Optimal Design of a Coaxial Magnetic Gear Pole Combination Considering an Overhang

Abstract

This paper presents a comprehensive design approach for optimizing the pole configuration of a coaxial magnetic gear (CMG) structure with an overhang to enhance torque characteristics. Five CMG models were designed, and their characteristics were analyzed. A three-dimensional finite element method analysis was conducted to account for axial leakage flux. To efficiently explore the design space, we utilized an optimal Latin hypercube sampling method to generate experimental points and constructed a kriging-based metamodel owing to its low root-mean-square error. We analyzed torque characteristics across the design variables to identify characteristic trends and performed a parametric sensitivity analysis to evaluate the influence of each variable on the torque. We derived an optimal solution that satisfied the objective function and constraints using the design variables. The characteristics of the proposed model were validated through electromagnetic field analysis, fast Fourier transform analysis of the air-gap magnetic flux density, and structural analysis. The optimal model achieved an average torque of 61.75 Nm, representing a 21.15% improvement over the initial model, while simultaneously reducing the ripple factor by 0.41%. These findings indicate that the proposed CMG design with an overhang effectively enhances torque characteristics.

1. Introduction

Magnetic gears serve the same fundamental purpose as mechanical gears but differ in their method of power transmission. In contrast to mechanical gears, which rely on the physical contact of gear teeth to transfer power, magnetic gears utilize the modulation of magnetic fields to achieve non-contact power transmission [1]. This unique approach offers several maintenance advantages over conventional mechanical systems, including reduced noise and vibration, elimination of lubrication requirements, and minimal wear from friction [2,3]. These attributes position magnetic gears as a promising technology for the development of sustainable systems. For example, they are currently being explored as advanced power transmission solutions in various applications, including wind, wave, and tidal power generation, electric vehicles, transportation systems, and robotic applications [4,5,6,7,8,9].

Previous magnetic gears had the limitation of a relatively low torque density, typically in the range from 40 to 80 kNm/m³, compared to mechanical gears. However, in 2001, Atallah proposed a coaxial magnetic gear (CMG) design featuring an inner and outer rotor, which significantly enhanced the torque density to 100 kNm/m³. This breakthrough demonstrated the potential of magnetic gears for high-torque applications [10]. Since then, research has focused on improving CMG performance through structural design. Previous studies have demonstrated that the torque characteristics of coaxial magnetic gears (CMGs) are influenced by various design factors, among which the gear ratio plays a particularly critical role [11]. Flux modulation techniques have been shown to enhance torque density by overcoming limitations inherent in certain gear ratio configurations. Additionally, the direction of the magnetic flux—classified into axial and radial types—affects both the structural and performance aspects of CMGs [12]. While axial-flux designs exhibit a steeper rise in volumetric torque density (VTD) with increasing outer radius, they also introduce asymmetric axial forces, leading to potential structural imbalances. In contrast, radial-flux CMGs offer superior mechanical stability due to the symmetrical distribution of radial magnetic forces, making them more suitable for practical applications [13].

To further improve torque performance, research has explored the optimization of multiple design parameters, including pole-pair combinations, pole piece geometry, and permanent magnet (PM) volume [14,15]. In particular, selecting appropriate pole shapes and angular configurations can significantly affect both torque output and ripple. Several studies have investigated how these parameters interact to simultaneously maximize torque density and minimize torque ripple, thereby enabling application-specific optimization of CMG performance [16,17,18,19]. Therefore, this study aims to maximize the torque characteristics by analyzing the effects of pole combinations and pole piece shapes using three-dimensional finite element method (3D FEM) analysis. Two-dimensional (2D) FEM is commonly employed to evaluate CMG characteristics; however, its ability to account for the axial magnetic flux is limited. In particular, although the 2D FEM is computationally efficient and adept at analyzing the radial flux, it fails to accurately capture the axial leakage flux.

