United States

In recent years we have been focused on the understanding of magnetism in ensembles of 4f metallic nanoparticles (NPs) (R = Tb,Nd, Gd-M = Cu, Al, Y, La). As a result, we have reported that the existence of an ordered magnetic core and a disordered magnetic shell is regularly found in those ensembles. Depending on the stoichiometry (in our case, RX&lt;sub&gt;2&lt;/sub&gt;) it is possible to easily tune the RKKY interactions to provoke the appearance of ferromagnetic or antiferromagnetic (AF) order within the NPs. Moreover, the fact that we are dealing only with thousand of moments per particle leads to use the super- prefix to define correctly the magnetic behaviour [1]. Besides, it is an upsetting fact that both the collective moment dynamics and the role of the crystalline electric field (CEF) in ensembles of NPs have barely been unearthed. To fill in this gap, we are reporting here a complete neutron-based study (Neutron Diffraction, ND, Small-Angle Neutron Scattering, SANS, and INS) on NdCu&lt;sub&gt;2 &lt;/sub&gt;NPs (&amp;lt;D&amp;gt;≈13nm) to depict the skyline of order and disorder at the nanoscale.&lt;br/&gt;&lt;br/&gt;ND measurements (LLB) point to that the bulk commensurate AF structure is retained within the core. SANS confirms the survival of the AF within the nanoscale, together with a strong emergence of disorder, in the form of a rise in the low-Q region (Q&amp;lt;0.665 Å&lt;sup&gt;-1&lt;/sup&gt;). Magnetic measurements indicate the interactions among the surface moments strong enough to establish a Spin Glass behaviour, leading therefore to a Superantiferromagnetic state. Regarding the CEF panorama, macroscopic specific heat analyses anticipate a broad Schottky contribution, with a maximum located at T∼25 K. Then, microscopic INS measurements (ILL) performed at T=10K (paramagnetic) and T=1.5K (magnetic state) reveal that the splitting of the CEF levels, as well as the spin–wave excitations that emerged below the Néel transition (T&lt;sub&gt;N&lt;/sub&gt; ≈ 6 K) in polycrystalline NdCu&lt;sub&gt;2&lt;/sub&gt; [2] are maintained in the NPs. Furthermore, we have been able to isolate the INS from the surface, which has allowed us to unravel the role of finite–size effects and microstrain. These two effects lead to a partial inhibition of the transitions from the ground state to the first excited level, as well as a stiffening of the magnon modes [3].

2022 Joint MMM-INTERMAG

Order and disorder : The magnetic and crystalline electric field skyline of NdCu2 nanoparticles

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INTERMAG 2021

In recent years we have been focused on the understanding of magnetism in ensembles of 4f metallic nanoparticles (NPs) (R = Tb,Nd, Gd-M = Cu, Al, Y, La). As a result, we have reported that the existence of an ordered magnetic core and a disordered magnetic shell is regularly found in those ensembles. Depending on the stoichiometry (in our case, RX2) it is possible to easily tune the RKKY interactions to provoke the appearance of ferromagnetic or antiferromagnetic (AF) order within the NPs. Moreover, the fact that we are dealing only with thousand of moments per particle leads to use the super- prefix to define correctly the magnetic behaviour [1]. Besides, it is an upsetting fact that both the collective moment dynamics and the role of the crystalline electric field (CEF) in ensembles of NPs have barely been unearthed. To fill in this gap, we are reporting here a complete neutron-based study (Neutron Diffraction, ND, Small-Angle Neutron Scattering, SANS, and INS) on NdCu2 NPs (&lt;D&gt;≈13nm) to depict the skyline of order and disorder at the nanoscale. ND measurements (LLB) point to that the bulk commensurate AF structure is retained within the core. SANS confirms the survival of the AF within the nanoscale, together with a strong emergence of disorder, in the form of a rise in the low-Q region (Q&lt;0.665 Å-1). Magnetic measurements indicate the interactions among the surface moments strong enough to establish a Spin Glass behaviour, leading therefore to a Superantiferromagnetic state. Regarding the CEF panorama, macroscopic specific heat analyses anticipate a broad Schottky contribution, with a maximum located at T∼25 K. Then, microscopic INS measurements (ILL) performed at T=10K (paramagnetic) and T=1.5K (magnetic state) reveal that the splitting of the CEF levels, as well as the spin–wave excitations that emerged below the Néel transition (TN ≈ 6 K) in polycrystalline NdCu2 [2] are maintained in the NPs. Furthermore, we have been able to isolate the INS from the surface, which has allowed us to unravel the role of finite–size effects and microstrain. These two effects lead to a partial inhibition of the transitions from the ground state to the first excited level, as well as a stiffening of the magnon modes [3].

