A novel auxetic stator winding to improve the performance of permanent magnet synchronous electric motors

High efficiency and torque density in permanent magnet synchronous motors (PMSMs) have contributed to their increasing popularity. Nonetheless, these advantages are compromised by higher vibration levels resulting from the torque ripple issue and magnetic flux density in the stator, causing magnetic forces on the stator surface. In this study, a new smart shape for the stator winding is proposed which reduces unwanted torque vibration and the overall magnetic flux density while keeping the same motor efficiency. The proposed windings shape is designed based on the auxetic principle and a locally resonant mechanism (LRM). Afterward, the proposed and original PMSM models are compared by looking at the average torque, total losses, torque ripple, flux density, output power, and motor efficiency under different speed operating conditions. In addition, the sensitivity analyses of the proposed model reveal the influence of auxetic structural parameters and initial mechanical angle on the system’s performance, which can be utilized to control the physical and mechanical properties of the system. According to the results, the designed model reduces torque ripple and magnetic flux density in the stator region by 41.38% and 4.70%, respectively, while the motor efficiency remains unaffected. The present work offers a potentially robust and affordable solution for regulating the vibration behavior of electric motors without impacting power efficiency.


Introduction
Over the past few years, electrification of engine vehicles and changing the infrastructures have been major trends in the automotive industry to move towards zero carbon emission.Although electric motors (e-motors) are quiet, noise, vibration, and harshness (NVH) remain a paramount concern in electric vehicles.In combustion engines, mechanical sources such as rotor dynamic, rotor unbalances, gearbox, and flexible shafts contribute to the higher level of NVH [1].A characteristic feature of electric powertrains (e-powertrains) is their high-frequency whining noise, torque ripple, and iron loss, which are mainly the result of electromagnetic forces rather than the low-frequency combustion and mechanical vibration of internal combustion engines (ICE) powertrains [2].Previous studies on NVH for e-motors confirm that they are mostly affected by tonal noise (harmonics of the rotor speed), commonly referred to as whistling noise, that is generated from the flux density forces [3].In general, there are two main complementary techniques to improve sound quality and design in automobiles, namely passive and active methods [4,5].Active control techniques are most effective in shaping low-frequency noises [6][7][8][9], while passive methods are proven to be efficient in dealing with high-frequency e-motor noises [10].NVH subjects in e-motors are practically investigated within three main concerns: the magnetic flux forces, torque ripple, and iron loss.In the following, we will explain each one of them and recent works to solve the relevant problems.
One way to reduce the motor whistling noise is to reduce the level of the magnetic forces at the source and eliminate the dominant resonance frequency of the powertrain structure by improving the design.Wang et al [11] investigated the electromagnetic force of a permanent magnet synchronous motor (PMSM) from a spatial and frequency order perspective.In addition, the motor's NVH performance was improved through the optimization of rotor slotting to minimize the generated electromagnetic force.Liu et al [12] proposed an optimization approach to the magnet shape to suppress the air gap flux density harmonic of a PMSM.In the case of parallel magnetization, the suggested method corrects the effective air gap length of the motor and calculates the magnetization length of the permanent magnet by considering the relative permeability of the permanent magnet.The magnet shape has been also used to optimize the air gap flux density.Liu et al [13] analyzed the PMSM radial and tangential forces using tooth harmonics in order to select suitable pole and slot combinations.
In addition, torque pulsation, or so-called torque ripple, is one of the most critical concerns in designing PMSMs, as it may result in unwanted byproducts such as vibrations and acoustic noise.Numerous scholars have studied techniques for torque pulsation mitigation.The available methods can be categorized into two main streams: control techniques, namely current profiling [14], and optimal design methods.This topic has been addressed by many strategies, such as rotor shape design using perforations [15], air gap profile design [16], rotor/ stator skewing [17,18], optimization of magnet shape and skewing [19], stator shape design [20], slot/pole combinations [21][22][23], and asymmetric flux barrier modeling [24].To name a few works; Liang et al [16] studied the influence of changing air gap profile on the flux density force, torque ripple, and average output torque to reduce NVH levels of an interior PMSM.Ma et al [25] structured a slotted rotor composition to reduce the radial force and torque ripple.Moreover, some other scholars investigated the stator windings (or stator tooth) designs inside the stator slots.The e-motors proposed by Demir et al [26] consider asymmetric stator windings that generate superior torque quality and moderately higher torque density levels.Reichert et al [27] analyzed the influence of stator windings shape of a PMSM on output torque and active/passive radial forces.Ni et al [28] added a short-circuited auxiliary three-phase winding into the same slots as the basic stator windings model.This new winding structure can generate a damping impact on the flux density harmonic elements, particularly highfrequency harmonics in the air gap area to diminish the NVH of the PMSM.Li et al [29] showed that the amplitude of electromagnetic force waves is influenced, by the order of tooth harmonic and magnetic field saturation level in a permanent magnet motor.It also demonstrated that a geometric optimization of the stator winding and iron core can effectively diminish the electromagnetic flux density amplitude and hamper electromagnetic vibration.