Hence, the implementation of 3D FEM is essential, despite its increased computational demands. Three-dimensional FEM analysis typically yields lower output torque and higher torque ripple compared with those obtained from 2D FEM analysis. This discrepancy arises from the inclusion of axial leakage flux in 3D FEM, which is often overlooked in 2D FEM. When the same CMG design is analyzed using both methodologies, 2D FEM yields faster results but sacrifices accuracy. In contrast, 3D FEM provides a more precise analysis of torque and magnetic flux density, albeit with longer analysis times [20,21,22]. To accurately evaluate torque performance and account for axial flux leakage, 3D FEM is essential, particularly in the early stages of CMG prototyping [23]. In this context, incorporating an overhang structure has been shown to partially recover torque loss and improve VTD, as demonstrated through 3D FEM simulations [24,25]. Achieving further performance gains within a fixed CMG volume requires a systematic optimization strategy, involving the selection of key design variables and the search for optimal configurations under design constraints [26]. As these design variables are interdependent and influence each other, they play a critical role in the overall system performance [27,28,29].

In this study, we employed the optimal Latin hypercube design (OLHD) method as a design of experiments (DOE) strategy to generate sample points with enhanced space-filling properties across the design domain [30,31,32]. While conventional Latin hypercube design (LHD) achieves uniform distribution, it may result in clustering of sample points, limiting metamodel accuracy [33,34]. To address this, the optimal LHD (OLHD) maximizes the spacing between samples, improving the quality of surrogate modeling for optimization tasks [35]. Since the metamodel provides only approximation, its optimal results are verified through high-fidelity 3D FEM simulations [36].

This paper proposes an advanced design methodology for CMGs that considers both pole combinations and overhang effects. Although CMGs possess a symmetric geometry that may initially suggest suitability for 2D FEM analysis, this approach inadequately captures the influence of the axial leakage flux. To address this limitation, a 3D FEM analysis was employed to obtain more accurate results. The optimization process commenced with the design of an initial CMG model, wherein the inner and outer diameters were fixed to streamline the analysis. Five initial models with distinct gear ratios were created based on different pole combinations. Five design variables were identified, and their respective ranges were established. Sample points were generated using OLHD, followed by FEM analysis for each point. Computational cost was reduced by constructing a metamodel based on the FEM results. The model with the best performance was selected to determine the optimal solution. The resulting optimal CMG design was validated through electromagnetic field analysis, air-gap magnetic flux density evaluation, and structural analysis. It was confirmed that torque characteristics differ depending on the difference in gear ratio within the same specification CMG, and that torque performance is improved through the overhang structure. The 3D FEM analysis results enabled us to effectively analyze magnetic flux leakage, and we expect that this type of magnetic gear can be used for connection with other power sources, including motors.

The remainder of this paper is organized as follows.

Section 2 presents the fundamental principles of magnetic gears and the initial design of the coaxial magnetic gear (CMG), including various pole-pair combinations and gear ratio analysis.

Section 3 describes the optimization methodology, including the design variables, optimal Latin hypercube sampling, metamodel construction, and optimization results.

Section 4 validates the optimal design through electromagnetic analysis, fast Fourier transform of air-gap flux density, and structural stress analysis.

Section 5 summarizes the key findings and contributions of the study and discusses directions for future work.

2. Magnetic Gear

2.1. Initial Design

In the context of PM-based magnetic gears, as mentioned earlier, magnetic flux can be categorized into two types based on its direction: radial flux, where the magnetic flux flows in a plane perpendicular to the shaft, and axial flux, where the flux flows parallel to the shaft axis. In both configurations, VTD generally tends to increase with the outer radius. While axial-flux CMGs exhibit a faster VTD increase than radial-flux CMGs, axial-flux CMGs tend to generate asymmetric axial magnetic forces, which can lead to structural imbalances. Conversely, radial-flux CMGs demonstrate a more balanced structure as the radial magnetic forces acting on the rotor symmetrically cancel each other out, resulting in a uniform force distribution and stable torque transmission. Consequently, this study adopts the radial flux configuration.

