In this paper we present a detailed theoretical analysis of lateral and angular misalignment effects in RF coils. Radio-frequency (RF) coils are used extensively in the design of implantable devices for transdermal power and data transmission. A design procedure is established to maximize coil coupling for a given configuration to reduce the effects of misalignment on transmission efficiency. Formulas are derived for the mutual inductance between all possible coil configurations including the coils of cross section, thin solenoids, pancakes and filamentary circular coils whose axes are laterally and angularly displaced. Coils are in air. In this approach we used the filament method and the mutual inductance between filamentary circular coils placed in any desired position. We completely describe all mathematical procedures to define coil positions that lead to relatively easy method for calculating the mutual inductance between previously mentioned coils. The practical coils in implantable devices fall into two categories: disk coils (pancakes) and solenoid coils. From the general approach for calculating the mutual inductance between coils of rectangular cross section with lateral and angular misalignments the mutual inductance between misalignment solenoids and disks will be calculated easily and accurately.
2. Snow, C., Formulas for Computing Capacitance and Inductance, Series: NBS circular 544, National Bureau of Standards, Washington, DC, Dec. 1954.
3. Dwight, H. B., Electrical Coils and Conductors, McGraw-Hill Book Company, Inc., New York, 1945.
4. Babic, S. I. and C. Akyel, "New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air," IEEE Trans. Mag., Vol. 42, No. 6, 1661-1669, Jun. 2006.
5. Babic, S. I. and C. Akyel, "Calculating mutual inductance between circular coils with inclined axes in air," IEEE Trans. Mag., Vol. 44, No. 7, 1743-1750, Jul. 2008.
6. Babic, S. I., F. Sirois, C. Akyel, and C. Girardi, "Mutual inductance calculation between circular filaments arbitrarily positioned in space: Alternative to Grover’s formulas," IEEE Trans. Mag., Vol. 46, No. 9, 3591-3600, Sep. 2010.
7. Akyel, C., 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, 287-301, 2009.
8. Babic, S. I., F. Sirois, and C. Akyel, "Validity check of mutual inductance formulas for circular filaments with lateral and angular misalignments," Progress In Electromagnetics Research M, Vol. 8, 15-26, 2009.
9. Conway, J. T., "Noncoaxial inductance calculations without the vector potential for axysimmetric coil and a planar coil," IEEE Trans. Mag., Vol. 44, No. 4, 453-462, Apr. 2008.
10. Conway, J. T., "Inductance calculations for noncoaxial coils using bessel functions," IEEE Trans. Mag., Vol. 43, No. 3, 1023-1034, Mar. 2007.
11. Conway, J. T., "Inductance calculations for coils of rectangular cross section using bessel and struve functions," IEEE Trans. Mag., Vol. 46, No. 1, 75-81, Jan. 2010.
12. Babic, S. I., C. Akyel, F. Sirois, G. Lemarquand, R. Ravaud, and V. Lemarquand, "Calculation of the mutual inductance and the magnetic force between a thick circular coil of the rectangular cross section and a thin wall solenoid (integro-differential approach)," Progress In Electromagnetics Research B, Vol. 33, 221-237, 2011.
13. Conway, J. T., "Mutual inductance between thin coils with parallel axes,", Private Communication, Jun. 2011.
14. Kamon, M., M. J. Tsuk, and J. White, "FASTHENRY: A multipole accelerated 3D inductance extraction program," IEEE Trans. Microw. Theory Tech., Vol. 42, No. 9, 1750-1758, Sep. 1994.
15. Zeirhofer, C. M. and E. S. Hochmair, "Geometric approach for coupling enhancement of magnetically coupled coils," IEEE Trans. Biomed. Eng., Vol. 43, No. 7, 708-714, Jul. 1996.
16. Puers, R., K. Van Schuylenbergh, M. Catrysse, and B. Hermans, "Wireless inductive transfer of power and data," Analog Circuit Design, 395-414, Springer, The Netherlands, 2006, ISBN: 978-1-4020-3884-6.
17. Soma, M., C. D. Galbraith, and R. White, "Radio-frequency coils in implantable devices: Misalignment analysis and design procedure," IEEE Trans. Biomed. Eng., Vol. 34, No. 4, 276-282, 1987.
18. Fotopoulou, K. and B. W. Flynn, "Wireless power transfer in loosely coupled links: Coil misalignment model," IEEE Trans. Mag., Vol. 44, No. 1, 453-462, Mar. 2011.
19. Jow, U. M. and M. Ghovanloo, "Design and optimization of printed spiral coils for efficient transcutaneous inductive power transmission," IEEE Trans. Biom. Circ. Sys., Vol. 1, No. 3, 193-202, Sep. 2007.
20. Kim, J.W., H. C. Son, D. H. Kim, K. H. Kim, and Y. J. Park, "Efficiency of magnetic resonance WPT with two off-axis self-resonators," 2011 IEEE MTT-S. Microwave Workshop Series on Innovative Wireless Power Transmission: Technologies, Systems, and Applications (IMWS), 127-130, 2011.
21. Heetderks, W. J., "RF powering of millimeter and submillimeter-sized neural prosthetic implants," IEEE Trans. Biomed. Eng., Vol. 35, 323-327, May 1988.
22. Waters, B., "High Q resonant coupling and RF-DC conversion for wireless power transfer,", http://students.washington.edu/bhw2114/pubs/Inductive Coupling + RF Wireless Power.pdf.
23. Zhong, W. X., C. K. Lee, and S. Y. R. Hui, "Wireless power domino-resonator systems with non-coaxial axes and circular structures," IEEE Trans. on Power Electronics, Vol. 27, No. 11, 4750-4762, Nov. 2012.
24. Zhong, W. X., C. K. Lee, and . Y. R. Hui, "General analysis on the use of Tesla’s resonators in Domino forms for wireless power transfer," IEEE Trans. on Industrial Applications, Vol. 60, No. 1, 261-270, Jan. 2013.
25. Gradshteyn, I. S. and I. M. Rhyzik, Tables of Integrals, Series and Products, Dover, New York, 1972.
26. Abramowitz, M. and I. A. Stegun, "Handbook of Mathematical Functions," Series 55, National Bureau of Standards Applied Mathematics, Washington DC, Dec. 1972.
27. Martinez, J., S. Babic, and C. Akyel, "On evaluation of inductance, DC resistance and capacitance of coaxial inductors at low frequencies," IEEE Trans. Mag., Vol. 50, No. 7, Jul. 2014, Doi: 10.1109/TMAG.2014.2303943.