Furthermore, efforts have been made to reduce iron loss so as to maximize e-motor efficiency and NVH behavior to meet users' expectations.The hysteresis loss and eddy current loss are the two main components of the iron loss in the e-motor, where eddy current loss is proportional to the square of the motor frequency, and hysteresis loss is proportional to motor frequency [30].The effect of utilizing different soft magnetic composite materials on the stator is studied in [31,32].They also compared the method with conventional laminated materials in iron loss and temperature rise according to the distribution of stator magnetic density.Wang et al [33] proposed a method that can simultaneously reduce the iron losses and magnetic noise of a PMSM considering various stator windings lengths and airgap surface sizes.The effect of topology parameters on NVH and iron loss is examined to evaluate the possibility of making adequate selections during the design process.Belli et al [34] studied the impact of altering stator winding's shape on reducing the eddy current loss for a surface-mounted PMSM.In order to minimize eddy current loss in magnets, this calculation is integrated with an optimization process by the genetic algorithm.Yamazaki et al [35] studied the shapes of stator teeth in a PMSM with scattered windings to reduce rotor loss.As reported in [35], two stator tooth shapes can be employed alternately along the circumferential direction to decrease the rotor loss.
It is worth noting that, it is often difficult to combine the lightweight design with improved NVH features.For example, materials with low density and high stiffness suffer from poor noise and vibration isolation properties.Hence, for a reasonable NVH level at lower frequencies, heavy and bulky additions are often required.There are also several avant-garde techniques, including utilizing porous liners [36], perforations [37], and auxetic structures [38], which have been mostly employed in the automotive industries to control NVH issues.Auxetic materials have an abnormal property called negative Poisson's ratio (NPR), allowing them to expand laterally under an axial tensile force.The idea of auxetic structures takes inspiration from nature.Some natural materials possess NPR properties, including the skin of certain animals (like cows and cats), some woods, and tendons.In recent years, the scientific community has shown a growing interest in auxetic metamaterials due to their exceptional mechanical properties, including improved resistance to indentation [39], shear resistance [40], vibration energy absorbing [41], enhanced stiffness [42], vibroacoustic filtration [43], and fracture toughness [44].These outstanding characteristics make auxetic structures a promising candidate for many engineering applications, such as vibration attenuation [45], vibration energy harvesting [46], structural health monitoring [47], and sensors [48].Figure 1 exhibits the practical use of auxetic structures in automotive industries.
Previous studies have indicated that although reducing torque ripple can have advantages, it may also result in a decrease in average torque, which can lead to a reduction in motor efficiency.As a result, the improvement of torque ripple has been found to mostly enclose a negative effect on motor efficiency and performance.Furthermore, researchers have usually concentrated solely on reducing magnetic forces or torque vibration.Also, conventional methods tend to add weight, complexity, maintenance, and manufacturing costs to the system.However, in the same line of research, the current paper proposes a novel design for the stator winding shape based on the principle of auxetic mechanism and the locally resonant mechanism (LRM), as a means to simultaneously reduce torque ripple and the magnetic flux density, while keeping the motor efficiency constant.In addition, the suggested stator winding shape diminishes total losses (iron and coil losses), where this phenomenon neutralizes the negative impact of average torque reduction on motor efficiency.The present method keeps the stator's weight unchanged, and uses the same materials as the basic PMSM model, without complex and costly manufacturing requirements.
First, the basic concepts and assumptions related to the stator models are presented.Then, some details on the numerical model used for the analysis are discussed and the two-dimensional (2D) proposed and original PMSM models are numerically analyzed using the finite element method (FEM).COMSOL Multiphysics software [49] is utilized for a comprehensive evaluation.Models are simulated operating an electromagnetic rotating machinery field under the time domain, stationary state, and time to harmonic frequency loss domain.As changes in the flux densities can cause substantial differences in the distribution and level of magnetic forces, this study investigates the influence of using the proposed auxetic shape stator winding on magnetic density reduction.Also, the torque ripple generated by the proposed and basic PMSM models is calculated.The models are compared, in terms of the torque ripple, average flux density, average torque, output power, iron and coil loss responses, and efficiency under different speed operating conditions.Meanwhile, we conducted sensitivity analyses to determine how the system's performance is affected by geometrical parameters and initial mechanical angle.By altering the matching scheme of structural parameters inferred from the sensitivity analysis, it is possible to control the physics and mechanical properties of the whole system.Finally, the paper is wrapped up with some concluding remarks.This study might provide insight into the development of e-motors using this new promising design approach.