An initial CMG shape was designed to optimize performance, considering both the pole combination and overhang structure. The shape is shown in Figure 1, and its dimensions and specifications are listed in

Table 1. Dimensions and specifications of the CMG.

Parameter	Value	Unit
Outer radius of Rotor 1	43.5	mm
Outer radius of Rotor 2	53.6	mm
Outer radius of Rotor 3	90	mm
Inner radius	12.5	mm
Axial length	40	mm
Air gap	1	mm
Input speed (Rotor 1)	1000	rpm
Permanent magnet material	N35	-
Core material	20JNEH1500	-

. The CMG comprises three rotors: Rotor 1, which receives the input torque; Rotor 2, which delivers the amplified output torque; and Rotor 3, which is fixed and serves as the structural support for the CMG. The outer and inner diameters of the CMG were fixed at 90 mm and 12.5 mm, respectively. The overall height was 40 mm, and the air gap between the rotors was 1 mm.

The torque ripple of a CMG is influenced by the gear ratio, which is a critical consideration during the design process. When the gear ratio is an integer, the torque ripple tends to increase, whereas the torque density decreases. Conversely, non-integer gear ratios typically result in reduced torque ripple and enhanced torque density. Therefore, the implementation of a non-integer gear ratio is an effective strategy for minimizing the torque ripple [12]. Torque ripples tend to be greater when the gear ratio is an integer or when the pole-pair numbers of Rotor 1 and Rotor 3 form an integer-multiple relationship. Minimizing ripple is often achieved through fractional or decimal gear ratios and pole-pair combinations.

When Rotor 3 is fixed, the gear ratio of the CMG is calculated as follows:

$$P_3{|}_{w_3=0}=\left\{\begin{array}{c}\left(G_i-1\right)P_1+1 for G_1{P}_1 odd \\ \left(G_i-1\right)P_1+2 for G_1{P}_1 even\end{array}\right.$$

(1)

where P₃ denotes the number of PM pole pairs on Rotor 3, ω₃ represents the angular velocity of Rotor 3, G_i represents the gear ratio arbitrarily defined for the CMG, and P₁ denotes the number of PM pole pairs on Rotor 1. The number of pole pieces is determined using Equation (2):

$$Q_2=P_1+P_3$$

(2)

where Q₂ represents the number of pole pieces in the CMG and is equal to the sum of the pole pairs of Rotor 1 and Rotor 3. Finally, the gear ratio of the CMG is determined as follows:

$$G{|}_{w_3=0}={\displaystyle \frac{W_1}{W_2}}={\displaystyle \frac{Q_2}{P_1}}$$

(3)

where ω₁ and ω₂ represent the angular velocities of Rotor 1 and Rotor 2, respectively.

2.2. Pole-Pair Combination and Gear Ratio

Given that torque characteristics are influenced by the gear ratio, we explored multiple combinations of pole pairs in the initial design of the CMG to achieve superior gear ratios. The number of pole pieces and corresponding gear ratios were determined using Equations (1)–(3). An arbitrary gear ratio G_i was set to 4, and five combinations of pole pairs were considered by varying P₁ from 3 to 7. Each of these five pole-pair combinations was implemented in the CMG design, resulting in five distinct models. The corresponding values of P₁, Q₂, and P₃, as well as the resulting gear ratios, are listed in

Table 2. Pole-pair combinations and gear ratios.

P1	Q2	P3	Gear Ratio
3	13	10	1:4.33
4	18	14	1:4.5
5	21	16	1:4.2
6	26	20	1:4.33
7	29	22	1:4.14

. Furthermore, the five designed CMG shapes are shown in Figure 2.

Three-dimensional FEM analyses were conducted to perform a detailed analysis of the five pole-pair combinations. To enhance the accuracy of the FEM results, we divided the CMG shape into approximately 110,000 mesh elements. In this setup, Rotor 3 was fixed, and the rated speed of Rotor 1 was set at 1000 rpm. The rated speed of Rotor 2 for each shape was subsequently calculated based on the gear ratios of the five pole-pair combinations using Equation (4).

$$S_2{|}_{w_3=0}={\displaystyle \frac{S_1}{Q_2/P_1}}$$

(4)

where S₁ and S₂ represent the rated speeds of Rotors 1 and 2, respectively.

