Design of a Six-Phase Surface Permanent-Magnet Synchronous Motor with Chamfer-Shaped Magnet to Reduce Cogging Torque and Torque Ripple for Large-Ship Propulsion

Abstract

Surface permanent-magnet synchronous motors (SPMSMs) have been widely adopted for ship propulsion due to their high power density and efficiency. However, conventional three-phase open-slot SPMSMs struggle to balance high efficiency with reductions in cogging torque and torque ripple. This paper proposes a design of an SPMSM with a six-phase winding configuration and a chamfer-shaped permanent magnet to reduce cogging torque and torque ripple. Electromagnetic performance is evaluated through finite element analysis (FEA). A reference three-phase interior PMSM and three-phase SPMSMs with different magnet shapes are first compared to identify a suitable basic design. Based on the basic machine, three pole–slot combinations for the six-phase winding are analyzed, and the most efficient configuration is selected. A final model is designed to minimize cogging torque and torque ripple for the chamfer-shaped permanent magnet. Finally, the effectiveness of the final model is validated through FEA by comparing its performance with that of the reference model.

1. Introduction

The 2023 IMO GHG Strategy tightened carbon-intensity targets to 40% by 2030 and 70% by 2050 (baseline 2008), requiring ship owners to comply with both the Energy-Efficiency Existing-ship Index (EEXI) and the Carbon Intensity Indicator (CII) rules [1]. Simultaneously, U.S.-flag vessels must meet MARPOL Annex VI Tier III NO_x limits in designated Emission Control Areas, creating a dual CO₂/NO_x challenge. Studies indicate that slow steaming can cut annual CO₂ emissions by roughly 32% without breaching new EEXI thresholds [1], while reactivity controlled compression ignition (RCCI) models predict NO_x reductions of up to 45% at maintained efficiency [2]. However, onboard measurements often reveal higher CO₂ factors than desk-based estimates [3], and variable-rate fuel injection struggles to meet Tier III at high load [4], underscoring the appeal of fully electric propulsion. Data-driven CII-prediction tools further indicate that electrification scenarios can improve a 2400 TEU container ship’s annual C-rating by an entire grade [5].

In large-ship propulsion applications, electrical machines are required for high efficiency, high power density, and high reliability [6,7,8]. Permanent magnet synchronous machines (PMSMs) with rare-earth magnets have attracted considerable attention for their superior efficiency and power density. Because large-ship propulsion involves operation at low speeds, surface PMSMs are widely used in large-ship propulsion applications due to their high performance [8,9,10]. To achieve high reliability, multi-phase electrical machines have been studied [11,12]. Specifically, six-phase winding, which involves dual three-phase winding, can operate through one inverter when a fault occurs in another inverter [11,13,14]. Furthermore, six-phase winding can reduce the torque ripple of SPMSMs through the phase shift of dual three-phase winding [15,16].

Because the slot-opening torque ripple of SPMSMs causes unacceptable vibration, acoustic noise, and poor control performance, designs to achieve torque ripple reduction have been studied; these have involved differing magnet pole arcs and ratios of slot width to slot pitch, as well as skewing [17,18,19,20,21,22,23,24]. In [17,18,19,20,21], a magnet shape design is proposed to reduce torque ripple through a sinusoidal-plus-third-harmonic shape, tapered shapes, and asymmetric designs. In [22,23,24], the skew method is employed to reduce torque ripple. However, this method poses challenges with regard to manufacturing and to achieving reductions in average torque. Alternatively, winding configurations to reduce torque ripple have been studied [25,26,27,28,29,30,31,32,33]. In [25,26,27,28,29,30,31], a fractional-slot configuration is proposed to reduce torque ripple. In [32,33], a multi-phase winding configuration is proposed to reduce torque ripple. Although this method can reduce electromagnetic torque ripple, it involves challenges in reducing cogging torque, which is determined by the magnet and stator shape. Achieving simultaneous reductions in cogging torque and torque ripple thus requires a combination of magnet shape design and winding configuration.