Models and methods
In this section, the prototype design of the proposed auxetic stator and the basic stator models are demonstrated.The auxetic stator is created by setting auxetic rotating rigid units which are formed by perforating holes with narrow slit patterns.In addition, the physics insight and the theory behind this novel structure design are explained.Finally, the FEM is utilized for a comprehensive evaluation of the PMSM models.The analysis includes torque ripple, flux density, output power, iron loss, coil loss, and motor efficiency under different speed operating conditions, using an electromagnetic rotating machinery field under the time domain, stationary state, and time to harmonic frequency loss domain.

Auxetic stator winding design
There is a unique class of synthetic materials known as auxetic materials.Amongst this class are objects exhibiting abnormal properties, such as negative Poisson's ratio (NGR), leading to their counter-intuitive ability to expand/shrink laterally when stretched/compressed longitudinally.The orthogonal expansion in the stretching direction (counter-intuitive deformation behavior under typically non-scale-dependent conditions) allows auxetic materials to maintain superior mechanical properties, including toughness, vibration isolation, and indentation resistance [50].The proposed auxetic stator winding shape is developed based on perforated and rotating-rigid auxetic form mechanisms. Through a locally resonant energy absorbing mechanism and the auxetic vibration isolation principle, the auxetic winding structure is expected to attenuate torque ripples and diminish magnetic flux density in the stator with flexural and torsional modes.Figure 2 illustrates the original and proposed stator models.The basic model is designed by COMSOL Multiphysics and used as a validation tool for the numerical results reported in the paper.The main geometrical parameters characterizing the stator windings shape modification are stator tooth length and stator's tooth width.Therefore, we consider two geometrical parameters L and W shown in figure 2, which are related to the stator's tooth length and width, respectively.It is worth noting that to study both models under the same condition, the value of the stator slot area in both models is considered constant (A = 225 mm 2 ).In addition, since the previous studies confirmed the superiority of 2D electromagnetic models in terms of high accuracy, simulation time, and complexity reduction [16][17][18][19]22], a 2D PMSM model for the simulation of the electromagnetic field is utilized.

Hypotheses
The principal sources of electromagnetic noise and vibration are the Maxwell forces (the radial and tangential magnetic flux forces acting on the surface model) and the torque ripple [51].Based on Ampère's circuital law, the magnetic flux density can be derived by: where the magnetic flux density B B B , r t ( )is related to the total current density J, magnetic permeability constant , 0 m and the electric field E. The radial and tangential magnetic flux density will determine the corresponding radial/tangential magnetic forces.Consequently, the motor torque can be calculated as follows : The main geometrical parameters characterizing the stator windings shape modification are stator tooth length and stator's tooth width.Therefore, we consider two geometrical parameters L and W shown in figure 2, which are related to the stator's tooth length and width, respectively.It is worth noting that to study both models under the same condition, the value of the stator slot area in both models is considered constant (A = 225 mm 2 ), in which the stator's weight is similar.
where W mag denotes the total stored magnetic energy, F q is the radial force, and R rotor stands for the radius of the external rotor.In addition, motor performance is significantly influenced by the iron loss level.Therefore, the total iron loss is governed by: in which P , h P , e and P a are hysteresis, eddy current, and anomalous loss with their corresponding coefficient of k , h k , e and k , a respectively.In equation (3), f denotes the motor frequency and B m stands for maximum magnetic induction intensity of core material [31].The auxetic structure has taken into account the total vibrational energy, which includes magnetic flux energy, iron loss, and torque ripple, as the incident excitement.The auxetic structure will attenuate the incoming vibrations and isolate most vibrations inside its auxetic region.It will prevent to transfer of the vibrations to the torque and rotor, and finally, the stored energy will disappear in the form of heat due to the local resonance and auxetic mechanisms.The asymmetric axial and flexural vibrations in a ring can be assumed as follows : where V denotes the transverse displacement, U denotes the in-plane displacement along the r-direction, r out and r in are the outer and inner radius of the ring, respectively, the surface thickness is shown by h, and the ring density is shown by .r The quantities F r and M r are the line force (bearing force) and line momentum operating across the entire surface of the ring.In addition, elastic wave propagation across the auxetic structure throughout the in-plane direction can be governed in the following manner: In equation (6), S ∆• is the responding deflection force, which originated from incoming magnetic forces, u i is the in-plane displacement vector in the r-direction, and w is the angular frequency.Also, the vibration response of the system can be evaluated by equation (7): where K and M show equivalent mass matrix and spring stiffness, respectively.According to equation (6), the local resonator acts as a spring, and the substrate acts as an equivalent mass to dampen vibrations coming from the responding deflection force and dissipate it as heat.Furthermore, the dissipated energy by the auxetic region is generally correlated with stress distribution on the surface, as shown in equation (8).
in which 11 ̅ s and 22 ̅ s exhibit the axial and lateral average stress in the auxetic stator.Conventional structures possess 11 ̅ s and 22 ̅ s with a different sign as they are laterally compressed under an axial tensile force.As a result, 11 ̅ s has a positive sign while 22 ̅ s has a negative sign.On the other hand, an auxetic structure has 11 ̅ s and 22 ̅ s with the same sign, simultaneously.Based on their auxetic feature, they can laterally expand under an axial tensile force, therefore, 11 ̅ s and 22 ̅ s have positive signs at the same time.Same sign average stresses allow the dissipated energy to maintain its maximum amount as determined in equation (8).The auxetic structure contains an auxetic region that traps vibration energy, resulting in even stress distribution along its edges and connections.Meanwhile, the local resonance acts as a spring and repeatedly converts the stored potential energy to kinetic energy until the stored energy is dissipated as heat.This section mathematically determines how the proposed auxetic stator can attenuate incident vibrations such as torque ripple, and magnetic forces.The following provides a numerical evaluation of auxetic potential, and the subsequent results are discussed.