The average torque and torque ripple for each combination, obtained through the 3D FEM analysis, are listed in

Table 3. Torque characteristics (3D FEM).

P1	Average Torque	Torque Ripple	Unit
3	47.55	2.21	Nm
4	50.38	7.16	Nm
5	50.97	2.10	Nm
6	44.65	3.66	Nm
7	43.93	3.37	Nm

.

Among the five combinations evaluated, the model with P₁ = 5 demonstrated the highest average torque of approximately 50.97 Nm. Additionally, it demonstrated the lowest torque ripple of approximately 2.10 Nm. Therefore, the model with P₁ = 5 was selected as the baseline design for further optimization.

3. Optimal Design

To enhance the torque characteristics of the CMG, we implemented an optimization process and identified the optimal values of the design variables within a constrained design space, ensuring compliance with the objective function and design constraints. The overall flow of the optimization process is shown in Figure 3.

The optimization process commenced with the selection of appropriate design variables and the definition of their feasible ranges. Subsequently, utilizing the OLHD method as part of the DOE, sample points were generated based on the established design variables. Each sample point was evaluated using FEM analysis to generate characteristic data, which were subsequently analyzed to discern the output tendencies of the design variables. If the results indicated that the initial variable range was insufficient, the process was iterated to reconstruct the variable ranges. Once a satisfactory range was established, parametric sensitivity analysis was conducted to assess the influence of each design variable on the output. Using the collected sample data, we constructed a metamodel to approximate the system characteristics.

Subsequently, the metamodel was employed to search for the optimal solution that satisfies the objective function and constraints. Finally, the validity of the identified optimal solution was confirmed through FEM-based analysis, ensuring the effectiveness and feasibility of the proposed optimal design.

The interrelationship among design variables significantly influences the characteristics of the model, rendering their optimization essential. In this study, we identified five adjustable design variables for optimization purposes.

First, in CMGs with various pole-pair combinations, a significant amount of axial magnetic flux leakage was observed. Hence, an overhang structure was introduced to mitigate this leakage. The overhang length was selected as a design variable to identify the optimal value.

Second, the torque characteristics were influenced by the shape of the pole pieces. Consequently, we included both the length and arc angle of the pole pieces as design variables to determine the most effective configuration.

Finally, the shape of the PMs also impacts the torque characteristics. To optimize characteristics while maintaining a constant PM volume, we selected the electrical angle of the PM arc as a design variable.

The five design variables utilized in this optimization are shown in Figure 4 and are defined as follows:

X₁: overhang length, X₂: pole piece length, X₃: PM electrical degree, X₄: arc angle of the outer pole piece, and X₅: arc angle of the inner pole piece.

The initial values and ranges of the five design variables for the optimization process are listed in

Table 4. Range of design variables.

Parameter	Initial	Minimum	Maximum	Unit
X1	0	0	20	mm
X2	10	7	13	mm
X3	180	100	180	degree
X4	180	100	180	degree
X5	180	100	180	degree

. These ranges were selected to prevent magnetic flux saturation within the operating conditions.

Specifically, the overhang length X₁ was defined within the range 0–20 mm, whereas the pole piece length X₂ was set between 7 and 13 mm. The electrical arc angle of the PM X₃, outer arc angle of pole piece X₄, and inner arc angle of pole piece X₅ were defined in electrical degrees, ranging from 100° to 180°.

