Since cogging torque (CT) leads to torque fluctuations, it needs to be diminished. As a result, researchers have developed numerous mitigation methods to reduce the CT of flux-switching permanent magnets (FSPMs). For instance, the effects of the short permanent magnet and flux bridges on the inner or outer side of the stator have been considered for CT mitigation [1]. Researchers have also investigated various teeth-notching schemes to eliminate torque fluctuation [2, 3]. The influence of rotor pole shaping on reducing the CT of both radial and axial flux FSPMs has been explored as well [4–6]. Furthermore, an experimental study employed the tooth chamfering method for CT minimization [7, 8]. In addition, the unequal width of the stator tooth has been applied to outer rotor permanent magnet machines for CT reduction [9]. Various core shapes, such as V-, C-, and E-shaped stator structures, have also been investigated [10, 11]. In this context, the magnetic permeance of the air gap has been observed in papers that adopted the subdomain [12, 13] and conformal mapping [14, 15] methods as mathematical approaches. Meanwhile, some studies used the co-energy [16, 17] and superposition [18] techniques to obtain the CT, while others comprehensively discussed more such techniques, including cos-chamfering [19, 20].

Notably, the majority of CT mitigation research has focused on machine modification, which involves changing the geometry of the stator or rotor around the air gap. As a result, implementing machine modification to achieve magnetic permeance manipulation is a conceivable option. In the case of FSPMs, the simplicity of their rotor side has helped researchers construct rotors for magnetic permeance distribution (MPD) manipulation to reduce CT.

In this paper, an MPD model is developed to determine the unequal rotor slot arc (URSA) method, following which the URSA technique is applied to achieve CT mitigation. Next, parametric optimization is implemented to diminish CT by conducting finite element analysis (FEA) simulations. Finally, the results obtained for chosen four designs and additionally three designs which proposed in the literature [2, 4, 19] are compared using the proposed URSA method.

The CT approach for a standard FSPM machine can be derived from the air gap MPD and the magnetomotive force (MMF) model, as depicted in Fig. 1 (See Table 1 for the nomenclature list of the parameters).

The CT (T_cogg) can be obtained by dividing the negative derivative of the stored magnetic co-energy in a machine by the angular rotor position along the air gap perimeter, as expressed in the following equation:

Here, θ refers to the angular position along the air gap perimeter. This calculation can be further simplified by presuming that the iron core has zero stored magnetic energy.

Fig. 1 depicts a typical 12s/10p FSPM machine structure [21]. A stator comprises three materials—core, winding, and magnets. The slots of the stator are placed between the magnets, while the coils cover them. Notably, since the salient poles in FSPMs are placed around the rotor, similar to the structure of switched reluctance machines, they possess a doubly salient architecture, as shown in Fig. 1(a). The MPD and MMF waveforms of an equal-slotted rotor FSPM machine are depicted in Fig. 1(b) and 1(c). The MPD in the air gap, as shown in Fig. 1(b), can be expressed in the Fourier series as follows:

Meanwhile, the MMF, presented in Fig. 1(c), can be obtained after Fourier analysis, formulated as follows:

Considering these inferences of the MMF function F², MPD function Λ², and the total stored energy in the air gap W, the torque equation can be reconsidered. Therefore, after implementing some mathematical adjustments, the CT approach can be formulated as follows:

where i and j are

The CT approach for standard FSPMs, as noted in Eq. (4), specifies its contents. Since the total MMF and total MPD in the airgap directly affect CT, most CT mitigation methods proposed in the literature have focused on examining this correlation. In this paper, the URSA technique is employed to manipulate this correlation.

Since the rotor side of FSPM machines is entirely made of steel, it is considered suitable for manipulating the MPD or MMF since its rotor side can be shaped more easily than other permanent magnet machines. Therefore, in this study, the focus of the CT mitigation process is especially on MPD and the rotor. The rotor slots were modified, as shown in Fig. 2(b), to change the MPD according to Eq. (4). As a result, the electromagnetic torque equation had to be modified to express the structure of the URSA method in Eq. (4).

Subsequently, the CT approach incorporating the URSA method was implemented. Fig. 2 depicts the rotor side configuration of FSPMs using the URSA technique in comparison to that in standard FSPMs.

Fig. 3 illustrates the change in MPD when using the URSA technique. Notably, this modification is bound to affect the MPD equation, denoted as Λ² in Eq. (2). To reflect this change, the expression of Λ² can be modified as follows:

Due to the modification of the MPD approach from Eq. (2) to (6), the Fourier interval changed from π/n_r to 2π/n_r. By resolving the analytical approach, the URSA method was found to be applicable for reducing CT in 12s/10p FSPMs. Therefore, a 12s/10p FSPM machine structure was utilized in the simulation stage.

