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Analytical and Numerical Methods for Determination of Electric Field Geometries

Severin Wiesmüller

The presented report exposes the basic methods of conformal mapping. Furthermore the wide range of features is displayed with particular mapping functions. In doing so the focus is on the Schwarz-Christoffel mapping and similar mapping functions which allow the smooth rounding of corners in polygons.
By the use of Exponential-, Joukowski- and Schwarz-Christoffel-Transformation the possibility of mapping the electric field of an infinitely wide parallel-plate capacitor into a field of an air gap between two rectangular shaped electrodes is pointed out. In this way the field in the capacitor can be calculated and its features can be transformed to the air gap. By the additional use of mapping functions derived from the Schwarz-Christoffel-Transformation this approach can also be used for modeling and analyzing the air gap between semicircular rounded electrodes.
Various air gaps are simulated with COMSOL Multiphysics for verifying the results and their capacity by calculation due to the simulated electric flux between the electrodes. Eventually the sensitivity of the capacity for variation of geometry of calculations based on conformal mapping and based on simulation can be compared.

Figure 1: The electric field of an air gap between two rectangular shaped electrodes derived from Schwarz-Christoffel mapping and out of simulation.

Figure 1: The electric field of an air gap between two rectangular shaped electrodes derived from Schwarz-Christoffel mapping and out of simulation.

June 06, 2011