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Vol.1 , No. 1, Publication Date: Jul. 19, 2016, Page: 62-66
 has been given by many authors (Spielrein, Grover, Dwight, Kalantarov, Kajikawa, Babic, Akyel, Yu, Conway, Luo) it is also the challenge in this time to obtain the simpler form of presented formulas. The expressions for calculating the self-inductance of the pancake are given over the convergent series, complete elliptic integrals or generalized hypergeometric functions. Definitely the closed form doesn’t exist except in the special cases. The propose of this paper is to give relatively simple and fast method comfortable for the potential users such as physicist and engineers. With the Mathematica code all proposed different formulas give the same results for all range of the parameters describing the disk coil (pancake). All expressions are obtained over the complete elliptic integral of the first kind and two terms given by the simple integrals which converge on the whole interval of integration. These integrals only drop to infinity (logarithmic singularities) when the radiuses of the pancake are the same and it is the case of the self-inductance of the circular loop of the negligible cross section. Numerical integration proposed by Matlab and Mathematica programming can have the significant influence in the treatment of the logarithmic singularities. We gave also the approximate formulas in the case of the circular loop which cross section can be considered not negligible.&picture=http://www.aascit.org/aascit/decorators/img/journal/825.jpg)
| [1] | Slobodan Babic, Département de Génie Physique, École Polytechnique, C. P. 6079 Succ. Centre Ville, QC H3C 3A7, Montréal, Canada. |
| [2] | Cevdet Akyel, Département de Génie Électrique, École Polytechnique, C. P. 6079 Succ. Centre Ville, QC H3C 3A7, Montréal, Canada. |
| [3] | Levent Erdogan, Faculty of Science, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, C1A 4P3, Canada. |
| [4] | Bojan Babic, Independent Consultant, 36 Gabrielle Roy, H3E 1M3, Montréal, Canada. |
Even though the self-inductance calculation of the disk coil (pancake) has been given by many authors (Spielrein, Grover, Dwight, Kalantarov, Kajikawa, Babic, Akyel, Yu, Conway, Luo) it is also the challenge in this time to obtain the simpler form of presented formulas. The expressions for calculating the self-inductance of the pancake are given over the convergent series, complete elliptic integrals or generalized hypergeometric functions. Definitely the closed form doesn’t exist except in the special cases. The propose of this paper is to give relatively simple and fast method comfortable for the potential users such as physicist and engineers. With the Mathematica code all proposed different formulas give the same results for all range of the parameters describing the disk coil (pancake). All expressions are obtained over the complete elliptic integral of the first kind and two terms given by the simple integrals which converge on the whole interval of integration. These integrals only drop to infinity (logarithmic singularities) when the radiuses of the pancake are the same and it is the case of the self-inductance of the circular loop of the negligible cross section. Numerical integration proposed by Matlab and Mathematica programming can have the significant influence in the treatment of the logarithmic singularities. We gave also the approximate formulas in the case of the circular loop which cross section can be considered not negligible.
Keywords
Self-Inductance, Thin Disk Coil (Pancake), Logarithmic Singularity, Numerical Integration
Reference
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| [02] | F. W. Grover, 'Inductance Calculations', New York: Dover, (1964), chs. 2 and 13. |
| [03] | H. B. Dwight, 'Electrical Coils and Conductors', McGraw-Hill Book Company, INC. New York, (1945). |
| [04] | Chester Snow, 'Formulas for Computing Capacitance and Inductance', National Bureau of Standards Circular 544, Washington DC, (1954). |
| [05] | P. L. Kalantarov et al., 'Inductance Calculations', Moscow, USSR/Russia: National Power Press, 1955. |
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| [09] | H. A. Wheeler, 'Simple inductance formulas for radio coils', Proceedings of the IRE, Vol 16, No. 10, October 1928. |
| [10] | J. T. Conway, 'Analytical Solutions for the Self and Mutual Inductances of Concentric Coplanar Disk Coil', IEEE Transactions on Magnetics, Digital Object Identifier: 10.1109∕TMAG.2012.2229287, 2012. |
| [11] | Y. Luo, 'Improvement of Self Inductance Calculations for Circular Coils of Rectangular cross Section', IEEE Transactions on Magnetics, Digital Object Identifier: 10.1109∕TMAG.2012.2228499, 2012. |
| [12] | Kazuhiro Kajikawa and Katsuyuki Kaiho, 'Usable Ranges of Some Expressions for Calculation of the Self-Inductance of a Circular Coil of Rectangular Cross Section', Cryogenic Engineering, Vol. 30, No. 7 (1995), pp. 324-332, (In Japanese). |
| [13] | M. Abramowitz and I. A. Stegun, 'Handbook of Mathematical Functions', National Bureau of Standards Applied Mathematics, Washington DC, (1972), Series 55, p. 595. |
| [14] | http://www.mathworks.com/matlabcentral/fileexchange/32-gaussq. |
