Coordinated Stator–Rotor Structural Optimization of an Automotive IPMSM for Improved Torque Performance

Abstract

Traditional optimization methods for interior permanent magnet synchronous motors (IPMSMs) often treat the stator and rotor as independent design domains, which limits the potential for suppressing torque fluctuations due to the neglected electromagnetic coupling between these components. This paper proposes a synergistic optimization strategy for a 120 kW IPMSM, aiming to overcome the inherent limitations of conventional unilateral optimization in design space exploration and achieve global performance enhancement through cross-domain collaboration. By establishing a unified surrogate model incorporating both stator slot geometries and rotor pole topologies, the collaborative effect of seven high-sensitivity design variables is systematically analyzed. The NSGA-II algorithm, coupled with a Kriging surrogate model, is employed to navigate the complex trade-offs among average torque, torque ripple, and cogging torque. Results demonstrate that the synergistic approach achieves a 28.1% reduction in torque ripple while maintaining high average torque, demonstrating superior improvement over conventional stator-only or rotor-only optimization schemes. Analysis based on Maxwell stress tensors and air-gap permeance functions reveals that the proposed method achieves simultaneous suppression of cogging torque and torque ripple by effectively harmonizing the 24th and 48th spatial harmonics. This study provides an efficient synergistic design methodology for the comprehensive performance enhancement of traction motors, offering practical reference value for the engineering development of high-performance electric vehicles.

1. Introduction

Permanent magnet synchronous motors (PMSMs) are widely employed in electric vehicle propulsion systems because of their high efficiency, power density, and wide operating range [1,2,3]. However, torque pulsations, primarily including cogging torque and load torque ripple, remain major factors affecting electromagnetic output smoothness, drivetrain stability, and vibration–noise behavior [4,5,6]. Existing studies have shown that torque-quality deterioration is not only reflected in output fluctuation itself, but also influences electromagnetic vibration through harmonic variation in the air-gap magnetic field [6]. Consequently, torque-quality optimization is directly related to both performance improvement and vibration suppression.

Existing PMSM torque-quality optimization strategies can generally be categorized into rotor-domain optimization, stator-domain optimization, and coordinated stator–rotor optimization. Rotor-domain optimization mainly modifies harmonic-generation mechanisms through rotor topology reconstruction, flux-barrier redesign, or auxiliary geometric adjustment [7]. In contrast, stator-domain optimization primarily regulates slot permeance harmonics through slot opening geometry, tooth configuration, or air-gap profile design [8]. Although both approaches can improve selected torque indicators, their mechanisms differ fundamentally. Rotor-side optimization mainly alters rotor magnetic-field modulation characteristics, whereas stator-side optimization primarily modifies stator-slot permeance distribution. Therefore, single-domain optimization typically suppresses torque pulsation by regulating only one side of the harmonic-coupling pathway.

In practical PMSMs, torque pulsation originates from the coupling interaction between stator-slot permeance harmonics and rotor magnetic-field harmonics rather than from either structural domain independently [9,10,11]. Previous studies on rotor-side optimization mainly regulate rotor magnetic-field modulation characteristics through rotor topology reconstruction or flux-barrier redesign [7], whereas stator-side optimization primarily modifies slot permeance harmonic distribution through slot opening or air-gap profile regulation [8]. This indicates that PMSM torque characteristics are fundamentally governed by a stator–rotor geometric coupling mechanism [9,10]. Under this framework, coordinated stator–rotor optimization differs from conventional single-domain optimization not merely by increasing the number of variables, but by simultaneously regulating both harmonic-generation sources and their coupling relationship. Such simultaneous regulation may provide broader harmonic redistribution capability and potentially improve the balance among cogging torque, torque ripple, and average torque more effectively than independent optimization [12,13]. Despite numerous studies on stator-side or rotor-side optimization, direct comparative investigations into this mechanism-level distinction remain limited [3,14].

Before large-scale coordinated optimization, reliable electromagnetic validation, waveform consistency verification, phase-angle matching, and parameter sensitivity analysis remain necessary to establish credible optimization objectives [15,16]. As design dimensionality increases, computational burden becomes a major challenge in coordinated optimization. Accordingly, surrogate-assisted models, including Kriging, radial basis function (RBF), response surface methodology (RSM), and machine learning-based predictive models, have been increasingly introduced to improve optimization efficiency [14,17,18,19,20]. Since predictive capability directly affects optimization reliability, comparative evaluation of surrogate models under unified datasets and criteria has become an important component of PMSM optimization workflow design [18]. Based on surrogate prediction, NSGA-II and related multi-objective optimization algorithms are widely employed to solve competing design objectives such as cogging torque, torque ripple, and average torque [21,22,23].

Therefore, this study focuses on coordinated stator–rotor optimization for an automotive interior PMSM from the perspective of torque-characteristic improvement. Unlike conventional approaches that separately optimize stator or rotor parameters, the present work emphasizes the coupling relationship between stator-slot geometry and rotor structural parameters, aiming to clarify their distinct and coordinated effects on torque characteristics through global sensitivity analysis, surrogate-assisted prediction, and NSGA-II-based multi-objective optimization.

2. Establishment and Analysis of the Motor Finite Element Model

In this study, a 120 kW interior permanent-magnet synchronous motor (IPMSM) with 48 stator slots and 8 poles is investigated. The stator and rotor geometries are parameterized to examine their effects on the air-gap magnetic field and torque metrics, including cogging torque, torque ripple, and average output torque. The comprehensive 3D structure of the motor is illustrated in Figure 1, which provides a visual representation of the stator slot configurations and the intricate topology of the rotor flux barriers. To balance computational efficiency and analytical depth, the subsequent electromagnetic performance simulations are conducted using a high-resolution 2D transient finite element model.