3.1. Sampling and Parametric Sensitivity

To enhance the torque characteristics of the CMG, we performed an optimization process utilizing the five selected design variables. Traditional optimization methods relying solely on FEM analysis can be computationally intensive. To mitigate this challenge, we first generated characteristic samples through FEM analysis, followed by the generation of a metamodel to facilitate the optimization process. To generate the metamodel, we employed the OLHD, a DOE method that optimizes the uniformity of the spacing between sample points within the design space. The number of sample points required for the design variables was determined based on the minimum number of experiments required as calculated using Equation (5). According to this equation, the minimum number of experiments required was approximately 31.5. Consequently, this study generated 40 sample points for five design variables [30,31].

$$nEXP \ge {\displaystyle \frac{1.5\times (nDV+1)\times (nDV+2)}{2}}$$

(5)

Subsequently, the torque characteristic samples for each design variable were generated using 3D FEM analysis based on the established sample points. The results, which demonstrate the torque characteristic trends for each variable, are shown in Figure 5. For X₁, the torque ripple decreased as the variable approached its maximum value. In contrast, for X₂, the torque ripple decreased as the variable approached its minimum value, whereas the average torque decreased as the variable approached its maximum range.

Based on the observed trends, the design variable ranges for X₁ and X₂ have been adjusted to enhance characteristics. Specifically, the range of X₁ was extended from 0 to 30 mm, whereas that of X₂ was modified from 4 to 10 mm. Although the characteristics for X₃, X₄, and X₅ generally improved as they approached their maximum values, their ranges remained unchanged. Given that these variables are defined in electrical degrees, they cannot exceed 180° per pole pair, and increasing the minimum range is deemed unnecessary. Consequently, new sample points were generated using OLHD within the updated variable ranges. The revised sample point ranges are listed in

Table 5. Range of reconstructed design variables.

Parameter	Initial	Minimum	Maximum	Unit
X1	0	0	30	mm
X2	10	4	10	mm
X3	180	100	180	degree
X4	180	100	180	degree
X5	180	100	180	degree

.

Torque characteristics generated through 3D FEM analysis, utilizing the newly adjusted sample points, are shown in Figure 6. The results reveal the presence of peak points within the ranges of the revised design variables, thereby validating the appropriateness of these adjustments for identifying the optimal design conditions.

The parametric sensitivity results for each model regarding the design variables are shown in Figure 7. Among the models, X₁ demonstrated the highest sensitivity. Notably, in the model designated as P₁ = 5, which served as the initial baseline model, X₁ accounted for 65.52% of the variation in average torque and 46.45% of the variation in the torque ripple, indicating its dominant influence. A comparative analysis of the torque characteristics for each pole-pair combination based on the sample points is presented in

Table 6. Comparison of torque characteristics based on pole combination.

P1	Average Torque [Nm]	Torque Ripple [Nm]	Ripple Factor [%]
3	45.06	2.87	6.36
4	50.86	5.04	9.91
5	51.70	1.97	3.82
6	49.37	4.18	8.46
7	46.99	3.04	6.47

. The initial model with P₁ = 5 demonstrated the highest average torque of 51.7 Nm and the lowest torque ripple of 1.97 Nm among the combinations. Here, the ripple factor is defined as the ratio of the torque ripple to the average torque.

3.2. Optimization

Based on the sample points detailed in the preceding section, a metamodel was designed to approximate the objective function and constraints. To generate this metamodel, we employed five distinct modeling methods: Kriging (KRG), multilayer perceptron (MLP), an ensemble of decision trees (EDT), polynomial regression (PRG), and radial basis function regression (RBF). The accuracy of each metamodel was evaluated by calculating the root-mean-squared error (RMSE) for both average torque and torque ripple characteristics. The metamodel demonstrating the lowest RMSE was identified as the most appropriate for the optimization process. The predictive performance of the metamodel is crucial, as it directly influences the reliability of the optimization results. Consequently, RMSE testing was conducted to evaluate the prediction accuracy. The RMSE quantifies the average deviation between predicted and actual values by calculating the square root of the mean squared differences. A lower (closer to zero) RMSE value indicates superior model performance.

To evaluate the predictive performance of each metamodel, we calculated the RMSE using Equation (6) [37], where n is the number of test points, y(X_i) is the actual function value, and $\widehat{y}$(X_i) is the predicted value:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{[y\left(X_i\right)-\widehat{y}(X_i)]}^2}$$

(6)

The resulting RMSE values for each metamodel are summarized in

Table 7. Metamodel performance evaluation (RMSE).

