On the conformal mapping of a circular disc with a curvilinear polygonal hole

1977 ◽  
Vol 85 (6) ◽  
pp. 406-414 ◽  
Author(s):  
P. N. Shivakumar ◽  
Kim Ho Chew
Keyword(s):  
Conformal Mapping ◽  
Circular Disc ◽  
Polygonal Hole

The conductivity of a sheet containing a few polygonal holes and/or superconducting inclusions

1992 ◽  
Vol 437 (1899) ◽  
pp. 215-227 ◽  
Keyword(s):  
Dipole Moment ◽  
Conformal Mapping ◽  
Computer Simulations ◽  
Equilateral Triangle ◽  
Unit Area ◽  
Reciprocity Theorem ◽  
Induced Dipole ◽  
Mapping Techniques ◽  
Polygonal Hole ◽  
Analog Experiments

I use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well-separated holes. The hole shapes studied are the equilateral triangle, square, pentagon and regular n -gons. I show that the conductivity can be written as σ / σ 0 = 1 – α n f + o ( f 2 ), where f is the area fraction of the inclusions and the coefficient α n = (tan (π/ n )/2π n ) Г 4 (1/ n )/Г 2 (2/ n ), which is 2.5811, 2.1884, 2.0878 for triangles, squares and pentagons, and tends to the circle limit of 2 as n →∞ . The coefficient α n is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping. I have also examined and compared the results for long thin needle-like holes in the shape of diamonds, rectangles and ellipses. In all cases the conductivity parallel to the needles has the limiting form σ / σ 0 = 1 – f , while for the perpendicular conductivity, I find that σ / σ 0 = 1 – n π a 2 , where 2 a is the length of the needle, and n is the number of needles per unit area. For thicker needles, the shape becomes important and I compare the results with recent analog experiments and computer simulations. Because of the reciprocity theorem, all the results found here apply equally well to superconducting inclusions.

A novel analytical calculation of magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping

2020 ◽  
pp. 1-15
Author(s):  
Jianqi Li ◽  
Yu Zhou ◽  
Jianying Li
Keyword(s):  
Magnetic Field ◽  
Conformal Mapping ◽  
Permanent Magnet ◽  
Flux Density ◽  
Analytical Method ◽  
Electromotive Force ◽  
Analytical Calculation ◽  
Air Gap ◽  
Permanent Magnet Machines ◽  
Peak Value

This paper presented a novel analytical method for calculating magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping. Firstly, flux density without slots and complex relative air-gap permeance of slotted air gap are derived from conformal transformation separately. Secondly, they are combined in order to obtain normalized flux density taking account into the slots effect. The finite element (FE) results confirmed the validity of the analytical method for predicting magnetic field and back electromotive force (BEMF) in the slotted air gap of spoke-type permanent-magnet machines. In comparison with FE result, the analytical solution yields higher peak value of cogging torque.

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