France



I. Introduction&lt;br/&gt;The conventional modulating ring in a coaxial magnetic gear is composed of separate pieces of laminated steel known as flux modulators. The assembly of the separate modulators is not easy because the laminations of each modulator must be firmly held together by using a clamping rod. A modulating ring with bridges that integrate the steel pieces into a whole laminated ring can greatly save the assembly efforts [1]. From the electromagnetic perspective, thin bridges are undoubtedly preferred as thick bridges compromise the modulating effect and hence downgrade the pull-out torque [2], [3]. However, thin bridges challenge the mechanical design of the modulating ring considering the stress due to the magnetic pull. Fernando and Saha investigated the effect of different numbers of flux modulators, thicknesses and positions of bridge on the shear stress in the modulating ring, and ended with a double-bridge design [3]. Due to the effectively thick bridges, this design has a large loss of torque capacity though the double-bridge structure is very robust mechanically.&lt;br/&gt;This digests proposes a design procedure of bridged modulators that reduces the mechanical stress at the cost of torque capacity as small as possible.&lt;br/&gt;II. Magnetic Gear With Bridged Modulators&lt;br/&gt;Fig. 1 shows the prototype magnetic gear with bridged modulators. This magnetic gear was originally presented with separate modulators, and more information about its construction can be found in [4]. For the study in this paper, the separate modulators are replaced with a modulating ring integrated by the bridges near the high-speed rotor (HSR), as shown Fig. 1. Such a position of bridges was proved to have larger torque capacity than any other positions [2], [3]. The modulating ring laminations are kept together by using the end plates and the clamping rods that are inserted into three of the modulators.&lt;br/&gt;III. Design of Modulating Ring Considering Stress&lt;br/&gt;A. With Different Thicknesses of Bridge&lt;br/&gt;Two modulating ring designs with different thicknesses of bridge and equal ratio of pole arc to pole pitch (&lt;i&gt;δ&lt;/i&gt;) (designs 1 and 2) are presented in Table I. The time-varying torques and stress distributions in these magnetic rings are calculated by three-dimensional (3-D) time-stepping finite element method (T-S FEM). The maximum mean torques extracted from the torque curves with different initial relative positions of the rotors, known as pull-out torques, are 15.70 Nm and 16.24 Nm at the outer rotors of designs 1 and 2, respectively. Table I shows the maximums of the fluctuating stress under approximately equal load condition. The stress of the thinner bridge is apparently higher than that of the thicker bridge. The maximum stress occurs at the junction of the bridge and the modulator that encloses a clamping rod.&lt;br/&gt;It should be noted that the modulating ring bears wildly fluctuating stress which is more prone to cause failure than a static stress of equivalent magnitude in a long-term operation, due to the metallic fatigue. Therefore, a sufficient margin of safety for yield must be reserved.&lt;br/&gt;B. With Different Ratios of Pole Arc to Pole Pitch&lt;br/&gt;Previous research indicates that the pull-out torque does not deviate significantly from the maximum for a certain large variation of &lt;i&gt;δ&lt;/i&gt; in the vicinity of &lt;i&gt;δ&lt;/i&gt;&lt;sub&gt;m&lt;/sub&gt; corresponding to the maximum pull-out torque [2]. This feature suggests that an appropriate value of &lt;i&gt;δ&lt;/i&gt; can reduce the stress without downgrading the torque capacity. For the bridge thickness of 0.8 mm, the calculated pull-out torques with the &lt;i&gt;δ&lt;/i&gt; values of 0.33, 0.375, 0.41, 0.45, 0.5 and 0.58 are 16.00 Nm, 16.33 Nm, 16.44 Nm, 16.53 Nm, 16.17 Nm and 15.11 Nm, respectively. When &lt;i&gt;δ&lt;/i&gt; equals 0.33 to 0.5, the torque capacity is not apparently different from the maximum. Table I shows the maximum stress in the modulating ring with that range of &lt;i&gt;δ&lt;/i&gt;, at approximately equal load. In comparison to the &lt;i&gt;δ&lt;/i&gt;&lt;sub&gt;m&lt;/sub&gt; design, the design with &lt;i&gt;δ&lt;/i&gt; of 0.33 reduces the maximum stress by 24.1% at the cost of only 3.2% lower torque capacity.&lt;br/&gt;C. With Different Fillet Radii at Junctions&lt;br/&gt;Filleting the sharp or right-angle corner is a commonly used mechanical design approach to reducing the stress concentration in a junction. Designs 1 to 6 all have right-angle junctions. Table I shows the maximum stresses with different &lt;i&gt;δ&lt;/i&gt; and fillet radii (designs 7 to 15) at approximately equal load. In those cases, the increase of fillet radius relieves the stress only when &lt;i&gt;δ&lt;/i&gt; equals 0.5. Filleting the junction may not relieve the stress in the modulating ring because the distribution of the magnetic load force changes together with the shape of bridge, unlike a mechanical load force.&lt;br/&gt;IV. Conclusion&lt;br/&gt;This digest proposes a design procedure for reduced mechanical stress in the bridged modulating ring of coaxial magnetic gear, at minimum cost of torque capacity. Step 1 of the design procedure is to place the bridges near the side of high-speed rotor and minimize the thickness of bridge for maximum torque capacity and acceptable stress. Step 2 is to vary the pole-arc-to-pole-pitch ratio of the modulating ring for reduced stress at acceptable cost of torque capacity. Step 3 is to fillet the junctions for the potential further reduction of stress. These steps could be repeated until the desired torque capacity and stress are achieved.&lt;br/&gt;Using more clamping rods could be another approach to reducing the stress. The full paper will investigate the effects of adding clamping rods on the stress and the eddy current loss in the modulating ring.

**References:**

[1] D. Z. Abdelhamid and A. M. Knight, &lt;i&gt;IEEE Trans&lt;/i&gt;. &lt;i&gt;Magn&lt;/i&gt;., vol. 53, no. 11, Art. no. 8112804 (2017).&lt;br/&gt;[2] Y. Zhan, &lt;i&gt;et al&lt;/i&gt;, &lt;i&gt;IEEE Trans. Magn.&lt;/i&gt;, vol. 55, no. 6, Art. no. 8103904 (2019).&lt;br/&gt;[3] N. Fernando and S. Saha, &lt;i&gt;IEEE Trans. Energy Convers.&lt;/i&gt;, vol. 33, no. 1, pp. 49–58 (2018).&lt;br/&gt;[4] Y. Ge, Q. Xin, and C. Nie, &lt;i&gt;Journal of Mech. Trans.&lt;/i&gt;, vol. 36, no. 4, pp. 5–8, 13 (2012).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/3d7ea0b425a5955b905b54bff4286211.jpg)
Fig. 1. Magnetic gear

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/8178c855ff5aa363505d14637e4d1e46.jpg)
Table I Load torques and stresses

INTERMAG 2021

 Design of Bridged Flux Modulators in Coaxial Magnetic Gear Considering Mechanical Stress

