This paper presents a method for the calculation of the self- and mutual inductance of coils surrounded by air, positioned on top of a ferromagnetic plate and placed in a cavity inside a ferromagnetic plate. The magnetic fields around an array of air-cored coils are obtained by means of the three-dimensional magnetic vector potential using Fourier Analysis. The current density distribution of a coil is modeled by four straight bars, three different configurations to position these bars against each other are presented as an alternative to model the round corners of a coil. An error of maximum 5% has been obtained in the analytically calculated flux density distribution compared to the values obtained from 3D finite element simulations. Self- and mutual inductances are calculated for each current density description and compared with finite element simulations and measurements, a good agreement has been found for coils modeled by four overlapping or four trapezoidal bars.