Current-voltage characteristics of dc corona

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Yuesheng Zheng,1,a) Bo Zhang,2,b) and Jinliang He2,c). 1College of Electrical Engineering and Automation, Fuzhou University, Fuzhou 350108, Fujian ...
Current-voltage characteristics of dc corona discharges in air between coaxial cylinders Yuesheng Zheng, Bo Zhang, and Jinliang He Citation: Physics of Plasmas (1994-present) 22, 023501 (2015); doi: 10.1063/1.4907234 View online: http://dx.doi.org/10.1063/1.4907234 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/22/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spatio-temporal characteristics of self-pulse in hollow cathode discharge Phys. Plasmas 22, 022106 (2015); 10.1063/1.4907236 Note: Measuring breakdown characteristics during the hot re-ignition of high intensity discharge lamps using high frequency alternating current voltage Rev. Sci. Instrum. 84, 046103 (2013); 10.1063/1.4801850 Radio frequency current-voltage probe for impedance and power measurements in multi-frequency unmatched loads Rev. Sci. Instrum. 84, 015001 (2013); 10.1063/1.4773540 Glow-to-arc transition events in H2-Ar direct current pulsed plasma: Automated measurement of current and voltage Rev. Sci. Instrum. 83, 015112 (2012); 10.1063/1.3678589 Experimental design for the determination of the injection barrier height at metal/organic interfaces using temperature dependent current-voltage measurements Rev. Sci. Instrum. 80, 033901 (2009); 10.1063/1.3090883

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PHYSICS OF PLASMAS 22, 023501 (2015)

Current-voltage characteristics of dc corona discharges in air between coaxial cylinders Yuesheng Zheng,1,a) Bo Zhang,2,b) and Jinliang He2,c) 1

College of Electrical Engineering and Automation, Fuzhou University, Fuzhou 350108, Fujian Province, China 2 Department of Electrical Engineering, Tsinghua University, Beijing 100084, China

(Received 9 January 2015; accepted 20 January 2015; published online 2 February 2015) This paper presents the experimental measurement and numerical analysis of the current-voltage characteristics of dc corona discharges in air between coaxial cylinders. The current-voltage characteristics for both positive and negative corona discharges were measured within a specially designed corona cage. Then the measured results were fitted by different empirical formulae and analyzed by the fluid model. The current-voltage characteristics between coaxial cylinders can be expressed as I ¼ C(U  U0)m, where m is within the range 1.5–2.0, which is similar to the pointplane electrode system. The ionization region has no significant effect on the current-voltage characteristic under a low corona current, while it will affect the distribution for the negative corona under a high corona current. The surface onset fields and ion mobilities were emphatically disC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907234] cussed. V

I. INTRODUCTION

High voltage dc transmission systems are competent for long distance and high capacity power transmission, especially at extra-high voltages. With the increase of the voltage level, corona discharges on high voltage dc overhead transmission lines can cause electromagnetic environment problems, such as radio interference and audible noise.1,2 Different from the alternating corona discharges, the dc corona discharges also generate the ion flow in the whole space and enhance the electric field at ground.3 Numerical simulations were efficient for predicting the ion current density and electric field intensity at ground induced by the corona discharges on high voltage dc overhead transmission lines.4–7 In these models, only the drift ions were considered and the ionization region was usually ignored. The boundary condition on the conductor surface was always replaced by Kaptzov’s assumption or empirical formulae deduced from experiments. Under the applied voltage above the corona onset level, the distributions of the ion current density and electric field intensity in space and at ground can be calculated. Mobility is defined as the proportionality coefficient between the drift velocity of a charged particle and the electric field.8 It is always assumed that the ion mobility in air is constant, independent of field intensity. The positive and negative ion mobilities are the basic parameters in the numerical model which can affect the ion flow distribution around the high voltage dc overhead transmission lines. However, the values of ion mobilities in air used by different researchers are much different for a wide range.9 The configuration of coaxial cylinders is one of the typical electrode systems used to investigate corona a)

Electronic mail: [email protected] Electronic mail: [email protected] c) Electronic mail: [email protected] b)

1070-664X/2015/22(2)/023501/6/$30.00

discharges.10–14 Taking advantage of the axial symmetry of the electrode system, the numerical calculation can be simplified. Based on Kaptzov’s assumption which says that the surface field keeps constant when the voltage is above the onset level, an analytic expression for the current-voltage characteristic can be deduced.8 However, the surface onset fields by different researchers are also much different for a wide range.15–17 Although the phenomena of corona discharges are very complex, the dc steady corona current-voltage relationship between coaxial cylinders can be characterized by a simple formula, namely, Townsend relation.8 The formula is also valid for the point-plane electrode system,18,19 which is given as I ¼ CUðU  U0 Þ;