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I. Introduction The conventional modulating ring in a coaxial magnetic gear is composed of separate pieces of laminated steel known as flux modulators. The assembly of the separate modulators is not easy because the laminations of each modulator must be firmly held together by using a clamping rod. A modulating ring with bridges that integrate the steel pieces into a whole laminated ring can greatly save the assembly efforts [1]. From the electromagnetic perspective, thin bridges are undoubtedly preferred as thick bridges compromise the modulating effect and hence downgrade the pull-out torque [2], [3]. However, thin bridges challenge the mechanical design of the modulating ring considering the stress due to the magnetic pull. Fernando and Saha investigated the effect of different numbers of flux modulators, thicknesses and positions of bridge on the shear stress in the modulating ring, and ended with a double-bridge design [3]. Due to the effectively thick bridges, this design has a large loss of torque capacity though the double-bridge structure is very robust mechanically. This digests proposes a design procedure of bridged modulators that reduces the mechanical stress at the cost of torque capacity as small as possible. II. Magnetic Gear With Bridged Modulators Fig. 1 shows the prototype magnetic gear with bridged modulators. This magnetic gear was originally presented with separate modulators, and more information about its construction can be found in [4]. For the study in this paper, the separate modulators are replaced with a modulating ring integrated by the bridges near the high-speed rotor (HSR), as shown Fig. 1. Such a position of bridges was proved to have larger torque capacity than any other positions [2], [3]. The modulating ring laminations are kept together by using the end plates and the clamping rods that are inserted into three of the modulators. III. Design of Modulating Ring Considering Stress A. With Different Thicknesses of Bridge Two modulating ring designs with different thicknesses of bridge and equal ratio of pole arc to pole pitch (δ) (designs 1 and 2) are presented in Table I. The time-varying torques and stress distributions in these magnetic rings are calculated by three-dimensional (3-D) time-stepping finite element method (T-S FEM). The maximum mean torques extracted from the torque curves with different initial relative positions of the rotors, known as pull-out torques, are 15.70 Nm and 16.24 Nm at the outer rotors of designs 1 and 2, respectively. Table I shows the maximums of the fluctuating stress under approximately equal load condition. The stress of the thinner bridge is apparently higher than that of the thicker bridge. The maximum stress occurs at the junction of the bridge and the modulator that encloses a clamping rod. It should be noted that the modulating ring bears wildly fluctuating stress which is more prone to cause failure than a static stress of equivalent magnitude in a long-term operation, due to the metallic fatigue. Therefore, a sufficient margin of safety for yield must be reserved. B. With Different Ratios of Pole Arc to Pole Pitch Previous research indicates that the pull-out torque does not deviate significantly from the maximum for a certain large variation of δ in the vicinity of δm corresponding to the maximum pull-out torque [2]. This feature suggests that an appropriate value of δ can reduce the stress without downgrading the torque capacity. For the bridge thickness of 0.8 mm, the calculated pull-out torques with the δ values of 0.33, 0.375, 0.41, 0.45, 0.5 and 0.58 are 16.00 Nm, 16.33 Nm, 16.44 Nm, 16.53 Nm, 16.17 Nm and 15.11 Nm, respectively. When δ equals 0.33 to 0.5, the torque capacity is not apparently different from the maximum. Table I shows the maximum stress in the modulating ring with that range of δ, at approximately equal load. In comparison to the δm design, the design with δ of 0.33 reduces the maximum stress by 24.1% at the cost of only 3.2% lower torque capacity. C. With Different Fillet Radii at Junctions Filleting the sharp or right-angle corner is a commonly used mechanical design approach to reducing the stress concentration in a junction. Designs 1 to 6 all have right-angle junctions. Table I shows the maximum stresses with different δ and fillet radii (designs 7 to 15) at approximately equal load. In those cases, the increase of fillet radius relieves the stress only when δ equals 0.5. Filleting the junction may not relieve the stress in the modulating ring because the distribution of the magnetic load force changes together with the shape of bridge, unlike a mechanical load force. IV. Conclusion This digest proposes a design procedure for reduced mechanical stress in the bridged modulating ring of coaxial magnetic gear, at minimum cost of torque capacity. Step 1 of the design procedure is to place the bridges near the side of high-speed rotor and minimize the thickness of bridge for maximum torque capacity and acceptable stress. Step 2 is to vary the pole-arc-to-pole-pitch ratio of the modulating ring for reduced stress at acceptable cost of torque capacity. Step 3 is to fillet the junctions for the potential further reduction of stress. These steps could be repeated until the desired torque capacity and stress are achieved. Using more clamping rods could be another approach to reducing the stress. The full paper will investigate the effects of adding clamping rods on the stress and the eddy current loss in the modulating ring.

**References:**

[1] D. Z. Abdelhamid and A. M. Knight, IEEE Trans. Magn., vol. 53, no. 11, Art. no. 8112804 (2017). [2] Y. Zhan, et al, IEEE Trans. Magn., vol. 55, no. 6, Art. no. 8103904 (2019). [3] N. Fernando and S. Saha, IEEE Trans. Energy Convers., vol. 33, no. 1, pp. 49–58 (2018). [4] Y. Ge, Q. Xin, and C. Nie, Journal of Mech. Trans., vol. 36, no. 4, pp. 5–8, 13 (2012).

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/3d7ea0b425a5955b905b54bff4286211.jpg)
Fig. 1. Magnetic gear

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/8178c855ff5aa363505d14637e4d1e46.jpg)
Table I Load torques and stresses

technical paper

I. INTRODUCTION Environment recognition is a crucial task for autonomous mobile systems such as robots and drones, and vison information obtained by a camera is important. These systems require a high spatial awareness while undergoing various disturbances in an unknown environment. However, there are some problems: degradation of recognition accuracy caused by vibration and increase in size of the structure using a multi-degree-of-freedom (DOF) driving mechanism. Therefore, we focused on a multi-DOF actuator. Various types of multi-DOF actuators have been proposed, which use principles based on synchronous motors [1], stepping motors [2]. Nevertheless, the conventional actuators have a large structure and control device, for high torque density and continuous rotation. For applying to a driving source of a camera attitude control system, an actuator that can be configured with a few components and driven by a simple control device is required. Therefore, a multi-DOF voice coil actuator (DOF-VCA) is discussed in this paper, which has high responsiveness because it has only a few components. Kim et al. proposed a two-DOF-VCA with four-coils and permanent magnets [3,4]. This actuator can generate two-DOF motion driven by two H-bridge circuits. Moreover, our group proposed a three-DOF-VCA with eight-coils driven by three H-bridge circuits [5]. However, further simplification of a structure and control device is required for robotic eyes in humanoid robots. In this paper, we propose novel multiple-DOF-VCAs: two-DOF actuator driven by a three-phase half-bridge circuit and three-DOF actuator driven by a four-phase half-bridge circuit. The proposed actuators can be driven by a smaller number of switching elements and coils compared to the conventional actuators. The proposed magnetic structure and driving principle are described. The torque characteristics are clarified by a magnetic field analysis using a three-dimensional finite element method (3D-FEM), and it is shown that the proposed actuator can be driven by the proposed principle. II. PROPOSAL OF NOVEL STRUCTURE AND PRINCIPLE Figs. 1(a) and (b) show the proposed two-DOF-VCA driven by three-phase and three-DOF-VCA driven by four-phase, respectively. The mover consists of three-coils arranged at intervals of 120 degrees. The stator consists of an outer stator yoke and an inner stator with a yoke and permanent magnets. This actuator can be driven two-DOF by combined torques generated in each coil. The coils are connected at a neutral point, and the coils are excited in a balanced state. Thus, the actuator can be driven by only a three-phase half-bridge circuit. The proposed three-DOF-VCA has a structure that coils and permanent magnets are arranged at intervals of 90 degrees, and the mover consists of four-coils. This actuator can be driven by combined torques in each coil connected at a neutral point and can be operated by a four-phase half-bridge circuit. III. TORQUE CHARACTERISTICS The evaluated models are designed for verifying the proposed principle. The static torques are calculated by the 3-D FEM when the mover is forcibly rotated. The torque characteristics of the proposed VCAs are shown in Fig. 2. Fig. 2(a) shows the calculated torque of the proposed two-DOF-VCA when the current is applied to generate torque in the x-axis direction. The torque in the y- and z-axis directions are not generated, and the torque in the x-axis can be only obtained. The torque is maximum at the origin position and decreases by increasing the rotational angle. This is because the centers of poles in permanent magnets and coils are close together, and the magnetically attracted torque decreases. Fig. 2(b) shows the torque characteristics of the proposed three-DOF-VCA. From the analyzed results, it is found that the torque can be generated by the proposed principle. The decrease of the torque by rotation is also occurred due to the same reason in that of the two-DOF-VCA. The maximum torque in the z-axis is times compared with that of the x- and y-axis. This is because the permanent magnets are arranged with tilts of 45 degrees. From this result, it is suggested that the maximum torque in each axis direction can be adjusted by changing the tilt angle of the permanent magnets. IV. CONCLUSION The novel multi-DOF-VCAs were proposed in this paper, which can be driven by a small number of switching elements and components. The structure and principle were shown, and it was clarified that the actuators can generate torque, through 3-D FEM analysis.