Numerical setup
The methodology plan utilized in the present study has been illustrated in figure 3.As a first step, the fully assembled PMSM models for the numerical simulation are constructed as depicted in figure 4. In this study, COMSOL Multiphysics has been employed using FEM to straitly calculate the desired characteristics, including average torque, torque ripple, magnetic flux density, and total loss.The initial parameters of the models used for the numerical simulations are listed in table 1. Besides, the materials used for the model components are shown in table 2. The main structural components of the PMSM model are explained in figure 5.According to the boundary conditions and initial numerical setup, the inward/outward magnetized regions, stator and rotor iron regions, and the three phases (A, B, and C) order arrangement are determined in figure 5.As depicted in figure 4, the red circle region follows the continuity boundary condition (CBC), while the blue marked areas use Ampère's circuital law.Furthermore, the black circle denotes the outer circle air region, where the magnetic insulation boundary condition is applied.The system is subjected to an incident excitation source with a rotational speed of 100 rpm, a peak current of 30 A, and an initial angle of 20 .
 The mesh convergence technique  (MRT) is utilized to find adequate mesh sizes and the models meshed using free triangular type considering the moving mesh feature for the rotary domains (rotor solid and air regions) with 10787 degrees of freedom (DoF) plus 9426 internal DoF for the basic model, and 11694 DoF plus 9492 internal DoF for the proposed model as shown in figure 6.Finally, to calculate the magnetic flux density, torque ripple, iron, and coil losses, the proposed and basic models are analyzed using an electromagnetic rotating machinery field under different studies, including the time domain, stationary state, and time to harmonic frequency loss domain.

Results and discussion
Comprehensive numerical analyses were conducted on the models to examine the effectiveness and potential of the proposed auxetic design in addressing the vibration problems in the PMSM without disturbing the motor efficiency.Consequently, magnetic flux density, torque ripple, and total iron loss are investigated using FEM under electromagnetic rotating machinery physics.Furthermore, the performance of the proposed and basic PMSM models are compared in terms of torque ripple, average torque, magnetic flux density, iron loss, coil loss, output power, and motor efficiency under different speed operating conditions.Also, a parametric study was performed on the designed PMSM model to evaluate the influence of structural parameters and mechanical angle on the motor performances.The main findings are represented and discussed in the following section.

