Axial Flux Permanent Magnet Synchronous Motor Cogging Torque Calculation Method Based on Harmonic Screening

Abstract

This paper proposes a harmonic screening-based method for calculating the cogging torque of the axial flux permanent magnet synchronous motor. The magnetic field energy in the air gap is derived from the air gap flux and the magnetomotive force of rotor. The cogging torque is then obtained using the energy-based method. Compared with finite element analysis, the proposed approach is significantly faster while maintaining high accuracy. It is particularly effective for scenarios involving stator staggering, which can facilitate quick calculation of cogging torques of many different staggering angles, offering rapid insights into motor performance during the initial design. The method achieves a similarity accuracy with FEA results and reduces computation time, demonstrating both its efficiency and reliability.

1. Introduction

The axial flux permanent magnet synchronous motor (AFPMSM), a high-performance motor type, offers several advantages, including high torque density, high efficiency, effective heat dissipation, and a compact axial profile. These features make it suitable for a wide range of applications in transportation, industrial manufacturing, and aerospace [1]. However, due to a relatively larger air gap magnetization compared to radial flux motors, AFPMSMs typically exhibit higher cogging torque for the same power level. As a result, cogging torque prediction and mitigation have long been recognized as critical issues in AFPMSM design. Extensive studies in the literature have established that three primary approaches are employed to predict the cogging torque of permanent magnet synchronous motors (PMSMs): simulation-based method via finite element analysis (FEA) [2,3], magnetic circuit method utilizing magnetic equivalent circuits [4,5,6,7], and analytical method based on energy calculations [8,9,10,11]. Among these methods, the simulation-based approach offers high accuracy but suffers from long computation times and substantial computational resource demands, particularly when applied to three-dimensional models such as those of AFPMSMs. The magnetic circuit method can achieve accuracy comparable to FEA, which has a high accuracy of 93.3% [12]. However, it involves a complex modeling process, limited model generality, and requires extensive expertise and experience in magnetic modeling.

The analytical method offers rapid computation and is well suited for gaining preliminary insights into motor performance during the early design stages of AFPMSMs. While extensive research has been conducted in recent years on cogging torque analysis using analytical methods—particularly for radial flux permanent magnet synchronous motors—applying such methods to AFPMSMs remains challenging due to their distinct structural and magnetic characteristics. These studies have addressed diverse design features, including different permanent magnet shapes (e.g., V- and U-shaped) [13], asymmetric stator and rotor eccentricity [14], and uneven air gap [15]. Similarly, numerous studies on the effect of stator–rotor configuration on cogging torque have also been carried out in AFPMSM, such as grouped stator permanent magnets [16], slanted poles and slots [17], various slot openings and tooth profiles [18], permanent magnet edge trimming [19], and unequal pole arc coefficients [20].

Nevertheless, the accuracy of the analytical method remains limited, as it often neglects critical harmonic components in physical quantities and introduces spurious harmonics during modeling, resulting in significant deviations from actual behavior. These works highlight that cogging torque mitigation in AFPMSMs is a well-established research area. However, many existing analytical models struggle to generalize across different AFPMSM structures—particularly complex ones such as staggered double-stator configurations—and often omit or misrepresent key harmonic interactions. As a result, improving the computational accuracy of cogging torque modeling and generalizing the calculation methodology continues to be a major challenge impeding the further development and practical deployment of this approach.

To address these challenges, this paper proposes an analytical cogging torque computation method based on harmonic screening, which systematically captures the interactions between magnetic harmonics and torque generation. The method is applicable to both single and double rotor AFPMSMs and enables further design optimization of the stator and rotor to accommodate additional performance requirements, such as reverse saliency. It achieves a level of accuracy comparable to that of the magnetic equivalent circuit method while offering significantly higher computational efficiency, requiring only one-tenth the computation time of FEA to attain over 90% of its accuracy. This method is thus particularly suitable for parametric studies and initial design iterations of AFPMSMs.

