A Study on Core Skew Considering Manufacturability of Double-Layer Spoke-Type PMSM

: The spoke-type permanent magnet synchronous motor (PMSM), which is a general ferrite magnetic ﬂux-concentrated motor, has a low portion of reluctance torque at the total torque magni-tude. Therefore, as a way to increase the reluctance torque, there is a double-layer spoke-type PMSM that can maximize the difference in inductance between the d-axis and the q-axis. However, in the double-layer spoke-type PMSM, cogging torque, torque ripple, and total harmonic distortion (THD) increase with reluctance torque, which is the main cause of vibration and noise. In this paper, a method is proposed that provides the same effect as skew without dividing stages of the permanent magnet by dividing the core of the rotor into two types so that it is easy to manufacture according to the number of stages, unlike extant skew methods. Based on the method, the reduction in cogging torque and THD was veriﬁed by ﬁnite element analysis (FEA).


Summary
The general interior permanent magnet synchronous motor (IPMSM) uses NdFeB magnets to take advantage of the high airgap magnetic flux density. However, it has an adverse impact on the cost of the motor because there are fluctuations in the cost from limited sources of heavy rare-earths. Therefore, for a stable supply of permanent magnets, much development of a spoke-type permanent magnet synchronous motor (PMSM), which is a structure capable of concentrating magnetic flux by using a ferrite magnet having a low magnetic flux density, has been conducted [1,2]. The spoke-type PMSM concentrates the magnetic flux density and can replace the NdFeB magnet used in the extant PMSM with a ferrite magnet.
However, in general spoke-type PMSM, the portion of reluctance torque is low because the difference in inductance between the d-axis and the q-axis is not large. In order to further utilize this reluctance torque, a double-layer spoke-type shape that can maximize the difference in inductance between the d-axis and the q-axis can be used to increase the reluctance torque to improve motor performance [3].
However, increasing the reluctance torque increases cogging torque, torque ripple, and total harmonic distortion (THD), which in turn leads to vibration and noise problems. Therefore, the double-layer spoke-type PMSM needs a design that can reduce cogging torque and THD [4][5][6]. As a representative method, it can be solved by applying skew to the rotor [7][8][9], but the conventional method of applying skew is not a good method in terms of productivity, because it is difficult to manufacture as it divides the stages of the permanent magnet. To ameliorate the limitation, the shape of the rotor core is divided and cross-stacked into stack1 and stack2 so that the permanent magnet can be inserted in one shape as the conventional shape. Therefore, the shape of the rotor core is divided into stack1 and stack2 so that the permanent magnet can be inserted in one shape as before and cross-stacked. The stack1 cuts the left part fixing the permanent magnet, and the stack2 cuts the right part fixing the permanent magnet to apply the same effect as skew. In this paper, this is expressed as a core skew, and through this core skew, the design to reduce cogging torque, torque ripple, and THD is conducted while considering productivity.

Conventional Model Specifications
In this paper, a study to reduce cogging torque, torque ripple, and THD for a doublelayer spoke-type motor was conducted by selecting a washing machine motor having two operating areas: Low speed and high speed.
Washing machine motors are classified into two types: Belt-type motor and direct drive type. The direct drive-type motor is generally mounted on the rear of the drum-type washing machine to operate directly, and the belt-type motor is operated by connecting the belt to the shaft of the motor. The belt-type electric motor is used for general washing machine types, and the belt-type electric motor is also selected as the target model in this paper. Figure 1 and Table 1 show the shape and specifications of the conventional model. cross-stacked into stack1 and stack2 so that the permanent magnet can be inserted in one shape as the conventional shape. Therefore, the shape of the rotor core is divided into stack1 and stack2 so that the permanent magnet can be inserted in one shape as before and cross-stacked. The stack1 cuts the left part fixing the permanent magnet, and the stack2 cuts the right part fixing the permanent magnet to apply the same effect as skew. In this paper, this is expressed as a core skew, and through this core skew, the design to reduce cogging torque, torque ripple, and THD is conducted while considering productivity.

