**5. Effect of Angle Shift on Cogging Torque**

In practice, magnet poles are neither made identically nor placed at the perfect location. There are a lot of methods available to obtain asymmetrical rotor structures. In this work, the method of shifting angle between the poles was adopted. Maintaining the L/τ ratio between 0.6 and 0.8, shifting angles were considered between 1◦ and 8◦ . From the analysis, it is understandable that the cogging torque is very sensitive to variations in the magnet angle. Figure 6 shows the procedure for finding the shifting angle with the minimum cogging torque.

In order to minimize cogging torque, a permanent magnet shift is considered here. A range of distinct angles were considered to analyze the effect of the angle shift on the rotor magnets. The permanent magnets on the rotor were shifted from 1◦ to 8◦ . The actual position of the permanent magnets was 90◦ . For every 1◦ change, four possible combinations were obtained. Table 4 shows the tabulation of the results of cogging torque with variation of magnet shifting from 1◦ to 8◦ . It shows that the shifting angle with 3◦ has the minimum cogging torque. The lowest cogging torque is 0.16 Nm when the angle between the poles A-B is 87◦ , B-C is 93◦ , C-D is 87◦ and D-A is 93◦ . Figure 7 shows the placement of the rotor magnet after applying the magnet shift to the rotor magnets. From the figure, it is clear that when the poles A, B, C, and D have a 90◦ difference with respect to each other, then each magnet has a 5 mm difference. Figure 7a shows shifting angle between the magnet is 1◦ . The spacing between A-B and C-D 4.44 mm and B-C and D-A is 5.56 mm. Figure 7b shows shifting angle between the magnet is 2◦ . The spacing between A-B and C-D is 5.58 mm and B-C and D-A is 4.42 mm. In Figure 7c the shifting angle between the magnet is 3◦ . The spacing between A-B and C-D is 3.32 mm and B-C and D-A is 6.68 mm. in Figure 7d the shifting angle between the magnet is 4◦ . The spacing between A-B and C-D is 7.24 mm and B-C and D-A is 2.76 mm. Figure 7e shows shifting

angle between the magnet is 5◦ . The spacing between A-B and C-D is 2.2 mm and B-C and D-A is 7.8 mm. Figure 7f shows shifting angle between the magnet is 6◦ . The spacing between A-B and C-D is 8. 38mm and B-C and D-A is 1.62 mm. In Figure 7g the shifting angle between the magnet is 7◦ . The spacing between A-B and C-D is 8.95 mm and B-C and D-A is 1.05 mm. When the shift angle becomes 8◦ in Figure 7h, the spacing between A-B and C-D is 9.52 mm, and the spacing between B-C and D-A is 0.48 mm. If we again increase the shift angle to 9◦ , the two magnets merge, and will act like a two-pole machine.

**Figure 6.** Flowchart for the procedure for finding shifting angle with minimum cogging torque.


**Table 4.** Tabulation of results of cogging torque with variation of magnet shifting from 1◦ to 8◦ .

> Table 5 shows the comparison of the 3D FEA results of cogging torque when shifting the magnetic pole angle from 1◦ to 8◦ . The graphical representation of variation in the cogging torque with magnet shifting is shown in Figure 8. From the above comparison, when the shifting angle is 3◦ , the lowest cogging torque of 0.16 Nm is obtained. The base rotor has a cogging torque of 0.64 Nm. Compared with the base rotor, the new asymmetrical rotor structure exhibits a 75% reduction in cogging torque.

**Figure 7.** Rotor magnet shift from 1◦ to 8◦ .



**Figure 8.** FEA results of cogging torque with varying shifting angle.

One of the main reasons for the existence of cogging torque is the change in magnetic flux density. Assessment of flux density and assessment of flux lines are two pivotal steps in FEA. Whenever the magnet displacement is applied to the rotor, the flux density is lower than that of the symmetrical structure. Though the flux density remains the same in many regions of the BLDC motor, the maximum flux density, indicated by yellow color, is attained in some parts of the stator (in addition to the permanent magnets). Red areas indicate undesirably high flux density, which may result in hot spots that could damage the motor. When applying magnet displacement in the rotor, the flux density decreases. This causes a reluctance to change that is comparatively better than the existing method, resulting in a reduction in cogging torque. Figures 9 and 10 show the flux plot distribution in symmetrical and asymmetrical rotors.

