France



I. Introduction&lt;br/&gt;The conventional modulating ring in a coaxial magnetic gear is composed of a sequence of ferromagnetic blocks. The rotating cylinder planetary gear (RCPG) proposed by Davey &lt;i&gt;et al&lt;/i&gt; borrowed the conventional topology but replaced the ferromagnetic blocks with free-spinning magnetized cylinders called planet gears [1]. The planet gears enhance the modulating effect and hence the torque density, since they themselves are a source of magnetomotive force (MMF). Davey&#39;s team reported 79% increase of torque density over a conventional coaxial gear of comparable dimension [1].&lt;br/&gt;In the operation of the RCPG, the planet gears move to new angular position corresponding to each updated relative position of the two rotors. It is a prerequisite to know these angular positions of planet gear for the calculation of the performance such as torque. Since the planet gears are unconstrained, they move to zero-torque positions. However, finding these positions is mathematically a difficult optimization with a large number of unknowns and strong internal coupling. A custom optimization algorithm combining splines, quadratic interpolation and response surface method was implemented for these positions that minimize the maximum torque on the planet gears. This algorithm was proved to be the most effective though it converged more slowly than the other algorithms that Davey&#39;s team had tried [1].&lt;br/&gt;Alternatively, Wang &lt;i&gt;et al&lt;/i&gt; proposed that the total magnetic energy stored in the gear should be minimum (or maximum magnetic coenergy) when all the planet gears settle in their equilibrium positions [2]. Wang&#39;s team found the positions of planet gear corresponding to the minimum stored magnetic energy, by using gradient-based Broydon–Fletcher–Goldfarb–Shanno (BFGS) and three preconditioning techniques, respectively. The BFGS algorithms converge apparently faster than Davey&#39;ss custom algorithm. However, two of the preconditioning techniques need flux-tracing finite element (FE) calculation with or without non-magnetized cylinders. The other technique seems analytical and consequently saves the preconditioning cost, but it was not described in detail [2].&lt;br/&gt;This digest proposes a BFGS algorithm with an alternative analytical preconditioning technique for the angular positions of planet gear corresponding to the minimum stored magnetic energy. The algorithm can quickly find these angular positions for each relative position of the two rotors in the operation of the gear.&lt;br/&gt;II.Description of Proposed Preconditioned BFGS&lt;br/&gt;The proposed preconditioning technique estimates the angular position of a planet gear depending on the polarities of the closest neighboring magnets on the rotors. This estimate can be easily formulated if the two closest neighboring magnets have different polarity, but it is difficult for some special relative positions of the planet gear and the two rotors. For example, the closest neighboring magnets have same polarity, as shown in Fig. 1 (a) and (b), or the distances from the planet gear to the two neighboring magnets on a rotor are equal, as shown Fig 1 (c) and (d). The angular position of planet gear (given by the direction of polarity, shown as the red arrow in Fig. 1) in each special case is found by the BFGS without preconditioning. The preconditioned BFGS relies on this result to formulate the presetting of angular position for each special case in the performance calculation of RCPG. The calculation of gradient in the BFGS algorithm was also improved to ensure the convergence to the global optimum. The formulation of the proposed preconditioning and the improved gradient calculation will be given in the full paper.&lt;br/&gt;III.Performance Calculation With Preconditioned BFGS&lt;br/&gt;The proposed preconditioned BFGS was used to calculate the torque for each relative position of the planet gears and the rotors in the running of a magnetic gear with 4 inner poles, 40 outer poles and 22 planet gears. Randomly initialized and flux-tracing preconditioned BFGS algorithms were also used respectively to calculate the torque. The latter algorithm makes two-dimensional (2-D) FE calculation of the flux distribution in the magnetic gear with the conventional modulators of the same dimension as the planet gears, and initializes each planet gear&#39;s angular position for the BFGS optimization according to the field direction in the center of the corresponding conventional modulator. Fig. 2 (a) compares the performances of the three BFGS algorithms for the optimization at a certain relative position of the planet gears and the rotors. The randomly initialized, flux-tracing preconditioned and proposed preconditioned BFGS algorithms made 21, 16 and 14 iterations, and calculated the stored magnetic energy by FE for 474, 348 and 279 angular position cases in total, respectively, until the convergence.&lt;br/&gt;The running of the magnetic gear with different initial angles of the inner rotor was simulated by FE and the proposed preconditioned BFGS. Fig. 2 (b) shows the torque-initial-angle curve given by this simulation.&lt;br/&gt;IV.Conclusion&lt;br/&gt;This digest proposes an analytically preconditioned BFGS algorithm for the performance calculation of RCPG. With the preset angular positions of planet gear, the BFGS converges fast and significantly saves the cost for the FE simulations of the full-range operation of RCPG. The full paper will discuss the application of other algorithms to the analysis of RCPG.

