Effect of Third Harmonic Flux Density on Cogging Torque in Surface-Mounted Permanent Magnet Machines

Cogging torque, inherent in permanent magnet machines, is closely related to the airgap flux density harmonic contents. The third harmonic can be utilized in combination with the fundamental one to increase the average torque, whilst its relationship with cogging torque is uncertain. The aim of this paper is to investigate the effect of third harmonic in airgap flux density on cogging torque in surface-mounted permanent magnet (SPM) machines considering different slot/pole (S/P) combinations. First, based on the simplified expressions of cogging torque in machines with sinusoidal plus third harmonic (Sine+3rd) airgap flux density, S/P combinations are classified into four cases for discussion. It is found that when N-s/GCD (N-s, N-p) >= 4 (N-s: slot number; N-p: pole number; GCD: great common divisor), third harmonic in airgap flux density has no effect on cogging torque. Otherwise, cogging torque will be generated. Furthermore, two approaches for obtaining the Sine and Sine+3rd airgap flux density, i.e., Sine/Sine-+3rd h(m), and Sine/Sine+3rd B-r, are validated, based on which finite-element analyses are carried out on four machines, i.e., 12-slot/10-pole (12S12P), 12S6P, 12S8P, and 12S10P to verify the conclusion. Finally, the 12S8P and 12S10P Sine+3rd SPM machines are prototyped and tested for validation.