Torque Density Enhancement of Magnetic Worm-Geared Motor with Half Skew Structure by Introducing Curved Tooth Shape

Haruki Yamanaka, Yukio Tsutsui, Akira Chiba, Kyohei Kiyota, Yusuke Fujii, Endo Mitsuru

International Electric Machines and Drives Conference (IEMDC) IEMDC2025

• T2R2

Background

To achieve more flexible manufacturing, collaborative robot actuators must be lightweight and highly backdrivable. A lightweight actuator requires a motor with high torque density for its weight. Robot actuators with high backdrivability also require a motor with high torque density to reduce the gear ratio while keeping output torque.

To develop a motor with high torque density and high backdrivability, the Magnetic Worm-Geared Motor (MWGM)[1] was proposed to improve the backdrivability of the Worm Drive Actuator (WDA)[2], but it has low torque density. To improve its torque density, claw poles[3] and a half skew structure[4] were introduced by the authors. As a result, the half skew structure improved the no-load flux density and torque density more than expected.

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Methods to investigate the cause of increased torque density

The torque of MWGM depends on the no-load flux linkage under equal armature current. Therefore, this section investigates the cause of increased no-load flux-linkage.

1. Simplified Magnetic Circuit

To compare the no-load flux-linkage, the no-load flux is first compared using the magnetic circuit. The flux path is composed of the stator core, air gap, permanent magnet, and rotor core. Let us ignore the reluctance of the iron core and assume the cross-sectional area of the flux path is equal to that of the claw yoke face. Figs. 1 (a) and (b) show the simplified circuits of one stator core of the full skew structure and the half skew structure, respectively. The $\mathcal{R}{\mathrm{PM}}$ and $\mathcal{R}{\mathrm{air}}$ are the reluctance of the air gap and the permanent magnet, respectively. The $\Theta_{\mathrm{PM}}$ is the magnetomotive force (MMF) of the permanent magnet. The reluctance is inversely proportional to the cross-sectional area, but the MMF of a permanent magnet is independent of the area[5].

Fig. 1 Simplified magnetic circuit of one stator core

Then, no-load flux-linkage is calculated from no-load flux and structure. Fig. 2 (a) and (b) show the layout of the U-phase stator and the permanent magnet under maximum no-load flux-linkage of the full skew structure and the half skew structure, respectively. The full skew structure has two separate U-phase stator cores and armature windings. On the other hand, the half skew structure has only one U-phase stator core and armature winding.

Fig. 2 The layout of the U-phase stator and the permanent magnet under maximum no-load flux-linkage

2. 3D Finite Element Analysis (3D-FEA)

Another 3D-FEA is carried out to verify the calculation using the simplified magnetic circuit. In this analysis, material of stator core is changed to ideal one whose relative permeance is 10,000. The large value means low reluctance and the constant value means magnetic saturation does not occur in stator teeth, so the condition of this analysis is similar to that of the comparison above.

Results of the methods

1. Simplified Magnetic Circuit

From Fig. 1, no-load flux $\psi$ through one stator core of full skew structure is expressed as: \(\psi = \frac{2\Theta_{\mathrm{PM}}}{2\mathcal{R}_{\mathrm{PM}}+2\mathcal{R}_{\mathrm{air}}} = \frac{\Theta_{\mathrm{PM}}}{\mathcal{R}_{\mathrm{PM}}+\mathcal{R}_{\mathrm{air}}}\) and no-load flux $\psi_{\mathrm{HS}}$ through one stator core of half skew structure is expressed as: \(\psi_{\mathrm{HS}} = \frac{2\Theta_{\mathrm{PM}}}{\mathcal{R}_{\mathrm{PM}}+\mathcal{R}_{\mathrm{air}}} = 2\psi\) It is found that the flux is theoretically increased two times.

Based on the relationship above, Figs. 3 (a) and (b) show the number of turns and flux linked with one-phase winding of the full skew structure and the half skew structure, respectively. $N$ is the number of turns per slot and $\phi$ is the flux linked with one armature winding of full skew structure.

Fig. 3 The number of turns and flux linked with one-phase winding

From Fig. 3, the flux-linkage of full skew strucuture $\lambda_{\mathrm{FS}}$ and half skew strucuture $\lambda_{\mathrm{HS}}$ is expressed as: \(\lambda_{\mathrm{FS}} = N \cdot \psi + N \cdot \psi = 2N\psi\) \(\lambda_{\mathrm{HS}} = 2N \cdot 2\psi = 4N\psi\) From the equations above, the ratio of the flux-linkage could be 2.