**References:**

[1] K. Davey, &lt;i&gt;et al&lt;/i&gt;, IEEE Transactions on Industry Applications, vol. 52, no. 3, pp. 2253–2260 (2016)&lt;br/&gt;[2] R. Wang, &lt;i&gt;et al&lt;/i&gt;, IEEE Magnetics Letters, vol. 7, Art. no. 1303805 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/24aefc2a2086278c9a4c40b19bcdde82.jpg)
Fig. 1. Special preconditioned cases

![](https://assets.underline.io/uploads/markdown_image/1/image/0e9bc3bec8e9ad352ffbbbf0931f3000.jpg)
Fig. 2. Performances of BFGS algorithm and torque capacity curve.

INTERMAG 2021

 Optimization Algorithms for Performance Calculation of Rotating Cylinder Planetary Gear

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I. Introduction The conventional modulating ring in a coaxial magnetic gear is composed of a sequence of ferromagnetic blocks. The rotating cylinder planetary gear (RCPG) proposed by Davey et al borrowed the conventional topology but replaced the ferromagnetic blocks with free-spinning magnetized cylinders called planet gears [1]. The planet gears enhance the modulating effect and hence the torque density, since they themselves are a source of magnetomotive force (MMF). Davey's team reported 79% increase of torque density over a conventional coaxial gear of comparable dimension [1]. In the operation of the RCPG, the planet gears move to new angular position corresponding to each updated relative position of the two rotors. It is a prerequisite to know these angular positions of planet gear for the calculation of the performance such as torque. Since the planet gears are unconstrained, they move to zero-torque positions. However, finding these positions is mathematically a difficult optimization with a large number of unknowns and strong internal coupling. A custom optimization algorithm combining splines, quadratic interpolation and response surface method was implemented for these positions that minimize the maximum torque on the planet gears. This algorithm was proved to be the most effective though it converged more slowly than the other algorithms that Davey's team had tried [1]. Alternatively, Wang et al proposed that the total magnetic energy stored in the gear should be minimum (or maximum magnetic coenergy) when all the planet gears settle in their equilibrium positions [2]. Wang's team found the positions of planet gear corresponding to the minimum stored magnetic energy, by using gradient-based Broydon–Fletcher–Goldfarb–Shanno (BFGS) and three preconditioning techniques, respectively. The BFGS algorithms converge apparently faster than Davey'ss custom algorithm. However, two of the preconditioning techniques need flux-tracing finite element (FE) calculation with or without non-magnetized cylinders. The other technique seems analytical and consequently saves the preconditioning cost, but it was not described in detail [2]. This digest proposes a BFGS algorithm with an alternative analytical preconditioning technique for the angular positions of planet gear corresponding to the minimum stored magnetic energy. The algorithm can quickly find these angular positions for each relative position of the two rotors in the operation of the gear. II.Description of Proposed Preconditioned BFGS The proposed preconditioning technique estimates the angular position of a planet gear depending on the polarities of the closest neighboring magnets on the rotors. This estimate can be easily formulated if the two closest neighboring magnets have different polarity, but it is difficult for some special relative positions of the planet gear and the two rotors. For example, the closest neighboring magnets have same polarity, as shown in Fig. 1 (a) and (b), or the distances from the planet gear to the two neighboring magnets on a rotor are equal, as shown Fig 1 (c) and (d). The angular position of planet gear (given by the direction of polarity, shown as the red arrow in Fig. 1) in each special case is found by the BFGS without preconditioning. The preconditioned BFGS relies on this result to formulate the presetting of angular position for each special case in the performance calculation of RCPG. The calculation of gradient in the BFGS algorithm was also improved to ensure the convergence to the global optimum. The formulation of the proposed preconditioning and the improved gradient calculation will be given in the full paper. III.Performance Calculation With Preconditioned BFGS The proposed preconditioned BFGS was used to calculate the torque for each relative position of the planet gears and the rotors in the running of a magnetic gear with 4 inner poles, 40 outer poles and 22 planet gears. Randomly initialized and flux-tracing preconditioned BFGS algorithms were also used respectively to calculate the torque. The latter algorithm makes two-dimensional (2-D) FE calculation of the flux distribution in the magnetic gear with the conventional modulators of the same dimension as the planet gears, and initializes each planet gear's angular position for the BFGS optimization according to the field direction in the center of the corresponding conventional modulator. Fig. 2 (a) compares the performances of the three BFGS algorithms for the optimization at a certain relative position of the planet gears and the rotors. The randomly initialized, flux-tracing preconditioned and proposed preconditioned BFGS algorithms made 21, 16 and 14 iterations, and calculated the stored magnetic energy by FE for 474, 348 and 279 angular position cases in total, respectively, until the convergence. The running of the magnetic gear with different initial angles of the inner rotor was simulated by FE and the proposed preconditioned BFGS. Fig. 2 (b) shows the torque-initial-angle curve given by this simulation. IV.Conclusion This digest proposes an analytically preconditioned BFGS algorithm for the performance calculation of RCPG. With the preset angular positions of planet gear, the BFGS converges fast and significantly saves the cost for the FE simulations of the full-range operation of RCPG. The full paper will discuss the application of other algorithms to the analysis of RCPG.

