An Electrostatic Approach to the Numerical Simulation of Combined Corona and Electrostatic Field in Three-Electrode System

Jei vainikinio islydžio elektrodų sistemą sudaro daugiau negu du elektrodai, visą lauko sritį galima padalyti į dvi skirtingų parametrų dalis. Pirmąją dalį sudaro elektrinis srautas tarp jonizuojanciųjų ir nejonizuojanciųjų elektrodų su paskirstytais erdviniais krūviais. Skaitmeniskai modeliuojant sią lauko dalį reikia spresti vainikinio islydžio elektrinio lauko diferencialinių lygcių sistemą. Antrąją lauko dalį sudaro elektrinis srautas tarp nejonizuojanciųjų elektrodų be erdvinių krūvių. Sis laukas yra kvazivienalytis ir jį analizuojant pakanka isspresti Laplaso lygtį. Siame straipsnyje elektrodų sistemos „laidas tarp dviejų lygiagrecių plokstumų“ skaitmeniniam lauko analizės modeliui sudaryti naudojama separatrisės tarp dviejų minėtų lauko dalių lygtis. Žinant separatrisės lygtį, nesunku apibrėžti sios ribos krastine sąlygą. Sios sąlygos nustatymo elektrostatinio metodo rezultatai palyginti su skaitiniu Laplace'o lygties sprendiniu visai nagrinėjamo lauko sriciai. Il. 2, bibl. 14, lent. 4 (anglų kalba; santraukos anglų ir lietuvių k.). DOI: http://dx.doi.org/10.5755/j01.eee.122.6.1832


Introduction
Two-electrode systems consisting of ionizing electrode, such as wire, needle or the other one with small radius of surface curvature, and a non-ionizing electrode are often used in many applications of corona field. Multi-electrode systems are of more complicated structure and more complex methods are needed for their analysis.
Numerical analysis of combined corona-electrostatic electric fields in wire-to-plane electrode system supplemented by a cylindrical non-ionizing electrode is presented in [1]. This third electrode is associated with the wire electrode in the dual corona-electrostatic arrangement. Electric field changes the sign in a point situated between the wire and cylindrical electrode. Electric field lines originating from the wire electrode will concentrate on a smaller region at the surface of the plane electrode, as compared with the case of a single wire element. Mathematical model of the field is similar to those commonly employed for the study of plain corona fields in wire-to-plane electrode systems.
Numerical investigation of the electric field in the domain with no free charges using the Multi Quadratic and Differential Quadrature methodology as a new meshless method is developed in [2]. Poisson equation includes a constant source term which does not affect the solution and therefore this technique can be used to solve Laplace and Poisson equations in the fields of complex geometries. But it isn′t suitable to solve the corona field system of equations comprising of Poisson and non-linear charge conservation equation.
Results of the analysis of electric field in a cylindrical triode corona charger used for charging aerosol nanoparticles are presented in [3]. Cylindrical corona charger consists of corona wire, perforated inner cylinder and grounded outer cylinder with coaxial position of all three electrodes. The inner zone of corona discharge and the outer particle charging zone are related with space charge distribution and there is no zone without space charges. The field is analyzed as unipolar corona field in all the computational area.
Potential distribution within a pipe through which charged gas is transported is analyzed in [4] by using Bessel functions. Boundary conditions at the surface between space charge and no charge regions are the equality of potentials and their derivatives normal to the surface. Uniform distribution of space charge is assumed in this analysis. This distribution is far from that in corona field, but the mentioned boundary conditions may be used for analysis of combined corona-and electrostatic fields.
Electric field of corona triode [5] is analyzed as the one comprising of two components. The first one, generated by the space charge, is governed by the Poisson equation with uniform Dirichlet boundary conditions and is solved using the finite element method. The second one, produced by the potentials of conductors, is described by the Laplace equation with non-uniform Dirichlet boundary conditions. It is solved by boundary element method. Computation of charge distribution on characteristic lines is performed by using the method of characteristics. There are no regions without the space charge and the method is the same as the one used for analysis of two-electrode corona field.
Electrostatic field of roll-type separator consisting of grounded drum, corona wire and a tubular non-ionizing high-voltage electrode [6] is analyzed by the charge simulation program based on boundary-element method. The field under study can be divided into region without space charges and the one with distributed space charges. Boundary conditions between these regions are of specific interest, but they are not determined there.
Computations and experiments are carried out in the [7,8] for a two-dimensional three-electrode system consisting of wire parallel to plane and a non-ionizing cylinder placed in opposite side of the wire. This configuration is characterized by the existence of singular point on the axis of symmetry where the electric field takes zero value, and this is a major difficulty for which none of the existing algorithms is able to provide a solution. To overcome this difficulty initial computational region D(z), z = x+jy, is transposed to a domain D(),  = +j, with simply defined boundary conditions by using a conformal transform (1) This enables to use finite difference method with a simple quadratic grid and to define numerically regions with space charge distribution and with no space charge. We develop there a concept of existence of the singular point with zero field strength on the axis of symmetry and of separatrix related to this point and dividing the field into region having no space charge and a region with space charge.