2.1. Fundamental Motor Parameters

The electromagnetic performance is analyzed using a high-resolution 2D transient finite element model. Stator and rotor cores utilize M250-35A electrical steel with non-linear B-H curves, while N30UH NdFeB with non-linear demagnetization characteristics is assigned to the magnets. The mesh comprises 62,062 second-order nonlinear triangular elements, yielding a total of 124,045 degrees of freedom. To ensure numerical robustness, the air gap is discretized into three uniform radial layers using a manual layering strategy, with local refinement applied to key regions such as stator tooth tips and magnet edges to improve local field-gradient resolution, solution stability, and the accuracy of force-density extraction. To isolate thermal effects, simulations are conducted at 25 °C using the Newton–Raphson algorithm. Temporal and spatial sampling parameters are configured based on the requirements of high-order harmonic decomposition: a 10 μs step ensures 250 sampling points per electrical cycle at a rotational speed of 6000 rpm, representing a total duration of 5 ms. Simultaneously, 1001 equidistant points are extracted along the air-gap circumference. This configuration minimizes spectral leakage, facilitating the quantitative identification of torque fluctuations and spatial harmonic attenuation. Main design parameters are summarized in

Table 1. Key design parameters of the motor.

Parameter Name	Parameter Value
Stator outer diameter/mm	198
Stator inner diameter/mm	132
Air gap length/(mm)	1
Rotor inner diameter/(mm)	80
Core length/(mm)	150
Number of slots	48
Number of pole pairs	4
Load velocity/(rpm)	6000
Load torque/(N·m)	191
Rated power/(kW)	120

.

After establishing the baseline model, feasibility checks and parametric sweeps are performed to determine the allowable ranges of optimization parameters [18]. For the rotor, the variable Y2 is analytically bounded by the centrifugal stress at the peak speed of 12,000 rpm where the angular velocity reaches 1256.6. While the theoretical minimum bridge thickness required to maintain a safety factor of 2.0 is 0.263, the design limit is set to 0.5 mm to comply with industrial stamping standards. Compared to the theoretical failure threshold of 0.263, this 0.5 boundary provides a 90.1% safety margin, ensuring structural reliability across the entire design space. Regarding the stator, the slot opening Bs0 (1.8–4.0 mm) is tailored to the 1.0–1.5 mm diameter of automated winding needles. This range ensures sufficient clearance to prevent insulation abrasion during mass production while stabilizing the slot fill factor within a robust window of 45–55%. The finalized optimization ranges, synthesized from these mechanical and fabrication constraints, are summarized in

Table 2. Rotor parameter value ranges.

Rotor Parameter Name	Parameter Range
X1/(mm)	−1.2–−0.5
Y1/(mm)	60.0–62.5
Y2/(mm)	62.5–64.2
Angle/(°)	2.4–3.5

and

Table 3. Stator parameter value ranges.

Stator Parameter Name	Parameter Range/(mm)
Slot Opening Height Hs0	0.5–1.2
Slot Core Height Hs1	0.21–0.42
Slot Width Height Hs2	15.5–18.5
Slot Opening Width Bs0	1.8–4.0
Slot Core Width Bs1	3.0–4.5
Slot Bottom Width Bs2	5.2–7.2
Slot Bottom Fillet Radius R	2.0–3.5

.

2.2. Motor Model Analysis and Validation

The magnitude and waveform of the no-load back electromotive force (EMF) are key indicators of the air-gap magnetic field characteristics, which fundamentally determine the torque performance of the motor. Based on the structural parameters listed in Table 1, a two-dimensional finite element model is established, and transient simulations under no-load conditions are carried out. With a DC bus voltage of 400 V and zero current excitation in the three-phase windings, the no-load back EMF and cogging torque are obtained using a transient magnetic field solver, and the corresponding EMF waveform is shown in Figure 2. The results indicate that the peak value of the no-load back EMF ranges from 175 V to 190 V. The waveform exhibits a quasi-sinusoidal profile with observable harmonic distortion, which is mainly attributed to slotting effects and magnetic nonlinearity. This indicates that the model is capable of capturing both the fundamental component and harmonic characteristics of the air-gap magnetic field. Since torque production and torque ripple are directly related to these magnetic field components, the accuracy of the torque analysis can be ensured. On this basis, load conditions are further simulated by applying current excitation. The phase relationship between the no-load back EMF and the load current is analyzed, as shown in Figure 2. The results indicate that the current waveform is well aligned with the back EMF. Such phase consistency is beneficial for stable electromagnetic energy conversion in PMSM drive systems and helps avoid additional torque fluctuations associated with improper control coordination [9]. Therefore, the developed finite element model provides a reliable foundation for subsequent analysis of torque characteristics and stator–rotor structural optimization.

The simulated no-load cogging torque is 1.1 N·m, with a peak-to-peak value $T_{\mathrm{c}\mathrm{o}\mathrm{g},\mathrm{p}\mathrm{k}}$ of approximately 2.2 N·m, as shown in Figure 3. To assess its effect on torque performance, the cogging torque is compared with the rated load torque. The results show that the cogging torque accounts for less than 5% of the rated load torque, indicating that it does not dominate the overall torque output. The electromagnetic torque is mainly governed by the fundamental component of the air-gap magnetic field, while the cogging torque appears as a periodic fluctuation component. Although the cogging torque amplitude is relatively low, its periodic variation can still induce torque pulsation, excite structural vibration, and affect operational stability. Therefore, further suppression of cogging torque is necessary to improve torque smoothness in automotive drive applications.