**References:**

[1] K. Davey, et al, IEEE Transactions on Industry Applications, vol. 52, no. 3, pp. 2253–2260 (2016) [2] R. Wang, et al, IEEE Magnetics Letters, vol. 7, Art. no. 1303805 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/24aefc2a2086278c9a4c40b19bcdde82.jpg)
Fig. 1. Special preconditioned cases

![](https://assets.underline.io/uploads/markdown_image/1/image/0e9bc3bec8e9ad352ffbbbf0931f3000.jpg)
Fig. 2. Performances of BFGS algorithm and torque capacity curve.

technical paper

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

**Images:**
![](https://assets.underline.io/uploads/markdown_image/1/image/803822cf89981ca73eec7659356111cd.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/27c20d466f0148f94a21c321290e6061.jpg)

 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:**



**Images:**





 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:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7b1c9d83771d07a1c7d8a2a737533d28.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/931b6ebbde46297ad75a3434b43e1d98.jpg)

 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:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e184980e50eb25832996607b5fbc0282.jpg)
Fig. 1 (a) torque and (b) axial force caused by angular displacement between the magnetic rings.

![](https://assets.underline.io/uploads/markdown_image/1/image/073c42b1fea45f07e5b3b6e362b3b838.jpg)
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:**

![](https://assets.underline.io/uploads/markdown_image/1/image/da90a1119a35e4bf7055dcd0b30f5edb.jpg)
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.

![](https://assets.underline.io/uploads/markdown_image/1/image/015e67cee587135f544e04ac077b1ccd.jpg)
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

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://assets.underline.io/uploads/markdown_image/1/image/3d7ea0b425a5955b905b54bff4286211.jpg)
Fig. 1. Magnetic gear

![](https://assets.underline.io/uploads/markdown_image/1/image/8178c855ff5aa363505d14637e4d1e46.jpg)
Table I Load torques and stresses

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

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:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2dd3339ff2411411f71aa3e80e14ad53.jpg)
Fig. 1 Specifications of the MGM used for examinations.

![](https://assets.underline.io/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.

 Influence of Magnetic Interaction on Power Factor and Efficiency of IPM-type Magnetic-Geared Motor

Tidal stream offers predictable electrical energy generation at higher power densities than other renewable energy sources, such as wind. However, compared to wind turbines, the tidal turbine technology is arguably still in its infancy, and various types of drivetrains are being considered [1][2]. These include permanent magnet (PM) generators and doubly-fed induction generators. The paper presents an investigation into the application of magnetically geared Pseudo Direct Drive (PDD) electrical machine to a floating tethered 6kW, 100rpm marine current turbine. The turbine is designed to harness the energy from tidal and ocean currents, in regions with relatively high-speed water flows at depths, that are a long distance from the seabed, and cannot be economically exploited by conventional fixed turbines. The marine current turbine is a robust, easy to service and cost-effective solution for regions with variable or fixed sea depth of more than 60 metres. Fig. 1 shows the prototype tidal current turbine and cross-sectional view of the PDD generator, where the magnetic gear element consists of a 4-pole pair high speed PM rotor, 31 ferromagnetic pole piece output rotor and a 27-pole pair stationary Halbach PM array. The high-speed rotor of the magnetic gear is magnetically coupled to the stator, which employs a double layer concentrated winding, forming the PM brushless machine element. The PDD generator was designed for a rated torque of 580Nm at 100rpm, while the pull-out torque requirement of the magnetic gear was fixed at 640Nm. Fig. 2 shows the PDD generator efficiency for a range of input load torque and speed points. It can be seen, that the 6 kW PDD generator is capable of an efficiency of 94-95% for a substantial range of operation. The paper will present the electromagnetic design and analysis of the PDD generator, and predicted performance such as efficiency, will be compared with dynamometer measurments on a prototype PDD generator. Furthemore, test results on a demonstrator marine turbine in a towing tank will also be presented.

