A General Mutual Inductance Formula for Parallel Non-coaxial Circular Coils
Keywords:
Filament method, Grover’s formula, Kim’s formula, magnetic vector potential, mutual inductance, Neumann’s formula, non-coaxial coilsAbstract
In this paper, we present a complete derivation of a new formula for calculating the mutual inductance between two parallel non-coaxial circular coils by using the magnetic vector potential. Although this problem has been studied by Kim et al, their formula is wrong when the two parallel circulars are non-coaxial. The newly derived formula can not only clarifies the origin of the error introduced in Kim’s work, but also can be reduced to the well-known F. W. Grover formula. Different from the Grove’s formula that has various cases of the singularities, the new formula enables to simplify all the singular cases into single situation. The singularity in our formula has the clearly geometrical significance, i.e., the point located in one projected circular coil overlapped with the center of another one, which can be easily tackled by the principal value integrals. In order to check the validity of our formula, we have investigated the mutual inductance of the circular coil of the rectangular cross section, and the results demonstrate the correctness of our formula as compared with the other approaches, such as the Neumann’s formula, Grover’s formula and other published data. In addition, our formula is capable of handling the topics involving the circular coils such as the coils of the rectangular cross section, wall solenoids, disk coils and circular filamentary coils etc. In conclusion, our newly derived formula is of great importance for the electromagnetic applications concerning the calculation of the mutual inductance of the two parallel non-coaxial circular coils.
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References
M. Brighton and M. J. English, “Calculation of optimum spacing for a three coil axially symmetric metal detector,” Electronics Letters, vol. 29, no. 10, pp. 838-839, 1993.
J. Acero, C. Carretero, I. Lope, et al., “Analysis of the mutual inductance of planar-lumped inductive power transfer systems,” IEEE Trans. Power Electron., vol. 60, no. 1, pp. 410-420, 2013.
M. Grzeskowiak, A. M. Diet, P. S. Diao, et al., “Pebbles tracking thanks to RFID LF multi-loops inductively coupled reader,” Progress in Electromagnetics Research C, 55, pp. 129-137, 2014.
M. Catrysse, B. Hermans, and R. Puers, “An induc tive power system with integrated bi-directional data-transmission,” Sensors & Actuators A Physical, vol. 115, no. 2, pp. 221-229, 2004.
Y. Ren, F. Wang, G. Kuang, et al., “Mutual inductance and force calculations between coaxial bitter coils and superconducting coils with rectangular cross section,” J. Supercond. Novel Magn., vol. 24, no. 5, pp. 1687-1691, 2011.
Y. Ren, G. Kuang, and W. Chen, “Inductance of bitter coil with rectangular cross-section,” J. Supercond. Novel Magn., vol. 26, no. 6, pp. 2159- 2163, 2013.
L. Yao, “An accurate high-speed method for mutual inductance calculations of coplanar disk coils using generalized hypergeometric function,” IET Electric Power Applications, vol. 31, no. 4, pp. 366-372, 2016.
K. B. Kim, Z. Zabar, E. Levi, et al., “In-bore projectile dynamics in the linear induction launcher (LIL) - Part 1: Oscillations, IEEE Trans. Mag., vol. 31, no. 1, pp. 484-488, 1995.
J. C. Maxwell, A Treatise on Electricity and Magnetism. Dover Publications Inc., New York, 1954 (reprint from the original from 1873).
F. W. Grover, “The calculation of the mutual inductance of circular filaments in any desired positions,” Proceedings of the IRE, vol. 32, no. 10, 620-629, 1944.
F. W. Grover, Inductance Calculations. Dover, New York, 1964.
J. T. Conway, “Inductance calculations for noncoaxial coils using bessel functions,” IEEE Trans. Mag., vol. 43, no. 3, pp. 1023-1034, 2007.
J. T. Conway, “Noncoaxial inductance calculations without the vector for axisymmetric coils and planar coils,” IEEE Trans. Mag., vol. 44, no. 4, 453-462, 2008.
C. Akyel, S. I. Babic, and M. M. Mahmoudi, “Mutual inductance calculation for non-coaxial circular air coils with parallel axes,” Progress in Electromagnetics Research, vol. 91, no. 4, pp. 287- 301, 2009.
S. Butterworth, “On the coefficients of mutual induction of eccentric coils,” Phil. Mag., Ser. 6, 31, pp. 443-454, 1916.
C. Snow, “Formulas for Computing Capacitance and Inductance,” National Bureau of Standards Circular 544, Washington DC, Dec. 1954.
K. B. Kim, E. Levi, Z. Zabar, et al., “Mutual inductance of noncoaxial circular coils with constant current density,” IEEE Trans. Mag., vol. 33, no. 5, pp. 4303-4309, 1997.
S. I. Babic, F. Sirois, and C. Akyel, “Validity check of mutual inductance formulas for circular filaments with lateral and angular misaligments,” Progress In Electromagnetics Research M, 8, pp. 15-26, 2009.
W. R. Smythe, Static and Dynamic Electricity. McGraw-Hill, New York, 1950.
S. I. Babic, F. Sirois, C. Akyel, et al., “New formulas for mutual inductance and axial magnetic force between a thin wall solenoid and a thick circular coil of rectangular cross-section,” IEEE Trans. Magn., vol. 47, no. 8, pp. 2034-2044, 2011.
H. B. Dwight, Electrical Coils and Conductors. McGraw-Hill, New York, 1945.
