IEEE !Zkansactions on Electrical Insulation Vol. 23 No. 3, June 1888 403

CO M M U N I CAT ION

An Expression for the Electric Field Distribution in Rod-Plane Gaps

Omar E. Ibrahim Dept. of Electrical Eng. & Comp. Science, University of Bahrain, Isa Town, Bahrain.

ABSTRACT

In this Communication a closed-form expression is derived to represent the electric field distribution along the axis of rod plane gaps in the ranges 0.1 < electrode radius < 1 cm and 2 < gap spacing < 12 cm. The expression is valuable in modeling discharge development in SF6 gaps.

INTRODUCTION HE hemispherically-capped rod-plane gap has been T extensively used for studying nonuniform field dis-

charge characteristics. However, the electric field distri- bution in this gap is only known numerically. A closed- form expression for the field in this gap will therefore be very useful for modeling leader breakdown, corona- stabilized breakdown and related features of nonuniform field breakdown. The paper develops an expression to represent the field distribution along the gap axis over a range of rod radii and gap distances of values as typi- cally used in compressed SFs studies.

METHOD OF DERIVATION HE surface charge simulation technique [1,2] was T used to compute the field distribution (per unit

applied voltage) in the rod plane gap. The present work considers the one-dimensional axial field, since it is known that a t low and a t high pressures breakdown channels follow the gap axis (31.

The following empirical procedure was used to de- velop the equation. Studying the variation of the field along the gap axis, a function was proposed to repre- sent this field. The function parameters were extracted through conventional curve fitting techniques. This re- sulted in a three-parameter expression. The variation of

these parameters on electrode radius r() and gap spac- ing d was then investigated and using multiple regression analysis, the dependence of these parameters on ro and d was obtained.

The proposed equation to represent the electric field (per unit applied voltage) along the gap axis in hemi- spherically-capped rod-plane gaps with rod radii in the range 0.1 < ro < 1 cm and gap distances in the range 2 < d < 12 cm is:

where E ( x ) is the field per unit applied voltage at a distance x cm along the gap ax is from the rod tip,

al(r0, d) = (1.125-0.09 In d)(0.2625r~+0.8555ro-0.017) (2) (3) az(r0,d) = d-1.39(0.25 + 0.24ro)

a3(ro, d) = 0.025exp(0.248d + 3 . 1 1 5 ~ ) (4)

Figures (1) and (2) and Table (1) compare the field values obtained from the numerical solution and those computed using the proposed function for configurations with varying degrees of field divergence. I t can be seen clearly that Equation (1) faithfully reproduces the field values over the entire length of the gap. A maximum

0018-0367/88/~00-403$1~00 @ 1088 IEEE

404 Ibrahim: The electric field distribution in rod-plane gaps

3.0

w/cnv,Kv

E (x)

2.a

1.1

1.1

0. '

1 2 3 4

d o n

Figure 1.

ro = 0.7 cm and d = 8 cm

4.0 1

1 2 3 4 5 6 7 8

d u n

Figure 2. Comparison of the field distribution given by equation (1) and the numerical solution for a gap with TO = 0.2 cm and d = 8 cm. -- numerical solution; 0 Equation (1).

Comparison of the field distribution given by equation (1) and the numerical solution for a gap with T O = 0.3 cm and d=4 cm. -- numerical solution; m Equation (1).

ranges and even then this is noted to be just ahead of the maximum field region. It is believed that this re- gion does not play any decisive role in the discharge development; since it is known that corona inception is determined by the high field region and it has been pro- posed that continuous discharge development and hence

Table 1. Comparison of the field values given by Equation (1) and the numerical solution [2] for a gap with

0.0 0 . 1 0 . 2 0 . 3 0.4 0 . 5 1 .o 1.5 2.0 2.5 3 . 0 3.5 4.0

Numerical s o l u t i o n

1.4510 1.1140 0.8855 0.7233 0.6041 0.5139 0.2765 0.1811 0.1337 0.1076 0.0927 0.0851 0.0828

E q u a t i o n (1 )

~~

1.4505 1.1107 0.8805 0.7191 0.6011 0.5120 0.2794 0.1877 0.1426 0.1173 0.1018 0.0916 0.0846

breakdown is controlled by the low field region [3].

REFERENCES [l] R. F. Harrington and J. Mautz, "Computation of

rotationally symmetric Laplacian potentials", Proc. IEE, Vol. 117, pp. 850-852, 1970.

[2] A. Kurimoto, "Non-uniform field impulse breakdown in Nz, air, SF6 and gas mixtures ", Ph.D. Thesis, University of Strathclyde, Glasgow, 1977.

[3] 0. Farish, 0. E. Ibrahim and A. Kurimoto, "Pre- breakdown corona processes in SF6 and SFGIN2 mixtures", 3rd Int. Symp. on HV Eng., Milan, paper 31.15, 1979.

Manuscript was received on 20 Feb 1988, in revised form 19 Apr 1988.

error of about 15% is found at the extreme value of the