Inductance Formula for Rectangular Planar Spiral Inductors with Rectangular Conductor Cross Section

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.