**Figure 9.** Flux plots in symmetrical rotor.

**Figure 10.** Flux plots in the asymmetrical rotor.

Figure 11a shows the symmetrical rotor structure with an angle difference of 90◦ , Figure 11b shows the asymmetrical rotor structure with 3◦ magnet shifting. There is a small difference in the spacing of the permanent magnet. In the symmetrical design, all the rotor magnets are placed equally at a distance of 5 mm from one another, and in the asymmetrical structure, the spacing between magnets A and B is 3.32 mm, and the spacing between magnets B and C is 6.68 mm. Table 6 shows a comparison of the cogging torque between the symmetrical and asymmetrical rotors.

**Figure 11.** Compared Rotor structure (**a**) Symmetrical rotor structure with an angle difference of 90◦ ; (**b**) asymmetrical rotor structure with 3◦ magnet shifting.

**Table 6.** Comparison of cogging torque for the symmetrical and the asymmetrical rotor.


Table 7 presents a comparison of cogging torque results between simulation and analytical method. When the shifting angle is 3◦ , both methods have almost the same results. Figure 12 shows a graphical representation of the FEA results and the analytical results.


**Table 7.** Comparison of simulation and analytical result.

**Figure 12.** Comparison of FEA and analytical results.

The magnet shifting can reduce the cogging torque effectively without deteriorating the trapezoidal shape of the back-EMF. Figure 13 shows the back-EMF of the SPMBLDC motor with different magnet shifting angles. In this figure, 0◦ represents all four magnets being placed at an exactly 90◦ phase difference from one another, and is represented using blue color. The yellow color, which is very close and similar to the 0◦ case, phase shifts the back-EMF curve with a 3◦ phase shift

**Figure 13.** Comparison of back-EMF.

Figures 14–17 show the transient 3D results and the performance characteristics of symmetrical and asymmetrical rotor magnets. Even when the rotor magnetic angles are shifted, the acceleration of the BLDC motor is almost constant, and the initial speed is also maintained at a constant level, meaning that the speed of the BLDC motor increases linearly with respect to time. Figure 13 shows the time vs. speed characteristic of symmetrical and asymmetrical rotor magnets. Orange color represents the speed of the motor with the base model and blue color shows the speed of the motor with a 3-degree phase shift. From the figure, it is evident that the motor with a 3-degree phase shift is able to achieve greater speed than the base model within the specified time. Figures 14–16 present comparisons of the magnetic torque, load torque and net torque of the symmetrical magnet and the asymmetrical magnet. From the above figures, it is clear that the rotor with the 3-degree magnet shift has excellent torque vs. time characteristics when compared with the base model.

**Figure 14.** Speed vs. time.

**Figure 15.** Magnetic force/torque vs. time.

**Figure 16.** Load force/torque vs. time.

**Figure 17.** Load force/torque vs. time.

Figures 18 and 19 represent the cogging torque and the voltage under transient conditions. For cogging torque, the transient starts at 0.15 and increases to 0.45 Nm for a particular period before settling down to 0.16 Nm. Considering the transient scenario for voltage, it starts from 0 V and increases to 0.24 V, before after a particular period decreasing to 0.05 V and settling at 0.084 V.

In reference [29], in order to reduce cogging torque, a magnet shifting technique was adopted. The authors considered a 4 pole 12 slot machine. Table 8 shows the comparative results of the existing and proposed design.

**Table 8.** Comparative results of the existing and proposed design.


**Figure 18.** Transient response of cogging torque.

**Figure 19.** Transient response of voltage.

On the basis of existing work, it is clear that the new proposed model offers a 60% reduction in cogging torque.

An undesirable effect that occurs in permanent magnet motors during shaft variation is torque ripple. This is a periodic increase and decrease in output torque. In BLDC motors, cogging torque is a crucial factor contributing to torque ripple. Figure 20 presents the resulting torque ripple waveform following a 3◦ magnetic shift. This is the difference between the maximum torque and the minimum torque compared to the average torque [30]. The rated torque of the motor was 1.424 Nm. The maximum, minimum and average values of electromagnetic torque were 1.18 Nm, 0.6 Nm and 1.1 Nm, respectively. Therefore, the degree of torque ripple was 52.7%.

**Figure 20.** Torque ripple waveform with 3◦ magnetic shift.