The remainder of this paper is organized as follows. Section 2 introduces the energy-based approach for cogging torque calculation and presents an in-depth investigation of the key factors influencing cogging torque magnitude, along with the theoretical foundation and methodology of harmonic screening. Section 3 details the practical computation methods for each influencing factor. Section 4 presents the numerical results for cogging torque in stator-staggered AFPMSMs using the proposed method and provides a thorough analysis of how stator staggering contributes to torque reduction. Finally, Section 5 summarizes the findings and concludes the paper.

2. AFPMSM Cogging Torque Calculation Method Based on Harmonic Screening

2.1. Cogging Torque Calculation with the Magnetic Field Energy Method

For AFPMSM, cogging torque can generally be calculated using the magnetic field energy method. The cogging torque in a no-load situation can be expressed as [21]:

$$T_c=-{\int }_0^{\mathrm{g}}{\displaystyle \frac{\partial W}{\partial \theta }}dz$$

(1)

where $\theta$ is the mechanical angle, g is the length of the gap, $T_{\mathrm{c}}$ is the cogging torque, and W is the total magnetic energy, which can be formulated as

$$W={\int }_V{\displaystyle \frac{{B_{\mathrm{g}}}^2\left(\theta ,z,t\right)}{2{\mu }_0}}d{V}_{\mathrm{z}}$$

(2)

where ${\mu }_0$ is the vacuum permeability, and V is the air gap area. The air gap flux density $B_{\mathrm{g}}(\theta ,z,t)$ is derived from the product of rotor magnetomotive force (MMF) $f_{\mathrm{r}}(\theta ,z,t)$ and the stator permeance ${\Lambda }_{\mathrm{s}}$:

$$B_{\mathrm{g}}(\theta ,z,t)={\displaystyle \frac{f_{\mathrm{r}}(\theta ,z,t)·{\Lambda }_{\mathrm{s}}(\theta ,z)}{S_{\mathrm{s}}}}$$

(3)

where $S_{\mathrm{s}}$ is the effective flux area.

2.2. Rotor MMF Distribution

The rotor MMF in AFPMSMs incorporates axial dependency due to dual-stator configurations. For a single rotor,

$$f_{\mathrm{r}}(\theta ,z,t)=\sum _{{\nu }_{\mathrm{r}}\in \mathrm{odd}}{F}_{\mathrm{r}}({\nu }_{\mathrm{r}},z)cos\left({\nu }_{\mathrm{r}}p(\theta -\omega t)\right)$$

(4)

where ${\nu }_{\mathrm{r}}$ is the odd integer of the harmonic order of rotor MMF, $F_{\mathrm{r}}({\nu }_{\mathrm{r}},z)$ is the amplitude of rotor MMF, which is determined by PM geometry and magnetization, $\omega$ is the mechanical angular speed of the rotor, and p is the pole pairs of the motor.

2.3. Stator Permeance Function

The stator permeance ${\Lambda }_{\mathrm{s}}(\theta ,z)$ is influenced by slot openings and axial airgap variations:

$${\Lambda }_{\mathrm{s}}(\theta ,z)=\sum _{{\nu }_{\mathrm{s}}=0}^{\infty }{A}_{\mathrm{s}}({\nu }_{\mathrm{s}},z)cos\left({\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta \right)$$

(5)

where ${\nu }_{\mathrm{s}}$ is the harmonic order of the slot-induced permeance, $A_{\mathrm{s}}({\nu }_{\mathrm{s}},z)$ is the amplitude of the slot-induced permeance, which depends on the axial position z, and $N_{\mathrm{s}}$ is the number of stator slots.