Conventional Model Specifications
In this paper, a study to reduce cogging torque, torque ripple, and THD for a doublelayer spoke-type motor was conducted by selecting a washing machine motor having two operating areas: Low speed and high speed.
Washing machine motors are classified into two types: Belt-type motor and direct drive type. The direct drive-type motor is generally mounted on the rear of the drum-type washing machine to operate directly, and the belt-type motor is operated by connecting the belt to the shaft of the motor. The belt-type electric motor is used for general washing machine types, and the belt-type electric motor is also selected as the target model in this paper. Figure 1 and Table 1 show the shape and specifications of the conventional model.
The motor design of the washing machine must consider two modes of operation. Figure 2 shows the torque-rotating speed characteristic graph of the conventional model, and the operating points of the washing load and dehydration load are indicated. As indicated by the operation point, it has two operation modes: Low speed (580 rpm), a washing operation mode; and high speed (15,660 rpm), a spin-dry operation mode. As the performance varies depending on the driving range, a design is required for two driving areas.  The motor design of the washing machine must consider two modes of operation. Figure 2 shows the torque-rotating speed characteristic graph of the conventional model, and the operating points of the washing load and dehydration load are indicated. As indicated by the operation point, it has two operation modes: Low speed (580 rpm), a washing operation mode; and high speed (15,660 rpm), a spin-dry operation mode. As the performance varies depending on the driving range, a design is required for two driving areas.

Governing Equation
The contents of the basic theory are as follows. First, the cogging torque is generated by the force that tries to move in the direction where the path of the magnetic field is minimized, while the magnetic flux generated in the rotor permanent magnet crosses the airgap to the stator. In other words, it occurs as a result of the combined impact of the magnetic field of a permanent magnet located at the rotor and stator with variable magnetic conductivity depending on an angle of rotation [10]. This can be represented by Equation (1) [11,12].
where α is the mechanical angle of the rotor and W is the magnetic energy of the apparatus. Magnetic energy means energy stored in a permanent magnet and is expressed as Equation (2). where µ is the permeability according to the material and B is the magnetic flux density. The cogging torque is expressed by the magnetic flux density generated in the permanent magnet and linkage to the teeth of the stator through the airgap.
In the electromagnetic field analysis based on finite element analysis (FEA), the torque equation is calculated using the Maxwell stress tensor method and is defined by Equations (3) and (4) [13].
Γ expressed in the above equation is the Maxwell stress tensor; B is the magnetic flux density; i and j are the components of the coordinate system; δ ij is the Kronecker delta, which has a value of 1 when i and j are the same and 0 if they are different. As this paper deals with a spoke-type PMSM, it will be dealt with in a cylindrical coordinate system. The Maxwell stress tensor matrix in a cylindrical coordinate system is shown in Equation (5).
Thus, Equations (6)-(8) express the force using a cylindrical coordinate system. If the force is expressed in the tangential direction in the above equation, the torque will be expressed. This is equivalent to Equation (7).
Additionally, if Equation (7) is expressed as a line integral equation, it can be expressed as Equation (9).
The torque equation can be defined using the Maxwell stress tensor method. In this paper, FEA is performed using Ansoft's Maxwell program and torque is calculated through the formula defined in Maxwell's calculator.

Concept of Core Skew
The core skew represents the stacking of only cores of opposite shapes when stacking the rotor cores of the motor. Figure 3 shows the flux lines of stack No. 1 and stack No. 2. Due to the shape of the rotor core, the phases of the airgap magnetic flux density of stack No. 1 and stack No. 2 are shifted, and when the two stacks are synthesized due to the phase of the shifted airgap magnetic flux density. As shown in Figure 4, the cogging torque decreases.