(1)

where I is the corona current, U is the applied voltage, U0 is the corona onset voltage, and C is a constant. Ferreira et al.20 found that Townsend relation was only valid for large gap distances in the point-plane electrode system and a relation for small distances was proposed, which was expressed as I ¼ CðU  U0 Þ2 :

(2)

21

Meng et al. suggested another kind of relation for the point-plane electrode system, which was expressed as I ¼ CðU  U0 Þm ;

(3)

where m is a constant that falls into a limited scope of 1.5–2.0. However, Townsend relation for the electrode system of coaxial cylinders is limited to the low corona current. With the increase of the corona current, the relationship is ambiguous. In this paper, a corona cage was specially designed to measure the current-voltage characteristics of dc corona discharges in air between coaxial cylinders. Then, the measured

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C 2015 AIP Publishing LLC V

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results were fitted by Eqs. (1)–(3) and analyzed by the fluid model. The surface onset fields and ion mobilities were emphatically discussed. II. MEASUREMENTS OF CURRENT-VOLTAGE CHARACTERISTICS A. Experimental setup

The schematic diagram of the experimental setup is shown in Fig. 1. The material for the U-type supporter was high density polyethylene and that for the corona cage and equalizing terminals was stainless steel. The corona cage consisted of three sections, which were all grounded. The equalizing terminals were specially designed for reducing the torque on the test conductor and the guard sections were specially designed for equalizing the electric field distribution within the measurement section. The distributions of electric potential and electric field within the corona cage by two-dimensional finite element analysis are shown in Fig. 2. It can be found that the electric potential and the electric field distributions within the measurement section were both uniform. The test conductor was a copper wire, which was polished by waterproof abrasive paper in axial direction and cleaned by industrial alcohol before testing. The pretreatment of the test conductor was done subject to the corona discharges at the voltage of 35 kV for about 1 min before measurement. The voltage was increased step by step after the current-voltage values were stable and the results were recorded every step. The conductor radius (r0) was 0.070 cm, the inner radius of the measurement section (R) was 10.35 cm, and the length of the measurement section (L) was 36.2 cm. All the measurements were carried out in air at room temperature (T) of 297.72 K, atmospheric pressure (p) of 101 kPa, and relative humidity (RH) of 65.62%. B. Experimental results

The measured data of current-voltage characteristics are shown in Fig. 3. When the applied voltage is low, the corona currents for positive and negative polarities are almost the same. With the increase of the applied voltage, the corona current for the negative corona is higher than that for the positive corona under the same voltage. The difference is significant when the corona current is relatively high. The maximum relative difference is larger than 7%.

FIG. 1. Schematic diagram of the experimental setup.

FIG. 2. Distributions of the electric potential (top) and the electric field (bottom) for the corona cage. The dotted box indicates the measurement section.

III. ANALYSIS OF EXPERIMENTAL RESULTS A. Linear fitting

Equation (1) can be rewritten as I=U ¼ CðU  U0 Þ:

(4)

It can be found that there is a linear relationship between I/U and U. Linear least square method was used to fit the experimental data. The measured data and the fitted data by applying Eq. (4) are compared in Fig. 4. The dependency of I/U verse U presents no good linear behaviours but upward curvatures for both positive and negative corona discharges. The extrapolated corona onset voltages are lower than the actual values. Equation (2) can be rewritten as I1=2 ¼ C1=2 ðU  U0 Þ:

(5)

It can be found that there is a linear relationship between I1/2 and U. The same linear least square method was used to fit the experimental data. The measured data and the fitted data by applying Eq. (5) are compared in Fig. 5. The dependency of I1/2 verse U presents no good linear behaviours but downward curvatures for both positive and negative corona discharges. The extrapolated corona onset voltages are higher than the actual values. Equation (3) can be rewritten as log10 I ¼ log10 C þ m log10 ðU  U0 Þ:

(6)

It can be found that there is a linear relationship between log10 I and log10 ðU  U0 Þ. If the corona onset voltages are

FIG. 3. Measured data of current-voltage characteristics.

