Advanced 3D Nonlinear Magnetic Equivalent Circuit Model for Overhang-Type WRSM Design

Abstract

The instability in rare-earth material supply and rising costs have driven research into rare-earth-free electric motors. Among various alternatives, wound rotor synchronous motors (WRSMs) stand out due to their adjustable excitation, enabling high torque at low speeds, and efficient field weakening at high speeds. Unlike permanent magnet synchronous motors (PMSMs), WRSMs offer greater operational flexibility and eliminate the risk of demagnetization. However, accurately modeling WRSMs remains challenging, especially when considering axial fringing flux and leakage components, which significantly affect motor performance. To address this challenge, this paper proposes a 3D nonlinear magnetic equivalent circuit (MEC) model that explicitly incorporates axial flux components and leakage paths in WRSMs with overhang rotor structures. Unlike conventional 2D MEC models, which fail to capture axial flux interactions, the proposed approach improves prediction accuracy while significantly reducing computational costs compared to full 3D finite element analysis (FEA). The model was validated through comparisons with 3D FEA simulations and experimental back-EMF measurements, demonstrating its accuracy and computational efficiency. The results confirm that the 3D nonlinear MEC model effectively captures axial flux paths and leakage components, making it a valuable tool for WRSM design and analysis. Future research will focus on further refining the model, incorporating hysteresis loss modeling, and developing hybrid MEC–FEA simulation techniques to enhance its applicability.

1. Introduction

The instability in rare-earth material supply and rising costs have accelerated research into rare-earth-free electric motors. Several alternatives have been explored, including switched reluctance motors (SRMs), induction motors (IMs), and wound rotor synchronous motors (WRSMs). However, each of these alternatives presents certain limitations. SRMs suffer from high torque ripple and acoustic noise, making them less suitable for applications requiring smooth operation. IMs, while widely used, exhibit lower efficiency and require higher excitation currents due to rotor losses, which can lead to increased energy consumption. These drawbacks necessitate the exploration of alternative solutions that balance efficiency, torque performance, and cost-effectiveness [1,2,3,4,5,6].

Among these alternatives, wound rotor synchronous motors (WRSMs) have gained significant attention due to their flux control capability, which enables high torque at low speeds and efficient field weakening at high speeds. Unlike permanent magnet synchronous motors (PMSMs), which rely on expensive rare-earth materials, WRSMs can achieve comparable performance while allowing for adjustable excitation, reducing dependence on fixed magnetic fields. Additionally, WRSMs eliminate the risk of demagnetization, a common concern in PMSMs operating under high temperatures or excessive currents. These characteristics make WRSMs a strong candidate for replacing PMSMs in various applications, particularly in automotive, industrial, and aerospace sectors, where both performance and material cost are critical considerations [7,8].

By directly comparing wound rotor synchronous motors (WRSMs) with permanent magnet synchronous motors (PMSMs), it becomes evident that while PMSMs offer high efficiency and power density, they are inherently limited by their reliance on rare-earth materials and fixed excitation. In contrast, WRSMs provide greater flexibility in flux control, allowing for improved performance across a wider operating range. Given these advantages, WRSMs emerge as a promising alternative to PMSMs, offering both technical and economic benefits for next-generation electric motor applications [9,10].

However, accurately modeling and analyzing WRSMs remains challenging, particularly when considering axial fringing flux and leakage components, which significantly affect motor performance. Existing studies on WRSM modeling primarily focus on 2D magnetic equivalent circuit (MEC) models due to their computational efficiency [6,9,10,11,12,13]. Also, conventional 2D MEC models fail to account for axial flux paths, leading to inaccuracies in performance predictions, especially in WRSMs with overhang rotor structures. While 3D finite element analysis (FEA) provides high accuracy, it is computationally expensive, making it impractical for iterative design evaluations. Despite the importance of capturing axial fringing flux and leakage components, research on 3D magnetic equivalent circuit (MEC) models for WRSMs remains unexplored, leaving a gap in computationally efficient modeling approaches that can effectively describe WRSM behavior in three dimensions.

To address this gap, this paper presents a 3D nonlinear magnetic equivalent circuit (MEC) model that explicitly incorporates axial flux components and leakage paths in WRSMs with overhang rotor structures. Unlike traditional 2D MEC models [6,9,10,11,12,13], which are inherently limited in representing axial flux interactions, the proposed approach enhances prediction accuracy by capturing 3D flux behavior, while significantly reducing computational time compared to full 3D FEA simulations. This is the first study to propose a 3D MEC framework for WRSMs, providing a novel method for evaluating axial flux paths and leakage components while maintaining computational efficiency.