2.4. Magnetic Energy Integration Considering the Axial Direction

The total magnetic energy W is computed by integrating $B_{\mathrm{g}}^2$ over the 3D airgap volume. To handle axial non-uniformity, the airgap is divided into $N_{\mathrm{z}}$ thin layers:

$$W=\sum _{z=1}^{N_{\mathrm{z}}}{\int }_0^{2\pi }{\int }_{R_{\mathrm{i}}}^{R_{\mathrm{o}}}{\displaystyle \frac{1}{2{\mu }_0}}{B}_{\mathrm{g}}^2(\theta ,z,t)rdrd\theta$$

(6)

where $R_{\mathrm{i}}$ and $R\mathrm{o}$ are the inside diameter and the outside diameter of the AFPMSM, $dr$ is the radial differential element, corresponding to the radial thickness of a circular ring, and $d\theta$ is the circumferential angle increment, representing the arc-wise span. Assuming radial uniformity ($d_{\mathrm{r}}\ll R_{\mathrm{avg}}$),

$$W=\sum _{z=1}^{N_{\mathrm{z}}}{\displaystyle \frac{\overline{R}·g}{2{\mu }_0{S}_{\mathrm{s}}}}{\int }_0^{2\pi }{B}_{\mathrm{g}}^2(\theta ,z,t)d\theta$$

(7)

where $\overline{R}$ is the average of $R_{\mathrm{i}}$ and $R_{\mathrm{o}}$. The Fourier expansions of $f_{\mathrm{r}}$ and ${\Lambda }_{\mathrm{s}}$ aere substituted into W:

$$W={\displaystyle \frac{\overline{R}·g}{2{\mu }_0{S}_s^2}}{\int }_0^{\mathrm{g}}{\int }_0^{2\pi }{\left(\sum _{{\nu }_{\mathrm{r}}\in \mathrm{odd}}{F}_{\mathrm{r}}({\nu }_{\mathrm{r}},z)cos\left({\nu }_{\mathrm{r}}p(\theta -\omega t)\right)\sum _{{\nu }_{\mathrm{s}}=0}^{\infty }{A}_{\mathrm{s}}({\nu }_{\mathrm{s}},z)cos\left({\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta \right)\right)}^{2}d\theta dz$$

(8)

Expanding the cross turns in (8), the following cross terms exist in the expansion:

$$\sum _{{{\nu }_{\mathrm{r}}}^{\prime },{{\nu }_{\mathrm{s}}}^{\prime }}\sum _{{\nu }_{\mathrm{r}},{\nu }_{\mathrm{s}}}\begin{array}{c}{F}_{\mathrm{r}}({\nu }_{\mathrm{r}},z)A_{\mathrm{s}}({\nu }_{\mathrm{s}},z){F_{\mathrm{r}}}^{\prime }({\nu }_{\mathrm{r}},z)A_{\mathrm{s}}({{\nu }_{\mathrm{s}}}^{\prime },z)\cos \left({\nu }_{\mathrm{r}}p(\theta -\omega t)\right)\cos \left({\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta \right)\cos \left({{\nu }_{\mathrm{r}}}^{\prime }p(\theta -\omega t)\right)\cos \left({{\nu }_{\mathrm{s}}}^{\prime }{N}_{\mathrm{s}}\theta \right)\hfill \end{array}$$

(9)

By applying trigonometric identities, the quadruple product can be reduced to a combination of sum and difference frequency terms. For example, these terms include

$$\cos \left({\nu }_{\mathrm{r}}p(\theta -\omega t)\right)\cos \left({\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta \right)={\displaystyle \frac{1}{2}}\left[\begin{array}{c}\cos ({\nu }_{\mathrm{r}}p(\theta -\omega t)+{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta )+\cos ({\nu }_{\mathrm{r}}p(\theta -\omega t)-{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta )\hfill \end{array}\right]$$

(10)

Among the above cross terms, a necessary and sufficient condition for the integral to be non-zero is

$${\nu }_{\mathrm{r}}p(\theta -\omega t)\pm {\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta =2k\pi ,k\in \mathbb{Z}$$

(11)