In this paper, a six-phase SPMSM with a chamfered magnet shape to reduce torque ripple is analyzed using finite element analysis (FEA). From a reference model, a three-phase interior PMSM (IPMSM), and a basic machine, a three-phase SPMSM, the winding configuration is converted from three-phase winding to asymmetric six-phase winding. For the asymmetric six-phase winding, the electromagnetic performance of various pole–slot combinations is investigated, and the superior pole–slot combination is selected. Based on the selected pole–slot combination, the chamfered magnet shape design is applied to minimize torque ripple and cogging torque. Results with respect to electromagnetic performance, including no-load and load characteristics, are compared to verify the effectiveness of the proposed design method. The final design model achieved a 28.49% reduction in cogging torque, a 1.19% reduction in torque ripple, and a 0.77% improvement in efficiency compared with the reference model, a three-phase IPMSM.

This paper is organized as follows: The characteristics of the six-phase SPMSM are analyzed in Section 2. In Section 3, the design of the six-phase SPMSM with chamfered magnet shape is proposed based on the three-phase machine. In Section 4, the performances of the reference and final models are compared to verify the effectiveness of the six-phase SPMSM with chamfered magnet shape. Section 5 gives the conclusions.

2. Characteristics of Six-Phase SPMSM for Large-Ship Propulsion

2.1. Machine Structure for Large-Ship Propulsion

Figure 1 presents a machine structure with integrated ventilation ducts for large-ship propulsion. For effective air cooling in high-power machines, the rotor is segmented into individual modules. This segmentation interrupts eddy current paths within the permanent magnets, thereby reducing associated losses, and permits separate machining and magnetization of each module, simplifying manufacturing. The motor can employ IPMSM or SPMSM. In SPMSM, a nonmagnetic protective sleeve encases each magnet to prevent separation under high centrifugal loads, as shown in Figure 1b. A rotor ventilation duct gap between modules guides cooling air from the inner magnet region to the housing, lowering thermal resistance and homogenizing temperature profiles without additional cooling systems. Interlocking laminations ensure structural integrity at high speed while providing defined radial airflow channels. Compared with liquid-cooled jackets, this air-cooled configuration adds negligible mass, preserving the power-to-weight ratio critical for ship propulsion applications.

2.2. Six-Phase Configuration

Figure 2a presents the conventional three-phase winding method. In a conventional three-phase system, the back electromotive force (BEMF) in each phase has a phase difference of 120°. There are two classifications of six-phase winding configuration: symmetric and asymmetric, as illustrated in Figure 2b and Figure 2c, respectively. The phase difference of each BEMF is 60° in symmetric six-phase winding, but the phase difference between the two three-phase groups (ABC and DEF), in which the phase difference is 120°, is 30° in asymmetric six-phase winding. Due to this characteristic of winding configuration, the BEMF can be expressed as follows:

(a)

BEMF of three-phase winding:

$$e_A={\displaystyle \sum _{n=odd}{E}_n\sin n{\omega }_{e}t},\hspace{0.17em}\hspace{0.17em}{e}_B={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-\frac{2}{3}n\pi \right)},\hspace{0.17em}\hspace{0.17em}{e}_C={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t+\frac{2}{3}n\pi \right)}$$

(1)

(b)

BEMF of symmetric six-phase winding:

$$\begin{array}{c}{e}_A={\displaystyle \sum _{n=odd}{E}_n\sin n{\omega }_{e}t},\hspace{0.17em}\hspace{0.17em}{e}_B={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-\frac{2}{3}n\pi \right)},\hspace{0.17em}\hspace{0.17em}{e}_C={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t+\frac{2}{3}n\pi \right)}\\ e_D={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-\frac{n\pi }{3}\right)},\hspace{0.17em}\hspace{0.17em}{e}_E={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-n\pi \right)},\hspace{0.17em}\hspace{0.17em}{e}_F={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t+\frac{n\pi }{3}\right)}\end{array}$$

