*4.2. Verification with Experimentation*

For the experimental verification, both models in Figure 6 were manufactured, one of each. Figure 10 is a picture of the produced motor. Then, the experiment process was performed as follows.

**Figure 10.** The manufactured motors for the experiment: (**a**) 6p/9s IPM and (**b**) 8p/12s IPM.


**Figure 11.** Gauss measurement of the magnet surface.



The results of cogging torque measurements are demonstrated in Figure 14. Figure 14a shows the result of the 6-pole/9-slot motor. Case A had a cogging torque of 56.7 mNmpk-pk, and Case B had 55.1 mNmpk-pk. In Figure 14b the 8-pole/12-slot motor showed 227.3 mNmpk-pk for Case A and 214.8 mNmpk-pk for Case B. As a result, although the shapes of the stator and rotor of the analyzed motors were already optimized for reducing cogging torque, the cogging torque could be improved more by using the proposed method. The main cause of this cogging difference between Case A and B is due to the slot harmonic component of Case B being smaller than Case A, as can be seen in the FFT result of each cogging torque in Figure 15. Consequently, the validity of the proposed method was confirmed again by the experimental results. Overall, since this method only affects the position of each magnet before assembly, it can be compatible with the conventional cogging torque reduction methods using the teeth curvature and rotor shape modulation.

**Figure 12.** PM position change according to the proposed method for the experimental verification for (**a**) 6p/9s IPM and (**b**) 8p/12s IPM.

**Figure 13.** Cogging torque measurement of each motor for (**a**) 6p/9s IPM and (**b**) 8p/12s IPM.

**Table 6.** Gauss measurement of each magnet surface of the 8p/12s model and the changes in magnet position according to the proposed method.


**Figure 14.** The measured cogging torque of each motor for (**a**) 6p/9s IPM and (**b**) 8p/12s IPM.

**Figure 15.** Cogging torque measurement of each motor for (**a**) 6p/9s IPM and (**b**) 8p/12s IPM.

#### **5. Discussion**

The reduction effect cannot be clearly seen in the peak-peak comparison of cogging torque in Figure 14. This is because the LCM component is much larger than the slot harmonic component in both cases. In this case, although the slot harmonic component was reduced, as shown in Figure 15, by the proposed method, the effect is not seen much. If the proposed method is applied to a model that is sensitive to the slot harmonic component, the cogging torque can be effectively mitigated, compared with the results of this paper. In other words, the proposed method has a different effect on the mitigation of cogging torque depending on which harmonic component is dominant.

Additionally, since there are some methods to measure the *Br* or flux density of PM, the real application for applying the proposed method can be manufactured. Among the measurement methods, the simplest example is using Helmholtz coil. As mentioned in the introduction, the proposed method is more appropriate for small quantity customized production than mass production because the Gauss value of each magnet should be measured before the assembly. In the case of mass production, it is possible that if the manufacturing tolerance of the magnetization yoke is adjusted based on the principle of the slot harmonic component mitigation condition, that the influence of the uneven magnetization can be alleviated.

As described above, there are some limitations to the proposed method. However, it is meaningful that we have dealt with the method to compensate manufacturing tolerance (Uneven PM) that has not been covered in the meantime. Furthermore, this method can prevent an increase in cogging torque caused by unevenly magnetized PMs of motors with a high number of poles. Since small scale customized manufacturing process, which adopts the method of the pre-magnetization of magnets before assembly, cannot adjust and compensate for the unevenness of the PMs, by using the proposed method, it will be possible to ensure the cogging performance of a manufactured motor.

#### **6. Conclusions**

In this study, a mitigation method of slot harmonic cogging torque caused by the unevenly magnetized magnet was proposed. This method was drawn through the qualitative analysis of the cogging torque from a macroscopic perspective. As shown in Figure 5, the main process of this method is arranging each magnet according to the non-slot harmonic condition described in Section 3. The validity of the proposed method was verified using FEA and experimentation. Here, the verification was performed by comparing the harmonic components of the cogging torque with and without the proposed method. In this process, it was confirmed that this method is sufficiently effective, even when considering the non-linear material characteristics of the ferromagnetic.

**Author Contributions:** Conceptualization, C.J. and D.L.; methodology, C.J.; software, D.L.; validation, C.J., and D.L.; formal analysis, C.J.; investigation, C.J. and D.L.; resources, C.J.; data curation, C.J.; writing—original draft preparation, C.J. and D.L.; writing—review and editing, C.J. and J.H.; visualization, C.J.; supervision, J.H.; project administration, J.H.

**Funding:** This research received no external funding.

**Acknowledgments:** This work was supported by the Incheon National University under Research Grant 2019-0254 (Corresponding author: Jin Hur).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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