Concept of Core Skew
The core skew represents the stacking of only cores of opposite shapes when stacking the rotor cores of the motor. Figure 3 shows the flux lines of stack No. 1 and stack No. 2. Due to the shape of the rotor core, the phases of the airgap magnetic flux density of stack No.1 and stack No.2 are shifted, and when the two stacks are synthesized due to the phase of the shifted airgap magnetic flux density. As shown in Figure 4, the cogging torque decreases.    Figure 3, fixed bars that fix permanent magnets can be made into each shape using counter punching in the same mold. Therefore, it is possible to maintain extant production methods and equipment for production, and it can

Concept of Core Skew
The core skew represents the stacking of only cores of opposite shapes when stacking the rotor cores of the motor. Figure 3 shows the flux lines of stack No. 1 and stack No. 2. Due to the shape of the rotor core, the phases of the airgap magnetic flux density of stack No.1 and stack No.2 are shifted, and when the two stacks are synthesized due to the phase of the shifted airgap magnetic flux density. As shown in Figure 4, the cogging torque decreases.    Figure 3, fixed bars that fix permanent magnets can be made into each shape using counter punching in the same mold. Therefore, it is possible to maintain extant production methods and equipment for production, and it can   Figure 3, fixed bars that fix permanent magnets can be made into each shape using counter punching in the same mold. Therefore, it is possible to maintain extant production methods and equipment for production, and it can overcome the limitations of the extant skew method's manufacturability by applying the core skew.  Through this core skew structure, research to reduce cogging torque, torque ripple, and THD is conducted. Most of the original skew research is conducted in 3D FEA, but as 3D FEA requires a lot of time, we first calculate the optimal model using 2D FEA and then proceed with 3D FEA according to the number of steps considering the 3D effect. Through this core skew structure, research to reduce cogging torque, torque ripple, and THD is conducted. Most of the original skew research is conducted in 3D FEA, but as 3D FEA requires a lot of time, we first calculate the optimal model using 2D FEA and then proceed with 3D FEA according to the number of steps considering the 3D effect.  Table 2 are settings for the first 2D FEA variables. First, the first core skew model is selected by deriving the optimal length through finite element analysis for the variable of the permanent magnet fixed bar. The variable range was set from 0 mm at the same point as the pocket opening of the permanent magnet of the existing model to 1.1 mm, which is the maximum pocket opening point at which the permanent magnet cannot be scattered. Through this core skew structure, research to reduce cogging torque, torque ripple, and THD is conducted. Most of the original skew research is conducted in 3D FEA, but as 3D FEA requires a lot of time, we first calculate the optimal model using 2D FEA and then proceed with 3D FEA according to the number of steps considering the 3D effect. Figure 6 and Table 2 are settings for the first 2D FEA variables. First, the first core skew model is selected by deriving the optimal length through finite element analysis for the variable of the permanent magnet fixed bar. The variable range was set from 0 mm at the same point as the pocket opening of the permanent magnet of the existing model to 1.1 mm, which is the maximum pocket opening point at which the permanent magnet cannot be scattered.

Parameter
Range Step Unit Length of Cutting1 0-1.   Figure 7 represents 2D FEA results for the washing and dehydrating loads. When the importance weight of each cogging torque, no-load THD, and torque ripple during lowand high-speed operation was applied equally, all models with length of cutting 1 of 0.9 and length of cutting 2 of 0.8 were satisfied.

Optimistic Design Based on Variables of Tapering
From the first core skew model, the second core skew model is selected through finite element analysis of the position and angle of tapering at the end of the pole arc once again.

Optimistic Design Based on Variables of Tapering
From the first core skew model, the second core skew model is selected through finite element analysis of the position and angle of tapering at the end of the pole arc once again. Figure 8 and Table 3 are the settings for the second 2D FEA variables. The design variables were set as variables for the position and angle of the tapering when applying the outer tapering and when applying the inner tapering in contact with the permanent magnet. Similarly, variables for tapering were set in consideration of the scattering of permanent magnets. Figure 7. Results of the first 2D finite element analysis at washing and dehydrating

Optimistic Design Based on Variables of Tapering
From the first core skew model, the second core skew model is selected element analysis of the position and angle of tapering at the end of the pole a Figure 8 and Table 3 are the settings for the second 2D FEA variables. The de were set as variables for the position and angle of the tapering when appl tapering and when applying the inner tapering in contact with the perm Similarly, variables for tapering were set in consideration of the scattering magnets.