P1	Average Torque	Torque Ripple
3	KRG (0.45)	KRG (0.65)
	RBF (1.31)	EDT (0.71)
	PRG (1.71)	MLP (0.93)
	MLP (1.95)	RBF (1.46)
	EDT (3.76)	PRG (1.47)
4	KRG (1.24)	KRG (1.18)
	PRG (2.17)	EDT (1.69)
	MLP (3.08)	MLP (1.89)
	RBF (3.72)	RBF (2.69)
	EDT (4.15)	PRG (3.66)
5	KRG (0.77)	PRG (0.38)
	PRG (1.66)	EDT (0.43)
	RBF (1.78)	MLP (0.49)
	EDT (1.96)	KRG (0.53)
	MLP (3.22)	RBF (0.70)
6	KRG (1.09)	KRG (2.19)
	RBF (1.16)	EDT (2.56)
	PRG (1.19)	MLP (2.62)
	MLP (1.71)	PRG (2.69)
	EDT (1.92)	RBF (3.21)
7	KRG (1.52)	PRG (0.28)
	EDT (1.69)	KRG (0.42)
	PRG (1.90)	EDT (0.48)
	RBF (2.55)	MLP (0.61)
	MLP (2.59)	RBF (0.95)

.

Based on the RMSE results calculated using Equation (6), the Kriging method yielded the lowest RMSE values among the five evaluated methods, signifying the highest prediction accuracy. Therefore, the Kriging method was selected for implementation in this study. This interpolation-based modeling technique is particularly effective as it passes precisely through the sampled data points, thereby facilitating accurate approximation without introducing random errors. The estimated function is designed to minimize the variance in prediction errors by eliminating bias [31,38].

The objective function and constraints for the optimal design are defined in Equations (7) and (8). The primary aim is to enhance the characteristics of the CMG, with the objective function established to maximize the average torque. Based on the 3D FEM analysis data for the initial model with P₁ = 5, as listed in Table 3, the average torque was 50.97 Nm, accompanied by a torque ripple of 2.103 Nm. To ensure that the optimal design surpasses the characteristics of the initial model, we defined the constraints such that the average torque must exceed 50.97 Nm, whereas the torque ripple must remain below 2.103 Nm.

$$\mathrm{O}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} = \mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} \mathrm{A}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{T}\mathrm{o}\mathrm{r}\mathrm{q}\mathrm{u}\mathrm{e}$$

(7)

$$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}=\left\{\begin{array}{c}\mathrm{Torque} \mathrm{Ripple}<2.103 \mathrm{Nm}\\ \mathrm{Average} \mathrm{Torque}>50.97 \mathrm{Nm}\end{array}\right.$$

(8)

The initial values of the design variables for the optimization process were derived from the parameters of the initial model. To facilitate a fair comparison, we performed optimization under identical conditions across all five models. A genetic algorithm [31] was employed to determine the optimal design variables based on the aforementioned objective function and the constraints. For each model, the optimal solution was achieved after 200 iterations. The convergence profiles of the design variables, along with the objective function and constraints for the selected model, are shown in Figure 8.

Based on the optimization, the initial model with P₁ = 5 achieved the highest average torque of 62.03 Nm, thereby satisfying the objective function. However, the models with P₁ = 3 and P₁ = 6 failed to satisfy the torque ripple constraint, rendering them invalid for optimal solutions. The optimization results for the five models are listed in

Table 8. Result of design optimization.

P1	X1 [mm]	X2 [mm]	X3 [degree]	X4 [degree]	X5 [degree]	Average Torque [Nm]	Torque Ripple [Nm]
3	17.9	8	164	100	153	51.03	2.56
4	12.2	8.4	158	133	152	60.46	1.96
5	11.1	8.6	164	156	174	62.02	2.04
6	21.4	5.2	155	131	147	53.58	2.46
7	9.9	4	141	180	180	52.75	2.08

.