(2)

(c)

BEMF of asymmetric six-phase winding:

$$\begin{array}{c}{e}_A={\displaystyle \sum _{n=odd}{E}_n\sin n{\omega }_{e}t},\hspace{0.17em}\hspace{0.17em}{e}_B={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-\frac{2}{3}n\pi \right)},\hspace{0.17em}\hspace{0.17em}{e}_C={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t+\frac{2}{3}n\pi \right)}\\ e_D={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-\frac{n\pi }{6}\right)},\hspace{0.17em}\hspace{0.17em}{e}_E={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t-\frac{5}{6}n\pi \right)},\hspace{0.17em}\hspace{0.17em}{e}_F={\displaystyle \sum _{n=odd}{E}_n\sin \left(n{\omega }_{e}t+\frac{n\pi }{2}\right)}\end{array}$$

(3)

where e_A, e_B, e_C, e_D, e_E, and e_F are the BEMFs of phases A, B, C, D, E, and F, respectively; E_n is the n-th harmonic component of the BEMF; n is the harmonic order; and ω_e is the electrical speed.

2.3. Cogging Torque

Cogging torque originates from the discontinuity in magnetic flux between the permanent magnet and the stator slots in the absence of armature current. Because it represents the attractive force between the magnet and stator teeth, this effect is governed by variations in slot reluctance and permanent-magnet flux. The cogging torque can be expressed as follows:

$$T_{cog}(\theta )=-{\displaystyle \frac{\partial W_{mag}(\theta )}{\partial \theta }}$$

(4)

where T_cog(θ) is the cogging torque, and W_mag(θ) is the magnetic storage energy. The magnetic storage energy is determined by the airgap magnetic flux density which is affected by the slot harmonic, and the magnetic flux density which is affected by the permanent magnet. The magnetic storage energy can therefore be expressed as a Fourier series, as shown in Equation (5).

$$W_{mag}(\theta )=W_0-{\displaystyle \sum _{n=1}^{\infty }{W}_n\cos \left(n{N}_{cog}\theta +{\varphi }_n\right)}$$

(5)

where W₀ is the DC component of magnetic storage energy, W_n is the harmonic component of magnetic storage energy, N_cog is the fundamental harmonic order of magnetic storage energy, and ϕ is the phase angle of magnetic storage energy. Because the magnetic storage energy is determined by the slot harmonic, and magnetic flux density by the permanent magnet, the fundamental harmonic order of magnetic storage energy is the least common multiple of the numbers of slots and poles. By substituting Equation (5) into Equation (4), the cogging torque can be expressed as follows:

$$T_{cog}(\theta )={\displaystyle \sum _{n=1}^{\infty }n{N}_{cog}{W}_n\cos \left(n{N}_{cog}\theta +{\varphi }_n\right)}$$

(6)

From Equation (6), cogging torque is determined by the flux density distribution and the slot/pole arrangement. The cogging torque coefficient W_n is governed by the interplay between slot count, pole count, and flux distribution. Slot asymmetry and mismatches between the numbers of slots and poles directly influence both the amplitude and harmonic order of the cogging torque.

2.4. Electromagnetic Torque

From the energy conversion principle, the electromagnetic torque can be expressed as follows:

$$T_e={\displaystyle \frac{{\displaystyle \sum e_k{i}_k}}{{\omega }_m}}$$

(7)

where T_e is the electromagnetic torque; e_k and i_k are the BEMF and current, respectively, of phase k; and ω_m is the mechanical speed. Assuming i_d = 0 control, the current of phase k can be expressed as follows:

(a)

Current of three-phase winding:

$$i_A=I_m\sin {\omega }_{e}t,\hspace{0.17em}\hspace{0.17em}{i}_B=I_m\sin \left({\omega }_{e}t-{\displaystyle \frac{2}{3}}\pi \right),\hspace{0.17em}\hspace{0.17em}{i}_C=I_m\sin \left({\omega }_{e}t+{\displaystyle \frac{2}{3}}\pi \right)$$

