Inductance Formula for Square Spiral Inductors with Rectangular Conductor Cross Section

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.