The magnetic flux density distributions for both the initial and optimal models are shown in Figure 9 and Figure 10, respectively. The isometric and front views of the initial model are shown in Figure 9a and Figure 9b, respectively, whereas the isometric and front views of the optimal model are shown in Figure 10a and Figure 10b, respectively. The overall flux density distribution of the entire CMG is shown in the isometric projections in Figure 9a and Figure 10a. Conversely, the front view in Figure 9b and Figure 10b highlights the axial flux leakage, a phenomenon not captured in conventional 2D FEM analysis. These findings demonstrate that the overhang structure of the optimal model significantly mitigates the axial flux leakage compared with the initial design. Unlike the initial model, which configured the permanent magnets with an electrical angle of 180 degrees, the optimal model, which uses the optimal solution of 164 degrees, reduces the magnetic flux interference caused by the magnets being in contact and concentrates the magnetic flux. Furthermore, the magnetic flux leakage in the axial direction is compensated for by the permanent magnet overhang.

4. Analytical Validation

To validate the optimal design, we performed an analytical evaluation based on the derived optimal solution using three methods: 3D FEM analysis, fast Fourier transform (FFT) analysis of the air-gap magnetic flux density, and structural analysis. The 3D FEM analysis provided an in-depth electromagnetic assessment of the CMG, incorporating the optimal solution. The FFT analysis focused on the magnetic flux density in the air gap between Rotor 1 and Rotor 2, where input and output interactions occur, thereby validating enhancements in magnetic field quality. Furthermore, the unique characteristics of the CMG, which include two air gaps, high- and low-speed rotors, and an overhang structure designed to account for axial leakage flux, necessitated a thorough structural analysis to ensure the mechanical stability of the design. Among the various configurations evaluated, the model with P₁ = 5 emerged as the most effective in satisfying the objective function and was subsequently selected for analytical validation. For completeness, models with P₁ = 4 and P₁ = 7, while not demonstrating optimal performance, were included in the analysis as they satisfied all relevant constraints.

4.1. Electromagnetic Analysis

To validate the electromagnetic characteristics of the optimal design, we conducted 3D FEM simulations based on the identified optimal solutions. An optimal model was designed by applying the optimal solutions to the design variables of the initial configuration. The CMG shape was divided into approximately 120,000 mesh elements to improve simulation accuracy. Rotor 3 was fixed, whereas the rated speed of Rotor 1 was set at 1000 rpm. Based on Equation (4), the corresponding rated speed of Rotor 2 was set at 238.09 rpm. In addition to the optimal model with P₁ = 5, the models with P₁ = 4 and P₁ = 7, which satisfied all the constraints, were analyzed for comparative purposes.

The results of the validation analysis of the three models are listed in

Table 9. Validation results of torque characteristics (3D FEM).

P1	Average Torque [Nm]	Torque Ripple [Nm]	Ripple Factor [%]
4	58.58	2.24	3.82
5	61.75	2.29	3.71
7	50.79	3.13	6.17

. The optimal model with P₁ = 5 achieved the highest average torque of 61.75 Nm. Although the P₁ = 4 model demonstrated the lowest absolute torque ripple of 2.24 Nm, the optimal model recorded the lowest ripple factor of 3.71%, indicating superior overall performance in terms of both torque output and stability.

A characteristic comparison was conducted to assess the torque characteristics of the initial and optimal models under identical operating conditions. The comparison results are presented in Figure 11. The optimal model demonstrated an average torque that was 10.78 Nm higher than that of the initial model, representing a 21.15% enhancement. Although the optimal model achieved a marginally higher torque ripple of 2.29 Nm, which was 0.19 Nm more than that of the initial model, the ripple factor decreased from 4.13% in the initial model to 3.71% in the optimal model, representing a decrease of 0.48%.