For sum frequency terms, when

$${\nu }_{\mathrm{r}}p(\theta -\omega t)+{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta =2k\pi \Rightarrow \left({\nu }_{\mathrm{r}}p+{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\right)\theta -{\nu }_{\mathrm{r}}p\omega t=2k\pi ,k\in \mathbb{Z},$$

(12)

i.e.,

$${\nu }_{\mathrm{r}}p+{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}=0$$

(13)

and

$${\nu }_{\mathrm{r}}p\omega t=2k\pi ,k\in \mathbb{Z}$$

(14)

The integral result is not zero. There always exists a value of t that satisfies (14). However, for (13), this condition cannot be met in practical situations. Therefore, the integral result for the sum-frequency term is consistently zero. In contrast, for the difference-frequency terms, a similar analysis applies. When

$${\nu }_{\mathrm{r}}p(\theta -\omega t)-{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta =2k\pi \Rightarrow \left({\nu }_{\mathrm{r}}p-{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\right)\theta -{\nu }_{\mathrm{r}}p\omega t=2k\pi ,k\in \mathbb{Z},$$

(15)

i.e.,

$${\nu }_{\mathrm{r}}p-{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}=0$$

(16)

and

$${\nu }_{\mathrm{r}}p\omega t=2k\pi ,k\in \mathbb{Z}$$

(17)

the integral result is non-zero. As previously analyzed, (17) always holds. For (16), it can be derived that

$${\nu }_{\mathrm{r}}={\displaystyle \frac{N_{\mathrm{s}}}{p}}{\nu }_{\mathrm{s}}$$

(18)

Therefore, the harmonic orders that contribute to the cogging torque can be identified by filtering the eligible harmonic components in the Fourier transforms of Fr and As according to (18). Consequently, (9) can be simplified as

$$\sum _{{{\nu }_{\mathrm{r}}}^{\prime },{{\nu }_{\mathrm{s}}}^{\prime }}\sum _{{\nu }_{\mathrm{r}},{\nu }_{\mathrm{s}}}\left[F_{\mathrm{r}}({\nu }_{\mathrm{r}},z)A_{\mathrm{s}}({\nu }_{\mathrm{s}},z){F_{\mathrm{r}}}^{\prime }({\nu }_{\mathrm{r}},z)A_{\mathrm{s}}({{\nu }_{\mathrm{s}}}^{\prime },z)·{\displaystyle \frac{1}{4}}\left[\cos ({\nu }_{\mathrm{r}}p(\theta -\omega t)-{\nu }_{\mathrm{s}}{N}_{\mathrm{s}}\theta )·\cos ({{\nu }_{\mathrm{r}}}^{\prime }p(\theta -\omega t)-{{\nu }_{\mathrm{s}}}^{\prime }{N}_{\mathrm{s}}\theta )\right]\right]$$

(19)

Continuing to simplify using the trigonometric constant gives

$$\sum _{{{\nu }_{\mathrm{r}}}^{\prime },{{\nu }_{\mathrm{s}}}^{\prime }}\sum _{{\nu }_{\mathrm{r}},{\nu }_{\mathrm{s}}}\left[F_{\mathrm{r}}({\nu }_{\mathrm{r}},z)A_{\mathrm{s}}({\nu }_{\mathrm{s}},z)F_{\mathrm{r}}({{\nu }_{\mathrm{r}}}^{\prime },z)A_{\mathrm{s}}({{\nu }_{\mathrm{s}}}^{\prime },z)·{\displaystyle \frac{1}{8}}\left[\begin{array}{c}\cos \left(\left(\left({\nu }_{\mathrm{r}}+{{\nu }_{\mathrm{r}}}^{\prime }\right)p-\left({\nu }_{\mathrm{s}}+{{\nu }_{\mathrm{s}}}^{\prime }\right)N_{\mathrm{s}}\right)\theta -\left({\nu }_{\mathrm{r}}+{{\nu }_{\mathrm{r}}}^{\prime }\right)p\omega t\right)\hfill \\ +\cos \left(\left(\left({\nu }_{\mathrm{r}}-{{\nu }_{\mathrm{r}}}^{\prime }\right)p-\left({\nu }_{\mathrm{s}}-{{\nu }_{\mathrm{s}}}^{\prime }\right)N_{\mathrm{s}}\right)\theta -\left({\nu }_{\mathrm{r}}-{{\nu }_{\mathrm{r}}}^{\prime }\right)p\omega t\right)\hfill \end{array}\right]\right]$$

