 ºC
Positive 1.6
P = 1012 mbar
RH = 60%
Air
Meng et al in this T = 23 ºC
Point-to-plane Negative 30 – 600 mm 50 µm 1.6
paper P = 760 mm Hg
RH = 60%
Air
6.35, 12.7, 19.05
Allibone et al [23] Point-to-plane Positive Other conditions not 1m 1.5
and 25.4 mm
provided

Figure 17. The effect of the exponent n of the formula (F-I) on the Figure 18. The effect of the exponent n of the formula (F-I) on the
COD value with the results in figure 2 (negative corona) and figure 5 COD value with the results in figure 1 (positive corona) of [23].
(positive corona) of [1]. The solid lines are obtained by least squares
fitting. inter-electrode distance on the optimal exponent n is negligible
and the formula (F-I) is applicable for the inter-electrode
distance ranging from 0 to 1 m. Furthermore, the effects of
of the formula (F-I) presents a descending tendency with the ambient conditions and corona polarity are not so noticeable.
point radius in the range of 1.5–2.0, and similar radii result
in similar exponents n (as shown in table 1). It is believed 5. Conclusions
that the effects of point radii on the optimal n result from the
strong dependence of the inception voltage V0 on point radii. With a point-to-plane geometry, experimental investigation
In addition, the diversities of the used tips in actual shapes seem of negative corona discharges demonstrates that all existing
no influence on the corona current–voltage characteristics, as empirical formulae met with some difficulties in describing the
referred by Henson [8]. From table 1, it is further confirmed current–voltage characteristics with the results of this study.
that for certain experimental conditions the influence of the Based on the experimental data and the knowledge of corona

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J. Phys. D: Appl. Phys. 41 (2008) 065209 X Meng et al

inception, a new general formula was developed in this paper to of coating polymer powders on their minimum ignition
uncover the phenomena. This formula suggests that the corona energies J. Loss Prevention Process Indust. 17 59–63
[4] Goldman M, Goldman A and Sigmond R S 1985 The corona
current can be expressed as a function of the voltage-difference
discharge, its properties and specific uses Pure Appl. Chem.
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with a power n. The scope of n is deduced in a range 1.5–2.0 [5] Dascalescu L, Sanuila A, Rafiroiu D, Iuga A and Morar R
and further confirmed with the results in earlier studies. The 1999 Multiple-needle corona electrodes for electrostatic
derivation and testing of the formula (F-I) suggest that there processes application IEEE Trans. Indust. Appl. 35 543–7
[6] Bhattacharyya S and Peterson A 2002 Corona
always exists an optimal exponent n in the range 1.5–2.0 for
wind-augmented natural convection: I. Single electrode
certain experimental conditions, which implies the best fittings studies J. Enhanced Heat Transfer 9 209–19
and can be determined by maximizing the value of R-square [7] Chang J-S, Lawless P A and Yamamoto T 1991 Corona
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the exponent n has a strong dependence on the point radius: [8] Henson B L 1981 A space-charge region model for
microscopic steady coronas from points J. Appl. Phys.
for microscopic points the exponent n is fixed as 2.0, for large
52 709–15
points with a radius of millimetres it holds as 1.5, and the [9] Yamada K 2004 An empirical formula for negative corona
exponent n decreases with the radius for points of micrometres. discharge current in point-grid electrode geometry J. Appl.
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polarities are not so noticeable. For this study, the optimal [10] Kip A F 1938 Positive-point-to-plane discharge in air at
atmospheric pressure Phys. Rev. 54 139–46
exponent n is determined around 1.6. As a result, the formula
[11] Giubbilini P 1988 The current-voltage characteristics of
(F-I) provides a very high accuracy on current prediction while point-ring corona J. Appl. Phys. 64 3730–2
the inter-electrode distance is within one metre, and the errors [12] Loeb L B, Kip A F, Hudson G G and Bennett W H 1941 Pulses
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45 85–120
new formula successfully explained the inconsistencies met [14] Trichel G W 1938 The mechanism of the negative point to
by other formulae, resulting from incorrectly weighing the plane corona near onset Phys. Rev. 54 1078–84
influence of voltage-difference on corona currents at given [15] Lama W L and Gallo C F 1974 Systematic study of the
experimental conditions. Consequently, its significance exists electrical characteristics of the ‘Trichel’ current pulses
from negative needle-to-plane coronas J. Appl. Phys.
not only in the practice but also in the scientific interest of
45 103–13
corona discharges. Potentially this formula may provide a [16] Parekh H, Salama M M A and Srivastava K D 1978
clue for more sophisticated studies of corona phenomena. Calculation of corona onset and breakdown voltage in short
point-to-plane air gaps J. Appl. Phys. 49 107–12
[17] English W N and Loeb L B 1949 Point-to-plane corona onsets
Acknowledgments J. Appl. Phys. 20 707–11
[18] Bandel H W 1951 Point-to-plane corona in dry air Phys. Rev.
The authors would like to thank Mr Xianzhong Zhu and 84 92–9
Mr Jianzhang Wen for their help and advice on the design [19] Weissler G L 1943 Positive and negative point-to-plane corona
of the experimental setup. in pure and impure hydrogen, nitrogen, and argon Phys.
Rev. 63 96–107
[20] English W N 1948 Positive and negative point-to-plane corona
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