(8)

(b)

Current of symmetric six-phase winding:

$$\begin{array}{c}{i}_A=I_m\sin {\omega }_{e}t,\hspace{0.17em}\hspace{0.17em}{i}_B=I_m\sin \left({\omega }_{e}t-{\displaystyle \frac{2}{3}}\pi \right),\hspace{0.17em}\hspace{0.17em}{i}_C=I_m\sin \left({\omega }_{e}t+{\displaystyle \frac{2}{3}}\pi \right)\\ i_D=I_m\sin \left({\omega }_{e}t-{\displaystyle \frac{\pi }{3}}\right),\hspace{0.17em}\hspace{0.17em}{i}_E=I_m\sin \left({\omega }_{e}t-\pi \right),\hspace{0.17em}\hspace{0.17em}{i}_F=I_m\sin \left({\omega }_{e}t+{\displaystyle \frac{\pi }{3}}\right)\end{array}$$

(9)

(c)

Current of asymmetric six-phase winding:

$$\begin{array}{c}{i}_A=I_m\sin {\omega }_{e}t,\hspace{0.17em}\hspace{0.17em}{i}_B=I_m\sin \left({\omega }_{e}t-{\displaystyle \frac{2}{3}}\pi \right),\hspace{0.17em}\hspace{0.17em}{i}_C=I_m\sin \left({\omega }_{e}t+{\displaystyle \frac{2}{3}}\pi \right)\\ i_D=I_m\sin \left({\omega }_{e}t-{\displaystyle \frac{\pi }{6}}\right),\hspace{0.17em}\hspace{0.17em}{i}_E=I_m\sin \left({\omega }_{e}t-{\displaystyle \frac{5}{6}}\pi \right),\hspace{0.17em}\hspace{0.17em}{i}_F=I_m\sin \left({\omega }_{e}t+{\displaystyle \frac{\pi }{2}}\right)\end{array}$$

(10)

where i_A, i_B, i_C, i_D, i_E, and i_F are the currents of each phase, and I_m is the magnitude of current. Based on Equation (7), the electromagnetic torque for three-phase, symmetric six-phase, and asymmetric six-phase winding can be expressed as follows:

(a)

Electromagnetic torque of three-phase winding:

$$\begin{array}{c}{T}_e=T_{avg}+{\displaystyle \sum _{n=1}{T}_{6n}\cos \left(6n{\omega }_{e}t+{\varphi }_n\right)}\\ T_{avg}={\displaystyle \frac{3}{2}}{\displaystyle \frac{E_1{I}_m}{{\omega }_m}},T_{6n}={\displaystyle \frac{3}{2}}{\displaystyle \frac{E_{6n-1}-E_{6n+1}}{{\omega }_m}}\end{array}$$

(11)

(b)

Electromagnetic torque of symmetric six-phase winding:

$$\begin{array}{c}{T}_e=T_{avg}+{\displaystyle \sum _{n=1}{T}_{6n}\cos \left(6n{\omega }_{e}t+{\varphi }_n\right)}\\ T_{avg}={\displaystyle \frac{3E_1{I}_m}{{\omega }_m}},T_{6n}={\displaystyle \frac{3\left(E_{6n-1}-E_{6n+1}\right)}{{\omega }_m}}\end{array}$$

(12)

(c)

Electromagnetic torque of asymmetric six-phase winding:

$$\begin{array}{c}{T}_e=T_{avg}+{\displaystyle \sum _{n=1}{T}_{12n}\cos \left(12n{\omega }_{e}t+{\varphi }_n\right)}\\ T_{avg}={\displaystyle \frac{3E_1{I}_m}{{\omega }_m}},T_{12n}={\displaystyle \frac{3\left(E_{12n-1}-E_{12n+1}\right)}{{\omega }_m}}\end{array}$$