2. 3D Finite Element Analysis (3D-FEA)

Fig. 4 shows the comparison of no-load flux-linkage. The left two bars show the no-load flux-linkage when the material of stator core is existing one with nonlinear $B-H$ curve. The right two bars show the no-load flux-linkage when the material of stator core is ideal one with constant relative permeance 10000 and linear $B-H$ curve. The ratio of no-load flux-linkage with ideal material is 1.93, close to 2. This means that the calculation above is valid and suppressing the magnetic saturation can improve the torque density.

Fig. 4 Comparison of no-load flux-linkage

Method to suppress the magnetic saturation

One solution to suppress the magnetic saturation is to increase the cross-sectional area of flux path. In the case of MWGM, suppressing it needs increasing circumferential length or axial thickness of stator teeth.

Figs. 5 (a) and (b) show the schematic image of proposed tooth shape from axial and circumferential directions, respectively. The area surrounded by red dotted lines show the area added to stator teeth in the proposed shape. The design of stator teeth uses $L_{\mathrm{linear}}(r)$, the linear approximation of $L(r)$ in Fig. 5 (a), because $L(r)$ and $L_{\mathrm{linear}}(r)$ have little difference. Then $d_{\mathrm{t}}(r)$ in Fig. 5 (b) is inversely proportional to $L_{\mathrm{linear}}(r)$ to keep cross-sectional area of flux path regardless of the radius $r$.

Fig. 5 Schematic image of proposed tooth shape

Result of the method

Another 3D-FEA is carried out to verify the effect of the proposed shape. This analysis compares straight tooth shape with curved tooth shape.

Table 1 shows the performance comparison between straight and curved tooth shape. The curved tooth shape improved the flux-linkage and torque density by almost 30%.

Table 1 Performance comparison between straight tooth shape and curved tooth shape

Item	Unit	Straight tooth	Curved tooth
Active mass	$\mathrm{kg}$	$5.94$	$6.02$
No-load flux-linkage	$\mathrm{Wb}$	$0.104$	$0.134$
Torque density	$\mathrm{N}\cdot\mathrm{m/kg}$	$0.921$	$1.20$
Power factor		$0.842$	$0.816$

Figs. 6 (a) and (b) show the flux density distribution of the straight tooth shape and the curved tooth shape under maximum no-load flux-linkage, respectively. The curved tooth shape improved magnetic saturation near the back yoke, but it did not improve the saturation at most of stator teeth. Further improvement of tooth shape is required to suppress it more.

Fig. 6 The flux density distribution under maximum no-load flux-linkage

Conclusions

In this paper, the torque density improvement of MWGM is investigated from the viewpoint of no-load flux-linkage. The result shows that a half skew structure could double the flux-linkage but the magnetic saturation occured at stator teeth worsens the ratio. Then, curved tooth shape for MWGM is proposed to suppress the magnetic saturation. Finally, 3D-FEA is carried out and analysis result shows the torque density is improved by 30%.

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References

[1] 山中・筒井・千葉・清田・藤井：「トルク密度の改善を目的とした磁気ウォームギヤドモータの提案」，令和6年電気学会全国大会，pp. 125-126 (2024)

[2] O. Efobi and Y. Fujimoto, “Design Considerations for a Radially Magnetized Permanent Magnet Worm Drive Actuator,” IEEE Conf. on A.I.M., pp. 1303-1308, Busan (2015)

[3] 山中・筒井・千葉・清田・藤井：「磁気ウォームギヤドモータのクローポールの導入によるトルク密度の向上」，電磁力関連ダイナミクスのシンポジウム, pp. 213-216 (2024)

[4] 山中・筒井・千葉・清田・藤井：「磁気ウォームギヤドモータにおける半スキュー構造の提案」，2024年電気学会産業応用部門大会 ヤングエンジニアポスターコンペティション (Y-26), p. 1 (2024)

[5] J. Pyrhonen, T. Jokinen and V. Hrabovcova, “Design of rotating electrical machines,” 2nd ed. Bognor Regis, UK: John Wiley and Sons, Ltd., 2014, p. 215

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