**References:**

[1] K. Kahlen, I. Voss, C. Priebe, and R. De Doncker, "Torque Control of a Spherical Machine With Variable Pole Pitch", IEEE Transactions on Power Electronics, Vol. 19, No. 6, pp. 1628-1634, 2004. [2] G. S. Chirikjian and D. Stein, "Kinematic Design and Commutation of a Spherical Stepper Motor", IEEE/ASME Transactions on Mechatronics, Vol. 4, No. 4, pp. 342-353, 1999. [3] H. Kim, H. Kim, D.Ahn, and D. Gweon, "Design of a new type of spherical voice coil actuator", Sensors and Actuators A: Physical, Vol. 203, pp. 181-188, 2013. [4] H. Y. Kim, H, Kim, D.-G. Gweon, and J. Jeong, "Development of a Novel Spherical Actuator With Two Degrees of Freedom", IEEE Transactions on Mechatronics, Vol. 20, No. 2, pp.532-540, 2015. [5] A. Heya, K. Hirata, and N. Niguchi, "Dynamic Modeling and Control of Three-Degree-of-Freedom Electromagnetic Actuator for Image Stabilization", IEEE Transactions on Magnetics, Vol. 54, No. 11, 8207905, 2018.

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/a311d6e64e9682e948dfa950b8094323.jpg)
Fig. 1. Proposed multi-DOF-VCAs

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/c9809c9091f2c198de95fa3470c5986a.jpg)
Fig. 2. Analysis results

 Proposal of Novel Multiple-Degree-of-Freedom Voice Coil Actuator

I.INTRODUCTION Tubular flux-switching permanent magnet machine (TFSPMM) inherits the advantages of tubular machine and flux-switching machine with simple structure, strong robustness and no radial force[1]. However, it is not practical to make a TFSPMM entirely from axially stacked silicon steel sheet in machine manufacturing, because the flux will pass vertically through the silicon steel at the yoke of stator and mover and cause eddy current loss. Soft Magnetic Composite (SMC) material can solve the problem of three-dimensional permeability due to its magnetic isotropy[2]. However, TFSPMM with a magnetic core made only by SMC have low thrust due to low permeability and saturation magnetic density of the material. Moreover, core loss of the SMC is very high at low frequency. In order to change the above situation, hybrid magnetic core combined silicon steel sheet and SMC is proposed in this paper, which combines the high permeability of silicon steel and the magnetic isotropy of SMC. In hybrid magnetic core, the silicon steel sheet and SMC have a parallel and series relationship in the magnetic field, so the distribution of the two materials will affect the performance of the machine, including the core loss. Therefore, core loss reduction is important in hybrid magnetic core. II. MAIN CONTENT Fig. 1 shows the configuration of the TFSPMM with hybrid magnetic core and stacking direction of the silicon steel sheet. The silicon steel sheets in hybrid magnetic core are stacked along several tangential directions, and each part of the silicon steel is of the same stack width. The remaining space on the circumference, which cannot be stacked by silicon steel sheet, will be filled by SMC. Fig. 2 (a) shows the position of the point on the yoke when the silicon steel sheet stack width is 21mm. Point 1 is on the end of the silicon steel sheet and point 5 is on the middle silicon steel sheet. Wss is the stack width of the silicon steel sheet. As the position of the point is farther from the end, the amplitude of By decreases. By is the component of magnetic density in the stacking direction of silicon steel sheet. According to Faraday's law of electromagnetic induction, eddy current loss will be generated on the silicon steel sheet affected By. The larger the amplitude of By, the larger the eddy current loss will be. Therefore, it is considered to reduce the stack width and increase the number of blocks with appropriate stack width. Fig. 2 (b) shows the By of point 1 with different stack widths and different silicon steel sheet blocks. 4 parts 12mm means number of silicon steel blocks is 4 and the stack width is 12mm and 8 parts 12mm means number of silicon steel blocks is 8 and stack width is 12mm shown in Fig. 1 (a). As can be seen, with the reduction of stack width, the amplitude of Bydecreases. When the stack width is 12mm, the amplitude of By of 8 silicon steel sheets is smaller than that of 4 silicon steel sheets. Compared with the silicon steel sheet located in the radial direction, the larger the stack width is, the larger By is on the silicon steel sheet at the end. Fig. 2 (c) shows the core loss of TFSPMMs with different hybrid magnetic cores. The results show that the loss difference is not significant, because the material is not assigned electrical conductivity, and the software does not calculate the eddy current loss at the end. If the silicon steel is set as a solid in the calculation, the solid loss calculated by the finite element method is shown in Fig. 2 (d). In practice, large eddy currents are not generated in the axial section because there is an insulating layer between the silicon steel sheets. But in the software, lamination setting cannot simulate the insulation layer. Therefore, solid manual partitioning is needed to approximate the actual situation. III. CONCLUSION In this paper, core loss of TFSPMMs with hybrid magnetic core are compared and analyzed. The stacking direction and width of silicon steel sheet as well as the match with SMC are described. Although the silicon steel sheet and the flux path are parallel to each other, the magnetic flux is not strictly limited to the plane of the silicon steel sheet. The component of magnetic density in the stacking direction of silicon steel sheet varies with the stack width. As the stack width of the silicon steel sheet decrease, By of point 1 will decrease. Since By is the magnetic density component perpendicular to the silicon steel sheet, the smaller By is, the smaller the eddy current loss generated on the silicon steel sheet will be. When the number of silicon steel blocks is 4, the eddy current loss at the end decreases with the reduction of stack width, but the reduction of the proportion of silicon steel in the hybrid magnetic core will reduce the flux linkage and reduce the thrust. Therefore, when the stack width is 12mm, the number of silicon steel blocks is increased to 8, so as to increase the proportion of silicon steel in the hybrid magnetic core, ensuring small eddy current loss at the end and increased thrust.