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FIG. 4. Comparison of the measured data and the fitted data by applying Eq. (4). The solid lines are least squares fitting lines.

given, the linear regressions can be carried out. Here, the coefficient of determination (COD) also called R-square is introduced to decide the corona onset voltages, which approaches unity as a regression presents a perfect fitting. The values of COD under different corona onset voltages by applying Eq. (6) are shown in Fig. 6. It can be found that the fitting results are the best when the values of COD reach the peaks for both the positive and negative polarities. Under the maximum values of COD, the measured data and the fitted data by applying Eq. (6) are compared in Fig. 7. The onset voltages 22.1 kV and 21.6 kV are for the positive corona and negative corona, respectively. It can be found that the dependency of log10 I verse log10 ðU  U0 Þ presents good linear behaviours for both positive and negative corona discharges. The results are better than Figs. 4 and 5 under low and high corona currents. Meanwhile, the values of m are 1.520 and 1.676 for the positive corona and negative corona, respectively. It can be found that the value of m also falls into the scope of 1.5–2.0, similar to the point-plane electrode system.22 The values of COD for linear fitting by applying different empirical formulae are compared in Table I. It can be

Phys. Plasmas 22, 023501 (2015)

FIG. 6. COD values under different corona onset voltages by applying Eq. (6).

FIG. 7. Comparison of the measured data and the fitted data by applying Eq. (6). The solid lines are least squares fitting lines.

found that the values of COD by Meng et al. relation are both maximum at the same polarity. Therefore, the results by Meng et al. relation are the best for the linear fitting. It can be found that the corona onset voltage at negative polarity is lower than that at positive polarity. B. Fluid model

The general fluid model contains the continuity equations of electrons, positive ions, and negative ions coupled with Gauss’s equation or Poisson’s equation. In the cylindrical coordinate system, the set of governing equations can be written as dðrne le EÞ ¼ 7ða  gÞne le E rdr

(7)

TABLE I. COD values. COD

FIG. 5. Comparison of the measured data and the fitted data by applying Eq. (5). The solid lines are least squares fitting lines.

Townsend’s relation Ferreira et al. relation Meng et al. relation

Positive corona

Negative corona

0.99876 0.99608 0.99942

0.99792 0.99712 0.99956

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  d rnp lp E

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¼ 6ane le E

(8)

dðrnn ln EÞ ¼ 7gne le E rdr

(9)

eðnp  ne  nn Þ dðrEÞ ¼6 ; e0 rdr

(10)

rdr

where r is the radial position; ne, np, and nn are the densities of electrons, positive ions, and negative ions, respectively; le, lp, and ln are the mobilities of electrons, positive ions, and negative ions, respectively; a and g are the electron ionization and attachment coefficients, respectively; e is the elementary charge; E is the local electric field; e0 is the permittivity in free space; and e is the elementary charge. The upper and nether signs of 7 and 6 are for positive corona and negative corona, respectively. In this paper, the boundary conditions are specified under a given corona current I.11 The boundary conditions of ion continuity equations for positive corona are np (r0) ¼ 0 and nn (R) ¼ 0. Similarly, the boundary conditions for negative corona are np (R) ¼ 0 and nn (r0) ¼ 0. The boundary conditions for electrons are decided by the law of charge conservation. Based on Kaptzov’s assumption, the boundary conditions for the electric field are specified under a given corona onset voltage by Eðr0 Þ ¼ E0 ¼ U0 =½r0 lnðR=r0 Þ;

(11)

where E0 is the surface onset field. The electrons are not considered in the ion flow model, which means that the source terms of Eqs. (7)–(9) are all equal to zero. Then the set of governing equations can be simplified as dðrni li EÞ ¼0 rdr

(12)

dðrEÞ eni ¼ ; e0 rdr

(13)

where i ¼ p and i ¼ n for positive corona and negative corona, respectively. The analytical expression of the local electric field can be obtained as "

   2 #1=2 I r02 E0 r0  1 2 þ : EðrÞ ¼ 2pe0 li r r

(14)

The relationships between the electric potential and the electric field for the general fluid model and the ion flow model are the same, expressed as du ¼ 7E; dr

(15)

where u is the electric potential with the boundary condition uðRÞ ¼ 0. Then, the applied voltage can be expressed as ðR EðrÞdr: (16) U ¼ 6½uðr0 Þ  uðRÞ ¼ r0

FIG. 8. Comparison of the measured data and the calculated data by the fluid model for the positive corona.

The parameters in the models and their relationships with the gas conditions are the same as those used in the corona onset criterion.23 In addition, the relationship between the mobilities lj (j ¼ e, p, n) and the gas conditions is assumed to be24 lj ¼ lj0 =d;