To validate the proposed method, the results are compared with 3D FEA simulations and experimental back-EMF measurements, demonstrating the method’s ability to accurately model WRSM performance while significantly reducing computational costs. This study contributes to the advancement of rare-earth-free WRSM technology by providing a computationally efficient and reliable analysis tool for WRSMs with overhang rotor structures. The proposed 3D nonlinear MEC model serves as an effective design methodology, enabling engineers to analyze WRSM performance with high precision while avoiding the excessive computational burden of full 3D FEA simulations.

The structure of the paper is as follows: Section 2 presents the analysis of the overhang-type WRSM, detailing its unique flux characteristics and how the overhang affects magnetic behavior. Section 3 introduces the 3D nonlinear MEC model for the overhang-type WRSM, explaining its formulation and methodological approach. Section 4 describes the nonlinear analysis process, outlining the iterative permeability updates and the techniques used to ensure numerical convergence. Section 5 provides a comparison between the 3D nonlinear MEC analysis and 3D FEA results, validating the proposed model’s accuracy and efficiency. Section 6 discusses the experimental setup and results. Finally, Section 7 concludes the study by summarizing the key findings and suggesting potential directions for future research.

This structured approach ensures that the study presents a comprehensive framework for 3D MEC modeling, demonstrating its practical application in WRSM design and analysis while addressing critical challenges in magnetic flux modeling and computational efficiency.

2. Analysis of Overhang-Type WRSM

The wound rotor synchronous motor (WRSM) is highly sensitive to flux density saturation in electrical steel due to variations in field current. Depending on the saturation level, the magnitude and distribution of the axially induced flux paths and leakage components vary significantly. To accurately account for these axial flux components, 3D finite element analysis (FEA) is essential. Moreover, WRSMs with an overhang rotor structure require extensive computation time for analysis and design when using 3D FEA.

In this section, the flux paths of an overhang-type WRSM are defined, and a mathematical model is developed to represent them.

2.1. Flux Path of Overhang-Type WRSM

The flux path of the WRSM with an overhang structure is illustrated in Figure 1. In the regions where the rotor and stator are aligned in parallel, the flux flows parallel from the rotor to the stator. However, in the overhang structure, the flux distribution deviates from this behavior. Figure 2 presents a detailed classification of the flux paths into four distinct paths (Path 1 to Path 4). In the overhang rotor structure, some flux components emerge parallel from the rotor and enter the upper part of the stator, forming Path 2. Meanwhile, flux emerging in the axial direction of the rotor and entering the upper stator region constitutes Path 1.

Additionally, flux paths directed toward the lateral side of the stator are identified. Path 4 represents flux that emerges parallel from the overhang rotor and enters the lateral side of the stator. In contrast, Path 3 corresponds to flux that emerges in the axial direction of the rotor and enters the lateral side of the stator. Therefore, the rotor-to-stator flux path in the overhang-type WRSM rotor can be categorized into four distinct paths.

2.2. Mathematical Modeling of the Magnetic Path in Overhang-Type WRSM

In Figure 2, the flux path can be mathematically represented using a permeance model, which describes how readily magnetic flux passes through a given path. To provide a clear understanding of the flux behavior, the paths are categorized as follows. The symbols and terms used in the equations mentioned in this section are described in

Table A1. Definition of variables in Section 2.

Variable	Description	Unit
X1	Length of the axial flux generated in the rotor and entering the stator.	mm
X2	Length of the flux emerging parallel from the rotor overhang and entering the stator in the axial direction	mm
X3	Length of the flux generated in the rotor overhang and entering the lateral side of the stator	mm
${\theta }_f$	Angle of the path where the axial flux is generated	deg
g	Air gap between the rotor and stator	mm
${\mu }_0$	$\mathrm{Permeability} \mathrm{of} \mathrm{air} (4\pi \times {10}^{-7}$)	H/m

of Appendix A.1.