technical paper

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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The effect of boundaries and how these can be used to influence the bulk behavior in geometrically frustrated systems are both long-standing puzzles, often relegated to a secondary role. In this talk, I will investigate boundary effects in a geometrically frustrated system, namely an artificial colloidal ice, a microscale soft matter analog of a frustrated nanoscale spin ice system. The artificial colloidal ice is realized by confining interacting paramagnetic colloids to a lattice of gravitational double wells [2]. With this system, both via numerical simulations and “proof of concept” experiments, I will demonstrate that boundaries can be engineered to control the bulk behavior in a colloidal artificial ice [3]. I will also show that an antiferromagnetic frontier forces the system to rapidly reach the ground state (GS), as opposed to the commonly implemented open or periodic boundary conditions. Further, strategically placing defects at the corners may be used to generate novel bistable states, or topological strings, which result from competing GS regions in the bulk. The presented results could be generalized to other frustrated micro- and nanostructures where boundary conditions may be engineered with lithographic techniques.

Topological Boundary Constraints in Artificial Colloidal Ice

Sleeve or bolting is applied to a permanent magnet synchronous generator (PMSG) to prevent defects caused by shattering. Bolting is more useful than the sleeve in manufacturing processes; however, the magnetic flux is reduced by the material characteristics of bolting. In addition, permanent magnet machines exhibit an overhang structure because the end effect causes leakage flux. Therefore, the axial length of the rotor must be considered in 3D analysis according to the overhang structure. However, the analysis is time-intensive for the 3D model with diverse and complex design variables. In this study, we performed a semi-3D analysis of a PMSG considering the bolting and overhang structure. The analysis time of a PMSG must be reduced because of the complex structure. Hence, we propose a semi-3D analysis for the 2D model by considering the operating point of the magnet in the 3D analysis. The operating point of the permanent magnet was calculated using magnetic energy analyses of the 2D and 3D models. Variations in the operating point were applied to the 2D analysis models, which were separated according to the permanent magnets and bolting regions. Then, a circuit parameter was derived using the semi-3D analysis and applied to the equivalent circuit of the generator. Characteristic analysis was performed using the superposition method. The results of the semi-3D model analysis were compared with those of the 3D model analysis. Figs. 1(a), (b), and (c) show the manufactured model. Fig. 2(a) shows the analysis model. Fig. 2(b) shows the variation in the operating point of the permanent magnet. Figs. 2(c) shows the output characteristic results from the comparison between the 3D and semi-3D models. We confirmed that the results of the semi-3D and 3D model analyses are congruent. Detailed data and experimental data is presented in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/b3d3b3536eccb58e234ae1715b043e4c.jpg)

Figs. 1. Manufactured model: (a) experimental set, (b) stator, (c) rotor

![](https://assets.underline.io/uploads/markdown_image/1/image/8a5fb512170e0ca742075f05c20bed37.jpg)

Figs. 2. (a) Analysis model, (b) demagnetization curve, and load power characteristic curve: (c) power-voltage curve

Semi 3D Analysis of a Permanent Magnet Synchronous Generator Considering Overhang Structure Using 2D Finite Element Analysis and Equivalent Magnetic Circuit

