A Preliminary Investigation of Computer-Aided Schwarz-Christoffel Transformation for Electric Machine Design and Analysis

An alternative method to finite element analysis (FEA) for electric machine design and analysis is presented that applies Schwarz-Christoffel (SC) conformal mapping theory using the SC toolbox for MATLABreg that has appeared in the previous literature. In this method, a two-dimensional (2D) developed machine cross-section domain is mapped via SC transformation to a concentric cylinder domain where solutions for the electromagnetic (EM) fields are known. These solutions are mapped back to the original domain, thus solving the original problem. All mapping is done via the SC toolbox. Examples are given in which the procedure is used to calculate the EM field in the air gap of and the force on the rotor of various 2D developed machine cross-sections. The numerical accuracy of the results is verified by comparing the solutions as the air gap gets small with magnetic equivalent circuit (MEC)-derived coenergy solutions

[1]  M. Markovic,et al.  Analytical Force Determination in an Electromagnetic Actuator , 2005, IEEE Transactions on Magnetics.

[2]  Lloyd N. Trefethen,et al.  Schwarz-Christoffel Mapping , 2002 .

[3]  Tobin A. Driscoll,et al.  Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping , 1996, TOMS.

[4]  Tobin A. Driscoll,et al.  Schwarz-Christoffel Toolbox User''s Guide , 1994 .

[5]  T. Broadbent.  Complex Variables , 1970, Nature.

[6]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[7]  A.B.J. Reece,et al.  Force and torque computation from 2-D and 3-D finite element field solutions , 1999 .

[8]  B. Hague,et al.  Electromagnetic problems in electrical engineering , 1929 .

[9]  S. J. Salon,et al.  Some aspects of torque calculations in electrical machines , 1997 .

[10]  J. Coulomb,et al.  Finite element implementation of virtual work principle for magnetic or electric force and torque computation , 1984 .

[11]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[12]  C. Steinmetz.  Theory and Calculation of Alternating Current Phenomena , 2008 .

[13]  Tobin A. Driscoll,et al.  Algorithm 843: Improvements to the Schwarz-Christoffel toolbox for MATLAB , 2005, TOMS.

[14]  J. L. Coulomb,et al.  A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness , 1983 .

[15]  Bernhard Arthur Behrend.  The Induction Motor A Short Treatise on its Theory and Design, with numerous Experimental Data and Diagrams , 1901, Nature.

[16]  B. Fahimi,et al.  Optimal excitation of permanent magnet synchronous machines via direct computation of electromagnetic force components , 2005, IEEE International Conference on Electric Machines and Drives, 2005..

[17]  Zhuoxiang Ren,et al.  Comparison of different force calculation methods in 3D finite element modelling , 1994 .

[18]  Tobin A. Driscoll,et al.  Numerical Conformal Mapping Using Cross-Ratios and Delaunay Triangulation , 1998, SIAM J. Sci. Comput..

[19]  Yao Xue-Biao,et al.  Distribution properties of the fringing field from a semi-infinite ring head with an arbitrary pole-angle , 1996 .

[20]  B. Fahimi,et al.  On the effect of stator excitation on radial and tangential flux and force densities in a permanent magnet synchronous machine , 2005, IEEE International Conference on Electric Machines and Drives, 2005..

[21]  Tobin A. Driscoll,et al.  A general conformal-mapping approach to the optimum electrode design of coplanar waveguides with arbitrary cross section , 2001 .

[22]  D. S. Betts.  Electromagnetism , 1977, Nature.

[23]  S. J. Salon,et al.  Finite element analysis of electrical machines , 1995 .

[24]  Lloyd N. Trefethen,et al.  Analysis and design of polygonal resistors by conformal mapping , 1984 .

[25]  F. W. Carter.  Note on air-gap and interpolar induction , 1900 .

[26]  N. Tesla,et al.  A new system of alternate current motors and transformers , 1888, Proceedings of the IEEE.

[27]  T. D. Howell,et al.  Exact field calculations for asymmetrical finite-pole-tip ring heads , 1988 .