Abstract
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.
Highlights
Slotless permanent magnet (PM) machines have been shown to provide interesting features and performances in several application areas
The perfect correspondence between the analytical solution and finite element analysis (FEA) clearly suggests that the optimized field formulation (OFF) was adopted in the simulations
Even though the solution proposed in [26] exhibits a better numerical behaviour than the raw field formulation (RFF), which has a condition number diverging to infinite at low harmonic orders, it clearly represents a sub-optimal solution if compared to the OFF
Summary
Slotless permanent magnet (PM) machines have been shown to provide interesting features and performances in several application areas. With a particular focus on the analytical field solution’s numerical performance, the paper aims to highlight an issue that tends to be overlooked in the current literature. This may lead to a lack of reproducibility of analytical frameworks proposed by the authors, they are theoretically exact. A. CONTRIBUTION OF THE PAPER In this article, different case studies for the slotless machine topology are used to address the described numerical issue. It is an extension of what is reported in [25], where only the PM field solution was studied.
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