(20)

Due to the presence of the sum-frequency terms in the first component of the expansion, their integral over one electrical period is always zero and can therefore be neglected. As a result, the cogging torque can be expressed as

$$T_{\mathrm{c}}={\displaystyle \frac{\pi \overline{R}·g\omega }{16{\mu }_0}}{\int }_0^g\sum _{{{\nu }_{\mathrm{r}}}^{\prime },{{\nu }_{\mathrm{s}}}^{\prime }}\sum _{{\nu }_{\mathrm{r}},{\nu }_{\mathrm{s}}}\left[F_{\mathrm{r}}({\nu }_{\mathrm{r}},z)A_{\mathrm{s}}({\nu }_{\mathrm{s}},z)F_{\mathrm{r}}({{\nu }_{\mathrm{r}}}^{\prime },z)A_{\mathrm{s}}({{\nu }_{\mathrm{s}}}^{\prime },z)·\left({\nu }_{\mathrm{r}}-{{\nu }_{\mathrm{r}}}^{\prime }\right)p\sin \left(\left({\nu }_{\mathrm{r}}-{{\nu }_{\mathrm{r}}}^{\prime }\right)p\omega t\right)\right]dz$$

(21)

The above derivation is based on the air gap magnetic density from a single stator layer (either the upper or lower) of the AFPMSM. To calculate the total cogging torque, including the contributions from both stator layers or to incorporate stator staggering for torque reduction, a staggering angle $\alpha$ can be introduced by shifting the angular position $\theta$ by $\alpha$. The total cogging torque is then obtained by summing the contributions from the upper and lower stator layers.

To accurately compute the cogging torque using this method, the waveforms of the MMF and air gap permeance must first be extracted. A Fourier transform is then applied to obtain their harmonic components and corresponding amplitudes. Unlike conventional approaches that rely on the fundamental amplitude of the air gap flux density $B_{\mathrm{g}}$, the proposed model explicitly reveals the relationship between cogging torque and the higher-order harmonics of $B_{\mathrm{g}}$. This allows for a quantitative evaluation of how each harmonic component of the rotor MMF and air gap permeance contributes to the cogging torque, thereby enabling targeted design optimizations, such as tuning the pole–slot ratio or employing harmonic injection strategies.

3. Reconstruction of MMF and Air Gap Permeability via FEA Post-Processing

According to the analysis in the previous part, if the cogging torque is to be calculated with consideration of harmonics, the waveforms of rotor MMF and air gap permeability must be obtained. This chapter presents a method to extract these quantities from the air gap flux density obtained through magnetostatic FEAs.

While the torque model is derived from the magnetic energy expression involving the product $f_{\mathrm{r}}(\theta ,z)·{\Lambda }_{\mathrm{s}}(\theta ,z)$, neither $f_{\mathrm{r}}(\theta ,z)$ nor ${\Lambda }_{\mathrm{s}}(\theta ,z)$ is directly accessible from standard finite element post-processing data. Most commercial solvers provide magnetic flux density $B_{\mathrm{s}}(\theta ,z)$, but without separating the underlying field sources. To address this limitation, a surrogate modeling strategy is employed based on two physically grounded approximations.

3.1. MMF Approximation via Field Difference

This approach involves two finite element simulations under identical excitation conditions:

• slotted (realistic) stator model;
• smooth (idealized) stator model.