I. Introduction and Machine Modeling Fig. 1 shows the topology of a 6/5 wound field flux-switching machine (WFFSM), which is simple and reliable [1],[2]. However, the armature reaction and the excitation magnetic field are seriously coupled in WFFSM, resulting in a complicated mathematical model. To solve this problem, the mathematical model of 6/5 WFFSM is derived based on the precise inductance parameters considering magnetic saturation calculated by the frozen permeability method in this paper. The average electromagnetic torque equation is as follows: Te(av) =3/2[Pr Mf Im if cos(β−θmb)] − 9/4[Pr Lm1 Im2 sin(2β−θb(2))] (1) In this equation, Te(av) is composed of constant components of the excitation torque and the reluctance torque, which are related to the current parameters Im , if , β , and the inductance parameters Mf , θmb , Lm1 , θb(2). II. Verification of Mathematical Model Results Fig. 2(a) and (b) show the comparisons of the derived torque model and finite-element analysis (FEA) results under different current densities and different electrical angles, respectively. It can be seen that the derived model results are in good agreement with the FEA results. Fig. 2(c) shows the error between the model and FEA results in average electromagnetic torque. In most areas, the error is within 0.2%. When the current density is 10 A/mm2, the error reaches a maximum of 0.42% due to magnetic saturation. Fig. 2(d) shows the comparison between the model and the finite-element interpolation of the maximum electromagnetic torque. The error between them is no more than 0.45%. More derivations and verifications of the mathematical model of 6/5 WFFSM will be shown in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/303c502f39e665f492642f770254637f.jpg)

Fig. 1. Topology of a 6/5 WFFSM.

![](https://assets.underline.io/uploads/markdown_image/1/image/20f8007d70d71600a0b4749b4befd1fd.jpg)

Fig. 2. Verification of mathematical model results.
(a) The torque under different current densities when β=0°. (b) The torque at different electrical angles when the current density is 5A/mm2. (c) The error between the model and FEA results in average electromagnetic torque under normal operation. (d) The electrical angle of maximum electromagnetic torque under different current densities.

Mathematical Modeling of Wound Field Flux-switching Machines Considering Magnetic Saturation

The doubly salient brushless DC generator (DSBLDCG) is suitable for aviation generator due to its simple and robust structure, and adjustable magnetic field [1]. The harsh environment of aviation operation puts forward a high demand on the thermal design of the generator winding. Hollow conductor with good heat dissipation performance can be used to replace conventional solid conductor in generator for winding cooling [2][3]. The harmonic content of the magnetic field in the slot of DSBLDCG is high and the radial component also exists, which makes its AC loss characteristics complex. Therefore, it is necessary to calculate the AC loss of the hollow conductor for DSBLDCG, analyze the influence of the conductor structural parameters on AC loss and optimize the structure of hollow conductor, so as to reduce the copper loss of the generator and facilitate the further cooling design. Fig. 1 shows the 2D finite element 1/2 section diagram of the 12-stator-pole/8-rotor-pole (12/8) DSBLDCG with hollow conductor. There are no windings or permanent magnets on the rotor, and the stator is wound with armature windings made of hollow conductors. Fig. 2 shows the magnetic flux density waveforms at different positions in the slot. The magnetic flux density can be decomposed into Bx in the x direction and By in the y direction. As the distance from the slot opening position increases, the amplitude of Bx decreases gradually, and By is mainly concentrated near the slot opening position. As shown in Fig. 3, the hollow conductor can be decomposed into four parts for the calculation of AC loss. The calculation formulas are shown as Equations (1) to (3). Fig. 4 shows the AC loss of hollow conductors at position I, II, III, and IV. It can be seen that as the distance from the slot opening position increases, the AC loss of the hollow conductor decreases. Take the hollow conductor at position I closest to the slot opening position as an example, the effect of conductor thickness on AC loss is compared and analyzed, as shown in Fig. 5. Considering the copper loss and cooling design of the generator comprehensively, 0.3mm section thickness of hollow conductor is more appropriate.

![](https://assets.underline.io/uploads/markdown_image/1/image/5474f84f7f9fda1adb57cdbc366a14cf.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/f157468289b7a84cbc9959820b16f0e1.jpg)

AC Loss Calculation and Analysis of Hollow Conductor for Doubly Salient Brushless DC Generator