**References:**

[1] R. Cao et al., “Quantitative comparison of linear flux-switching permanent magnet motor with linear induction motor for electromagnetic launch system,” IEEE Trans. Ind. Electron., vol. 65, no. 9, pp. 7569_7578, Sep. 2018. [2] C. Liu et al., “Performance evaluation of an axial flux claw pole machine with soft magnetic composite cores,” IEEE Trans. Appl. Supercond., vol. 28, no. 3, Apr. 2018, Art. no. 5202105.

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/9d355e97588dc96892a4bbf22b162f14.jpg)
Fig. 1 (a) Configuration of TFSPMM with hybrid magnetic core; (b) point 1 on stator yoke and stacking direction of silicon steel sheet.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/e16de32d628da9877cbdc10f448ee469.jpg)
Fig. 2 (a) point on stator yoke; (b) By at point 1 under different stack widths; (c) core loss of TFSPMMs with different hybrid magnetic cores; (d) solid loss of TFSPMMs with different hybrid magnetic cores.

 Core Loss Reduction of Tubular Flux-Switching Permanent Magnet Machine with Hybrid Magnetic Core

Tubular linear permanent magnet machines (TLPM) have gained increasing popularity over the past years due to their versatile nature. Whether used as actuators for linear motion applications or as generators for energy production applications, these machines can provide accurate and precise motion with relatively compact structures and simpler control systems [1]-[3]. Like any other mechanical application, TLPMs might suffer from different structural deformations during either operation or assembly. The moving part of the machine called the translator is the most vulnerable to such mechanical deformations. One commonly occurring deformation happens at the level of the translator shaft and is usually called a translator eccentricity where the translator shaft is no longer aligned perfectly with the central axis of the machines's cylindrical structure. Eccentricity can take different forms and has been studied previously by some authors [4]-[7]. In [4] [5], eccentricity analysis was done using analytical and 3D FEM modeling and the calculated results were compared and validated by experimental results. In [6], eccentricity effects were studied for a surface mounted TLPM structure using the 3D finite element method (3D FEM). The study was continued in [7] by analyzing the eccentricity effects on another TLPM structure, also using 3D FEM, and comparing the eccentricity effects on the two structures. The authors of [6] [7] noted the extensive amount of time and effort required to compute some of the 3D FEM models and generate the results. Hence, an alternative modeling technique for eccentricity analysis that can provide relatively good results with less computation time is needed. The aim of this paper is to present a quasi-3D modeling technique that can represent the 3D problem using the superposition of multiple 2D models. It will be shown that the proposed model takes much less computation time and effort and produce relatively accurate results in comparison to the 3D model. Translator eccentricity unbalances the radial electromagnetic forces acting on the translator by forming a non-uniform air gap between it and the primary. Static eccentricity, which is a common eccentricity form, is the state of having the translator axis displaced by a constant distance ε from the primary axis while both axes remain parallel to each other. This work only focuses on the static eccentricity form leaving other forms for future works. The proposed quasi-3D model is achieved by dividing the 3D structure in the tangential direction (θ direction) into a certain number of sections (see Fig. 1). Since the air gap is non-uniform around the translator in the tangential direction, there is a different air gap value e (Fig. 1) for each θ along the tangential direction where θ∈[0, 2π]. Hence, each section in the tangential direction is modeled as a standalone 2D linear machine having an air gap equal to the air gap value at the central angle of the corresponding section. Then, for each 2D linear machine model, the magnetic induction in the air gap is calculated where it will have two components x and y corresponding to the axial (z-direction) and normal (r-direction) cylindrical components of the 3D model respectively. The tangential component can be associated with a z component which is impossible to determine in a 2D model that has only x and y directions, and hence it will be neglected. After gathering the results from all the 2D modeled sections, using the Maxwell stress tensor the electromagnetic forces acting on the translator in the axial and radial directions are calculated. Both the 2D and 3D models used are finite element method based models, and for this work, they were constructed using the FLUX commercial software. The quasi-3D model will be used to study two TLPM structures having static eccentricity. The first one is a surface-mounted PM structure, the second one is an inserted PM flux focusing structure, and the two have similar dimensions. Both open circuit and on load operations will be studied, and the two cases of linear and non-linear ferromagnetic materials will be considered as well. Studying the different cases will provide an insight into the accuracy and efficiency of the proposed quasi-3D model. Fig. 2 shows a comparison between the quantities (cogging force and attraction force) calculated using the quasi-3D and the 3D FEM models for the surface mounted TLPM having an eccentricity value ε equal to 0.75mm (75% of the air gap value). For this comparison, the ferromagnetic materials were defined in their linear case with a constant μr = 10000. Also, the relative permeability of the PM was defined as μr = 1 and its magnetic remanence as Br = 1T. Noting that the attraction force computed is acting along the axis aligned with the angles having the minimum and maximum air gap values. The results shown in Fig. 2 are relatively accurate and acceptable especially that the solution of 65 time steps of the quasi-3D model with 32 tangential divisions took only around 15 minutes with much less construction effort, while it took around 3 days to solve 33 time steps of the 3D FEM model with much more construction and computation efforts. Computation time differences will be even more noticeable and remarkable when solving non-linear models that will be presented and compared in the full paper release.

**References:**

[1] N. Bianchi, S. Bolognani, D. D. Corte and F. Tonel, "Tubular linear permanent magnet motors: an overall comparison," in IEEE Transactions on Industry Applications, vol. 39, no. 2, pp. 466-475 (2003) [2] J. Wang, W. Wang and K. Atallah, "A Linear Permanent-Magnet Motor for Active Vehicle Suspension," in IEEE Transactions on Vehicular Technology, vol. 60, no. 1, pp. 55-63 (2011) [3] L. Szabo, C. Oprea, I. Viorel and K. A. Biro, "Novel Permanent Magnet Tubular Linear Generator for Wave Energy Converters," IEEE International Electric Machines & Drives Conference, pp. 983-987 (2007) [4] F. Marignetti, P. Cancelliere, and M. Scarano, “Analytical formulation of the no-load magnetic field in IPM tubular machines with translator eccentricity,” in IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics, pp. 829–835 (2006) [5] L. Cappelli, Y. Coia, F. Marignetti and Z. Q. Zhu, "Analysis of Eccentricity in Permanent-Magnet Tubular Machines," in IEEE Transactions on Industrial Electronics, vol. 61, no. 5, pp. 2208-2216 (2014) [6] D. Sarr Lo, H. Lawali Ali, Y. Amara, G. Barakat, and F. Chabour, “3D Finite Element Analysis of Eccentricity in a Tubular Linear Permanent Magnet Machine,” in 2018 XIII International Conference on Electrical Machines (ICEM), Alexandroupoli, p. 1996–2001 (2018) [7] H. Diab, Y. Amara and G. Barakat, "Comparison of Two Tubular Linear Permanent Magnet Machines with Translator Eccentricity," 2020 International Conference on Electrical Machines (ICEM), Gothenburg, pp. 612-618 (2020)

**Images:**

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/2fedcffd641e7ac4c796e19d941f5391.jpg)
Fig.1 A 3D TLPM structure divided in the tangential direction with a 2D axial cross-section view with dimensions

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/094e73a6fefce9ab02b34bf22374e05b.jpg)
Fig.2 Comparison of the cogging and attraction forces generated by the quasi-3D and 3D FEM models

 Translator Eccentricity Analysis in Tubular Linear Machines using Quasi-3D Finite Element Method Modeling