(13)

From Equations (11) and (12), it may be seen that the torque ripple of three-phase winding and symmetric six-phase winding is primarily influenced by the (6n − 1)th and (6n + 1)th harmonic components of BEMF. However, the torque ripple of asymmetric six-phase winding is determined by the (12n − 1)th and (12n + 1)th harmonic components of BEMF because the phase difference eliminates the contribution of the (6n − 1)th and (6n + 1)th harmonic components of BEMF. As a result, the torque ripple of asymmetric six-phase winding can be lower than that of three-phase winding or symmetric six-phase winding. In the next section, a chamfer-shaped SPMSM with asymmetric six-phase winding is designed using FEA.

3. Design of Six-Phase SPMSM with Chamfered Magnet Shape

In this section, we investigate the electromagnetic performance of a three-phase SPMSM with different magnet shapes, including arc-shaped, arc-tapering-shaped, bread-shaped, and bread-tapering-shaped, to select a basic design. Based on the basic design, a six-phase SPMSM with a pole–slot combination and chamfered magnet shape is analyzed. Electromagnetic performance is evaluated based on FEA using ANSYS Maxwell 2024R2.

3.1. Basic Design Specification

To implement the six-phase winding, a basic three-phase machine is selected through comparison of the reference model (IPMSM) and four shapes of SPMSM, including arc-shaped, arc-tapering-shaped, bread-shaped, and bread-tapering-shaped, as illustrated in Figure 3. The specifications of these machines are summarized in

Table 1. Specifications of reference model.

Parameter		Value	Unit
Rated power		2	MW
Rated speed		56	rpm
Rated torque		341	kNm
Maximum	Three-phase	2850	Arms
current	Six-phase	1425	Arms
Stator outer/inner diameter		2450/2100	mm
Rotor outer/inner diameter		2086/1740	mm
Stack length with/without ducts		1210/1354	mm
Airgap length		7	mm
Number of poles/slots		24/180	-
Rotor core material		50PN600	-
Stator core material		50PN400	-
Magnet material		N35UH	-
Copper and magnet temperature		80	℃

. All models are designed with the same numbers of poles and slots, using a fractional-slot configuration to achieve low torque ripple. Their electromagnetic performances are compared using FEA, as shown in

Table 2. Comparison of electromagnetic performances of IPMSM and SPMSMs.

Parameter	Reference Model (IPMSM)	SPMSM				Unit
		Arc-Shaped	Arc-Tapering-Shaped	Bread-Shaped	Bread-Tapering-Shaped
Total series turns	30	30	30	30	30	-
Parallel circuit	12	12	12	12	12	-
No-load BEMF	236.75	288.16	274.19	285.25	267.93	Vrms
Power	2	2	2	2.05	1.99	MW
Torque	341.33	343.46	341.78	349.79	340.15	kNm
Torque ripple	1.63	2.11	1.13	1.92	1.09	%
Current	2700	2380	2485	2450	2530	Arms
Copper loss	54.8	42.54	46.38	45.08	48.07	kW
Eddy current loss	0.02	11.31	14.8	9.94	14	kW
Iron loss	5.5	4.8	4.41	4.83	4.34	kW
Efficiency	97.07	97.16	96.82	97.16	96.77	%
Power factor	80.76	95.38	94.38	94.13	92.86	-
Magnet total area	0.167	0.166	0.164	0.163	0.164	m2

. To ensure comparability, total numbers of series turns and parallel circuits are the same for all models, so that they may be compared under the same conditions. Although the magnet total area is similar among the models, the no-load BEMF of SPMSM is higher than that of IPMSM, indicating greater torque capability. Under maximum torque per ampere (MTPA) control, the SPMSMs demonstrate superior performance compared with the reference IPMSM. While electromagnetic performance is comparable across different magnet shapes, the bread-shaped magnet is preferable from a manufacturing perspective, because both the magnet and core surfaces are flat. Therefore, the bread-shaped SPMSM is selected as the basic design for converting three-phase winding to six-phase winding.