Range
Step Tapering Figure 9 shows the results of the second 2D FEA of the washing load and dehydrating load. Like the first 2D FEA, the importance weight of cogging torque, no-load THD, and torque ripple is equal, but in this second 2D FEA, the weight of satisfying the specifications of the conventional model is considered high priority.
In the case of the dehydrating load, which is a high-speed operation area, the magnetic flux density of the airgap through the tapering decreases, so the models that satisfy the torque compared to the conventional model are very limited. The model has a value of 0.8 mm for Tapering Pos_Out, 50 • for Tapering Ang_Out, 0.2 mm for Tapering Pos_In, and 90 • for Tapering Ang_In. The results of the first core skew model in the previous section and the second core skew model in this section are summarized in Table 4.
In the case of the second core skew model compared to the conventional model, the cogging torque decreased 82%, the no-load THD decreased 1.12% p , and the torque ripple in the high-speed operation range decreased 61% p . However, it can be seen that the torque ripple in the low-speed operation range decreased compared to the first core skew model, but increased in the second core skew model. Energies 2021, 14, x FOR PEER REVIEW torque ripple is equal, but in this second 2D FEA, the weight of satisfying the s tions of the conventional model is considered high priority.  In the case of the second core skew model compared to the conventional m cogging torque decreased 82%, the no-load THD decreased 1.12%p, and the torq in the high-speed operation range decreased 61%p. However, it can be seen that th ripple in the low-speed operation range decreased compared to the first core ske but increased in the second core skew model.