2.2.1. Description of Paths 1 and 3 (Axial Flux)

Paths 1 and 3 in Figure 2 correspond to the axial flux components and are further detailed in Figure 3. The axial flux emerges from the overhang rotor and enters the stator, following a trajectory governed by its respective permeance. For Path 1, the permeance is obtained by summing the infinitesimal widths of each differential flux path, leading to the integral formulation expressed in Equation (1):

$$P_{p1}={\int }_0^{X_1}{\displaystyle \frac{1}{g+{\theta }_f{X}_2+2{\theta }_{f}x}}dx={\displaystyle \frac{1}{2{\theta }_f}}\ln \left(1+{\displaystyle \frac{2{\theta }_f{X}_1}{g+{\theta }_f{X}_2}}\right)$$

(1)

where $P_{p1}$ represents the permeance along Path1, $\mathrm{g}$ is the air gap, and ${\theta }_f$ is angle of axial flux.

However, Path 3 follows a more complex trajectory. The axial flux first crosses the air gap, then re-enters the stator in the negative axial direction before reaching the lateral side of the stator. Due to this bidirectional movement, the mathematical representation must account for the two-stage flux transition. This results in a double integral formulation, as expressed in Equation (2):

$$P_{p3}={\int }_0^{X_3}{\int }_0^{X_1}{\displaystyle \frac{1}{g+{\theta }_f{X}_2+2{\theta }_f{x}_1+{\theta }_f{x}_3}}d{x}_{1}d{x}_3={\displaystyle \frac{{\mu }_0}{2{\theta }_f^2}}({\alpha }_3-{\beta }_3-{\gamma }_3+{\delta }_3)$$

(2)

$${\alpha }_3=(\mathrm{g}+{\theta }_f{X}_2+2{\theta }_f{X}_1+{\theta }_f{X}_3)\ln (g+{\theta }_f{X}_2+2{\theta }_f{X}_1+{\theta }_f{X}_3)$$

(2a)

$${\beta }_3=(\mathrm{g}+{\theta }_f{X}_2+{\theta }_f{X}_3)\ln (g+{\theta }_f{X}_2+{\theta }_f{X}_3)$$

(2b)

$${\gamma }_3=(\mathrm{g}+{\theta }_f{X}_2+2{\theta }_f{X}_1)\ln (g+{\theta }_f{X}_2+2{\theta }_f{X}_1)$$

(2c)

$${\delta }_3=(\mathrm{g}+{\theta }_f{X}_2)\ln (g+{\theta }_f{X}_2)$$

(2d)

where $P_{p3}$ represents the total permeance for Path 3, incorporating the variation in flux density as it transitions through multiple regions.

2.2.2. Description of Paths 2 and 4 (Parallel Flux)

Paths 2 and 4 from Figure 2 are illustrated in Figure 4, where they represent the parallel flux components emerging from the overhang rotor and entering the stator. Similar to the axial flux case, the permeance for Path 2 is determined by integrating the infinitesimal contributions along the flux path, leading to the expression in Equation (3):

$$P_{p2}={\int }_0^{X_2}{\displaystyle \frac{1}{g+{\theta }_{f}x}}dx={\displaystyle \frac{1}{{\theta }_f}}\ln \left(1+{\displaystyle \frac{{\theta }_f{X}_2}{g}}\right)$$

(3)

Additionally, Path 4, much like Path 3, follows a double integral formulation due to its complex trajectory through multiple regions before reaching the stator. This is mathematically represented in Equation (4):

$$P_{p4}={\int }_0^{X_3}{\int }_0^{X_2}{\displaystyle \frac{1}{g+{\theta }_f{X}_2+{\theta }_f{x}_3}}d{x}_{2}d{x}_3={\displaystyle \frac{{\mu }_0}{2{\theta }_f^2}}({\alpha }_4-{\beta }_4-{\gamma }_4+{\delta }_4)$$

(4)

$${\alpha }_4=(\mathrm{g}+{\theta }_f{X}_2+{\theta }_f{X}_3)\ln (g+{\theta }_f{X}_2+{\theta }_f{X}_3)$$

(4a)

$${\beta }_4=(\mathrm{g}+{\theta }_f{X}_3)\ln (g+{\theta }_f{X}_3)$$

(4b)

$${\gamma }_4=(\mathrm{g}+{\theta }_f{X}_2)\ln (g+{\theta }_f{X}_2)$$

(4c)

$${\delta }_4=\mathrm{g}\ln g$$

(4d)

where $P_{p4}$ accounts for the multi-layered flux transition that occurs before reaching its final destination.