Sliding electrical contact (SEC) exists extensively in the electromagnetic (EM) launchers (e.g., railguns). In an EM railgun, the armature slides on the rails at hypervelocity, and the real contact interface is never perfectly smooth. Due to the elevations and depressions of the contact interface, the sliding pair only contacts at discrete asperities, called ‘a-spots’ by Holm, at microscale [1]. The contact microscopic morphology of the armature and the rail is shown in Fig. 1. The fact that the current is only flowing through the a-spots results in larger electrical resistance in the contact zone than bulk electrical resistance and the generation of separation force due to current constriction effects [2]. Thus, it is crucially important to build a detailed model of SEC considering contact resistance to compute the interfacial current density and temperature distributions accurately. A 2-D EM railgun with a C-type armature model is simulated by the multiphysics coupling finite element method. The calculation method of contact resistance is based on Holm’s electric contact theory to simulate the real contact zone. Due to symmetry, only a half of the model is modeled, whose finite element mesh on the rail and armature is shown in Fig. 2. The copper rail and aluminum armature are used in the simulation. The driving current waveform is with a peak current of 100 kA and a rising time of 0.5 ms. An initial contact pressure of 10 MPa is applied on the apparent contact area. The armature is initially stationary and then accelerated up to 1000 m/s. The start-up behavior of the railgun is modeled to study the distribution of initial contact pressure, and then to analyze the influence of the unevenly distributed contact pressure on the interface current and temperature distributions. In this paper, the current and temperature distributions in an EM raligun have been studied by a theoretical analysis and numerical simulation. As the result shows, the skin effect and the influence of contact resistance is quite obvious when velocity is high, which provide a guidance for reducing the wear and damage of the contact surface.

![](https://assets.underline.io/uploads/markdown_image/1/image/28cc6b9adae3fc3940de66d9eaa13b65.jpg)

Fig. 1. Contact interface at the microscopic scale.


![](https://assets.underline.io/uploads/markdown_image/1/image/94aebe19ddc01fd3bee6f33998f658de.jpg)

Fig. 2. Finite element mesh of the rail and armature.

Electromagnetic Simulation of Current and Temperature Distribution in Railgun Based on Moving Contact Resistance Model

The hybrid electric vehicle composed of diesel engine and generator can greatly alleviate the “range anxiety” problem [1]. The concentrated winding PM motor with high power and small volume is adopted for the large transport or mining machinery vehicles. In order to solve the heat dissipation problem of concentrated winding PM motor and realize efficient cooling, the heat dissipation structure of stator core with auxiliary teeth without any additional facilities is introduced [1-4]. In this paper, the influence of auxiliary teeth in stator core on concentrated winding PM motor is studied from electromagnetic and thermal aspects. It can be used as an effective scheme for the thermal design of concentrated winding PM motor. Different from the slot type of general motors, a part of the stator core is extended into the slot to form the auxiliary teeth, as shown in Fig. 1. The purpose of adding the auxiliary teeth is to make the heat inside the slot winding have more paths to transfer to the outside. The stator core is mainly divided into three structures, as shown in Fig. 2. Fig. 5 shows the waveforms of the air-gap flux density. Fig. 7 shows the back EMF. The cogging torques are shown in Fig. 8. Fig. 9 shows the flux density contour. Fig. 10 shows a comparison of the torques of the three models. It can be seen the loss change from Fig. 11 and Fig. 12. Fig. 13 shows the change of torque from model 2 to model 3. Fig. 14 and Fig. 15 show the loss changes from model 2 to model 3. Fig. 16 shows the temperature distribution under the rated condition. The actual stator core with auxiliary teeth (Model 2) is shown in Fig. 17(a). In this paper, the influence of the size of stator core auxiliary teeth on concentrated winding PM motor is studied from electromagnetic and thermal aspects. The results show that the auxiliary teeth will weaken the performance of the motor and make the stator temperature drop and the rotor temperature rise. The auxiliary cooling teeth of stator core provide a new guide for the thermal design of concentrated winding PM motor.