3.2. Pole–Slot Combination for Six-Phase Winding

Based on the basic machine, which is the bread-shaped SPMSM, the pole–slot combination for six-phase winding is investigated. Three pole–slot combinations are selected for consideration of torque, as mentioned in Section 2.4; these are 36pole-216slot, 84pole-144slot, and 120pole-144slot, as illustrated in Figure 4a. For pole–slot combinations, the number of turns is designed to provide a similar no-load BEMF. In motor design, increasing the number of poles reduces the magnetic flux per pole, which decreases the stator yoke thickness but increases the leakage flux, thereby reducing the harmonic component of BEMF. As shown in Figure 4b, a higher pole count results in lower BEMF and a more sinusoidal waveform. Figure 5 shows the harmonic components of the no-load BEMF, obtained through Fast Fourier Transform (FFT) analysis. In the case of the high-pole machine, the harmonic component of BEMF is lower, compared to the other machines. However, the fundamental component of BEMF also decreases, which reduces the average torque. From Equations (11) and (13), the fifth and seventh BEMFs contribute to the sixth torque harmonic, while the eleventh and thirteenth BEMFs contribute to the twelfth torque harmonic. Although 36pole-216slot exhibits higher fifth and seventh harmonic components compared with the other models, these do not contribute the torque characteristics shown in Equation (13). Moreover, the average torque of this model is higher than those of the other models.

Table 3. Comparison of electromagnetic performances of three-phase SPMSM and six-phase SPMSM with different pole-slot combinations.

Parameter	Basic Machine	Six-Phase			Unit
		36poles-216slots	84poles-144slots	120poles-144slots
Poles/slots	24/180	36/216	84/144	120/144	-
Power	2.05	2.06	2.09	2	MW
No-load BEMF	285.25	306.7	270.36	260.61	Vrms
Current	2450	1160	1355	1356	Arms
Cogging torque	2885.6	2500.5	24.63	93.73	Nm
Torque	349.78	352.45	359.22	345.04	kNm
Torque ripple	1.92	2.74	16.27	23.02	%
Copper loss	45.08	34.22	89.54	68.1	kW
Eddy current loss	9.94	3.6	15.76	27.86	kW
Iron loss	4.83	7.1	1627	23.02	kW
Efficiency	97.16	97.87	94.5	94.73	%

shows a comparison of the electromagnetic performances of the basic machine and the six-phase SPMSM with different pole–slot combinations. As the number of poles increases, iron loss, eddy current loss, and copper loss increase, leading to lower efficiency. Although 36pole-216slot has high torque per current and high efficiency, the torque ripple of this model is higher than that of other models due to the higher cogging torque. To reduce the torque ripple, a chamfered magnet shape design is introduced; this is discussed in the next section.

3.3. Chamfered Magnet Shape Design

Although the 36poles-216slots machine with six-phase winding demonstrates superior performance, it suffers from high torque ripple caused by increased cogging torque. To address this issue, a chamfered magnet shape design is proposed. The design variables for the chamfered magnet shape are defined as chamfer shape thickness (T_chamfer) and chamfer ratio (α_chamfer), as illustrated in Figure 6. The objective function of the design is to minimize cogging torque and torque ripple, and the constraints are the magnetic thickness and the magnet pole ratio, as shown in Figure 6. For the design variables, electromagnetic performance results, including power, efficiency, cogging torque, and torque ripple, are shown in Figure 7. The final model of the six-phase SPMSM is selected to minimize cogging torque and torque ripple. As a result, the design variables of the selected final model are 5.2mm of T_chamfer and 0.66 of α_chamfer, and the final model is shown in Figure 8.

4. Performance Comparison

In this section, a performance comparison of the reference model (IPMSM) and the six-phase SPMSM final model is carried out for no-load and load characteristics. Through performance comparison, the effectiveness of the final model is validated.