Finally, the axial flux rotor-to-rotor component is illustrated in Figure 5, and its mathematical modeling is represented in Figure 6. The permeance between the rotors is given by Equation (5), and, ultimately, the permeance can be expressed in terms of magnetic reluctance, as formulated in Equation (6) in [9,10,12,13,14].

$$P_{R-R}={\int }_0^X{\displaystyle \frac{1}{g+{2\theta }_{f}x}}dx={\displaystyle \frac{1}{{2\theta }_f}}\ln \left(1+{\displaystyle \frac{{2\theta }_f\mathrm{X}}{g}}\right)$$

(5)

$$\mathrm{R}={\displaystyle \frac{1}{{\mu }_{0}P{L}_{stk}}}$$

(6)

3. 3D Nonlinear MEC Model for Overhang-Type WRSM

The flux path of the 12-pole/36-slot wound rotor synchronous machine (WRSM) with an overhang rotor structure is illustrated in Figure 7. This figure provides a detailed visualization of how the magnetic flux circulates through the stator, rotor, and air gap in the presence of an overhang rotor. The overhang structure introduces additional flux paths that contribute to the overall magnetic behavior of the machine.

To analyze this system mathematically, a 3D magnetic equivalent circuit (MEC) is derived from the flux path representation, as depicted in Figure 8. This equivalent circuit represents the magnetic system using an electrical analogy, where magnetic reluctances correspond to electrical resistances and magneto-motive forces (MMFs) correspond to voltage sources. Figure 8 specifically models the closed-loop flux paths considering the interactions between the stator, rotor, and air gap, allowing for the formulation of governing equations based on magnetic circuit principles.

The closed-loop flux generated by the overhang rotor exhibits symmetry with respect to the pole demarcation line. This symmetry ensures that the flux distribution satisfies the mathematical relationships expressed in Equations (7) and (8), which describe the balance of magnetic potentials in the system.

To establish the mathematical model, Kirchhoff’s Voltage Law (KVL) is applied to the closed-loop flux paths in the MEC model. Specifically, for the flux loops that include points ${\varnothing }_{al1}$ and ${\varnothing }_{ar1}$, the governing equations are formulated as Equations (9) and (10). These equations describe how the MMF is distributed along different flux paths in the presence of magnetic reluctances. Similarly, all other closed-loop flux paths in the system can be expressed in a similar manner, ensuring a comprehensive representation of the flux interactions.

$${\mathrm{\varnothing }}_{al(n)}=-{\mathrm{\varnothing }}_{al\left(n+7\right)},n=\mathrm{1,2},\mathrm{3,4},\mathrm{5,6},7$$

(7)

$${\mathrm{\varnothing }}_{ar1}=-{\mathrm{\varnothing }}_{ar2}$$

(8)

$$-R_{rl1}{\mathrm{\varnothing }}_{rl1}+\left(R_{rl1}+R_{al1}\right){\mathrm{\varnothing }}_{ar1}+R_{rl1}{\mathrm{\varnothing }}_{ag7}=0$$

(9)

$$-R_{ag1}{\mathrm{\varnothing }}_{ag1}-R_{ag1}{\mathrm{\varnothing }}_{ag7}+\left(R_{ag1}+R_{al1}\right){\mathrm{\varnothing }}_{al1}=0$$

(10)

The introduction of the rotor overhang leads to 16 additional closed-loop flux paths, which account for the increased magnetic reluctance in the system. However, due to the inherent symmetry of the structure, the number of independent equations required to fully describe the system is reduced to eight additional equations, effectively simplifying the model without losing accuracy.

To solve the system mathematically, the flux matrix and the magneto-motive force (MMF) matrix are constructed and expressed in Equations (11) and (12). These matrices define the relationship between the MMF sources, magnetic reluctances, and the resulting flux distributions. The final mathematical representation of the system is consolidated into a matrix equation, as shown in Equation (13), which forms a 19 × 19 matrix structure that encapsulates the full set of magnetic circuit equations for the given WRSM configuration.

$$\phi ={\left[\begin{array}{c}{\varnothing }_{st1},{\varnothing }_{st2},{\varnothing }_{st3},{\varnothing }_{rt1},{\varnothing }_{ar1}{\varnothing }_{ag1},{\varnothing }_{ag2},{\varnothing }_{ag3},{\varnothing }_{ag4},{\varnothing }_{ag5},{\varnothing }_{ag6},{\varnothing }_{ag7},\\ {\varnothing }_{al1},{\varnothing }_{al1},{\varnothing }_{al3},{\varnothing }_{al4},{\varnothing }_{al5},{\varnothing }_{al6},{\varnothing }_{al7}\end{array}\right]}^T$$