To evaluate the effect of the air-gap magnetic field on torque performance, transient finite-element simulations are conducted under load conditions. A rotational velocity of 6000 rpm is applied, and the current excitation is imposed with an appropriate phase angle. The resulting torque waveform is shown in Figure 4. The motor produces an average torque of 191.2 N·m and a peak torque of approximately 215 N·m, satisfying the design requirements. According to Equation (5), the torque ripple is calculated as 21.04%, indicating a noticeable electromagnetic torque fluctuation.

This fluctuation is mainly related to the harmonic components of the air-gap magnetic field. To clarify this mechanism, the spatial harmonic spectrum of the air-gap flux density under load conditions is analyzed, as shown in Figure 5. In addition to the dominant fundamental component, several low-order spatial harmonics, such as the 12th- and 20th-order components, show relatively large amplitudes. These harmonics distort the air-gap magnetic field distribution and cause periodic variations in electromagnetic torque. Therefore, low-order spatial harmonics are an important source of torque ripple.

Based on this analysis, structural optimization aimed at regulating the air-gap magnetic field distribution is necessary and feasible for suppressing parasitic harmonic components and reducing torque ripple.

3. Mathematical Model Analysis

3.1. Mathematical Model of Cogging Torque

The presence of stator slots in a permanent magnet synchronous motor leads to a non-uniform distribution of air-gap reluctance. As the rotor rotates, the air-gap reluctance varies periodically, resulting in fluctuations in the air-gap magnetic field. These periodic variations in the magnetic field produce alternating electromagnetic forces, which give rise to cogging torque. Cogging torque is an inherent torque component determined by the stator–rotor structural configuration. From an energy perspective, cogging torque can be defined as the negative derivative of the magnetic field energy with respect to the rotor position angle [24,25,26], as expressed in Equation (1).

$$T=-{\displaystyle \frac{dW}{d{\theta }_m}}$$

(1)

In the equation, $T$ represents cogging torque, $W$ represents magnetic field energy, and ${\theta }_m$ represents the position angle.

Assuming a high magnetic permeability of the armature core and neglecting the magnetic reluctance of the core, the magnetic field energy stored in the air-gap and permanent-magnet regions can be expressed as [24,25,27]

$$\begin{array}{l}W=W_{sirpap+PM}\\ ={\displaystyle \frac{1}{2{\mu }_0}}{{\displaystyle \int }}_V{B}^{2}dV\\ ={\displaystyle \frac{1}{2{\mu }_0}}{{\displaystyle \int }}_V{B}_r^2(\theta ){\left[{\displaystyle \frac{h_m(\theta )}{h_m(\theta )+\delta (\theta ,{\theta }_m)}}\right]}^{2}dV\end{array}$$

(2)

In Equation (2), $W_{sirpap+PM}$ represents the magnetic field energy stored in the air-gap and permanent-magnet regions, $B_r\left(\theta \right)$ denotes the radial air-gap flux density of the equivalent slotless machine, $h_m\left(\theta \right)$ is the magnetization-direction length of the permanent magnet, $\delta (\theta ,{\theta }_m)$ is the effective air-gap length varying with rotor position, and ${\mu }_0$ is the vacuum permeability. The term ${\left[h_m\left(\theta \right) / \left(h_m\left(\theta \right) +\delta \left(\theta ,{\theta }_m\right) \right)\right]}^2$ represents the squared relative permeance effect caused by stator slotting and rotor position variation. $V$ denotes the corresponding integration volume.

The cogging-torque formulation adopted in this study is based on the magnetic-energy method and the relative-permeance model of the slotted air gap [24,25,28]. For IPM/IPMSM machines, previous analytical studies have considered the influence of interior magnet topology, flux barriers, and air-gap field harmonics on cogging torque [26,27,29].

In this framework, the stator-slotting effect is represented by the relative-permeance harmonic coefficient, while the rotor topology is reflected in the equivalent slotless radial air-gap flux-density term. Therefore, for the investigated IPMSM, the effects of buried magnets, flux barriers, magnetic bridges, and leakage-flux paths are included in the harmonic coefficients of $B_r^2(\theta )$. This treatment allows the energy-based cogging-torque expression to be applied to the present 48-slot/8-pole IPMSM.

Based on the magnetic-energy method, relative-permeance modeling [24,25,28], and analytical cogging-torque models for IPM/IPMSM machines [26,27,29], the squared equivalent slotless radial air-gap flux-density term and the squared relative-permeance term are expanded into Fourier series. According to the orthogonality of trigonometric functions, only the harmonic components satisfying the slot–pole matching condition contribute to the rotor-position-dependent magnetic field energy, where the matching relationship is $2ip=jz=n{N}_c$. Here, $p$ is the number of pole pairs, $\mathrm{z}$ is the number of stator slots, and $N_c=LCM(2p,\mathrm{z})$. Therefore, the corresponding harmonic indices are $i=n{N}_c/(2p)$ and $\mathrm{j}=n{N}_c/z$. Substituting these matched harmonic components into Equations (1) and (2), and then taking the negative derivative of the magnetic field energy with respect to ${\theta }_m$, gives the cogging-torque expression:

$$T_{\mathrm{cog}}\left({\theta }_m\right)=-{\displaystyle \frac{\pi L_a}{4{\mu }_0}}\left(R_2^2-R_1^2\right){\displaystyle \sum _{n=1}^{\infty }n}{N}_c{G}_n{(B_r)}_{{\displaystyle \frac{n{N}_c}{2p}}}^2\sin \left(n{N}_c{\theta }_m\right)$$