Analysis of Result Data
In the previous section, the optimal models were calculated by applying each variable for the core skew model. For this, the components for each harmonic order were analyzed for cogging torque, torque ripple, and no-load THD. Figures 10 and 11 show the analysis of cogging torque. In Figure 10, it can be seen that the two stacks have cogging torques of different phases, and the peak value decreases compared to the conventional model during the synthesis of the two stacks. Figure 11 shows the spectrum of cogging torque for each harmonic order. In the case of the conventional model, it can be seen that most of the components are composed of sixth harmonics. It can be seen that the sixth harmonic is greatly reduced in the two models selected in the previous section. In particular, in the case of the second model, tapering can be applied to increase the third harmonic and decrease the peak value of the sixth harmonic to reduce cogging torque. compared to the conventional model during the synthesis of the two stacks. Figure 11 shows the spectrum of cogging torque for each harmonic order. In the case of the conventional model, it can be seen that most of the components are composed of sixth harmonics. It can be seen that the sixth harmonic is greatly reduced in the two models selected in the previous section. In particular, in the case of the second model, tapering can be applied to increase the third harmonic and decrease the peak value of the sixth harmonic to reduce cogging torque.   Figures 12 and 13 show the torque ripple waveform and torque spectrum in the highspeed operation range, respectively. It can be seen that the dominant component of the torque ripple in the conventional model and the first core skew model, as in cogging torque, is the sixth harmonic. However, in the case of the second core skew model, it can be seen that the sixth harmonic is significantly reduced and the third harmonic is increased. It can be seen that by applying the tapering structure, the third harmonic is increased, the sixth harmonic is reduced, and the torque ripple is reduced. harmonics. It can be seen that the sixth harmonic is greatly reduced in the two selected in the previous section. In particular, in the case of the second model, can be applied to increase the third harmonic and decrease the peak value of harmonic to reduce cogging torque.   Figures 12 and 13 show the torque ripple waveform and torque spectrum in speed operation range, respectively. It can be seen that the dominant compone torque ripple in the conventional model and the first core skew model, as in torque, is the sixth harmonic. However, in the case of the second core skew mod be seen that the sixth harmonic is significantly reduced and the third harmon creased. It can be seen that by applying the tapering structure, the third harmo creased, the sixth harmonic is reduced, and the torque ripple is reduced. Figure 11. Spectra of cogging torque for each model. Figures 12 and 13 show the torque ripple waveform and torque spectrum in the highspeed operation range, respectively. It can be seen that the dominant component of the torque ripple in the conventional model and the first core skew model, as in cogging torque, is the sixth harmonic. However, in the case of the second core skew model, it can be seen that the sixth harmonic is significantly reduced and the third harmonic is increased. It can be seen that by applying the tapering structure, the third harmonic is increased, the sixth harmonic is reduced, and the torque ripple is reduced.     Figure 14 shows the back electromotive force between lines at no model reduced the seventh harmonic and increased the fifth harmonic co conventional model, but it was confirmed that the fifth harmonic decrease was reduced in the second model to which tapering was applied.  Figure 15 shows that optimal models selected in 2D FEA are model steps. In the case of skew, as the 3D effect must be considered for each step, analyzes the 3D effect for each step through 3D FEA. Table 5 represents re FEA. It provides the fact that both the extant model and the core skew mod cogging torque, torque ripple, and no-load THD compared to the 2D FEA 3D effect is considered, these factors decrease based on a circumstance magnetic flux density decreases due to the generation of the leakage magn axial direction.  Figure 14 shows the back electromotive force between lines at no load. The first model reduced the seventh harmonic and increased the fifth harmonic compared to the conventional model, but it was confirmed that the fifth harmonic decreased and the THD was reduced in the second model to which tapering was applied.   Figure 14 shows the back electromotive force between lines at no model reduced the seventh harmonic and increased the fifth harmonic co conventional model, but it was confirmed that the fifth harmonic decrease was reduced in the second model to which tapering was applied.  Figure 15 shows that optimal models selected in 2D FEA are model steps. In the case of skew, as the 3D effect must be considered for each step, analyzes the 3D effect for each step through 3D FEA. Table 5 represents re FEA. It provides the fact that both the extant model and the core skew mod cogging torque, torque ripple, and no-load THD compared to the 2D FEA 3D effect is considered, these factors decrease based on a circumstance magnetic flux density decreases due to the generation of the leakage magn axial direction.  Figure 15 shows that optimal models selected in 2D FEA are modeled from 2 to 8 steps. In the case of skew, as the 3D effect must be considered for each step, core skew also analyzes the 3D effect for each step through 3D FEA. Table 5 represents results of the 3D FEA. It provides the fact that both the extant model and the core skew model all decreased cogging torque, torque ripple, and no-load THD compared to the 2D FEA results. As the 3D effect is considered, these factors decrease based on a circumstance that the airgap magnetic flux density decreases due to the generation of the leakage magnetic flux in the axial direction. Compared to the conventional model, the core skew optimization model confirmed that by 3D FEA, the cogging torque decreased by 89%, the no-load THD by 1.15%p, and the torque ripple in the high-speed operation range by 62%p.  Figures 16 and 17 show the waveform of the cogging torque from 3D FEA steps. It is almost similar from 2 to 8 steps numerically, and it can be confirmed that it has decreased significantly compared to the conventional model. From some facts from Figure 17, the high sixth harmonic of the conventional model can be confirmed, and the optimal models show that the sixth harmonic is greatly reduced, and the 12th harmonic is also reduced by more than 50%, indicating that the cogging torque is greatly reduced.   Compared to the conventional model, the core skew optimization model confirmed that by 3D FEA, the cogging torque decreased by 89%, the no-load THD by 1.15% p , and the torque ripple in the high-speed operation range by 62% p . Figures 16 and 17 show the waveform of the cogging torque from 3D FEA steps. It is almost similar from 2 to 8 steps numerically, and it can be confirmed that it has decreased significantly compared to the conventional model. From some facts from Figure 17, the high sixth harmonic of the conventional model can be confirmed, and the optimal models show that the sixth harmonic is greatly reduced, and the 12th harmonic is also reduced by more than 50%, indicating that the cogging torque is greatly reduced. Compared to the conventional model, the core skew optimization model con that by 3D FEA, the cogging torque decreased by 89%, the no-load THD by 1.15% the torque ripple in the high-speed operation range by 62%p.  16 and 17 show the waveform of the cogging torque from 3D FEA ste almost similar from 2 to 8 steps numerically, and it can be confirmed that it has de significantly compared to the conventional model. From some facts from Figure high sixth harmonic of the conventional model can be confirmed, and the optimal show that the sixth harmonic is greatly reduced, and the 12th harmonic is also redu more than 50%, indicating that the cogging torque is greatly reduced.  The torque ripple in the high-speed operation range is also significantly reduced. can be checked from the amplitude of the waveform in Figure 18. Although fundamenta components slightly decrease in the high-speed region, which can be seen in Figure 19, i the case of the sixth harmonic or twelfth harmonic, the core skew models are significantl reduced compared to the conventional model. Likewise, in the case of no-load THD, th core skew models have reduced harmonics of each order compared to the conventiona model. It can be seen in Figure 20.   The torque ripple in the high-speed operation range is also significantly reduced. It can be checked from the amplitude of the waveform in Figure 18. Although fundamental components slightly decrease in the high-speed region, which can be seen in Figure 19, in the case of the sixth harmonic or twelfth harmonic, the core skew models are significantly reduced compared to the conventional model. Likewise, in the case of no-load THD, the core skew models have reduced harmonics of each order compared to the conventional model. It can be seen in Figure 20. The torque ripple in the high-speed operation range is also significantly reduced. I can be checked from the amplitude of the waveform in Figure 18. Although fundamenta components slightly decrease in the high-speed region, which can be seen in Figure 19, i the case of the sixth harmonic or twelfth harmonic, the core skew models are significantl reduced compared to the conventional model. Likewise, in the case of no-load THD, th core skew models have reduced harmonics of each order compared to the conventiona model. It can be seen in Figure 20.   The torque ripple in the high-speed operation range is also significantly reduced. I can be checked from the amplitude of the waveform in Figure 18. Although fundamenta components slightly decrease in the high-speed region, which can be seen in Figure 19, i the case of the sixth harmonic or twelfth harmonic, the core skew models are significantl reduced compared to the conventional model. Likewise, in the case of no-load THD, th core skew models have reduced harmonics of each order compared to the conventiona model. It can be seen in Figure 20.