The smooth stator serves as a reference, representing a system with uniform air gap permeance. By comparing the flux densities at corresponding spatial locations, the differences can be attributed solely to the slotting effects on the MMF.

Specifically, subtracting the air gap flux density of the smooth stator, $B_{\mathrm{s}}(\theta ,z)$, from that of the slotted stator, $B_{\mathrm{t}}(\theta ,z)$, effectively eliminates baseline components and isolates the harmonic contributions induced by the stator slots. Assuming negligible fringing effects and linear magnetic material properties in the air gap, the magnetic field strength can then be estimated as

$$H\left(\theta ,z\right)={\displaystyle \frac{B\left(\theta ,z\right)}{{\mu }_0}}$$

(22)

Thus, the local MMF across the air gap becomes

$$f\left(\theta ,z\right)={\displaystyle \frac{H\left(\theta ,z\right)}{g}}$$

(23)

Substituting (22) into (23) can obtain

$$f_r(\theta ,z)={\displaystyle \frac{B_t(\theta ,z)-B_s(\theta ,z)}{{\mu }_0·g}}$$

(24)

3.2. Permeance Approximation via Field Ratio

The primary source of all harmonic components in the air gap permeance arises from the slotting of the stator surface. Assuming identical applied MMF in both configurations, the permeance can be related through the ratio of the magnetic flux densities:

$${\displaystyle \frac{B_{\mathrm{t}}(\theta ,z)}{B_{\mathrm{s}}(\theta ,z)+\epsilon }}={\displaystyle \frac{\Lambda (\theta ,z)}{{\Lambda }_0}}\Rightarrow \Lambda (\theta ,z)={\Lambda }_0·{\displaystyle \frac{B_{\mathrm{t}}(\theta ,z)}{B_{\mathrm{s}}(\theta ,z)+\epsilon }}$$

(25)

Here, ${\Lambda }_0$ denotes the constant baseline air gap permeance in the smooth stator model, serving as a reference for evaluating the effects of slotting. The term $ϵ$ is a small positive constant introduced to avoid division by zero and to ensure numerical stability during the computation. The complete workflow of the proposed cogging torque calculation method, based on harmonic screening and surrogate modeling of MMF and permeance, is illustrated in Figure 1. This framework captures the essential electromagnetic interactions and enables an efficient yet accurate prediction of cogging torque, particularly under conditions involving potential staggering.

4. Simulation and Analysis

To validate the accuracy of the proposed computational method, a series of simulations are conducted to verify the principal conclusions established in the theoretical analysis. The simulations involve both slotted and smooth AFPMSMs, one with stator slotting (slotted) and one without (smooth). Importantly, the two models are identical in all other design parameters, thereby isolating the effect of slotting on the electromagnetic behavior and cogging torque. The detailed machine parameters are listed in

Table 1. Structural parameters of the AFPMSM.

Parameters	Value
Outer diameter (mm)	110
Inner diameter (mm)	80
Airgap length (mm)	1
Magnet thick(mm)	8
Slot opening width (mm)	0.86
Slot opening height (mm)	0.6

, and the structure of the AFPMSM is shown in Figure 2.

4.1. Effects of Stator Staggering

Based on the preceding theoretical derivation, it is evident that cogging torque originates from high-order harmonics induced by stator slotting. Mechanically, introducing stator staggering corresponds to a relative angular displacement between the upper and lower stator teeth. In the frequency domain, this manifests as a phase shift in the slot-induced harmonics, while their amplitudes remain unaffected. Consequently, to evaluate the impact of staggering on cogging torque, it is sufficient to compute the cogging torque of a single stator layer, apply an appropriate phase shift corresponding to the staggering angle, and superimpose the contributions from both stator layers.