![](https://assets.underline.io/uploads/markdown_image/1/image/e693d60f4b87e0da9aeb708253acfaa4.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/fa13280bf506c6232a888a058cef06ff.jpg)

Electromagnetic Thermal Coupling Optimization of Concentrated Winding PM Motor with Auxiliary Teeth for Electric Vehicle

Turbomachineries have large single-unit power and high efficiency, they play an important role in many areas, such as ship propulsion and thermal power generation and so on [1]. With the increase in demand, high power is an important development direction for turbomachineries in the future. However, now mechanical bearings are most used in turbomachineries, and the wear and loss at high speed severely restrict the speed. For this reason, the magnetic bearing (MB) is chosen to support the rotor to overcome the problems caused by the direct contact with the rotor [2]. However, limited by the operating principle of the steam turbine, the axial magnetic bearing (AMB) needs to overcome the large axial thrust generated by the steam flowing in turbomachineries. The traditional topology and design methods will cause the AMB to be too large to be integrated with the steam turbine. Therefore, this paper proposes an asymmetric axial magnetic bearing (AAMB) topology with large bearing capacity in a single direction. As shown in Fig. 1(a), the AAMB has a pair of stators of differential bearing and a stator of unidirectional bearing, both of which are fixed on the bearing seat and share a thrust plate. In normal operation, the unidirectional bearing works in the saturation zone of the material, it generate force and overcome the main axial thrust generated by steam. The differential bearing works in the linear region of the material, it is used to adjust the bearing capacity and overcome the remaining thrust. As shown in Fig. 1(b), The reluctance network method is used to establish the magnetic circuit model of the AAMB. And then the load capacity, dimensions and other constraints are considered, the shortest axial length is taken as the optimization object to design the AAMB, and the design scheme is obtained, and compared with the traditional differential bearing scheme, the main parameters of AAMB are shown in Fig. 2(a). Finally, The finite element model is built as shown in Fig. 2(b), the bearing capacity of the AAMB is verified by finite element method (FEM). The results show that the AAMB meets the bearing capacity requirements, and the overall axial length is less than half of the traditional design scheme.

![](https://assets.underline.io/uploads/markdown_image/1/image/3cf0b6ae6d5cd3f342de8321298524e9.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/4a7cbd7cb78b70bb7fd45e911c113de7.jpg)

Asymmetrical Axial Magnetic Bearings for Turbomachineries

Asymmetrical Axial Magnetic Bearings for Turbomachineries

This work paper presents a methodology based on discrete multi-material topology optimization (TO) for the design of electrical machine cores using finite element analysis. Our methodology is based on the Bi-directional Evolutionary Structural Optimization (BESO) heuristics to remove inefficient elements [1] or add magnets from a meshed model based on how they increase the magnetic energy. Determining the average torque with the calculation of its instantaneous values can be very expensive. The proposed methodology improves the average torque density while formulating cost function on calculated energy difference in the d-axis and q-axis. While usual multi-material topology optimization is formulated using a density-based method [2], this paper determines magnets distribution discretely. The magnetization magnitude can be considered as a discrete variable, and the challenge is to determine its direction. Thus, we deduce magnetization direction from the flux density distribution in the rotor; the magnet magnetization direction is chosen to be normal to B. The main advantage of the proposed methodology is that no “gray” materials are created, and the manufacturing of the machine with the proposed distribution of magnets can be realistically considered. The optimization problem is formulated in order to maximize the magnetic energy difference subject to xe = 0 or 1, xm= 1 or 0, ∑(xe.Ve)-Viron0 ≤ 0 and ∑(xm.Vm)-Vmagnets0 ≤ 0 where, xe,Ve are the iron presence and volume of an iron element e, and xm ,Vm have the same definitions for the magnets. Viron0, Vmagnets0 are the constraints on volumes. The basic diagram is presented in the figure below. The methodology is applied to design several stator structures as given in Fig. 1. The final optimization design of each structure is a classical synchro-reluctant machine with inserted magnets is presented in Fig. 2. The computation time is less than 10 min with 8640 mesh elements.

![](https://assets.underline.io/uploads/markdown_image/1/image/e1674d7217687fe9a3bb5451d806095b.jpg)

Figure 1. Methodology chart

![](https://assets.underline.io/uploads/markdown_image/1/image/67fc42d85f9f647459216a39a05335fd.jpg)

Figure 2. Optimized 36 slots electrical machines, flux density distribution and vectors in the flux barriers, magnets distribution

Rapid multi-material discrete topology optimization for the design of electrical machines