Study of magnetic flux density
This section investigates the effect of using an auxetic stator windings shape on the magnetic flux density saturation.Reducing the magnetic flux density in the stator can substantially diminish the level of radial/ tangential magnetic forces in the stator, which are influential for vibration problems [16,52].Figure 7 represents the magnetic flux density distribution for the proposed and basic PMSM models.It can be observed that the auxetic structures can effectively reduce the magnetic flux density levels across the channels between the stator teeth.Nevertheless, the magnetic flux density saturation in the rotor region has remained unchanged as the same rotor and air gap structures for both models are considered.In addition, the variation of magnetic flux density as a function of time and mechanical degree has been studied in this section.Figure 8 illustrates the average magnetic flux density in relation to time in the stator region.It can be seen that the proposed PMSM model exhibits lower average magnetic flux density with fewer fluctuations than the basic model.Also, figure 9 depicts the average magnetic flux density changes in the stator region as a function of mechanical degree.It is demonstrated that the magnetic flux density levels in the proposed model are under those of the basic model at all mechanical angles.The analysis indicates that the average flux density in the stator has been reduced by 4.70%.As a result, this study confirms that the proposed PMSM model generates lower magnetic flux density and substantially lower magnetic forces in the stator surface.

Study of torque ripple
This section explores the effect of using an auxetic stator windings shape on the torque vibration.The variation of the rotor torque as a function of motor angle is exhibited in figure 10 for the proposed and basic PMSM models.It is evident that the torque saturation levels are remarkably lower in the proposed model compared to the original model under both full rotation and electrical period conditions.In addition, table 3 provides the outcomes of torque waveform analysis.There is a significant reduction in torque ripple from 1.45% to 0.85%, representing a reduction of 41.38%, while average torque is reduced by 7%.Previous results confirm that improving torque ripple may have led to an average torque reduction.More specifically, the results in [52] found that skewing the magnet had a significant positive impact on torque ripple.However, unlike this positive effect, it can be seen that the torque averages have been negatively affected.In [53], by using a Halbach array and permanent magnet (PM) tapering parameter, a method to reduce the bearing force  ripple and torque ripple of an integrated magnetic bearing motor was proposed.However, it was found that the torque ripple reduction will diminish the average torque from 8.85 Nm to 8.25 Nm.Furthermore, the torque waveform results of the present study are compared with those of two previous works.As can be seen in table 4, the proposed auxetic stator would considerably improve the torque ripple.

Study of iron loss
The influence of utilizing the designed auxetic stator winding on total iron-coil loss is further investigated.Figure 11 illustrates the iron loss density distribution for both the proposed and basic PMSM models.The darker red color shows a lower iron loss level.It can be observed that the proposed auxetic winding structure would considerably reduce the iron loss level in the stator surface, especially in the throat spaces between the stator windings (stator teeth).This improvement in iron loss can naturalize the negative impact of average torque reduction on motor efficiency.In addition, table 5 provides the percentage of changes in principal parameters, including coil loss, iron loss, flux density, average torque, and torque ripple.It can be concluded that despite previous studies, the proposed auxetic structure can positively and notably mitigate torque ripple and magnetic density distribution while keeping the motor efficiency unaffected.

Comparing characteristics of the proposed and basic PMSM models
The performance of the original and proposed PMSM models are listed in table 6 under different speed operating conditions, including low, moderate, and high speeds.The average magnetic forces in the stator region can be substantially reduced by decreasing the average flux density.Also, the total loss is lowered since both the average coil loss and iron loss gradually decrease.Additionally, the torque ripple is significantly reduced.However, this decrease in average torque ultimately results in a drop in output power.Nevertheless, the motor efficiency remains almost unchanged as the positive impact on the total loss counteracts the negative effect on the average torque.
Comparing the average flux density in the stator surface, and torque ripple of the proposed PMSM model with those of the original one reveals how well the auxetic winding shape can improve the vibration performance according to the above-mentioned indices.The outstanding vibration behavior of the proposed auxetic winding structure can be explained based on the LRM and auxetic principle.Due to twisted routes and multiple  scatterings of the incident and reflected waves, auxetic structures with high tortuosity significantly boost vibration-absorbing performance [54].The auxetic design allows the dissipated energy to maintain its maximum amount as determined in equation (8).The auxetic structure comprises a special region called the auxetic spot that has the ability to capture and contain vibration energy.This unique feature allows for uniform stress distribution throughout its edges and connections.Meanwhile, the local resonance acts as a spring and repeatedly transforms the stored potential energy into kinetic energy until the energy is finally dissipated as heat.
It is shown that such structures will exhibit high energy absorption and vibration mitigation, thereby reducing the total iron loss, torque ripple, and magnetic flux density on the stator surface.The suggested auxetic design absorbs a significant portion of incoming vibrations and confines them within the auxetic winding area.This prevents the transmission of the vibrations to the rotor and torque.Eventually, the energy stored in the auxetic region will be released as heat and decrease by the LRM. Figure 12 displays the overall performance of both the proposed and basic PMSM models, comparing critical factors such as torque ripple, magnetic flux density, and motor efficiency.It can be observed that the proposed model can effectively reduce the torque ripple and magnet flux density, all while maintaining the motor's efficiency.Furthermore, the main contributions of the current work can be explicitly explained as follows: • According to the existing literature, previous studies have shown that improving the vibration performance of e-motors would negatively affect the power output and, consequently, the motor efficiency.Besides, researchers have usually concentrated solely on reducing magnetic forces or torque vibration.However, the  present work improves the vibration performance of the system, more specifically the whistling noise and torque ripple, by reducing torque pulsation and magnetic flux density while the motor efficiency is unaffected.
• The motor efficiency is almost unchanged as the positive impact on the total loss acts against the negative effect on the average torque.
• The suggested auxetic design enhances vibration-absorbing performance due to its twisted routes and multiple scatterings of the incident and reflected waves.This novel design features high tortuosity, which enables it to absorb a significant portion of incoming vibrations and isolate them within the auxetic winding area.As a result, the vibrations are prevented from being transmitted to the rotor and torque.Eventually, the energy stored in the auxetic region is released as heat and dissipated by the local resonance mechanism.
• Conventional methods often make systems heavier, more complex, costly to maintain, and expensive to manufacture.In contrast, the proposed method focuses solely on creating smart designs, without altering the weight or material used in the system.This approach maintains the system's simplicity and avoids the need for expensive manufacturing processes.As a result, the proposed method is an affordable, effective, reliable, and robust solution.

