csm_capacitance_exlent - 240 J Gomolln et al Calculation of...

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J. Á. Gomollón et al.: Calculation of 3D-Capacitances with Surface Charge Method by means of the Basis Set of Solutions 1070-9878/10/$25.00 © 2010 IEEE 240 Calculation of 3D-Capacitances with Surface Charge Method by means of the Basis Set of Solutions Jesús Á. Gomollón, Emilio Santomé University of La Coruña Escuela Politécnica Superior, C/Mendizábal s/n 15403 Ferrol, Spain and Roberto Palau I.E.S. Fernando Wirtz Suárez C/ Caballeros 1 15006 La Coruña, Spain ABSTRACT This paper shows how to use the solution of electrostatic field problems by means of the basis set of solutions for the determination of the three dimensional partial capacitances between the electrodes of a given configuration. The Surface Charge Method and the Boundary Element Method are used to carry out the field calculation and determine the charge distributions on the electrode surfaces. The integration of the charge densities for the basis vectors of potentials allows the direct determination of the capacitance matrix of the problem. The difference between integrating surface charge densities and adding up fictitious charges for the determination of capacitances is also shown. Special attention is given to the computation in the presence of electrodes at floating potential. Index Terms Capacitance, electrostatic field calculation, surface charge method, charge simulation method, boundary element method, basis solution, floating potential, boundary condition. 1 INTRODUCTION THE determination of partial capacitances for a generic configuration of electrodes and dielectric boundaries is a well known theoretical issue of field theory and was already stated by Maxwell [6], but until now very few references deal with the implementation of this theory into program codes for three dimensional field calculation. In this paper we will show how the use of the basis set of solutions of a field problem, as stated in [2], can help in the determination of the capacitance matrix of a field problem. This will be illustrated by analytical and practical examples. 2 BASIS SET OF SOLUTIONS FOR A FIELD PROBLEM [2] When the solution of a field problem for a given geometry, including electrode and dielectric boundary surfaces, can be stated as an equation system of the form [3,10,11,12] [ ][ ] [ ] φ = σ A (1) where [A] is a matrix of potential or field coefficients, [ σ ] is the unknown vector of charges or charge densities, and [ φ ] is a vector of potentials (or zeros when the boundary condition of a row comes from a dielectric boundary), it is possible to group the electrodes of the configuration into N groups of electrodes. Within each group, all the electrodes have the same voltages and phases. The equation system can be rewritten as: [ ][ ] [ ] [ ] [ ] N N A 1 ... 1 1 2 2 1 1 φ + + φ + φ = σ (2) where φ i is the voltage of the group of electrodes i and [1] i is a vector containing ones in the rows of the equation system corresponding to the statement of the boundary conditions for points situated on electrodes set at voltage φ i and zeros elsewhere.
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