In magnetic coupling resonant wireless power transfer devices, planar spiral coils are often used as the coupling mechanism. In such application area, the coils are often wound sparsely with wide screw pitches and often under complex relative position. An accurate model of mutual inductance for planar spiral coils wound either densely or sparsely is established in this article. Utilizing Euler Angles, an exact expression of mutual inductance for a couple of planar spiral coils under arbitrary relative position is proposed. Based on this expression and geometric mean distance principle, a self-inductance expression for planar spiral coils is derived. Based on numerical calculation of the expressions proposed in this article, the influence of precession and spin angle on mutual inductance is studied; the method of determining radius of each turn when using traditional concentric circular filaments cluster approximation model for self- and mutual inductance calculation is studied; the difference of calculation results between the accurate model and the traditional approximation model is compared. Finite element simulation and experimental results verify the correctness of the proposed models.

Considering the helicity of Archimedean spiral coils, this paper proposes accurate expressions of mutual inductance and their numerical calculation methods, which can be applied in the wireless power transmission field, etc. Accurate expressions of mutual inductance are deduced respectively for two coils that are coaxial, laterally misaligned, or non-parallel, and numerical calculations are performed using Gaussian integration as well. In the case of coaxial coils, the calculation results are verified by the 3D finite element method (3D FEM) and compared with the results gained by the traditional method that approximates two spiral coils to two clusters of series-connected circular coils ignoring helicity. The comparison of the three methods shows that results achieved by the proposed expression are close to that of 3D FEM, while there is increasing error with the screw pitches of the coils when using the traditional circular coil approximation method. The influence of relative position on the mutual inductance of the two coils is also studied and it is further explained through magnetic field distribution. Finally, the validity of the proposed expressions of mutual inductance is verified by experimental results.