I. INTRODUCTION Ferromagnetic materials have complex nonlinear hysteresis characteristics, which play a decisive role in simulating electrical equipment's magnetic properties, such as the power loss and the magnetic remanence of the core in transformers and motors[1]. Therefore, the accurate and efficient hysteresis model is fundamental to improving the power equipment's operational efficiency. As a fast and precise hysteresis model, the Preisach model is widely used in analyzing the magnetic properties of the electromagnetic device coupled with the FEM method. However, some problems remain to exist in the current Preisach model parameter identification methods, such as the convergence speed and the accuracy need to be further improved[2,3]. To develop a more efficient parameter identification method of the Preisach model, a hybrid algorithm combining conjugate gradient method (CGM) and velocity-controlled particle swarm optimization (VCPSO) algorithm is proposed in this paper: Firstly, a one-dimensional magnetic property test system is built to measure the hysteresis loop data of B30P150 steel sheet under quasi-static conditions. Secondly, the parameters of the Preisach model with closed-form Everett function are identified based on the proposed hybrid optimizing algorithm. Finally, in the full paper, the proposed method's accuracy is discussed by comparing the convergence speed and calculation accuracy of the proposed hybrid algorithm with other single intelligent algorithms. II. THE IMPLEMENT AND IDENTIFICATION OF PREISACH MODEL A. The Preisach model based on the closed-form Everett function The traditional Preisach model is implemented by the double integral of the distribution function in the Preisach plane, which is time-consuming and inefficient. To overcome the complex parameters identification process of the traditional Preisach model, the distribution function is simplified to the product of two single-valued functions, and a numerical Preisach model based on the closed-form Everett function is established. The improved model only needs a series of basic operations and a few parameters to be identified, which implement efficiency is much improved. B. Parameters identification of the Preisach model by combing the CGM and VCPSO algorithm A hybrid intelligent algorithm based on the conjugate gradient method (CGM) and a velocity-controlled particle swarm optimization (VCPSO) algorithm is proposed to overcome the low convergence speed and easily fall into a locally optimal solution. The identification process of a hybrid algorithm is shown in Fig 1. Firstly, according to the fast optimization of the VCPSO algorithm, finding the optimal solution range of parameters. Secondly, the searching range of VCPSO is taken as the initial parameter of the conjugate gradient method. Then the optimal solution is determined by local search. Finally, the optimal parameters are obtained by CGM. The measured and simulated hysteresis loop are compared after the identification process. The efficiency of the hybrid algorithm with other single optimal identification algorithms will be discussed in the full paper. III. RESULT AND DISCUSSION The simulated and measured hysteresis loops of the B30P150 silicon steel sheet are shown in Fig 2. It can be found that the model parameters identification results can well simulate the hysteresis loop of the B30P150 silicon steel sheet. The error is due to the approximate substitution of the Everett function, and increasing the number of relevant terms of the Everett function can improve the Model accuracy. What’s more, considering the reversible and irreversible magnetization, the simulation accuracy at the model's turning point can also be improved. IV. CONCLUSION In this paper, a hybrid optimization algorithm combining a velocity-controlled particle swarm optimization algorithm and the conjugate gradient method is proposed to identify the parameters of the Preisach model. The hybrid algorithm is an effective means for parameter identification of the Preisach model and contributes to the design and upgrading of electrical equipment.

**References:**

[1]Szabo Z, Fuezi J,Journal of magnetism and magnetic materials, 406(may):251-258(2016). [2] N. Duan et al., IEEE Trans. Magn. 51, 7300904 (2015) [3]Stoleriu L, Andrei P, Journal of Applied Physics, 103(7p.3):132-134(2018).

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Fig. 1 Identification process based on the hybrid algorithm of CGM and VCPSO

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Fig. 2 Comparison between the measured and the simulated hysteresis loops

 Parameter identification of Preisach model based on the conjugate gradient method and velocity-controlled particle swarm optimization

I. Permanent magnet (PM) synchronous motors have become excellent alternatives for asynchronous motors as well as DC motors in numerous applications in recent years. Therefore, torque density and efficiency improvements in development of PM synchronous motors has become a significant issue. In this study, various PM synchronous motor alternatives are proposed with both non-oriented and grain-oriented silicon steel material in the same stator in order to increase the power density and the efficiency. Detailed performance comparison using 2D/3D FEA is completed and a feasible stator designs manufactured. The proposed design is also compared with a conventional PM motor with a non-oriented stator. As a result of the investigations, it is observed that the proposed motor with both materials in the stator have higher torque density and efficiency. II. INTRODUCTION In this study, detailed investigation on the stator design with both non-oriented and oriented electrical steel is conducted so as to increase the performance of conventional permanent magnet synchronous motors. Different motors with the same stator dimensions and the same rotor are examined with six different cases as in Fig.1:Case-1 (Entire stator made of non-oriented silicon steel), Case-2 (Entire stator made of grain oriented silicon steel), Case-3 (Entire stator made of grain oriented silicon steel with 1/3rd segments), Case-4 (Stator yoke made of non-oriented silicon steel and the conventional teeth made of grain-oriented steel), Case-5 (Stator yoke made of non-oriented silicon steel and the Screw shaped teeth made of grain-oriented steel),Case-6 (Stator yoke made of non-oriented silicon steel and the wedge shaped teeth made of grain-oriented steel). Detailed FEA simulations of all of the 6casesare performed and compared. Detailed performance tests of the promising prototype motors are performed. Torque output, losses and efficiencies of the motors are obtained and compared in order to illustrate the benefits of the proposed solution. III. FEA of PROPOSED SKEWED MODELS AND EXPERIMENTAL VERIFICATION Detailed FEA is performed for both proposed and reference PM motors. Since the flux density in stator teeth is higher than that of stator yoke, stator teeth are modeled and manufactured with grain-oriented material. The model structure of the two motors is shown in Fig 2. Detailed analyses including cogging variations, back EMF, losses, torque-speed curves and output torque of both motors are performed. Flux densities and flux lines obtained by both simulations are obtained. A prototype motor is manufactured and tested to show the promising benefits. It is seen that the torque and losses shows the potential performance benefits of the proposed motor. III. CONCLUSION In this paper, the design of innovative PM motors with two different materials in the stator is proposed. Promising alternatives are investigated and summarized. Detailed FEA results are presented. The motor with promising oriented silicon steel material is manufactured and tested. Detailed test results will be provided in the final version of the paper. Based on the initial results of the analyses, the output torque is increased by 1.88% and the stator tooth losses is decreased by 51.1%. The test results and comparisons will also be given in the last version of the paper.