I. Introduction Hybrid excited machines have attracted wide attention due to several advantages, such as easy adjustment of air gap magnetic field, high power density and high efficiency in wide speed range [1]. Nowadays, hybrid excited multitooth flux-switching permanent magnet (HE-MFSPM) machines have become a research hotspot [2]. Based on the literature [3], three novel HE-MFSPM machines are proposed in this paper. Fig. 1 shows the topologies of the proposed three HE-MFSPM machines, called Model I, Model II and Model III respectively. The three models are basically the same except that PM with radial magnetization is added in each dummy slot in Model II, while a fault tolerant tooth is employed and wound with field winding in Model III. Significantly, the amount of PM of the Model I is the least, and the amount of PM of Model II is equal to that of Model III. II. Performance comparisons Fig. 2 (a) presents the overload capability of three proposed HE-MFSPM machines supplied with the same excitation current density (Jf), i.e., 5A/mm2. It can be seen that when the current density of armature winding (Ja) increases from 2A/mm2 to 10A/mm2, the torque of Model III increases the most, showing the best saturation characteristics. Fig. 2 (b) shows the speed range of the three proposed HE-MFSPM machines. It should be noted that all the three structures run at the same speed and the same rated voltage, i.e., 700r/min and 380V. It can be observed that Model III has the highest average torque in the constant torque region and has the widest speed range in the constant power region. Overall, all the proposed three HE-MFSPM machines exhibit good ability of flux regulation. But Model III is superior to Model I and Model II since it exhibits best overload capability, highest average torque and widest speed range. Moreover, fault-tolerant ability and modularity exist in Model III because of the existence of fault-tolerant teeth. The other performance, such as back-EMF, cogging torque and electromagnetic torque, will be further discussed in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/2dadc16756df96c8c823592f1ae3ce2c.jpg)

Fig.1. Topologies of three HE-MFSPM machines. (a) Model I. (b) Model II. (c) Model III.


![](https://assets.underline.io/uploads/markdown_image/1/image/4adae74452ff0be4d22d493315020533.jpg)

Fig.2. Performance comparisons of three models. (a)Overload capability. (b) Speed range

Comparative Study on Three Novel Hybrid Excited Multitooth Flux-switching Permanent Magnet Machines

Torque ripple is the main cause of noise and vibration [1]. Mutual electromagnetic torque ripple is produced by a mismatch between excitation current and the shape of back-EMF [2]. So far, the most popular technology to reduce the cogging torque and torque ripple is skewing, including skewing of stator slots or magnet poles [3]. Based on the above analysis, this paper proposes a segment-skew U-shaped permanent magnet motor. The PMSM rotor topology used for vibration and noise research in this paper is shown in Fig. 1. PMSM parameters are shown in Table 1. Table 2 shows the torque ripple of the motor under different number of skew pole segments and angles. The data are the maximum and minimum values of torque ripple at peak current. According to Table 2, we choose a scheme with 5 segments and 5 degrees of skew angle in this paper. The spatial order of electromagnetic force is shown in Table 3. Only the major force wave orders are listed in the Table 3. Where is v the harmonic frequency of armature reaction magnetic field; is harmonic frequency of rotor permanent magnetic field. Fig. 2 compares the torque ripple at the optimum Angle under different segments of the rotor under peak current. Fig. 3 compares the cogging torque of the rotor with and without segmented skew. Fig. 4 compares the line back electromotive force (EMF). Fig. 5 compares the torque ripple of the rotor at rated current and peak current. Fig. 6 compares the torque current curves of the rotor under MTPA and Id=0 control. Fig. 7 compares the external characteristic curves of the motor. Fig. 8 compares the torque ripple under the magnetic weakening characteristics. Fig. 9 compares the radial electromagnetic force harmonics of rotors. Fig. 10 shows the comparison of noise harmonic response curves of the motor at 5000rpm. Fig 11 and Fig 12 show the experimental equipment and results. In this paper, the characteristics of segmented skew and non-segmented skew rotor motors are studied. It is proved that the segmented skew rotor can make the Back-EMF curve more close to the sine wave, suppress the torque ripple, reduce the electromagnetic force generated by the first order tooth harmonics, and reduce the electromagnetic noise.

![](https://assets.underline.io/uploads/markdown_image/1/image/84bc25c69b293c37f062e882d38e8158.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/7b54b4d2e62db22ee1c07b7e5c485139.jpg)

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