Improved fictitious charge method for calculations of electric potential and field generated by point-to-plane electrodes

In the framework of standard tip-to-plane electrode geometry favorable to corona streamer discharge development at atmospheric pressure, this work is devoted to the improvement of fictitious charge method for calculations of electric potential and field repartition when the tip is powered by a DC voltage. It is in fact dedicated to implement the image charge method (generally used in plane-to-plane electrodes) in the case of a point-to-plane geometry. The numerical method is based on the solution an open system of n equations with m unknowns (n >> m) where m is the number of fictitious charges and n the number of contours at the surface of the tip electrode defining the boundary conditions. This numerical technique can accurately interpolate the shape of the electrode tip whatever its geometry and hence allows us to accurately calculate the electric potential and field even at a position very close to the electrode. It is noteworthy that the solution of such open system of equations cannot be obtained from conventional techniques (Cramer, Gauss, matrix inversion, etc.). We used the method of least squares which enables us to close the equation systems and to find the optimal solution fulfilling all the required boundary conditions. The present method is therefore based on the coupling between the conventional method of fictitious charges using image charge method and the optimization by the Least Squares Method. The results of simulation show that the punctual fictitious charges have given the most accurate results when the electrode has symmetry of revolution like the present geometry of a pen shape anode cylinder ended by a sharp tip set in front of cathode plane.