**References:**

[1] S. Benlghali, M. E. Benbouzid, J. F. Charpentier, IEEE Transactions on Oceanic Engineering, Vol. 37 (2012) [2] K. Touimi. M. Benbouzid and P. Tavner, IEEE International Power Electronics and Application Conference and Exposition (PEAC), (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bd094d2afbc6f9458474c8afa56df111.jpg)
Fig. 1 Prototype tidal current turbine and PDD generator.

![](https://assets.underline.io/uploads/markdown_image/1/image/a6d8d36d60a6cd3fffebf64d92c3b836.jpg)
Fig. 2 Variation of PDD generator efficiency with torque.

 Pseudo Direct Drive Electrical Machine for a Floating Marine Current Turbine

Currently, subsea remote operated vehicles (ROVs) are equipped with hydraulically or electrically powered thruster units. For the former, the surface powered on-board electric motors are only used to drive hydraulic pumps, which distribute hydraulic power to hydraulically driven thrusters and auxiliary toolings [1]. For the elecrically powered thruster, direct-drive brushless permanent magnet (PM) machines are normally employed [2][3]. A novel all-electric ROV, which employs 25kW electric thruster units, exhibits increased system efficiency, in addition to a 20% reduction in weigh/size of the power delivery and drive systems, when paired with a high voltage DC supply umbilicals. Thus, enabling operation from smaller vessels. This paper presents an investigation into the electromagnetic design and analysis of a magnetically geared thruster motors, which can operate in closed loop control providing accurate tracking of designated routes or position holding of the ROV. Additionally, the novel thrusters provide 20% more thrust with 50% fewer moving parts, reducing the maintenance requirement, while providing hot-swap capability to reduce down-times of the ROV. The torque limit capability of the magnetic gear unit can passively isolate the propeller from the electric motor and re-engage normal operation when the excess load, on the output shaft, has been removed. In addition, the hermetically sealed magnetic gear topology, isolates the active electrical part of the thruster from the passive output stage of the magnetic gear. Thus, sea water ingress will not affect normal operation and the oil leak severity is reduced, since the output stage of the gear only requires a small quantity of oil for pressure compensation. Fig. 1 shows the main components of the novel magnetically geared thruster, and a cross-sectional view of the magnetic gear connecting the high-speed PM motor to the propeller, which is driven by the low speed rotor (LSR) of the magnetic gear. Furthermore, high-speed side of the thruster is hermetically sealed from the output LSR by the stationary pole piece structure. The high speed rotor (HSR) of the magnetic gear is rigidly connected to the rotor of the PM motor, which is designed for a rated output of 25kW. Fig. 2 shows the predicted efficiency map of the magnetically gear thruster. It can be observed that a high system efficiency is possible, despite the entire thruster being filled with mineral oil for pressure compensation. The paper will present the electromagnetic design and analysis of the magnetic gear and PM motor, and predictions are compared with experiments undertaken on a prototype. It is shown for the magnetic gear an airgap shear stress in excess of 100kPa can be achieved.

**References:**

[1] F. Zhou et al. OCEANS, 2013 [2] D. Uygun et al. IEEE 5th International Conference on Power Engineering, Energy and Electrical Drives (POWERENG), 2015. [2] H. Tian, Z. Mao, B. Song and Y. Li, OCEANS, 2018.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/07a2ec71341290e2b0ff97f4389d3748.jpg)
Fig. 1 a) Magnetically geared thruster topology (b) Magnetic gear cross-section

![](https://assets.underline.io/uploads/markdown_image/1/image/39402114b3ca13f31cd673dc540843e5.jpg)
Fig. 2 Efficiency map of the thruster system