(3)

In Equation (3), $L_{\alpha }$ denotes the axial length of the armature core, $R_2$ and $R_1$ represent the outer and inner radii of the air-gap integration region, respectively, $n$ is the harmonic order, $G_n$ is the $n$-th Fourier coefficient of the squared relative permeance term, and ${(B_r)}_{{\displaystyle \frac{n{N}_c}{2p}}}^2$ is defined as the corresponding Fourier coefficient of the squared equivalent slotless radial air-gap flux-density term $B_r^2(\theta )$, rather than the square of a single flux-density harmonic amplitude.

For the investigated 48-slot/8-pole IPMSM, $Z=48$, $2p=8$ and $N_c=48$. Therefore, the effective rotor-field harmonic index is $n{N}_c/(2p)=6n$, and Equation (3) can be further written as

$$T_{\mathrm{cog}}\left({\theta }_m\right)=-{\displaystyle \frac{\pi L_a}{4{\mu }_0}}\left(R_2^2-R_1^2\right){\displaystyle \sum _{n=1}^{\infty }4}8n{G}_n{(B_r)}_{6n}^2\sin \left(48n{\theta }_m\right)$$

(4)

A dimensional consistency check is further performed for Equation (4). In this equation, the constant coefficient, $n$, $G_n$, and the sine term are dimensionless, where $G_n$ is a Fourier coefficient of the squared relative-permeance term. The coefficient ${(B_r)}_{6n}^2$ has the unit of $T^2$, and ${\mu }_0$ has the unit of N/A². Since 1 T = N/(A·m), the unit of $B_r^2/{\mu }_0$ is N/m². Meanwhile, $L_a\left(R_2^2-R_1^2\right)$ has the unit of m³. Therefore, the right-hand side of Equation (4) has the unit of N·m, which is consistent with the physical unit of cogging torque.

Equation (4) indicates that the cogging torque of the investigated motor is governed by the coupled effect of the stator-slot permeance harmonic $G_n$ and the equivalent rotor-field harmonic ${(B_r)}_{6n}^2$. Thus, stator-slot parameters mainly affect $G_n$, while rotor bridge and flux-barrier parameters mainly affect ${(B_r)}_{6n}^2$. This relationship provides the theoretical basis for coordinated stator–rotor structural optimization.

3.2. Mathematical Model of Torque Ripple and Average Output Torque

Torque ripple refers to the fluctuation of steady-state electromagnetic torque, resulting from the combined effects of cogging torque and harmonic components in the air-gap magnetic field.

The torque ripple ratio is commonly used to quantify the smoothness of motor torque output and is defined as

$$Ripple={\displaystyle \frac{T_{\max }-T_{\min }}{T_{\mathrm{avg}}}}\times 100\%$$

(5)

In Equation (5), $T_{max}$ and $T_{min}$ denote the maximum and minimum instantaneous torque within one electrical period, respectively, and $T_{avg}$ represents the average torque over the same period.

The average output torque corresponds to the mean component of the instantaneous electromagnetic torque over one steady-state period. From the Maxwell stress perspective, the tangential electromagnetic force is related to the product of the radial and tangential air-gap flux-density components, $B_{\mathrm{r}}{B}_t$ [30]. To maintain consistency with the rotor-position-based cogging-torque formulation, the air-gap flux-density components are expressed as functions of the circumferential position $\theta$ and the rotor mechanical position ${\theta }_m$. The average output torque can be expressed as

The expression for the average output torque is given as

$$T_{avg}={\displaystyle \frac{L_a{r}^2}{{\mu }_0}}{\displaystyle {\int }_0^{2\pi }{B}_r}(\theta ,{\theta }_m)B_t(\theta ,{\theta }_m)d\theta$$

(6)

In Equation (6), $r$ denotes the mean air-gap radius, $B_{\mathrm{r}}\left(\theta \right)$ and $B_t\left(\theta \right)$ represent the radial and tangential air-gap flux-density components, respectively, and the overbar denotes the average value over one steady-state period. Equation (6) indicates that average torque and torque ripple are both related to the spatial and temporal harmonic contents of the air-gap magnetic field. Reducing the relevant air-gap flux-density harmonics can decrease the fluctuating torque component and improve torque smoothness [23,30].

4. Parameter Optimization Design

4.1. Optimization Variable Selection

As indicated by Equations (3) and (4), the cogging torque of the investigated 48-slot/8-pole IPMSM is related to the coupled harmonic term $G_n{B}_{r,6n}^2$, where $G_n$ is associated with the stator-slot permeance distribution and $B_{r,6n}^2$ is associated with the equivalent rotor-field distribution. Therefore, the selected variables should cover both the stator-side permeance term and the rotor-side field term.