To validate this principle, a simulation-based verification approach is employed. The air gap flux density waveform from the smooth stator is regarded as the fundamental component. A Fourier transform is applied to the air gap flux density of the slotted stator, and the corresponding components from the smooth stator are subtracted to isolate the slot-induced harmonics. These harmonics are then phase-shifted according to the staggering angle and added back to the fundamental harmonics of the smooth stator. An inverse Fourier transform is subsequently performed to reconstruct the full air gap flux density waveform. This reconstructed waveform is compared against the FEA results of the staggered stator configuration. Figure 3 shows the results of the Fourier transform of the air gap magnetic flux density using the no staggering, FEA, and calculation methods. The prediction error between the two is illustrated in Figure 4a, and the Pearson correlation coefficient between the reconstructed and simulated waveforms reaches 0.9984, demonstrating excellent agreement. This confirms that stator staggering primarily introduces phase shifts in the high-order harmonics associated with slotting.

Given that cogging torque arises solely from slot-induced harmonics, staggering only affects the phase of these components. Therefore, incorporating staggering into the model simply requires applying a phase shift to the upper stator’s cogging torque waveform and summing it with the unshifted contribution from the lower stator. Figure 4b presents a comparison between the FEA results and the computed cogging torque waveforms under both non-staggered and 3.75° of staggered conditions. The Pearson correlation coefficients are 0.9918 and 0.9290 for the non-staggered and staggered cases, respectively. The strong consistency between the computed and simulated results further validates the accuracy and robustness of the proposed method.

4.2. Efficiency of Calculations and Accuracy

To verify the accuracy and efficiency of the proposed harmonic screening method, cogging torque calculations are performed for AFPMSMs with various staggering angles ranging from 0° to 7.5° using both the harmonic screening approach and FEA. The results are compared in terms of computational accuracy and calculation time. The computer configurations used for each method are summarized in

Table 2. Computer configurations used for each method.

	FEA	Harmonic Screening
CPU Model	i7-9750H	R5 3500U
CPU Threads	12	8
CPU Frequency (GHz)	2.6	2.4
RAM Capacity (GB)	32	20
System Version	Windows 10 Enterprise 21H2
Software Version	Maxwell 2023R2	MATLAB R2023b

. Notably, high-performance computing resources are utilized across all software platforms.

Figure 5 shows the peak values of the cogging torque for staggering angles ranging from 0° to 7.5°, along with the motor structure and cogging torque waveforms at staggering angles of 1° and 5°. The similarity between these two waveforms reaches a Pearson correlation coefficient of 0.9863. Comparing the curves reveals that changes in the staggering angle have a significant impact on the cogging torque, further confirming the high consistency between the harmonic screening method and the FEA results.

Moreover, there is a substantial difference in computational time between the two methods. While the full FEA simulations required 3.88 hours to complete, the harmonic screening method accomplished the same analysis in only 0.32 hours, achieving a 91.75% reduction in computation time. This remarkable efficiency stems from the harmonic screening method needing magnetostatic FEA solely to extract air gap flux densities of the smooth and slotted stators, greatly reducing both computational time and hardware resource demands.

5. Conclusions

In this paper, a cogging torque calculation method based on harmonic screening is proposed. The torque is calculated by analyzing the harmonic orders and amplitudes of MMF and air gap permeance. The main contributions of this work are summarized as follows:

• The proposed harmonic screening-based method provides accurate and efficient prediction of cogging torque in AFPMSMs. Compared with conventional FEA, it achieves a 91.75% reduction in computation time without demanding excessive hardware resources and reaches over 0.9 of the waveforms’ similarity. This makes it particularly suitable for rapid evaluation of motor performance during the early design phase;
• The method establishes a clear relationship between cogging torque and the harmonic frequencies and amplitudes of key electromagnetic parameters, offering valuable insights for future studies aimed at cogging torque reduction;
• The approach reveals the connection between the harmonics of air gap magnetic density and stator staggering, thereby laying a solid foundation for further research into the effects of stator staggering on other motor performance types;
• It is necessary to further analyze the torque fluctuations of AFPMSM under load conditions using the energy method. The results of this study contribute to enriching the application of the energy method in torque prediction and calculation.