**References:**

V. References: [1] Stefano Cicalé, Luciano Albini, Francesco Parasiliti, Marco Villani, Design of a Permanent Magnet Synchronous Motor with Grain Oriented Electrical Steel for Direct-Drive Elevators, IEEE, 2012, 978-1-4673-0142-8 [2] Nuwantha Fernando, Fuad HaninMagnetic, Materials for Electrical Machine Design and Future Research Directions: A Review, IEEE, 2017, 978-1-5090-4281-4 [3] Damian Kowal, Peter Sergeant, Luc Dupré, Alex Van den Bossche, Comparison of Nonoriented and Grain-Oriented Material in an Axial Flux Permanent-Magnet Machine, IEEE, 2010, 0018-9464 [4] Andreas Lindner, Ingo Hahn, Investigation of a large air-gap E-core flux-switching machine with arbitrary magnet shape and grain-oriented material, IEEE, 2016, 978-1-5090-2538-1

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 Design of Innovative Radial Flux Permanent Magnet Motor Alternatives with Non-Oriented and Grain-Oriented Electrical Steel for Servo Applications



Additive manufacturing technology carries its potential to develop high-performance electrical machines enabling direct printing of complex shapes and incorporating the complex sequence of feedstock materials in a single build to optimize the demanding physical parameters such as low weight, cost, energy loss, and high mechanical strength and corrosion resistance. We printed Fe-3%Si stator core using selective laser melting and machined it down to thin laminates to assemble a prototype brushless permanent magnet electric motor. The magnetic and electrical properties of the thin heat-treated laminates and back emf of the electrical motors are studied. The thin laminates exhibit maximum relative permeability of 7494 with an approximate 0.8 T applied field and core-loss of 0.473 watts/lb at 60 Hz. We will report in detail about the magnetic and mechanical properties of Fe3Si soft magnets and their influence on the electrical motor. The designed machine is analyzed using finite element analysis and compared with measured electrical parameters.

**References:**



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 Additively manufactured Fe-3Si stator for high-performance electrical motor

This paper proposed a novel hybrid axial magnetic bearing (HAMB) for a vertical system, which can produce a unidirectional magnetic force. The principle of the new HAMB is analyzed. When the rotor is suspended in the equilibrium position, the bias force offsets the gravity of the rotor and the current is zero, which can reduce the copper loss. According to the magnetic field distribution, an equivalent magnetic circuit is established. Based on the analytical model, the curve of force-current, curve of force-displacement and bearing capacity are calculated. The results of analytical are consistent with the ones of the finite element method (FEM). Index Terms—Hybrid, axial magnetic bearing, magnetic circuit, finite element. I. Introduction Magnetic bearings (MBs) are widely used in high-speed rotating machinery, such as compressors, flywheel energy storages for the merits of contactless property and long life[1]. To reduce copper loss, hybrid MB has become the research feature combing the characteristics of active MB and passive MB [2]-[4]. Several axial MB are significant in vertical levitation rotor systems. A novel hybrid axial MB is proposed in [5], which reduces the reluctance of the control path and power loss of axial MB and through constructing the second air gap. In [6], an improved magnetic circuit model is introduced for a novel axial hybrid MB, which provides a reference for the MB design. In [7], an equivalent magnetic circuit with the conformal mapping method and Laplace’s equation is developed for axial MB with multiple air gaps. The results of analysis are consistent with the ones of finite element analysis. In this paper, a novel hybrid axial magnetic bearing (HRMB) is proposed for the vertical system, which produces a unidirectional electromagnetic force to replace the traditional bidirectional force. There is no need for the current to control the MB when the rotor is in the equilibrium position, which reduces the copper loss greatly. Then, the equivalent magnetic circuit method is adopted to analyze the performances of the HRMB. Ultimately, the accuracy of the analytical model is validated by the finite element method (FEM). II. Principle of Hybrid Axial Magnetic Bearing A. Principle of HAMB The structure of HAMB is shown in Fig.1. The blue solid line represents the bias flux, and the red dotted line represents the control flux. B. Magnetic circuit model The equivalent magnetic circuit models of the bias flux and control flux are established. III. Analysis of HRMB The variation of force with current and displacement can be obtained in Fig.2. The analytical results are in agreement with the results of FEM. The bias force and maximal force are calculated and compared with the finite element method. The margin of error is within 10%. IV. Conclusion A novel hybrid axial magnetic bearing is proposed in this paper and the principle of HAMB is analyzed. The structure has low copper loss when the rotor is stable in the suspension position. According to magnetic field distribution, an equivalent magnetic circuit is established. The curve of force-current, curve of force-displacement and bearing capacity are calculated by analytical and FEM respectively. The accuracy of the analytical model is confirmed by FEM.

**References:**

[1] J. X. Zhou, S. Q. Zheng, B. C. Han, et al. “Effects of Notch Filters on Imbalance Rejection With Heteropolar and Homopolar Magnetic Bearings in a 30-kW 60000-r/min Motor.” IEEE Trans. Ind. Electron., vol. 64, no. 10, pp. 8033-8041, Oct. 2017. [2] K. D. Bachovchin, J. F. Hoburg, and R. F. Post, “Magnetic fields and forces in permanent magnet levitated bearings,” IEEE Trans. Magn., vol. 48, no. 7, pp. 2112–2120, Jul. 2012. [3] W. Zhang and H. Zhu, “Improved model and experiment for AC-DC three-degree-of-freedom hybrid magnetic bearing,” IEEE Trans. Magn., vol. 49, no. 11, pp. 5554–5565, Nov. 2013. [4] B. C. Han, S. Q. Zheng, X. Wang, et al. “Integral design and analysis of passive magnetic bearing and active radial magnetic bearing for agile satellite application,” IEEE Trans. Magn., vol. 48, no. 6, pp. 1959–1966, Jun. 2012. [5] J. C. Fang, J. J. Sun, Y. L. Xu, et al. “A New Structure for Permanent-Magnet-Biased Axial Hybrid Magnetic Bearings.” IEEE Trans. Magn., vol. 45, no. 12, pp. 5319–5325, Dec. 2009. [6] X. X. Liu, J. Y. Dong, Y. Du, et al. “Design and Static Performance Analysis of a Novel Axial Hybrid Magnetic Bearing.” IEEE Trans. Magn., vol. 50, no. 11, pp. 1–4, Nov. 2014. [7] K. Wang, D. Wang, H. Y. Lin, et al. “Analytical Modeling of Permanent Magnet Biased Axial Magnetic Bearing With Multiple Air Gaps.” IEEE Trans. Magn., vol. 50, no. 11, pp. 1–4, Nov. 2014.

**Images:**

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 A Novel Hybrid Axial Magnetic Bearing that Produces a Unidirectional Electromagnetic Force