The stator slot parameters Hs1, Bs0, Bs1, and Bs2 are selected because they affect the slot-opening geometry and the spatial distribution of air-gap permeance. The rotor parameters X1, Y1 and Y2, shown in Figure 6, are selected because they are related to the flux-barrier and magnetic-bridge geometry and can modify the equivalent rotor-field distribution. According to Equations (5) and (6), torque ripple and average torque are also affected by the air-gap flux-density harmonics and the Maxwell-stress product $B_{\mathrm{r}}{B}_t$. The selected variables are therefore consistent with the key terms in the torque-related analytical expressions and enable coordinated optimization of cogging torque, torque ripple, and average torque [30].

Latin hypercube sampling is applied to the full set of stator and rotor structural parameters listed in Table 2 and Table 3 to construct the sample space for surrogate modeling. Sensitivity analysis is then performed to identify high-sensitivity variables, which are selected as optimization variables for coordinated stator–rotor design. This enables the investigation of the relationship between key structural parameters and torque characteristics.

To further define the initial design space, 90 design points are generated for the seven stator variables and 60 design points for the four rotor variables using Latin hypercube sampling, resulting in a total of 150 samples. Additional samples are introduced when necessary to improve surrogate model accuracy, supporting subsequent sensitivity analysis and torque-oriented optimization. The overall optimization procedure is shown in Figure 7.

4.2. Sensitivity Analysis

Sensitivity analysis $G\left(x_i\right)$ is used to assess the contribution of each parameter to the optimization objectives and, in turn, allows for a reduction in the number of experiments and alleviation of computational complexity [13].

$$s\left(x_i\right)={\lambda }_1{\displaystyle \frac{T_{\mathrm{cog}}^{\prime }}{T_{\mathrm{cog}}\left(x_i\right)}}+{\lambda }_2{\displaystyle \frac{T_{\mathrm{pk}}\left(x_i\right)}{T_{\mathrm{pk}}^{\prime }}}+{\lambda }_3{\displaystyle \frac{T_{avg}^{\prime }}{T_{avg}\left(x_i\right)}}$$

(7)

$$G(x_i)={\lambda }_1{S}_{\mathrm{Tcog}}+{\lambda }_2{S}_{\mathrm{Tpk}}+{\lambda }_3{S}_{Tavg}$$

(8)

$$S(x_i)={\displaystyle \frac{\partial s\left(x_i\right)}{\partial x_i}}={\displaystyle \frac{\frac{\Delta s\left(x_i\right)}{s\left(x_i\right)}}{\frac{\Delta x_i}{x_i}}}$$

(9)

To optimize computational efficiency, sensitivity analysis $G\left(x_i\right)$ is implemented to refine the design variable set. Weighting factors ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are assigned as 0.5, 0.3, and 0.2 for cogging torque, torque ripple, and average torque, respectively, with a total sum of 1. Cogging torque is prioritized because, as an intrinsic structural excitation, it remains immune to active control compensation. As demonstrated by the analytical results in Figure 8, seven parameters including Hs1, Bs0, Bs1, Bs2, X1, Y1, and Y2 are retained as their weighted sensitivity indices exceed the 10% threshold. Specifically, while Y2 exhibits limited sensitivity toward torque ripple, its dominant influence on cogging and average torque results in a total weighted sensitivity of 16%, thereby justifying its retention. In contrast, Angle is excluded due to its sub-threshold aggregate sensitivity of 9.2%. Furthermore, parametric scanning confirms that Angle contributes less than 0.1 N·m to torque ripple variation, validating the reliability of this dimensionality reduction. Consequently, this process streamlines Kriging model training while preserving the potential for global optimization.

4.3. Development and Validation of the Kriging Surrogate Model

Kriging is a Gaussian process-based spatial interpolation technique that constructs a statistical model to predict responses in unexplored regions while quantifying uncertainty, offering high computational efficiency and prediction accuracy.

For a dataset consisting of n sample points x1, x2, …, xn, the prediction of the Kriging surrogate model at an arbitrary point x can be formulated as

$$\stackrel{\wedge }{y}\left(x\right)=f{\left(x\right)}^T\beta +z\left(x\right)$$

(10)

In this expression, $\beta$ denotes the vector of unknown regression coefficients, and $f\left(x\right)$ represents the vector of polynomial basis functions used in regression. $Z\left(x\right)$ is a stochastic process assumed to follow a normal distribution and must satisfy

$$E\left[z\left(x\right)\right]=0$$

(11)

$$Var\left[z\left(x\right)\right]={\sigma }^2$$

(12)

$$Cov\left[Z\left(x^i\right),Z\left(x^j\right)\right]={\sigma }^{2}R\left[R\left(\theta ,x^i,x^j\right)\right]$$

(13)

In this expression, $E$, $Var$ and $Cov$ denote the mean, variance, and covariance, respectively.

Full stator optimization, full rotor optimization, and coordinated stator–rotor optimization are performed under identical constraints, optimization algorithms, surrogate models, and objective functions. Kriging surrogate models are constructed based on Latin hypercube sampling to describe the relationships between cogging torque, torque ripple, average torque, and the design variables X1, Y1, Y2, Hs1, Bs0, Bs1, and Bs2 Model accuracy is evaluated using the coefficient of determination R², where values closer to 1 indicate higher prediction accuracy [19]. Each variable is discretized into 120 levels within its range to support surrogate modeling across the design space.

A total of 20 samples are selected as an independent validation set, while the remaining samples are used for training the Kriging surrogate model. As shown in Figure 9, the coefficients of determination R² for cogging torque, torque ripple, and average torque are 0.908, 0.994, and 0.996, respectively. These results indicate that the surrogate model provides sufficient prediction accuracy for subsequent optimization analysis.