Parametric analysis
A suitable choice of geometrical parameters is decisive when it comes to industrial design.It ensures the model possesses a better probability of succeeding.In this case, the model will be technically suited to achieving the desired results.For this purpose, the influence of auxetic structure parameters on the system's performance has been investigated as the next step.Two geometrical parameters considered here are L and W , which are related to the stator teeth length and the stator teeth width, respectively.Figure 13 displays the effect of changing L on the average coil loss, iron loss, magnetic flux density, and torque properties.The graph indicates that the coil loss and average torque are exponentially increased by increasing L. However, the iron loss and flux density are increased until L = 1.5 mm, and then they are gradually decreased.The torque ripple fluctuates as L increases, reaching its maximum torque ripple when L = 2 mm.Besides, it can be seen that increasing the value of L can lead to a rise in average torque while torque ripple fluctuates irregularly.In addition, table 7 presents the system's performance as a function of stator teeth size.It is observed that increasing L leads to progressive decreases in Additionally, increasing the value of L can lead to higher average torque, but also results in irregular torque ripple rises.
motor efficiency.This is because the coil loss considerably increases in proportion to the L size, which would negatively affect the motor efficiency.Furthermore, the influence of W on the average coil loss, iron loss, magnetic flux density, and torque properties is investigated in figure 14.It can be observed that the coil loss is exponentially increased by increasing W . Also, iron loss is directly proportional to the shift in W whereas flux density is inversely proportional to the shift in W . Regarding torque characteristics, it is evident that as W increases, the average torque rises progressively until it reaches its maximum amount at W = 3.5 mm.However, during this process, the torque ripple will dynamically increase while reaching its minimum at W = 3.1 mm.Table 8 summarizes the system performance, based on altering the aforementioned properties as a function of W .It can be seen table 8 that increasing W would negatively affect motor efficiency.This phenomenon occurred as a result of a significant rise in total loss, which overcomes the benefits of increased average torque.
The variation of two main system characteristics, namely output power, and efficiency, are plotted in figures 15(I) and (II) as a function of the geometrical parameters L and W .It can be seen that increasing the size of these two parameters would gradually increase the output power until it reaches an ultimate value while the efficiency would exponentially decrease.This is because increasing the dimensions of the stator can result in larger coils, which can improve the motor's torque and output power.However, such an increase in size can also lead to an increase in torque ripple and total iron loss, ultimately reducing the motor's efficiency.It can be perceived that increasing the value of W would exponentially increase the coil loss.Additionally, the iron loss is directly proportional to growing W, while the flux density is inversely proportional to its growth.In terms of torque characteristics, it is evident that the average torque progressively rises until it reaches its maximum at W = 3.5 mm.However, during this process, the torque ripple dynamically increases while it has a minimum value at W = 3.1 mm.Furthermore, the influence of using different initial mechanical degrees on the average torque and torque ripple is investigated in figure 15(III).It is evident that employing the initial angle of 20  under operating conditions would yield the highest average torque and lowest torque ripple.Based on the outcomes of the sensitivity analysis, the structural parameters were adjusted to their optimal values: L = 1 mm, W = 3.1 mm, and an initial mechanical angle of 20 .