(11)

$$\mathrm{F}={\left[\mathrm{0,0},0,F_{rt1}-F_{rt2},\mathrm{0,0},\mathrm{0,0},\mathrm{0,0},\mathrm{0,0},\mathrm{0,0},\mathrm{0,0},\mathrm{0,0},0\right]}^T$$

(12)

$$A_R\phi =\mathrm{F}$$

(13)

4. Nonlinear Analysis Process for Overhang-Type WRSM

Figure 9 illustrates the 3D mesh-based nonlinear analysis process for the magnetic equivalent circuit (MEC), as described in references [9,15]. This process is essential for accurately modeling the nonlinear magnetic behavior of the wound rotor synchronous machine (WRSM) with an overhang structure, ensuring a more precise flux distribution analysis.

The process begins with the construction of the magnetic equivalent circuit model, where the WRSM flux paths are represented using magnetic reluctances and magneto-motive forces (MMFs). The presence of the rotor overhang significantly influences both axial and parallel flux components, making it necessary to incorporate this structural characteristic into the model. For each closed-loop flux path, a corresponding matrix equation is formulated to express the relationships between MMF, reluctance, and flux.

Once the model is established, the initial relative permeability of the electrical steel is assigned based on its material properties. This permeability value is used to determine the initial magnetic reluctance matrix, which plays a crucial role in defining the flux distribution within the machine. Following this initialization, the closed-loop flux is computed by solving the matrix equations derived from the MEC model. The computed flux values are then used to construct the flux matrix, which serves as a fundamental component in the nonlinear analysis process.

Since the permeability of electrical steel is nonlinear and varies with flux density, it is necessary to determine the differential permeability for each region of the model. The obtained flux values allow for the calculation of the corresponding differential permeability, ensuring that the material’s nonlinear properties are accurately represented. Using these updated permeability values, the reluctance matrix is revised to reflect the new magnetic characteristics. Additionally, a Jacobian matrix is formulated to enhance numerical stability, facilitating more efficient convergence of the iterative solving process.

With the reluctance and Jacobian matrices updated, the closed-loop flux is recalculated. If the newly computed flux deviates beyond an acceptable tolerance compared to the previous iteration, the analysis proceeds with further updates to the permeability values. This iterative process continues until the flux variation falls within the tolerance limit, at which point the nonlinear analysis is considered to have converged and the computation terminates.

By following this structured approach, the nonlinear behavior of the WRSM with an overhang rotor can be effectively analyzed, ensuring that the magnetic equivalent circuit model accounts for material-dependent variations in permeability. Figure 9 provides a visual representation of this iterative computational process, illustrating how permeability updates influence flux distribution and how convergence is achieved through repeated refinements of the reluctance matrix. The material properties of the electrical steel used in this study can be found in Figure A1 of Appendix A.2.

5. Comparison of 3D Nonlinear MEC Analysis and FEA Results

To evaluate the accuracy of the proposed mathematical model for the WRSM with an overhang rotor structure, a comparison was conducted with finite element analysis (FEA) results. Figure 10 illustrates the FEA model, and its detailed specifications are provided in

Table 1. Specifications of overhang-type WRSM.

	Value	Unit
Material	50PN470	-
Poles/slots	12/36	-
External diameter of stator	144	mm
External diameter of rotor	100	mm
Stack length	50	mm
Diameter of shaft	15	mm
Turns per stator tooth	13	-
Turns per rotor tooth	60	-
Overhang length	4	mm

.

The 3D nonlinear magnetic equivalent circuit (MEC) analysis and 3D finite element analysis (FEA) were performed for different overhang lengths.

Table 2. Back-EMF results of FEAs and 3D MEC analysis according to overhang length.

Overhang Length [mm]	3D FEA [V1st]	3D MEC [V1st]	Error Ratio of 3D FEA and MEC [%]
2	16.524	16.162	2.19
3	16.686	16.165	3.12
4	16.848	16.173	4.01
5	17.010	16.241	4.52
6	17.172	16.438	4.01

presents the back-EMF results at an excitation current of 50 A. The discrepancy between the different analysis methods was found to be within 5%, validating the accuracy of the proposed model.

