An Analytical Solution of a Multi-Winding Coil Problem with a Magnetic Core in Spherical Coordinates

Nowadays, technology is advancing rapidly in parallel with developments. The traditional concept of machine loses its function and is replaced by particular devices with spherical geometry such as spherical electric motors and brain stimulation systems. Consequently; the calculation of self-inductance (L-ii) and mutual inductance (M-ij) coefficients in the spherical coordinate system with analytical or semi-analytical methods has become one of the major research topics in recent years. In this study, the terms of B, E, and A for multi-winding coils were calculated analytically by using the single-winding coil approach. The same geometries were calculated with the assumption of axial symmetry in ANSYS Maxwell using finite element analysis (FEA). The obtained results with the FEA and analytical calculations were compared. Finally, two concentric coil geometries with magnetic core with radius r1 were determined, the variation of the self-inductance (L-ii) and mutual inductance (M-ij) coefficients of the determined geometries based on the gamma angle in spherical coordinates were calculated analytically. Simulation studies were conducted by creating 3D models of coil geometries in ANSYS Maxwell program. An experimental setup that can be produced with 3-dimensional (3D) printers has been designed and constructed compatibly with the determined geometries. The variation of L-ii and M-ij coefficients with. was studied experimentally after the production of necessary coil windings. At the end of the study, it was observed that the analytical results collected for the mutual inductance M-ij and the self-inductance coefficient L-ii were consistent with each other by comparing the FEA and experimental results.