Conclusion
One of the main factors influencing customer satisfaction in the automotive industry is the interior sound design of the car.Consequently, the vibration performance of the car plays a key role in influencing and persuading the customers to purchase a specific electric motor vehicle, which performs 'silence as a luxury'.The next generation of e-motors must have higher power/torque and be lightweight in order to meet both customer expectations and current policies determined by the electric motor companies.However, both of these parameters would negatively impact the whistling noise and torque ripple issues in electric vehicles.
Based on the available literature, prior investigations have demonstrated that enhancing the vibration performance of electric motors could have an adverse effect on their power output and, in turn, their overall efficiency.Additionally, most researchers have focused on reducing either magnetic forces or torque vibration, Figure 15.The variation of output power and motor efficiency as a function of (I) L, and (II) W .It is noticeable that gradually increasing the size of L, and W would lead to an increase in output power until it reaches a maximum value, while the efficiency would exponentially decrease.In addition, (III) depicts the variation of average torque and torque ripple as a function of the initial angle.It is demonstrated that using an initial angle of 20  under operating conditions would result in the highest average torque and the lowest torque ripple.without exploring other comprehensive potential solutions.Likewise, the slight negative impact on the output torque is the limitation of the current method.However, the present work improves the vibration performance of the system, more specifically the whistling noise and torque ripple, simultaneously.In this study, we proposed a passive method to improve the PMSM performance in terms of noise and vibration by studying average flux density in stator surface and torque ripple while keeping motor efficiency unchanged by enforcing the iron and coil losses to be reduced.Accordingly, a PMSM model with an auxetic winding shape is designed based on the auxetic principle and the LRM.The physical insight behind modeling such a novel winding design is that auxetic structures can add some beneficial features to the system, including indentation resistance, high vibration isolation, and energy absorption.The suggested auxetic design absorbs a significant portion of incoming vibrations and isolates them within the auxetic winding area.This prevents the transmission of vibrations to the rotor and torque.Eventually, the energy stored in the auxetic region will be released as heat and attenuated by the LRM.As a result, torque ripple and magnetic forces in the stator would be significantly diminished.
The result shows that the auxetic stator winding model can positively decrease the average magnetic density to 0.81T at teeth regions.Besides, the torque ripple response of both proposed and conventional types is compared.The study determines that the auxetic winding model has profoundly lowered torque ripple by a 41.38% reduction compared to the original PMSM sample.However, average torque has been negatively affected, which was also reported by previous studies.
In addition, the present work indicates that the proposed model has also resulted in a reduction in both iron and coil loss.Unlike previous works, results demonstrate that the total loss reduction would neutralize the negative impact of average torque reduction on the motor efficiency, as a result, the motor efficiency remains unaffected while the vibration performance is improved in terms of torque ripple and magnetic forces.
Conventional methods often make systems heavier, more complex, costly to maintain, and expensive to manufacture.In contrast, the proposed method focuses solely on creating smart designs, without altering the weight or material used in the system.This approach maintains the system's simplicity and avoids the need for expensive manufacturing processes.As a result, the suggested method is an affordable, effective, reliable, and robust solution.
Sensitivity analyses are conducted on the designed PMSM model for the auxetic geometrical parameters and initial mechanical angle to control the physical and mechanical properties of the system.By following the design procedure proposed in this study, one can unlock a fundamental approach that can be applied practically.This study might provide insights into the future development of electric motors (e-motors) by utilizing such advanced designs while the NVH issue is alleviated.Auxetic materials and structures might find a place in a variety of promising fields, such as noise attenuation and vibration energy harvesting in e-motors and the automotive industry.

Future perspective
In future work, coupled 3D E-powertrain and 2D electromagnetic FEM analyses will be investigated, where the auxetic stator winding has been employed in the PMSM model to control the vibration behavior of the system.In addition, we will numerically and experimentally examine the complex system in terms of NVH and powerextracted performances.

Figure 1 .
Figure 1.Practical applications of auxetic structures in the automotive engineering field.

Figure 2 .
Figure 2. Illustration of the stator models: (a) the proposed stator model with auxeticity, and (b) the basic stator model created by COMSOL Multiphysics.The main geometrical parameters characterizing the stator windings shape modification are stator tooth length and stator's tooth width.Therefore, we consider two geometrical parameters L and W shown in figure2, which are related to the stator's tooth length and width, respectively.It is worth noting that to study both models under the same condition, the value of the stator slot area in both models is considered constant (A = 225 mm 2 ), in which the stator's weight is similar.

