Inductance Formula for Rectangular Planar Spiral Inductors with Rectangular Conductor Cross Section
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Abstract
In modern technology, inductors are often shaped in the form of planar spiral coils, as in radio frequency integrated circuits (RFIC’s), 13.56 MHz radio frequency identification (RFID), near field communication (NFC), telemetry, and wireless charging devices, where the coils must be designed to a specified inductance. In many cases, the direct current (DC) inductance is a good approximation. Some approximate formulae for the DC inductance of planar spiral coils with rectangular conductor cross section are known from the literature. They can simplify coil design considerably. But they are almost exclusively limited to square coils.
This paper derives a formula for rectangular planar spiral coils with an aspect ratio not exceeding a value between 2.5 and 4.0, depending on the number of turns, and having a cross-sectional aspect ratio of height to width not exceeding unity. It is valid for any dimension and inductance range.
The formula lowers the overall maximum error from hitherto 28 % down to 5.6 %. For specific application areas like RFIC’s and RFID antennas, it is possible to reduce the domain of definition, with the result that the formula lowers the maximum error from so far 18 % down to 2.6 %. This was tested systematically on close to 140000 coil designs of exactly known inductance. To reduce the number of dimensions of the parameter space, dimensionless parameters are introduced. The formula was also tested against measurements taken on 16 RFID antennas manufactured as PCB’s.
The derivation is based on the idea of treating the conductor segments of all turns as if they were parallel conductors of a single-turn coil. It allows the inductance to be calculated with the help of mean distances between two arbitrary points anywhere within the total cross section of the coil. This leads to compound mean distances that are composed of two types of elementary ones, firstly, between a single rectangle and itself, and secondly, between two displaced congruent rectangles. For these elementary mean distances, exact expressions are derived. Those for the arithmetic mean distance (AMD) and one for the arithmetic mean square distance (AMSD) seem to be new.
The paper lists the source code of a MATLAB® function to implement the formula on a computer, together with numerical examples. Further, the code for solving a coil design problem with constraints as it arises in practical engineering is presented, and an example problem is solved.
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References
J. Chen and J. J. Liou, On-Chip Spiral Inductors for RF Applications: An Overview, Journal of Semiconductor Technology and Science, vol. 4, no. 3, 149-167, 2004.
D. Paret, RFID and Contactless Smart Card Applica-tions, John Wiley & Sons, Ltd, West Sussex, 2005.
D. Paret, Antenna Designs for NFC Devices, ISTE Ltd, London, 2016.
R. A. Potyrailo, C. Surman, S. Go, Y. Lee, T. Sivavec, and W. G. Morris, Development of radio-frequency identification sensors based on organic electronic sens-ing materials for selective detection of toxic vapors, Journal of Applied Physics, vol. 106, 124902-1 - 124902-6, 2009.
J. M. Park, K. B. Lee, and K. H. Baek, Apparatus and Method for Wireless Charging, United States Patent US 9,590,446 B2, 2017.
C. Panchal, S. Stegen, an J. Lu, Review of static and dy-namic wireless electric vehicle charging system, Engi-neering Science and Technology, an International Jour-nal, vol. 21, no. 5, 922-937, 2018.
T. P. Theodoulidis and E. E. Kriezis, Impedance evalu-ation of rectangular coils for eddy current testing of pla-nar media, Ndt & E International, vol. 35, no. 6, 407-414, 2002.
R. J. Ditchburn and S. K. Burke, Planar rectangular spi-ral coils in eddy-current non-destructive inspection, Ndt & E International, vol. 38, no. 8, 690-700, 2005.
H. M. Greenhouse, Design of Planar Rectangular Mi-croelectronic Inductors, IEEE Trans. on Parts, Hybrids, and Packaging, vol. 10, no. 2, 101-109, 1974.
H. A. Aebischer, Comparative Study of Analytical In-ductance Formulae for Square Planar Spiral Inductors, Advanced Electromagnetics, vol. 7, no. 5, 3-48, 2018.
J. Crols, P. Kinget, J. Craninckx, and M. Steyaert, An Analytical Model of Planar Inductors on Lowly Doped Silicon Substrates for High Frequency Analog Design up to 3 GHz, IEEE Symposium on VLSI Circuits, Honolulu, Digest of Technical Papers, 28-29, 1996.
S. S. Jayaraman, V. Vanukuru, D. Nair, and A. Chakra-vorty, A Scalable, Broadband, and Physics-Based Model for On-Chip Rectangular Spiral Inductors, IEEE Trans. on Magnetics, vol. 55, no. 9, 8402006, 2019.
S. S. Mohan, The design, modeling and optimization of on-chip inductor and transformer circuits, Ph.D. disser-tation, Dept. Elect. Eng. Stanford University, CA, USA, 2000.
http://cc.ee.nchu.edu.tw/~aiclab/public_htm/VCO/Theses/1999mohan.pdf
H. A. Aebischer, Inductance Formula for Square Planar Spiral Inductors with Rectangular Conductor Cross Section, Advanced Electromagnetics, vol. 8, no. 4, 80-88, 2019.
H. A. Aebischer and B. Aebischer, Improved Formulae for the Inductance of Straight Wires, Advanced Electro-magnetics, vol. 3, no. 1, 31-43, 2014.
M. Kamon, M. J. Tsuk, and J. K. White, FASTHENRY: A Multipole-Accelerated 3-D Inductance Extraction Program, IEEE Trans. on Microwave Theory and Techniques, vol. 42, no. 9, 1750-1758, 1994.
W. M. Haynes, Th. J. Bruno, and D. R. Lide, CRC Handbook of Chemistry and Physics, 95th ed., Internet Version 2015, p. 12-41, 2015.
C. R. Paul, Inductance, John Wiley & Sons, Hoboken NJ, 2010.
E. B. Rosa, The Self and Mutual Inductances of Linear Conductors, Bulletin of the Bureau of Standards, vol. 4, no. 2, 301-344, Washington, 1908.
F. W. Grover, Inductance Calculations: Working For-mulas and Tables, Dover Publications, New York, 2004, first published by D. Van Nostrand Co., New York, 1946.
E. B. Rosa, Calculation of the Self-Inductance of Single-Layer Coils, Bulletin of the Bureau of Standards, vol. 2, no. 2, 161-187, Washington, 1906.
H. A. Aebischer and H. Friedli, Analytical Approxima-tion for the Inductance of Circularly Cylindrical Two-Wire Transmission Lines with Proximity Effect, Ad-vanced Electromagnetics, vol. 7, no. 1, 25-34, 2018.
W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, Resistive and Inductive Skin Effect in Rectangular Con-ductors, IBM J. Res. Develop., vol. 23, no. 6, 652-660, 1979.
A. Gray, The theory and practice of absolute measure-ments in electricity and magnetism, MacMillan & Co., London and New York, vol. II, part I, 1893.
E. B. Rosa, On the Geometrical and Mean Distances of Rectangular Areas and the Calculation of Self-Induct-ance, Bulletin of the Bureau of Standards, vol. 3, no. 1, 1-41, Washington, 1907.
T. J. Higgins, Formulas for the Geometric Mean Dis-tances of Rectangular Areas and of Line Segments, Journal of Applied Physics, vol. 14, 188-195, 1943.
J. C. Maxwell, A Treatise on Electricity and Magnetism, vol. 2., Dover Publications, New York, 1954, una-bridged 3rd ed. of 1891.