Conclusions
In this paper, as the model selected as the target model was a belt-type wash chine motor, it was necessary to consider bidirectional operation instead of unidir operation during the washing load. Therefore, the core skew model was derived thesizing the symmetrical model by calculating the optimal model in 2D finite analysis. First, the optimum model was selected by using the variable setting for manent magnet fixed bar, and then the optimum model was selected by applying ing to the end of the rotor. Finally, in order to consider the 3D effect of each num final model was selected through final verification through 3D finite element ana Compared to the conventional model, it was confirmed that the component harmonic of the torque in the high-speed operation range in the core skew mod reduced, resulting in a significant reduction in cogging torque and torque ripple. THD also confirmed a remarkable performance improvement through componen sis for each harmonic through FFT. In particular, the fifth and seventh harmonic are the main harmonic components of the back EMF, were reduced.
In addition, when the proposed structure is applied to an application of u tional operation, the cogging torque or torque ripple can be converged to almost z a skew shape in which the cogging torque and torque ripple of Stack No1 and St are in opposite phases to each other. Finally, the proposed core skew principle only provide various core types, but is also expected to have a good influence o ture electric motor industry.

Conclusions
In this paper, as the model selected as the target model was a belt-type washing machine motor, it was necessary to consider bidirectional operation instead of unidirectional operation during the washing load. Therefore, the core skew model was derived by synthesizing the symmetrical model by calculating the optimal model in 2D finite element analysis. First, the optimum model was selected by using the variable setting for the permanent magnet fixed bar, and then the optimum model was selected by applying a tapering to the end of the rotor. Finally, in order to consider the 3D effect of each number, the final model was selected through final verification through 3D finite element analysis.
Compared to the conventional model, it was confirmed that the components of each harmonic of the torque in the high-speed operation range in the core skew model were reduced, resulting in a significant reduction in cogging torque and torque ripple. No-load THD also confirmed a remarkable performance improvement through component analysis for each harmonic through FFT. In particular, the fifth and seventh harmonics, which are the main harmonic components of the back EMF, were reduced.
In addition, when the proposed structure is applied to an application of unidirectional operation, the cogging torque or torque ripple can be converged to almost zero with a skew shape in which the cogging torque and torque ripple of Stack No1 and Stack No2 are in opposite phases to each other. Finally, the proposed core skew principle can not only provide various core types, but is also expected to have a good influence on the future electric motor industry.