To validate the Kriging method, Response Surface Methodology (RSM) and Radial Basis Function (RBF) are introduced for benchmarking. All models are trained and evaluated on identical datasets to ensure a rigorous comparison of their generalization capabilities. This benchmarking quantitatively demonstrates the advantages of the Gaussian process-based Kriging approach in characterizing the high-dimensional nonlinearities inherent in motor optimization. The comparative metrics are listed in

Table 4. Comparison of prediction models foreach optimization objective.

Surrogate Model	Cogging Torque (Tcog)		Torque Ripple (Tpk)		Average Torque (Tavg)
	R2	MAE/%	R2	R2	MAE/%	R2
Kriging	0.908	5.58	0.994	0.95	0.996	0.03
RSM	0.882	6.52	0.978	0.99	0.999	0.02
RBF	0.812	8.86	0.925	1.09	0.979	0.79

.

As indicated in Table 4, the Kriging model exhibits superior predictive performance compared to RSM and RBF, particularly for highly nonlinear indicators. For cogging torque Tcog, which is notoriously difficult to predict due to its high sensitivity to slot-opening geometry, the Kriging model maintains an R² of 0.908 and a minimum MAE of 5.58%. In contrast, the R² of the RBF model drops to 0.812, with a significantly higher MAE of 8.86%. This demonstrates the Kriging model’s advantage in utilizing the variogram to capture local variations in the air-gap magnetic field. Furthermore, for torque ripple and average torque, the Kriging model achieves near-ideal fitting with MAE values as low as 0.95% and 0.03%, respectively, ensuring a high-fidelity surrogate mapping for the subsequent optimization process.

To further verify whether the surrogate models accurately capture the underlying physical laws, a single-factor parametric analysis is conducted. By using the validated models to predict response trends across the design space, the influence of individual structural variables can be visualized. As shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, the predicted trends are highly consistent with electromagnetic theory, which not only validates the model’s reliability in unexplored regions but also provides a clear basis for narrowing down the search space. Based on these insights, the optimal intervals for the design variables are summarized in

Table 5. Optimal interval of optimization design variables.

Optimization Design Variable	Optimal Interval/mm
Hs1	0.25–0.40
Bs0	2.25–2.70
Bs1	4.50–5.09
X1	−1.20–−0.80
Y1	61–62
Y2	63–64

, laying the foundation for the subsequent multi-objective optimization.

The influence of stator tooth-shoulder height Hs1 on torque characteristics is presented in Figure 10, cogging torque Tcog increases slightly with increasing Hs1. This is attributed to the change in air-gap permeance caused by the variation in tooth tip thickness, although the effect on permeance fluctuation is limited. As Hs1 increases from 0.20 mm to 0.45 mm, the average torque Tavg decreases slightly from 195.37 N·m to 195.27 N·m. The increase in tooth tip thickness leads to a marginal rise in magnetic reluctance, resulting in minor variation in air-gap flux and torque. Smaller Hs1 values are therefore preferred. An optimal range of 0.25–0.40 mm is selected for Hs1.

The effect of slot opening width Bs0 on torque characteristics is shown in Figure 11, cogging torque Tcog increases with increasing Bs0, with a more pronounced rise at larger values. Local reductions are observed in the ranges of 2.1–2.4 mm and 3.9–4.1 mm. This trend is attributed to the increase in slot opening width, which enhances the air-gap permeance variation and amplifies cogging torque. As Bs0 increases from 1.5 mm to 4.0 mm, the average torque Tavg decreases from 193 N·m to 181 N·m, indicating a reduction in electromagnetic torque capability. An optimal range of 2.25–2.7 mm is selected for Bs0.

Due to the strong geometric coupling between the slot bottom width Bs2 and the slot center width Bs1, variations in Bs2 are reflected through Bs1. Therefore, only Bs1 is considered in the parametric analysis. As shown in Figure 12, cogging torque Tcog varies non-monotonically with Bs1. It decreases from 1.05 N·m to 0.97 N·m as Bs1 increases from 3.0 mm to 4.0 mm, followed by a slight increase and stabilization. When Bs1 exceeds 4.8 mm, Tcog decreases further to approximately 0.85 N·m. This behavior is attributed to the variation in slot geometry, which modifies the air-gap permeance distribution and its harmonic components. The average torque Tavg increases monotonically with Bs1, rising from 154 N·m to above 195 N·m as Bs1 increases from 3.0 mm to 5.1 mm. An optimal range of 4.5–5.09 mm is selected for Bs1.

The torque response to the rotor structural parameter X1 is shown in Figure 13, cogging torque Tcog increases as the absolute value of X1 decreases, rising from 0.725 N·m to 0.88 N·m as X1 varies from −1.2 mm to −0.5 mm. This behavior is attributed to the radial displacement of the permanent magnets, which alters the air-gap flux distribution and enhances the harmonic components of the magnetic field, leading to increased cogging torque. As the absolute value of X1 decreases, the average torque Tavg decreases from 197 N·m to 190 N·m. An optimal range of −0.8 to −1.2 mm is selected for X1.

The effect of rotor parameter Y1 on torque characteristics is presented in Figure 14, cogging torque Tcog varies non-monotonically with Y1, reaching a maximum of 0.89 N·m at Y1 is 60.5 mm and then decreasing to 0.84 N·m. This behavior is attributed to the variation in rotor geometry, which modifies the air-gap flux distribution and its harmonic components. The average torque Tavg increases with Y1, rising from 193.25 N·m to 195.5 N·m as Y1 increases from 60.0 mm to 62.5 mm. Considering the mechanical constraints under high-velocity operation, excessively large Y1 values are avoided. An optimal range of 61–62 mm is selected for Y1.

