Abstract
Planar spiral coils are used as inductors in radio frequency (RF) microelectronic integrated circuits (IC’s) and as antennas in both radio frequency identification (RFID) and telemetry systems. They must be designed to a specified inductance. From the literature, approximate analytical formulae for the inductance of such coils with rectangular conductor cross section are known. They yield the direct current (DC) inductance, which is considered as a good approximation for inductors in RF IC’s up to the GHz range. In principle, these formulae can simplify coil design considerably. But a recent comparative study of the most cited formulae revealed that their maximum relative error is often much larger than claimed by the author, and too large to be useful in circuit design.
 This paper presents a more accurate formula for the DC inductance of square planar spiral coils than was known so far. It is applicable to any design of such coils with up to windings. Owing to its scalability, this holds irrespectively of the coil size and the inductance range. It lowers the maximum error over the whole domain of definition from so far down to . This has been tested by the same method used in the comparative study mentioned above, where the precise reference inductances were computed with the help of the free standard software FastHenry2. A comparison to measurements is included. Moreover, the source code of a MATLAB® function to implement the formula is given in the appendix.
Highlights
Planar spiral inductors are used in radio frequency (RF) microelectronic integrated circuits (IC’s) [2] and both as 13.56 MHz radio frequency identification (RFID) [3] and telemetry antennas [4]
An improved formula for the direct current (DC) inductance of square planar spiral coils with rectangular conductor cross section has been derived. It is based on purely physical principles on one part, and on a correction factor on the other
Since the correction factor only depends on dimensionless parameters, the derived formula remains scalable, i.e. it is valid irrespectively of the coil size and the inductance range
Summary
Planar spiral inductors are used in radio frequency (RF) microelectronic integrated circuits (IC’s) [2] and both as 13.56 MHz radio frequency identification (RFID) [3] and telemetry antennas [4]. Many researchers have worked on finding approximate inductance formulae that explicitly depend on the design parameters Using such a formula is by far the easiest and fastest way to calculate coil inductance, for solving inverse problems. Subsequent inspection of the extensive data collected during the course of [1] has revealed that the relative errors of the inductance formula of [9], after correcting an erroneous equation, seem to correlate with the number of windings N and the filling factor ρ (which is a normalized measure of the extent to which the area taken by the coil is used up or covered by its windings) This observation has spurred the hope for improving the formula by amending it by a correction factor depending on N and ρ.
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