Figure 3 .
Figure 3. Process flowchart for the methodology plan.

Figure 4 .
Figure 4.The complete-assembled PMSM models.(a) the proposed FEM model.(b) the basic FEM model.The red circle region follows the continuity boundary condition (CBC), while the blue marked areas use Ampère's circuital law.Furthermore, the black circle denotes the outer circle air region, where the magnetic insulation boundary condition is applied.

Figure 5 .
Figure 5.The schematic view of different PMSM components.According to the boundary conditions and initial numerical setup, the inward/outward magnetized regions, stator and rotor iron regions, and the three phases (A, B, and C) order arrangement are determined.

Figure 6 .
Figure 6.The meshed PMSM models.(a) the proposed model.(b) the basic model.The mesh convergence technique (MRT) is used to find adequate mesh sizes and the models meshed using free triangular type considering the moving mesh feature for the rotary domains (rotor solid and air regions).

Figure 7 .
Figure 7. Magnetic flux density saturation plots.(a) corresponding to the proposed model.(b) corresponding to the basic model.It is shown that the auxetic structures can effectively reduce the magnetic flux density levels across the channels between the stator teeth.Nevertheless, the magnetic flux density saturation in the rotor region has remained unchanged as the same rotor and air gap structures for both models are considered.

Figure 8 .
Figure 8. Magnetic flux density as a function of time in the stator region.The blue diagram corresponds to the proposed PMSM model, while the red diagram corresponds to the basic PMSM model.It can be observed that the designed PMSM model exhibits lower average magnetic flux density with fewer fluctuations than the basic model.

Figure 9 .
Figure 9. Magnetic flux density distribution as a function of mechanical degree in the stator region.The blue diagram corresponds to the proposed PMSM model, and the red diagram corresponds to the basic PMSM model.It is demonstrated that the magnetic flux density levels in the proposed model are under those of the basic model at all mechanical angles.Therefore, it can be deduced that the proposed auxetic design can effectively reduce the magnetic flux density forces, which will considerably improve the NVH performance, especially whistling noise.

Figure 10 .
Figure 10.Demonstration of Rotor torque responses as a function of motor angle for full rotation and electrical period: (I) corresponds to the proposed PMSM case study.(II) corresponds to the basic PMSM case study.It can be sobered that the torque saturation levels are remarkably lower in the proposed model compared to the original model under both full rotation and electrical period conditions.

Figure 11 .
Figure 11.Iron loss density map on; (a) the proposed model.(b) the basic model.The darker red color represents a lower iron loss level.It can be observed that the proposed auxetic winding design would significantly reduce the iron loss level in the stator surface, especially in the throat spaces between the stator windings (stator teeth).

Figure 12 .
Figure 12.Comparison of the proposed and original PMSM models in critical terms of torque ripple, magnetic flux density, and motor efficiency.It is evident that the proposed model effectively reduces torque ripple and magnet flux density while maintaining motor efficiency.

Figure 13 .
Figure 13.Influence of L on the (a) coil loss, (b) iron loss, (c) flux density, and (d) torque ripple.The graph shows that as the value of L increases, the coil loss and average torque will grow exponentially.However, the iron loss and flux density increase until L reaches 1.5 mm, and then, start decreasing gradually.The torque ripple fluctuates as L increases, reaching its maximum value at L = 2 mm.Additionally, increasing the value of L can lead to higher average torque, but also results in irregular torque ripple rises.

Figure 14 .
Figure 14.Influence W on the (a) coil loss, (b) iron loss, (c) flux density, and (d) torque ripple.It can be perceived that increasing the value of W would exponentially increase the coil loss.Additionally, the iron loss is directly proportional to growing W, while the flux density is inversely proportional to its growth.In terms of torque characteristics, it is evident that the average torque progressively rises until it reaches its maximum at W = 3.5 mm.However, during this process, the torque ripple dynamically increases while it has a minimum value at W = 3.1 mm.

Table 1 .
The dimensions and parameters of PMSM models.

Table 2 .
The materials used for PMSM components.

Table 5 .
The output of the proposed PMSM model.

Table 3 .
Comparison of the basic and proposed model torque waveforms.

Table 4 .
Comparison of the results of the presented work with available ones.

Table 6 .
Comparison of the output of the basic and proposed PMSM models.

Table 7 .
System's performance as a function of L.

Table 8 .
System's performance as a function of W .