Problem formulation
We analyze the electrostatic field in three-electrode system consisting of two parallel non-ionizing plane electrodes and a corona wire between them (Fig. 1).
is defined as the superposition of the field in the electrode system "a wire between two parallel grounded plates" and the homogeneous electrostatic field [9]. Potential function is the real part of the complex potential (2) and the field-line function is the imaginary part of the function (2) Meanings of the symbols used in the equations (3)-(10) are the following: u 1 , u 2 , u 3 are the electrode potentials, r 0 is the radius of the wire, h 1 and h 2 are the distances from the center of the wire to the upper and lower planes, correspondingly.
Equation of the separatrix can be derived from the field-line function (4) equality to zero (11)

Analysis of separatrix boundary condition
Results of analysis of the field separatrix correspond to the following values of field parameters: r 0 = 0,025 mm, h 1 = 10 mm, h 2 = 40 mm, u 1 = -10000 V, u 2 = 0, u 3 = -5000 V. Coordinates of separatrix points calculated by using (6)-(11) formulas are given in Table 1 for the step Δx = 1 mm. The part of the field area under study is presented in Fig. 2. Lower plain electrode of potential u 3 isn′t shown here. There are two regions of the shown field area: the region on the left of separatrix, comprising the flux of field lines between electrodes 1 and 3 with no space charge, and the region on the right of separatrix with field lines directed from the wire to the upper plane electrode and with the distributed space charge. The first region corresponds to Laplacian field, and the second one is occupied by corona field. The field in the first region is quasi-homogeneous, the grid in Cartesian coordinates can be used for solution of Laplace equation [10]. Potentials of the points given in Table 1 and calculated by using equations (3) and (6)-(9) are given in Table 2. Polar grid is used in [10] for computation of corona field in wire-to-plane electrode system similar to that located in the area to the left of separatrix in Fig. 1. Position of points on separatrix coinciding with points on lines of polar grid denoted from j  5 to j  13 can be determined from the system of equations: Coordinates of points common to the separatrix and lines of polar grid corresponding to the values of j from 5 to 12 are given in Table 3. Potentials of that points determined by using the equation (3) and (6)-(9) are also given here. Other boundary conditions of the field under study are given in [11].
Numerical analysis of an electrostatic field in the system under study is carried out by using the finite difference method in MATLAB environment. Laplace equation is solved as a test problem by using the Cartesian coordinate system with the grid size Δ  1 mm. Values of potentials at the several points of separatrix are compared in the Table 4 with the potential values of numerical analysis at the neighbor nodes of Cartesian grid. All potential values given in Table 4 are negative. Numerical values of potentials corresponding to points of separatrix are determined by linear interpolation of potential values at neighbor nodes. Differences of compared values are equal to 0 at the ends of separatrix and are less than 1 % at the middle points of this line. Discussed data are the result of analysis of the electrostatic field. Computation of corona field in the region with bounds formed by straight lines x  10 mm, y  0 and separatrix can be performed under assumption that the shape of separatrix doesn′t depend on the presence of the space charge in the region. This assumption corresponds to the conclusion of [8] that the space charge density is 0 in all points of separatrix. This is in agreement with the results of [1] where presence of the point with zero field strength and zero space charge density on the symmetry axis is stated. The mentioned statement is similar to Deutsch assumption successfully used for analysis of corona fields in various electrode systems [12], although some restrictions of this assumption were made in [13] and it was criticized in [14]. Our hyphothesis of independence of separatrix position on the space charge density is not such rigorous as the Deutsch assumption because the component of electric field produced by space charge is much smaller than the field of electrodes [5].

Conclusions
Numerical analysis of combined corona and electrostatic field can be performed for two related regions: a region without and with space charge. If the separatrix between the regions is known from the analysis of the electrostatic field boundary condition for potential can be determined easily. The field in the region without space charge can be assumed as quasi-homogeneous and the finite-difference method in Cartesian coordinates can be used to solve the Laplace equation. The field in the region related to emitting electrode is strongly inhomogeneous with distributed space charge. Analysis of the field in this region can be performed by numerical solution of corona field equations in polar coordinates. Assumption of independence of the separatrix position on the space charge must be used. Analytical and numerical results of the separatrix parameters determination coincide well, maximum differences between the values of potentials doesn′t exceed 0,7 %