As shown in Figure 15, cogging torque Tcog increases when Y2 exceeds 63.78 mm. This behavior is attributed to the variation in rotor geometry, which modifies the air-gap flux distribution and enhances its harmonic components, leading to increased cogging torque.

The average torque Tavg exhibits a non-monotonic trend with Y2, decreasing to 193.5 N·m at 62.5 mm and then increasing to 195.3 N·m as Y2 continues to increase. Considering mechanical constraints under high-velocity operation, an optimal range of 63–64 mm is selected for Y2.

Design variables are selected based on their influence on the air-gap permeance. Although parametric scanning focused on Bs1, sensitivity analysis confirms that Bs2 also significantly affects electromagnetic performance. To ensure comprehensive geometric characterization, the Kriging surrogate model is constructed using seven independent parameters. During the optimization stage, to maintain a consistent slot taper and manufacturing feasibility, Bs2 is constrained to vary proportionally with Bs1 based on their initial design ratios. This approach streamlines the search space while preserving the modeling accuracy of the stator geometry. The optimized design intervals for the primary variables are summarized in Table 5.

4.4. Construction of Multi-Objective Optimization Model Based on NSGA-II

In this study, the NSGA-II algorithm combined with a Kriging surrogate model is employed to perform the coordinated stator–rotor structural optimization. To ensure the optimized IPMSM satisfies rigorous performance requirements, the multi-objective optimization problem is formulated by imposing specific constraints on the objective functions [3,23]. The optimization aims to reduce no-load cogging torque and load torque ripple while maintaining or improving average output torque. It is mathematically expressed as follows:

$$\left\{\begin{array}{cc}\mathrm{find}& x_i={[X1,Y1,Y2,Hs1,Bs0,Bs1,Bs2]}^T\\ \min & f_1(x_i)=T_{cog}(x_i)\\ \min & f_2(x_i)=T_{p\mathrm{k}}(x_i)\\ \max & f_3(x_i)=T_{avg}(x_i)\\ \mathrm{s}.\mathrm{t}.& T_{cogpk}<2.2 \mathrm{N}\cdot \mathrm{m}\\ & T_{rip}<21.04\%\\ & T_{avg}>191.2 \mathrm{N}\cdot \mathrm{m}\\ & x_{i,\min }\le x_i\le x_{i,\max },i=0,1,\dots ,6\end{array}\right.$$

(14)

By iteratively evolving the seven design variables within their optimal constrained ranges, the NSGA-II algorithm effectively identifies the optimal structural parameters. This process balances the complex trade-offs between conflicting torque characteristics. The detailed optimization procedure, including the population evolution and convergence steps, is illustrated in the flowchart in Figure 16.

According to the sampling analysis results, the population size is initialized to 300 to guarantee adequate convergence, and the number of generations is set to 200 to balance efficiency and accuracy, with a mutation probability of 0.1 and a crossover probability of 0.9 to preserve promising solutions.

The effectiveness of the optimization is evidenced by the convergence behavior and the distribution of the Pareto optimal set. As shown in Figure 17, the objective functions Tavg, Tcog, and Tpk gradually stabilize and reach a consistent plateau after the 150th generation. This trend demonstrates that the choice of 300 individuals and 200 generations is well-suited to the complexity of the 7-parameter design space, ensuring an exhaustive search without premature convergence. Furthermore, the 3D Pareto front in Figure 18 illustrates the physical trade-offs between average output torque and cogging torque.

Figure 17 shows the convergence histories of the three optimization objectives during the NSGA-II process. In the early generations, the objective values fluctuate markedly because the population is widely distributed in the design space. As the evolution proceeds, inferior individuals are gradually eliminated, and the solutions move toward the favorable objective region. After approximately 100 generations, the average torque becomes nearly stable, while the cogging torque continues to decrease slightly. After about 150 generations, the variations in all three objectives are small, indicating that the optimization process has reached a stable state. This trend demonstrates that the choice of 300 individuals and 200 generations is well-suited to the complexity of the 7-parameter design space, ensuring an exhaustive search without premature convergence [31,32]. A compromise solution was then selected according to the motor performance requirements and operational constraints, as shown in Figure 18. The final solution provides a cogging torque of approximately 0.744 N·m, a peak-to-peak torque fluctuation of 29.53 N·m, and an average output torque of 194.12 N·m.

As shown in

Table 6. Comparison between optimization result and simulation result.

Parameter	Optimization Results	Simulation Results
Tcog/(N·m)	0.744	0.75
Tpk/(N·m)	29.53	29.37
Tavg/(N·m)	194.12	194.23

, the relative errors for cogging torque, torque ripple, and average output torque were 0.8%, 0.54%, and 0.57%, respectively. All discrepancies remained below 1%, demonstrating the reliability of the proposed multi-objective optimization method for the structural design of the motor.

Overall, coordinated optimization provides the best comprehensive torque performance among the evaluated strategies. The detailed structural parameters before and after optimization are summarized in

Table 7. Parameter values before and after optimization.