To achieve the force/torque control feature for robotic applications, some robotic devices such as robotic grippers or robotic arms use servo motors’ current or the structures’ mechanical strain to measure torques and forces. Additional commercial torque sensors are also commonly used to detect the output forces. Although these sensors can detect motor torques or contact forces, they are too stiff to render a soft stiffness [1]. If the load environment is stiff and the actuator is also stiff, the force controller will lead to low accuracy, as any tiny movements will cause a large force variety and components are rigid. To make robotic actuators perform more compliant with torque control and stiffness control, a series elastic actuation method was proposed [2]. The series elastic actuator (SEA) is constructed with an actuator and an elastic spring which can be added between the motor and the load to provide compliant features [3]. With displacement sensors attached at two sides of the spring, the output torque can be computed [4] by measuring the relative movement. With the SEA, a force control can then be transferred into a position control whatever the load environment is stiff or not. Magnetic coupling is a component that transmits torque from the motor to load without any direct mechanical contact, as attractive forces between N and S poles are used to transmit the force. When there is a relative angle difference between N and S poles, the attractive forces, which can be separated into axial forces and shear forces, generate a relative force that can be utilized as a virtual spring in the SEA system. The torsional stiffness of magnetic coupling was reported in [5] to be sensitive to strength of magnetic forces and air gap, axial displacement, and angular misalignment. When the load excesses maximum torque that can be transmitted by the magnetic coupling, it then switches to pole-slipping mode to avoid possible permanent deformation in structure [6]. To form a magnetic SEA (mSEA) system, an elastic magnetic coupling must be placed within the actuator and the load. This configuration of SEA connects the elastic component to the load directly, which apparently is sensitive to changes of the external load. By using the magnetic coupling, the attraction forces between two magnets can be transferred to the rotary torque when angular rotation occurs. With the angle solver attached to the load, the angular displacement of mSEA can be measured by the angle difference between the load’s angle solver and the motor’s encoder. The mathematic relation between output torque and rotating angle is the stiffness of magnetic coupling, which is an important variable for a mSEA controlling system. In this study, a 4-pole axially magnetized NdFeB ring magnet (Magtech Corp., TW) was chosen. The magnet’s outer and inner diameters are 24 and 13mm, respectively. The magnet’s thickness 3mm with flux density of 3530G. Because of its flat ring structure, the mSEA is more compact in size than other types of magnetic coupling. Beyond the size and material of the magnet, some other parameters affecting characteristics of the mSEA, like airgap length, pole number, and angular displacement were studied and analyzed in this paper toward understanding the torsional stiffness and attractive force for the axial magnetic coupling. The relation between airgap length and angular displacement is shown in Fig. 1(a), from which one can observes that the torque is a parabola curve with a saturation region between 30 o and 60o, with decreasing of torque when angular displacement is over 60 o. The maximum torque is observed to decrease significantly when the airgap, indicated as “e” in the figure, becomes larger. On the other hand, the axial force renders a linear relationship with the angular displacement as shown in Fig. 1(b). When the angle between the two magnets goes larger, the axial force decreases linearly, leading to zero value where the two magnets are separated. Along with these results, the airgap is found to be provide significant influence in axial attractive force and pull-out torque. With higher airgap, attractive force and pull-out torque both go lower. From this study, the torque for the magnetic couple is also found to be affected by the angular displacement of two magnets and the number of pole pairs. When the angular displacement goes higher, the pull-out torque follows a parabola curve, reaching to the maximum torque value 150 mN-m around 15 o with the airgap is 1mm as showed in Fig. 1. The range of angular displacement beyond the saturation region is the magnet coupler’s effective region. By using the model 4-pole axially magnetized ring magnet, the transmitting torque for the coupler can be then controlled, as the torsional stiffness can be rendered by the model of magnetic coupling. The proposed mSEA can render various magnitudes of torsional stiffness and the rendered results are showed in Fig. 2. The mSEA system can be controlled to perform desired motion when a virtual spring stiffness is set between 50 to 300mN-m/rad with ~5% stiffness rendering error compared to ideal stiffness. The maximum output torque of the mSEA is 130mN-m with force resolution of 0.1mN and control error less than 1mN. With the proposed mSEA put between actuator and load, the output torque and stiffness can certainly be controlled for robotic applications with fragile objects or human.

**References:**

[1] Y. Yu and C. Lan, "Design of a Miniature Series Elastic Actuator for Bilateral Teleoperations Requiring Accurate Torque Sensing and Control," IEEE Robotics and Automation Letters, vol. 4, no. 2, pp. 500-507 (2019) [2] J. Pratt, B. Krupp, and C. Morse, "Series elastic actuators for high fidelity force control," Industrial Robot: An International Journal, vol. 29, pp. 234-241, 06/01 (2002) [3] C. Lee, S. Kwak, J. Kwak, and S. Oh, "Generalization of Series Elastic Actuator Configurations and Dynamic Behavior Comparison," Actuators, vol. 6, p. 26, 08/22 (2017) [4] S. Oh and K. Kong, "High-precision robust force control of a series elastic actuator," IEEE/ASME Transactions on Mechatronics, vol. 22, no. 1, pp. 71-80 (2016) [5] T. Lubin, S. Mezani, and A. Rezzoug, "Simple Analytical Expressions for the Force and Torque of Axial Magnetic Couplings," IEEE Transactions on Energy Conversion, vol. 27, no. 2, pp. 536-546 (2012) [6] R. Montague, C. Bingham, and K. Atallah, "Servo Control of Magnetic Gears," IEEE/ASME Transactions on Mechatronics, vol. 17, no. 2, pp. 269-278 (2012)

**Images:**

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Fig. 1 (a) torque and (b) axial force caused by angular displacement between the magnetic rings.

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Fig. 2 Measured and predicted torque for the model mSEA.

 Design of a Stiffness Control Actuator Utilizing Magneto-Elastic Actuation

With the increasing focus on energy efficiency for the green transition, there is a desire to identify and reduce all losses possible. For rotary machines losses occur as friction at the bearings, and therefore frictionless magnetic bearings can be an important upgrade to high-speed machinery. However, as proven by Samuel Earnshaw in 1842 magnetic bearings are inherently unstable. Typically, magnetic bearings are stable in the lift (often vertical) direction. This results in instability in the radial direction and therefore some kind of stabilizing force is required in this direction [1]. For a magnetic bearing, the radial stiffness determines the active force needed to keep the bearing stable. Thus there is a compromise in designing a magnetic bearing which can both lift as high a load as possible and at the same time have a low and constant radial instability [2-5]. Here we describe a new magnetic bearing aiming to fulfill these requirements [6]. The bearing consists of two nested conical axi-symmetric permanent magnet rings with quadratic cross-sections, as shown in Fig. 1. This configuration is a superposition between a radially magnetized bearing and an axially magnetized bearing. The rotor and stator are centered on the same axis, and when load is applied on the rotor bearing this will move along the z-axis. The stator has a magnetization direction that is along the inwards normal to its surface, while the rotor has a magnetization direction outwards normal to its surface. Thus, there is repulsion between the rings. The idea behind this design is that the magnetic forces will both have a component providing a lifting force along the z-axis, but also a component along the radial direction, which might allow the stiffness along this direction to be controlled. Such a design of tilted conical rings of permanents magnets can easily be made of cubes of permanent magnets placed in a ring. We consider a design where there is a minimum distance between rotor and stator of 0.2 mm and where the conical rings that make up both the rotor and the stator have quadratic cross-section side lengths of 3 mm. The radius of the center of the tilted ring in the stator is fixed at 24 mm, while it is varied for the rotor. We consider different tilt angles of the stator/rotor rings in the bearing and we also consider ring geometries with twice the cross-section and different magnetization directions. The “single” bearing consists of two tilted concentric rings with opposite magnetization as described above, the “double” bearing has two magnets in each tilted ring, and finally the “alternating” bearing has the magnetization of one part of the ring reversed in both the stator and rotor. All configurations are illustrated in Fig. 1. We assume that the permanent magnets in the bearing have a remanence of 1.4 T and a relative permeability of 1. We determine the force and stiffness of the bearing using a magnetostatic finite element (FEM) axi-symmetric model in Comsol Multiphysics. As the simulations are 2D axi-symmetric a very high mesh resolution can be used, eliminating the known convergence issues with bearing models in 3D [7]. We have conducted a variation study to determine the optimal geometry of the bearing as function of the tilt angle and the bearing type. Tilt angles from 0 to 90 degrees in steps of 15 degrees were modelled. Combined with the three bearing types studied, this gives a total of 21 bearings modelled. For each of these we use an adaptive routine to determine the optimal rotor radius to a precision of 0.1 mm. We show that the optimal value of the rotor radius is almost identical for the different bearing geometries considered, with decreasing rotor radius for increasing tilt angle. Following this we determine which bearing would be easiest to control, i.e. has a constant stiffness value, but which also provides a high lifting force. We investigate this by calculating the relative standard deviation for both the axial force and stiffness. This is shown in Fig. 2 for all studied bearings which are stable in the axial direction. Choosing the best bearing depends on the desired characteristics of the bearing. If one desires a bearing that is easy to control, i.e. has a mostly constant stiffness, for a 3 mm axial displacement the double geometry with a tilt angle between 60-75 degrees is clearly the best. When a smaller axial displacement of 1 mm is considered, all bearings with a tilt angle of 60 degrees display almost constant stiffness. This is a remarkable property, as such a bearing is very easy to control with an active bearing. In this bearing the exact load on the rotor is not of importance, as long as it is between the minimum and maximum lift, and at the same time the stiffness is constant regardless of the load. We thus show that by varying the tilt angle and the bearing geometry it is possible to tailor a bearing to the desired characteristics and achieve an easily controllable bearing with a high lifting force.