 Magnetically Geared Propulsion Motor for Subsea Remote Operated Vehicle

Axial flux magnetic-geared double-rotor machines (AMGDRMs) are characterized as a potential core component of the power splitting system in hybrid electric vehicles (HEVs). Conventional AMGDRMs suffer from low mechanical strength due to high axial magnetic force. In this paper, an interior-modulating-rotor axial flux magnetic-geared double-rotor machine (IMR-AMGDRM) is proposed to solve this problem. An analytical model of the IMR-AMGDRM is proposed to predict the magnetic field distribution and electromagnetic performances. Since analytical model is based on 2-D geometry, a 2-D approximation of IMR-AMGDRM which has similar output performance to 3-D finite element analysis (FEA) is presented based on the permanent magnet (PM) magnetic energy equivalent principle. Utilizing subdomain method, the magnetic field distribution and electromagnetic performances are calculated. Finally, the accuracy of the proposed analytical model is validated by 3-D FEA. Index Terms—Analytical model, axial flux, magnetic-geared machine, double-rotor machine. I. Introduction Axial flux magnetic-geared double-rotor machines (AMGDRMs), which enable high torque density, have been considered a promising candidate for power splitting in hybrid electric vehicles (HEVs) [1]. However, conventional AMGDRMs suffer from low mechanical strength due to high axial magnetic force. To enhance mechanical strength, an interior-modulating-rotor axial flux magnetic-geared double-rotor machine (IMR-AMGDRM) which can reduce axial magnetic force dramatically is proposed in this paper. Fig. 1 (a) and (b) illustrate the configuration of the proposed IMR-AMGDRM. As axial flux type machine, IMR-AMGDRMs have complex 3-D magnetic circuits. 3-D finite element analysis (FEA) can offer accurate results by taking into account the nonlinear factors such as magnetic saturation and flux leakage. But the major drawback of 3-D FEA is serious time cost. Analytical model which is much lower time-consuming provides an alternative for electromagnetic performance prediction in the initial design procedure. In this paper, an analytical model of the IMR-AMGDRM is proposed. The geometry of analytical model is derived based on PM magnetic energy equivalent principle to guarantee similar output performance to 3-D FEA. The solution to analytical model is calculated by subdomain method. To evaluate the accuracy of the proposed analytical model, the results computed by analytical model and that of 3-D FEA are compared. II. Configuration of IMR-AMGDRM In conventional configuration, the inner and outer rotors are the permanent magnet (PM) rotor and the modulating rotor, respectively. Since the modulating rotor is closer to the PM rotor than stator, the modulating rotor suffers from high axial magnetic force. Furthermore, the modulating rotor is hollow and thus the low mechanical strength of modulating rotor becomes the major drawback of conventional configuration [1]. To solve this problem, the IMR-AMGDRM which switches positions of the PM rotor and the modulating rotor is proposed. In interior-modulating-rotor configuration, as shown in Fig. 1 (a) and (b), the PM rotor is between the modulating rotor and stator, i.e. the inner and outer rotors are the modulating rotor and the PM rotor, respectively. Owing to that the PM rotor suffers from reverse axial magnetic forces from both sides simultaneously, the total axial magnetic force is dramatically reduced. Hence, although the outer rotor is hollow in both configurations, the axial magnetic force of interior-modulating-rotor configuration is significantly decreased, leading to higher mechanical strength and reliability. III.Geometry of 2-D Approximation Generally, the unrolled section of axial flux machine at mean radius is used as 2-D approximation. However, few references are found to evaluate the electromagnetic performance equivalence of such approximation. To make the electromagnetic performance of 3-D model and 2-D approximation equal, the amount of magnetic energy of the PM should remain unchanged after approximation, i.e. PM magnetic energy equivalent principle [2]. A 2-D approximation of IMR-AMGDRM which has identical output performance is presented by using PM magnetic energy equivalent principle. IV. Analytical Calculation of IMR-AMGDRM The proposed analytical model is illustrated in Fig. 1 (c). It can be seen that the model consists of four subdomains. Region i, I, II and III represent the ith slot of the modulating rotor, the inner air gap, the PM rotor and the outer air gap, respectively. Utilizing subdomain method, the magnetic field distributions in the inner and outer air gaps are obtained by the solution of Laplace’s and Poisson’s equations. Then, the electromagnetic performances such as no-load back EMF, torque and axial force can be derived from the calculated magnetic field distributions. Comparisons of computation accuracy between analytical model and 3-D FEA is shown in Fig. 2. As can be observed, the results of analytical model are well consistent with that of 3-D FEA, validating the proposed method.

**References:**

 [1] M. Q. Wang, D. C. Tong and P. Zheng, IEEE Trans. Ind. Electron., vol. 66, no. 1, pp. 806-817( 2019). [2] Kim J S, Lee J H, Song J Y, and Kim D W, IEEE Trans. Magn., vol. 53, no. 6, 8105304(2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6f461fd509eebd7cd34868edda2a46ab.jpg)
Fig. 1. IMR-AMGDRM. (a) Overall view of configuration. (b) Exploded view of configuration. (c) Analytical model.

![](https://assets.underline.io/uploads/markdown_image/1/image/464feedb2617c359e0d07874766cd2b5.jpg)
Fig. 2. Comparisons between analytical model and 3-D FEA. (a) Axial flux density distribution in the inner air gap. (b) Axial flux density distribution in the outer air gap. (c) The static torque characteristic. (c) The static axial force characteristic.

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