Furthermore,

Table 3. Back-EMF results of FEAs and 3D MEC analysis according to field current.

Field Current [ADC]	3D FEA [V1st]	3D MEC [V1st]	Error Ratio of 3D FEA and MEC [%]
10	4.69	4.51	3.94
20	9.32	8.96	3.86
30	13.42	12.88	3.96
40	15.86	15.25	3.87
50	16.85	16.17	4.01
60	17.36	16.67	3.97

compares the analysis results for different excitation current levels. The results confirm that as the excitation current increases, the rate of back-EMF increment decreases. The 3D nonlinear MEC analysis demonstrates a similar trend, confirming its reliability in accurately representing the magnetic characteristics of the WRSM.

6. Experiment

Figure 11 shows the manufactured WRSM along with the test bench setup used for performance evaluation. The fabricated WRSM was initially designed using the 3D nonlinear magnetic equivalent circuit (MEC) proposed in this study during the preliminary design stage. Using the test bench in Figure 11, back-EMF measurements were conducted, and the results are presented in Figure 12 and

Table 4. No-load back-EMF measurement results of the manufactured motor.

Field Current [ADC]	Test [Vrms]	3D FEA [Vrms]	Error Ratio of 3D FEA and MEC [%]
10	2.89	2.86	−0.98
20	5.87	5.78	−1.50
30	8.69	8.63	−0.75
40	10.88	10.67	−1.95
50	11.79	11.87	0.73
60	12.47	12.63	1.27

. Figure 12 shows the back-EMF test waveforms under no-load conditions with a field current of 5 A. Table 4 compares the 3D finite element analysis (FEA) results of the optimized model with the measured back-EMF performance of the manufactured motor under different excitation currents. The discrepancy between the two results was found to be within 3%, demonstrating the accuracy of the proposed method. These findings confirm that the 3D nonlinear MEC analysis used in the initial motor design is an effective and reliable tool for optimizing motor performance before final implementation.

7. Conclusions

This study proposed a 3D nonlinear magnetic equivalent circuit (MEC) analysis method for designing and evaluating a wound rotor synchronous machine (WRSM) with an overhang rotor structure. The method was developed to efficiently capture the complex magnetic interactions within the machine while significantly reducing computational costs compared to full 3D finite element analysis (FEA). The proposed approach was applied to the initial design process, providing a systematic method for analyzing the impact of the overhang rotor on flux distribution and machine performance.

To validate the accuracy of the proposed model, its results were compared with 3D FEA simulations. The back-EMF analysis at an excitation current of 50 A showed a discrepancy of less than 5%, confirming the reliability of the MEC model. Additionally, the back-EMF increase rate exhibited a similar trend in both the MEC and FEA methods, further verifying the validity of the proposed approach. To enhance verification, a prototype WRSM was fabricated, and experimental back-EMF measurements were conducted using a dedicated test bench. The experimental results showed an error within 3% compared to 3D FEA, demonstrating that the proposed method provides a high level of accuracy while significantly reducing computational complexity. The findings confirm that 3D nonlinear MEC analysis effectively predicts WRSM performance, particularly in capturing axial flux paths and leakage components, making it a valuable tool for motor design and performance evaluation.

Despite its advantages, the proposed method has certain limitations. The MEC model relies on permeability approximations, which can introduce deviations under high magnetic saturation conditions. While the model effectively captures major flux paths, certain localized flux variations and higher-order harmonics may require a more detailed approach for improved precision. These limitations suggest the need for further refinement to enhance the model’s applicability to a wider range of operating conditions.

Future research can focus on improving the method’s accuracy by incorporating hysteresis loss modeling into the MEC framework, allowing better prediction of core losses in practical WRSM applications. Additionally, developing a hybrid MEC–FEA model could optimize computational efficiency by applying full FEA in critical regions while using MEC for less complex areas. Further experimental validation under variable speed and load conditions would also help assess the robustness of the model in real-world applications.

In conclusion, the 3D nonlinear MEC method presented in this study provides a computationally efficient and accurate tool for WRSM design and analysis, offering a significant reduction in computational cost compared to conventional 3D FEA while maintaining high accuracy. By incorporating further refinements such as hysteresis modeling, hybrid simulation techniques, and broader experimental validation, this approach can contribute to the advancement of electrical machine analysis and design methodologies, enabling more efficient WRSM development.