Arbitrarily shaped coils' inductance simulation based on a 3-dimensional solution of the Biot-Savart law

The analytical calculation of self-inductance is possible for ideally shaped long coils only. To obtain the inductance of a short helical coil there are some approximation formulas based on empirical studies. In this paper we propose an algorithmic solution to the spatial magnetic flux density-or magnetic induction - of arbitrarily shaped current paths based on the law of Biot-Savart. Knowing the non-uniform magnetic induction the corresponding magnetic flux through each mesh cell inside the coil is calculated. Therefore, the coil and its confined volume are discretized and fed to the algorithm. The fast and stable calculation process offers parameterization of the underlying mesh grid and can therefore be driven to better convergence. Once the magnetic flux is obtained the resulting self-inductance of the coil is determined. To give proof to the algorithm's quality, its output is compared to the analytical inductance of a long cylindrical coil. Furthermore, a short coil's inductance based on empirical studies is obtained and used for verification of the proposed algorithm. The output is also compared to a solution obtained by a commercially available solver resulting in conformity of the calculated values.