**References:**

[1] J. P. Yonnet, IEEE Transactions on Magnetics, Vol. 17, p.1169-1173 (1981) [2] J. G. Detoni, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 228, p.1829-1844 (2014) [3] J. Sun, D. Chen and Y. Ren. , IEEE Transactions on Instrumentation and Measurement, Vol. 62, p.2960-2965 (2013) [4] L. Zhang, H. Wu, P. Li, Y. Hu and C. Song, Mathematical Problems in Engineering, Vol. 2019, p.4265698 (2019) [5] K. K. Nielsen, C. R. H. Bahl, N. A. Dagnaes, I. F. Santos and R. Bjørk, IEEE Transactions on Magnetics, 10.1109/TMAG.2020.3042957 (2020) [6] R. Bjørk and C. R. H. Bahl, “A conical passive magnetic bearing with constant stiffness”, submitted for publication (2021) [7] N. A. Dagnæs-Hansen and I. F. Santos., Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 233, p.5280-5293 (2019)

**Images:**

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Fig.1: The bearing geometry considered. The stator, shown in green, and the rotor, shown in grey, each consists of two tilted rings of permanent magnets with a quadratic cross-section. Their direction of magnetization is also indicated. On the right is shown a cross-section of the bearing, with the three types of magnet configurations considered. The bearing illustrated here has a tilt angle of 30 degrees.

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Fig.2: The relative standard deviation of the stiffness in the z-direction in percent as function of the relative standard deviation of the force in the z-direction in percent.

 A constant stiffness magnetic bearing

The electrification of vehicles and ships has been required to reduce the carbon dioxide emissions on global environment. In many cases, a general drive system of electric vehicles and ships consists of an electric motor and a mechanical gearbox to obtain desired speed and torque in terms of reducing size and weight. However, the mechanical gearbox has inherent problems caused by mechanical contacts, e.g., acoustic noise, wear and tear, vibration, and periodical maintenance. To solve the above problems, a magnetic-geared motor (MGM) has attracted attention [1]. The MGM consists of a permanent magnet (PM) motor and a flux-modulated type magnetic gear (MG), and they can transmit the torque without any contacts by an electromagnetic force. In addition, the MGM has the structure that a PM motor and an MG are combined magnetically and mechanically as one electric machine so that it can offer the higher torque density than the conventional drive system. Hence, the MGM is expected to be put into practical use. In previous studies, Refs. [2] and [3] have reported about a magnetic interaction of the MGM caused by magnetically coupling between a PM motor part and an MG part. These studies have discussed about the relationship between the magnetic interaction and the output torque of MGM. The magnetic interaction may influence only the output torque, but also the efficiency and the power factor. However, the influence of the magnetic interaction on the efficiency and the power factor have not been clarified. This paper investigates that the influence of the magnetic interaction on the efficiency and the power factor of the MGM by using a three-dimensional finite element method (3D-FEM). Furthermore, in order to verify the validity, the experimental results are compared to the calculated ones. Fig.1 shows the specifications of the MGM used for examinations. It consists of a PM motor part and an MG part, which share one high-speed rotor (HSR). The PM motor part consists of an armature and the shared HSR, while the MG part consists of the shared HSR, a magnetic modulator called pole-pieces (PP), and a low-speed-rotor (LSR). The numbers of pole-pairs of the shared HSR and the LSR are 4 and 23, respectively. Hence, the gear ratio is 5.75 which is given by the pole-pair ratio of the shared HSR and the LSR. First of all, the phase of the armature voltage is calculated by 3D-FEM in various load angles, namely, in various load torques, to clarify the influence of the magnetic interaction on the armature voltage. In 3D-FEM, the rotational speed and the current phase are kept constant. Fig. 2(a) indicates the calculated waveforms of the armature voltage at various load angles of 0 (no-load), 30, 45, and 90 degrees. In addition, an input current is also indicated in the figure as a reference of the phase. It is understood that the phase of the armature voltage is gradually shifted by increasing the load angles, i.e. load torque. The reason why the phase of the armature voltage is shifted is discussed based on the rotational coordinate system synchronized with HSR. Fig. 2 (b) shows the vector diagram of MGM based on the rotational coordinate system of the HSR when a load angle is δ and a current phase is β. In the figure, the no-load induce voltage is ea, the armature voltage is Va, the armature current is ia, the phase difference between Va and ia is Φ, d- and q-axis inductance are Ld and Lq, the magnetic flux of HSR magnet is ψh, the modulated magnetic flux of LSR magnet is ψml due to the magnetic interaction, respectively. The sum of four magnetic fluxes, which are Ldid, Lqiq, ψh, and ψml, is represented as the combined magnetic flux ψo. The phase of ψml changes with the load angle δ, thereby the phase of ψo changes too. Hence, the phase of Va shifts due to the magnetic interaction when the load angle changes, because the phase difference between Va and ψo is always 90 degrees. Based on the above discussions, it is considered that power factor also changes due to the magnetic interaction. Fig. 2 (c) indicates the calculated results of the displacement power factor of MGM in various load torques and current phases. The figure reveals that the displacement power factor changes with the load torques. Moreover, it is understood that the maximum point of the displacement power factor varies in each load torque. Therefore, it is concluded that if the current phase is advanced in accordance with increasing the load torque, the MGM can be a high-power-factor drive. Fig. 2 (d) indicates the measured results of the displacement power factor of the prototype MGM. The figure reveals that the measured displacement power factor changes with the load torque in the same way as the calculated ones shown in Fig. 2(c). Hence, the validity of the calculated results is proved.

**References:**

[1] K. T. Chau, Z. Dong, J. Z. Jiang, L. Chunhua, and Z. Yuejin, IEEE Trans. Magn., Vol. 43, pp. 2504-2506 (2007) [2] H. Suzuki, K. Hirata, N. Niguchi, A. Kohara and K. Takahara, Proc. of 19th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering (2019) [3] K. Ito and K. Nakamura, IEEE Trans. Magn., doi: 10.1109/TMAG.2020.3020121 (2020)

**Images:**

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Fig. 1 Specifications of the MGM used for examinations.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/aee100ae06c85f4490a7c1b2f434fe0c.jpg)
Fig. 2 Calculation and experimental results, (a) Calculated waveforms of the armature voltage at various load angles, (b) Vector diagram considering the magnetic interaction, (c) Calculated results of the displacement power factor, (d) Measured results of the displacement power factor.

Design of Bridged Flux Modulators in Coaxial Magnetic Gear Considering Mechanical Stress

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Design of Bridged Flux Modulators in Coaxial Magnetic Gear Considering Mechanical Stress

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