To evaluate the impact of the optimal design on the total harmonic distortion (THD), we analyzed the frequency components of the air-gap magnetic flux density for both models. The comparison results are shown in Figure 12. The air-gap magnetic flux densities of the two models and the results of the FFT analysis are shown in Figure 12a and Figure 12b, respectively. From the FFT results, the fundamental frequency components accounted for 59.61% and 68.66% of the initial and optimal models, respectively, representing a 147.48% increase in the fundamental component for the optimal design. Furthermore, the analysis showed increases in the 3rd, 5th, 7th, and 9th harmonic components by 101.46%, 115.18%, 134.35%, and 16.89%, respectively.

Consequently, the THD of the optimal model was reduced to 17.2% compared with 23.9% of the initial model, corresponding to a 6.7% decrease. These findings indicate that the air-gap flux density waveform in the optimal model has been substantially enhanced compared with that of the initial model.

4.2. Structural Analysis

To validate the structural stability of the designed CMG, which incorporates an overhang structure to account for the axial magnetic flux leakage, we performed a structural analysis. In this analysis, stress was employed as a key evaluation metric for stiffness and durability, as it represents the internal force per unit area that arises when an external load induces a deformation. Such deformations can lead to mechanical damage, resulting in reduced efficiency. Permanent failure occurred when the deformation exceeded the yield strength of the material. Therefore, the stress levels should be maintained below the yield strength limit [39].

The results of the structural analysis for the optimal model are shown in Figure 13. The stress distribution in the core is shown in Figure 13a. The core is composed of JFE_Steel_20JNEH1500, a material with a yield strength of 374 MPa. It was analyzed at a rated rotational speed of 1000 rpm for Rotor 1. The simulation results confirmed that the von Mises stress remained within the yield-strength range of the material.

The stress distribution in the PMs is shown in Figure 13b. These PMs were composed of NdFeB (Arnold-N35), which possesses a yield strength of 85 MPa. Given the overhang structure, PMs protrude outward, rendering them more susceptible to potential structural deformation; hence, their mechanical stability is critical. The analysis results indicated that the von Mises stress in the PMs remained below the yield strength threshold, thereby validating the structural stability.

In conclusion, the maximum stress levels recorded in the core and PMs were 0.82 MPa and 0.25 MPa, respectively. These values are significantly lower than their corresponding yield strengths, affirming that the optimal design of the CMG is structurally stable under the specified operating conditions.

5. Conclusions

In this study, an optimal design of a CMG was proposed to enhance its torque characteristics by considering the overhang structure and pole-pair combinations. Five initial CMG models, each featuring different gear ratios, were designed based on various pole combinations. To accurately assess axial flux leakage, we conducted a 3D FEM analysis and incorporated an overhang structure to mitigate the effects of leakage flux. Five adjustable design variables were selected for optimization, with their ranges carefully defined to prevent magnetic flux saturation. Within these specified ranges, 40 sample points were generated using the OLHD. Subsequently, five types of metamodels were constructed to approximate the system response. Among these, the Kriging method model demonstrated the highest accuracy, as determined by the RMSE results, and was thus selected for optimization. Utilizing the Kriging method model, the optimal values of the design variables were obtained while adhering to both the objective function and constraints. Electromagnetic field analysis, FFT analysis of the air-gap magnetic flux density, and structural analysis were performed to validate the optimal design. Electromagnetic analysis revealed that the optimized model achieved an average torque of 61.75 Nm, representing a 21.15% improvement over that of the initial model. The torque ripple increased by 9.17% to 2.29 Nm, but the ripple factor decreased from 4.12% to 3.71%, indicating improved torque characteristics. Furthermore, the FFT analysis of the air-gap flux density indicated a 6.7% decrease in THD, further validating the superior characteristics of the magnetic flux waveform. Moreover, we performed structural analysis to ensure mechanical stability. The results indicated that both the core and PMs operated within their respective yield strengths, thereby affirming the structural stability of the optimal design. The proposed optimization method effectively enhanced the torque characteristics of the CMG by refining pole-pair combinations and incorporating an overhang structure. Therefore, magnetic gear is considered promising for future applications in various fields, such as electric vehicles, mobile robots, and surgical robots.