4.1. No-Load Characteristics

For the no-load characteristics, the no-load BEMF and the cogging torque of the reference and final model are compared. The waveform and FFT results of no-load BEMF for Phase A are shown in Figure 9 and Figure 10. As shown in Figure 9, the no-load BEMF of the reference model exhibits a more sinusoidal waveform than that of the final model. However, as shown in Figure 10, the fundamental component of the final model is higher than that of the reference model. Because the reference model has the more sinusoidal waveform, the harmonic component of the final model is higher than that of the reference model. Although the fifth and seventh harmonic components of the final model are higher than those of the reference model, they do not contribute torque ripple in the final model due to the asymmetric six-phase winding, as illustrated in Equation (13). By contrast, the fifth and seventh harmonic components of the reference model can contribute the sixth torque harmonic. Because the period of the cogging torque is the least common multiple of the numbers of poles and slots from Equation (6), the periods of the reference and final models are 12deg and 30deg, respectively, as shown in Figure 11. The cogging torque of the reference model is higher than that of the final model.

4.2. Load Characteristics

The load characteristics of the reference and final models are compared under the MTPA control. Because the average torque of the six-phase machine is doubled under the same BEMF and current conditions, the current magnitudes of the reference and final models are set to 2700 A and 1350 A, respectively. For these current magnitudes, the current phase angles of MTPA in the reference and final models are 15deg and 0deg, respectively. Figure 12 illustrates the torque waveforms of both models, showing that the peak-to-peak torque of the final model is reduced by approximately 72.63%. Furthermore, as shown in Figure 13, the dominant torque ripple orders in the reference and final models correspond to the sixth and twelfth harmonics, respectively. Figure 14 shows the radial forces of the reference and final models. The maximum radial force of the reference model is higher than that of the final model, as shown in Figure 14a. The radial force period is determined by the number of poles, as shown in Figure 14b. Since the reference and final models employ 24 and 36 poles, their fundamental radial forces are the 24th harmonic and 36th harmonic, respectively. The overall performances of the reference and final models are summarized in

Table 4. Comparison of reference model and final model.

Parameter	Reference Model (Three-Phase IPMSM)	Final Model (Six-Phase SPMSM)	Unit
Pole/Slot	24/180	36/216	-
Power	2	2.02	MW
Cogging Torque	1.86	1.33	kNm
No-load BEMF	236.75	257.86	Vrms
Current	2700	1350	Arms
Power Factor	80.76	93.7	%
Torque	341.33	346.16	kNm
Torque Ripple	1.63	0.44	%
Copper Loss	54.8	34.06	kW
Eddy Current Loss	0.02	3.79	kW
Iron Loss	5.5	6.81	kW
Efficiency	97.07	97.84	%
Total cost	134.03	124.46	k$

. It can be seen that the final model has better performance than the reference model, and that the effectiveness of the final model is validated through FEA results.

5. Conclusions

In this paper, an SPMSM with asymmetric six-phase winding and a chamfered magnet shape is proposed to mitigate cogging torque and torque ripple. Based on a reference model (three-phase IPMSM), a basic machine, a three-phase SPMSM, is designed and evaluated through FEA with different magnet shapes. From various pole and slot combinations, a 36poles-216slots model is selected, and a chamfered magnet shape design is applied to minimize cogging torque and torque ripple. The final model achieves 28.49% lower cogging torque, 1.19% lower torque ripple, and 0.77% higher efficiency, compared with the reference model. These results demonstrate the effectiveness of the proposed design in enhancing electromagnetic performance, including cogging torque, torque ripple, and efficiency. Future research will explore the applicability of this design method to machines of different sizes and extend the study to multi-physics analysis, including evaluations of electromagnetic, thermal, and structural factors, as well as noise, vibration, and harshness (NVH). Furthermore, experimental validation will be conducted through prototype fabrication and testing of the motor drive system.