For the optimization study, a 12s/10p structure was employed owing to the analytical restrictions of the URSA technique mentioned in the previous section. The basic FSPM machine parameters are presented in Table 2. Notably, these characteristics were kept constant for each of the designs analyzed in this study to present the results of the URSA method in a fair manner. The FSPM machine model was designed and verified based on [21], which was experimentally proven. Fig. 4(a) depicts the rotor of the 12s/10p FSPM machine consisting of equal rotor slot arcs. Fig. 4(b) shows the average torque waveform of the FSPM machine, which achieved an average value of 8.88 Nm. Furthermore, the waveform depicts a peak-to-peak torque ripple value of 0.819 Nm (9.2%). Fig. 4(c) illustrates the back EMF waveform at 1,200 rpm, and Fig. 4(d) shows the magnetic flux density and flux lines of the basic FSPM machine, which were confirmed results in the literature. In the current study, this basic FSPM machine with equal rotor slot arcs was titled Design 0 (D0) and was considered the reference design. The operational parameters of the simulations conducted in this study are presented in Table 3.

Fig. 5 illustrates the arc angle parameters of the optimization. For parametric optimization, two different angles, titled θ_a and θ_b, that could freely move the rotor pole in a radial direction were defined as the optimization parameters.

Presuming that the angles of the equal rotor slot arcs of a basic FSPM machine is zero, the angles of the unequal rotor slot arcs were set within the range of ±2° as constraints to prevent the rotor poles from overlapping. Furthermore, the URSA technique was utilized for the two consecutive poles of the rotor. In other words, θ_r1 and θ_r2 angles in Fig. 5 were modified by ±2°.

These two angle parameters affected the rotor slot arcs, as shown in Eq. (7), which directly affected the MPD in Eq. (6).

After performing more than 2,000 FEA simulations, five rotor slot arcs were chosen as the best results to depict the advantages of the URSA technique. Furthermore, three more methods were simulated on the FSPM model—cos-chamfering [19], notching [2], and adding a flange [4]—all of which involve shaping the rotor pole of FSPM machines.

The average torque waveforms of the nine designs examined in this study, including that of the basic FSPM machine, are illustrated in Fig. 6. The designs that incorporated the URSA technique displayed a lower peak-to-peak torque ripple. Furthermore, Table 3 presents the rotor slot arc differences of D1– D5. Overall, the designs that used the URSA method exhibited significant torque ripple mitigation.

The results of the spectral analyses performed on all designs are demonstrated in Fig. 7, while Fig. 8 traces the back EMF waveforms of the nine designs.

Fig. 9 shows the simulated CT waveforms for the nine designs. Among these, the five designs using the proposed URSA method exhibited a substantial reduction in CT. While D0 attained a 2 Nm peak-to-peak CT value, D1 registered the lowest CT value, achieving a 0.36 Nm peak-to-peak value. Notably, by implementing the URSA method on D1 according to the specifications D0, an 82% reduction in the CT was achieved.

Furthermore, magnetostatic FEA was conducted on the simulations for each design. Fig. 10 demonstrates the magnetic flux lines and flux density distributions of the nine designs. The electromagnetic torque characteristics of each design are summarized in Table 4.

The acceleration and climbing abilities of machines rely heavily on their overload capacity. In Fig. 11, the electromagnetic torque averages of the nine machines obtained through two-dimensional FEA are presented against the current densities. Notably, D4 demonstrated the highest torque outputs across the entire current density spectrum, displaying superior torque capabilities even under overload conditions.

Furthermore, the rated current was 3.8 A, as shown in Table 4, which was kept constant for all simulations. However, when the current changed from 0 to 6, the average torque varied along with the change in current. Fig. 11 depicts the outcomes of the torque density (overload capability) simulations, showing the varying cycles of the URSA method. The average torque of the designs was compared to find that those using the URSA method did not have high average torque deflection, meaning that they could be reasonably applied. Meanwhile, Fig. 12 illustrates the output power in terms of changes in the current, with the design that incorporated the URSA method exhibiting reasonable outcomes for mitigating CT.

In this study, a novel CT mitigation method is evaluated, and an analytical approach aimed at understanding the effects of MPD on CT expression is proposed. To calculate the CT reduction, the MPD was manipulated using the analytical model for CT. Furthermore, an iterative FEA analysis was performed for the simulations, using nine different designs to conduct a comparison. According to the simulation and optimization outcomes, D1 achieved the best CT value, attaining 82% CT mitigation and only 4.1% average torque loss. These findings prove that the URSA technique has immense potential for CT reduction in FSPM machines. Moreover, this approach makes FSPM machines quieter and less prone to vibration, rendering them suitable for use in highly sensitive applications, such as in airplanes, electric vehicles, and air conditioners.