IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 34, NO. 2, MARCH/APRIL 1998 301 Improved Calculation of Finite-Element Analysis of Bipolar Corona Including Ion Diffusion Zakariya M. Al-Hamouz, Member, IEEE, Mazen Abdel-Salam, Fellow, IEEE, and Anwar Mufti, Senior Member, IEEE Abstract— This paper presents an iterative method for the analysis of bipolar corona associated with the ionized field around high-voltage bipolar direct-current (HVDC) transmission-line conductors. A new finite-element technique (FET) is proposed to solve Poisson’s equation where the constancy of the conductors’ surface field at the corona onset value is directly implemented in the finite-element formulation. Satisfying the current continuity condition and updating the space–charge density are based on the application of Kirchoff’s current balance law at each node of the finite-element grid and take the ion diffusion into account. In order to investigate the effectiveness of the proposed method, a laboratory model was built. It has been found that the calculated – characteristics and the ground plane current density profiles agreed well with those measured experimentally. The simplicity in writing the computer program, in addition to the low number of iterations required to achieve convergence, characterize the new method of analysis. Index Terms— Bipolar corona, dc transmission lines, finite- element analysis, ion diffusion, ionized fields, space–charge mod- ified fields. I. INTRODUCTION T HERE is an ongoing interest in the analysis of a bipolar ionized field in different geometric configurations. The prospects for the widespread use of high-voltage bipolar direct- current (HVDC) transmission underlie the great interest in the evaluation of corona power loss on bipolar transmission lines [1]. A major difficulty in electrostatic precipitators is the “back corona” which results in a bipolar ionized field that seriously affects the precipitator operation [2]. Bipolar spray charging, as opposed to monopolar charging, was implemented to eliminate corona discharges active at the leaf tips of the crops being sprayed [3]. A bipolar ionized field includes the generation of positive and negative ions at the coronating conductors and the per- tinent recombination between ions of different polarity. This makes the analysis of a bipolar ionized field more complicated than the monopolar one [4]. Paper MSDAD 97–26, presented at the 1996 Industry Applications Society Annual Meeting, San Diego, CA, October 6–10, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electrostatic Processes Committee of the IEEE Industry Applications Society. The work of Z. M. Al-Hamouz was supported by King Fahd University of Petroleum and Minerals. Manuscript released for publication September 19, 1997. Z. M. Al-Hamouz is with the Department of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. M. Abdel-Salam is with the Electrical Engineering Department, Assiut University, Assiut 71518, Egypt. A. Mufti is with the Department of Electrical and Computer Engineering , King Abdul Aziz University, Jeddah 21413, Saudi Arabia. Publisher Item Identifier S 0093-9994(98)02176-8. II. EQUATIONS DESCRIBING THE BIPOLAR IONIZED FIELD The equations describing the bipolar ionized field are as follows: (1) (2) (3) (4) (5) Equations (1)–(5) are, respectively, Poisson’s equation, the positive and negative current density vectors, , the con- tinuity condition of , the total current density vector , and the continuity condition of . and are the mobilities of positive and negative ions, and are the positive and negative space–charge density values, and are the diffusion coefficients of the positive and negative ions, is the ion recombination coefficient in air, and is the electron charge. The solution of (1)–(5) requires the following boundary conditions. 1) The potentials on the positive and negative coronating conductors are equal to the applied voltages, and , respectively. 2) The potential on the ground plane is zero. 3) The magnitude of the electric field at the surface of the positive and negative coronating conductors is related to the onset values, and , respectively. 4) The potential values of the nodes on the artificial bound- ary defining the finite-element bounded region are up- dated and utilized in the finite-element formulation. The exact solution of (1)–(5) is extremely difficult, due to their nonlinear nature. All attempts reported before were based on some simplifying assumptions. The most common ones are the following. 1) The space is full of charges of both polarities. The thickness of the ionization layer around the conductor is so small as to be neglected with respect to the interelectrode spacing. 2) The space charge affects only the magnitude and not the direction of the electric field (Deutsch’s assumption). 0093–9994/98$10.00  1998 IEEE