Comparative study of the accuracy of analytical inductance formulae for square planar spiral inductors

Abstract

In the design of radio frequency (RF) microelectronic integrated circuits (IC’s) and of antennas for short-wave radio frequency identification (RFID) and telemetry systems, planar spiral coils are important components. Many approximate analytical formulae for calculating the inductance of such coils can be found in the literature. They all approximate the direct current (DC) inductance. Its applicability to high frequencies depends on the ratio of the conductor width to the skin depth. In many cases, the DC inductance is a good approximation in RFID and telemetry systems at 13.56 MHZ, and in RF IC’s up to about 4 GHz. The formulae can simplify the problem of designing inductors to a predefined inductance considerably. But the error statistics given by different authors cannot be compared because they are based on different or unknown domains of definition. Hence, it is not possible to decide which formula is best in a given case by merely studying the literature. This paper compares the maximum relative errors of six of some of the most cited formulae in the literature. Five of them are scalable, which means that they are valid for all coil sizes and all inductance ranges. To all formulae, the same domains of definition are applied. Each of them spans all four dimensions of the parameter space. Precise inductances are obtained numerically with the help of the free scientific and industrial standard software FastHenry2 and used as reference values to calculate the errors of the formulae. It has been found that the alleged maximum errors reported by some authors are far too optimistic. Only two formulae feature small enough errors to be useful in circuit design. The method and the domains of definition applied in the present study may also prove useful for the assessment of future formulae.