Vibroacoustic Analysis of Radial and Tangential Air-Gap Magnetic Forces in Permanent Magnet Synchronous Machines

This paper analyzes Maxwell tensor tangential and radial magnetic forces in permanent magnet synchronous machines in the no-load case. Using matrix notation in the complex domain, a simple expression of the Fourier harmonics of both radial and tangential forces is derived, including all space and time harmonics. These expressions prove that both the frequency content of cogging torque and zeroth-order radial forces are linked to the least common multiple between the stator slot number and the rotor poles number, and that the optimal pole arc to pole pitch ratio to reduce cogging torque is also optimal for the reduction of average radial magnetic forces. It is also shown that both the smallest non-zero spatial order of tangential and radial force harmonics are given by the greatest common divider of the number of slots and the number of poles. These results can be used during the design stage when choosing the pole and slot numbers combination. These analytical results are then compared with calculations using MANATEE vibroacoustic and electromagnetic simulation software. Finally, some variable speed acoustic noise simulations are carried out on three different designs to analyze the efficiency of different vibroacoustic design rules on the slot and pole numbers combination. An attempt to formulate a new vibroacoustic design rule choice is detailed. It is concluded that no simple analytical design rule can be used to evaluate noise and vibrations induced by magnetic forces, and that numerical simulation is necessary.