(17)

where d is the relative air density and lj0 are the mobilities at d ¼ 1. The first values of lj0 for electrons, positive ions, and negative ions were set as 500 cm2 V1 s1, 1.5 cm2 V1 s1, and 1.8 cm2 V1 s1, respectively.25 The values of U0, lp0, and ln0 were determined when the sum of squared differences between the measured and calculated onset voltages was minimum. The measured data and the calculated data by the fluid model for the positive corona at U0 ¼ 22.9 kV and lp0 ¼ 1.9 cm2 V1 s1 are compared in Fig. 8. It can be found that the calculated results by the general fluid model and the ion flow model are almost the same under different corona currents. The calculated results by the two models are both close to the measured results. It means that the ionization region with the boundary a ¼ g has no significant effect on the current-voltage characteristic of the positive corona. The measured data and the calculated data by the fluid model for the negative corona at U0 ¼ 23.1 kV and ln0 ¼ 2.0 cm2 V1 s1 are compared in Fig. 9. It can be found that the calculated results by the general fluid model and the ion flow model are close when the corona current is lower than 10 lA/cm. The calculated results by the general fluid model are more close to the measured results when the corona current is higher than 10 lA/cm. The difference will be significant under relatively high corona currents. The ionization region of the negative corona is wider than the boundary a ¼ g,11,26 which will affect the current-voltage characteristic under a high corona current. IV. DISCUSSION

The surface onset fields for cylindrical electrodes can be described by a general formula, expressed as17

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Phys. Plasmas 22, 023501 (2015) TABLE III. Ion mobilities. Researchers

lp0 (cm2 V1 s1)

ln0 (cm2 V1 s1)

1.5 1.82 3.4617 1.4 1.4 2.43

1.5 1.82 3.4617 1.9 3.3 2.7

Abdel-Salam et al.27 Long et al.28 Seimandi et al.29 Aliat et al.11 Nikonov et al.30 Kang et al.31

FIG. 9. Comparison of the measured data and the calculated data by the fluid model for the negative corona.

E0 ¼ Ad0 ð1 þ B=

pffiffiffiffiffiffiffiffiffi d0 r0 Þ;

(18)

where A and B are constants, and d0 is the relative air density with a reference temperature of 298 K. The values of A and B were suggested by different researchers. Whitehead’s equation15 and Peek’s equation16 are the most widely used. The surface onset fields obtained by different methods are compared in Table II. It can be found that the surface onset fields deduced by the linear fitting method are lower than those deduced by the fluid model. The surface onset field deduced by the fluid model presents that the corona discharges are incepted along the whole line. It seems that the surface onset field for the first corona discharge on the line can be detected by the linear fitting method, so the surface onset field is lower than the fullscale corona onset field. It also can be reflected by Eq. (3) with the m value in the range 1.5–2.0, which is similar to the point-plane electrode system. The surface onset field in the fluid model is important for predicting the current-voltage characteristics of corona discharges. The surface onset field for the positive corona is slightly lower than that for the negative corona under our experimental conditions, which is consistent with Whitehead’s equation. To some extent, the values deduced by the fluid model are close to those calculated by the equations by Whitehead and Peek, respectively. Therefore, Whitehead’s equation and Peek’s equation are widely used for practical applications base on Kaptzov’s assumption. The calculated results by the general fluid model are almost the same as those by the ion flow model under low corona currents. This special characteristic can be used to TABLE II. Surface onset fields. E0 (kV/cm) Linear fitting Fluid model Whitehead’s equation Peek’s equation

Positive corona

Negative corona

63.190 65.478 64.475 67.134

61.761 66.050 67.124 67.134

measure the ion mobilities in air under different conditions. The ion mobilities used by different researchers are significantly different, as shown in Table III. It can be found that the used air mobilities for positive ions are smaller than or equal to those for negative ions. The values of lp0 and ln0 deduced by the fluid model are 1.9 cm2 V1 s1 and 2.0 cm2 V1 s1, respectively. A difference of about 0.1 cm2 V1 s1 is detected by the method in this paper. For further analysis, the effect of the air humidity should be considered. V. CONCLUSIONS

The current-voltage characteristics for both positive and negative corona discharges in air between coaxial cylinders were measured within a specially designed corona cage. The current-voltage characteristics between coaxial cylinders can be expressed as I ¼ C(U  U0)m, where m is within the range 1.5–2.0, which is similar to the point-plane electrode system. The surface onset fields deduced by the linear fitting method are lower than those deduced by the fluid model, which may indicate the partial corona onset and fullscale corona onset along the line. The surface onset fields deduced by the fluid model are close to those calculated by Whitehead’s equation and Peek’s equation. The ionization region has no significant effect on the current-voltage characteristic under a low corona current, while it will affect the distribution for the negative corona under a high corona current. A difference of about 0.1 cm2 V1 s1 between the positive and negative ion mobilities in air can be detected by the method in this paper. ACKNOWLEDGMENTS

This work was partly supported by the National Basic Research Program of China (973 Program) under Grant No. 2011CB209401 and National Natural Science Foundation of China under Grant No. 51237004. 1

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