An Approach for Optimal Pre-Conditioning of the Analytical Field Solution of Slotless PM Machines

Analytical modeling of electrical machines has the advantage of remarkable computational efficiency when compared with finite element analysis (FEA). This is especially important for slotless topologies, as they are well suited for 2-D analytical field solutions. Nevertheless, the analytical techniques are doubtlessly non-trivial; besides, the ill-conditioned nature of the problem comes along with the mathematical complexity. The numerical issues are shown to be loosely assessed or even ignored in some parts of the literature, which causes a lack of replicability and low practical usability of analytical approaches. Although researchers often adopt a numerically optimized formulation in their works, the mathematical manipulations that make the solutions to be well-posed and efficiently exploitable are often hidden from the reader since the focus is rather on the theoretical derivation and the exact solutions. This paper shows how the direct field solution of the magnetic field problem, named the raw field formulation (RFF), can lead to significant errors throughout the domain of slotless SPM machines, which vary significantly with the machine geometry. Then, an approach to reach the numerically optimal form of the field solution, named optimized field formulation (OFF), is proposed, comprehensibly described, and discussed. Finally, the closed-form expression and the optimal pre-conditioner underneath are explicitly presented and shown to outperform the accuracy of other pre-conditioned formulations used in the literature (including RFF). The OFF's performance is significant, especially at higher harmonic orders