Parameter	Initial Value /mm	Stator-Only/mm	Rotor-Only /mm	Optimized Value/mm
Hs0	1	0.934	-	-
Hs1	0.373	0.277	-	0.335
Hs2	17.541	18.101	-	-
Bs0	3	2.729	-	2.457
Bs1	4.314	4.503	-	5.094
Bs2	6.614	5.832	-	6.733
Rs	2	1.548	-	-
X1	−1	-	−0.841	−1.133
Y1	61.992	-	60.625	61.914
Y2	63.992	-	64.136	63.895
Angle	2.732	-	2.963	-

.

5. Simulation-Based Analysis of Optimization Results

The initial design is taken as the reference. Three optimized parameter sets, including stator-only, rotor-only, and coordinated stator–rotor optimization, are implemented in the finite element model under identical no-load and rated load conditions, as shown in Figure 19.

The electromagnetic performance metrics in Figure 19 are analyzed based on the cogging-torque formulation in Equation (4) and the Maxwell-stress-based torque expression in Equation (6), to clarify the relationship between structural optimization and torque stability.

As shown in Figure 19a, the initial design exhibits periodic cogging-torque fluctuation caused by slot–pole harmonic coupling. Stator-only optimization leads to a limited reduction in the cogging-torque peak because it mainly modifies the stator-slot permeance-related coefficient $G_n$, while the rotor-side equivalent field component remains largely unchanged. Rotor-only optimization produces a more visible reduction in the torque peak by changing the magnetic bridge and flux-barrier geometry, which affects ${(B_r)}_{6n}^2$. However, the stator-slot permeance harmonics are not adjusted simultaneously, so residual cogging-torque components remain. In contrast, coordinated stator–rotor optimization reduces the coupled contribution of $G_n{(B_r)}_{6n}^2$, leading to a lower cogging-torque amplitude. As shown in

Table 8. Target parameter values before and after optimization.

Optimization Objective	Tcog/(N·m)	Trip/(%)	Tavg/(N·m)
Before Optimization	1.10	21.04	191.20
After Optimization	0.75	15.12	194.23

, the cogging torque decreases from 1.10 N·m to 0.75 N·m, corresponding to a reduction of 31.8%.

Regarding the load torque in Figure 19b, the torque ripple can be interpreted using Equation (6). According to the Maxwell stress formulation, electromagnetic torque is related to the product of the radial and tangential air-gap flux-density components, $B_{\mathrm{r}}{B}_t$ Therefore, harmonic components in the air-gap magnetic field lead to periodic torque fluctuation. Stator-only optimization has a limited effect on the load torque waveform because local slot-geometry adjustment mainly changes the permeance distribution and cannot sufficiently suppress rotor-field-related harmonic components. Rotor-only optimization improves the torque trajectory by modifying the magnetic bridge and flux-barrier geometry, but residual fluctuations remain due to uncompensated slot-permeance harmonics. The coordinated design reduces both the rotor-field and slot-permeance harmonic contributions, resulting in a smoother torque waveform.

The air-gap flux-density spectrum in Figure 20 further supports this interpretation. After coordinated optimization, the 24th and 48th spatial harmonics are reduced, which weakens the harmonic interaction in the Maxwell-stress product $B_{\mathrm{r}}{B}_t$ and decreases the fluctuating component of electromagnetic torque. As listed in Table 8, the torque ripple ratio decreases from 21.04% to 15.12%, corresponding to a reduction of 28.1%, while the average torque increases from 191.20 N·m to 194.23 N·m. These results indicate that coordinated stator–rotor optimization improves torque stability by reducing both cogging-torque-related harmonic coupling and load-torque harmonic fluctuation.

The visualized foundation for these enhancements is further substantiated by the transient magnetic field distributions in Figure 21. In the baseline design, severe flux crowding occurs at the stator tooth tips and rotor magnetic bridges, where the local flux density peaks at 2201.183 mT. This localized saturation distorts the air-gap permeance and exacerbates high-order torque components. Coordinated optimization moderates this peak to 2188.097 mT, effectively alleviating saturation in critical magnetic paths. This homogenization ensures that the magnetic harmonic energy is strategically redistributed into the fundamental torque-producing component, validating the superior comprehensive performance of the synergistic design.

This study develops and validates a synergistic stator–rotor optimization framework to mitigate torque pulsations in an automotive IPMSM. The primary scientific findings and manufacturing considerations are as follows:

(1)

Sensitivity analysis confirms that torque performance is a coupled output of the entire magnetic circuit, where stator tooth-width Bs0, Bs1 effects are modulated by rotor magnet positioning X1, Y1;

(2)

By filtering parameters via a 10% sensitivity threshold and coupling stator slot variables, the design space was reduced to 7 independent variables, enhancing optimization efficiency and ensuring assembly simplicity;

(3)

The synergistic configuration effectively suppressed the 24th and 48th spatial harmonics, reducing Tcog and torque ripple rate Trip by 31.8% and 28.1%, respectively, without compromising the 90.1% mechanical safety margin.

In conclusion, the proposed coordinated stator–rotor optimization approach is effective for mitigating torque pulsations and improving torque performance in automotive IPMSMs. It should also be noted that the finite-element simulations in this study were conducted under a constant ambient temperature of 25 °C, and the present work mainly focuses on electromagnetic and structural-parameter optimization. Thermal effects, such as winding temperature rise, permanent-magnet demagnetization risk, and temperature-dependent material-property variations, were not fully coupled into the optimization process. In addition, although the optimized rotor geometry was constrained by manufacturability and basic mechanical feasibility, further experimental validation and multiphysics coupled verification are still required. These aspects will be considered in future work to further improve the operational reliability and industrial applicability of the proposed optimization method.