Cogging Torque Reduction Techniques in Axial Flux Permanent Magnet Machines: A Review

Abstract

Axial flux permanent magnet machines have garnered significant attention in recent years due to their numerous advantages in various applications, including electric vehicles, wind turbines, and robotics. However, one of the critical challenges associated with these machines is the presence of cogging torque, which can hinder their efficiency and performance. This review article provides a comprehensive overview of the state-of-the-art techniques employed for cogging torque reduction in Axial Flux Permanent Magnet Machines. Different techniques are described, encompassing geometric optimization, magnet placement, and skewing methods. Firstly, the significance of Axial Flux Permanent Magnet Machines is described, as well as the issue of the cogging torque. In the methods section, a review of the strategies for the reduction of cogging torque is described from various articles, and finally, in the discussion section, a list of actions is presented for cogging torque reduction for different topologies. The novelty of the study is that it combines strategies for cogging torque reduction in a single article.

1. Introduction

Axial flux permanent magnet machines (AFPMM) are growing more and more popular due to their simple construction, high power density, efficiency, and compactness [1,2]. Depending on the application, AFPMM has various topologies, such as:

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single-sided (one stator and one rotor), shown in Figure 1,

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double-sided with single stator and double rotor, shown in Figure 2. or single rotor with double stator, shown in Figure 3,

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and multistage (multiple rotors and stators), shown in Figure 4,

Each topology has its own strengths and weaknesses. Topologies without stator cores are used for low and medium-power generators and have advantages such as the absence of cogging torque, linear torque-current characteristics, high power density, and compact construction.

The topologies with stator cores offer higher values of torque and Electromotive force (EMF) but also require dealing with cogging torque.

Cogging torque in electric motors results from the interaction between the rotor’s permanent magnets and the stator slots in a permanent magnet machine. It is also referred to as no-current torque, which varies with the rotor’s position, showing periodicity per revolution determined by the magnetic poles and stator teeth count. Cogging torque is an undesirable element in motor operation, particularly noticeable at lower speeds, as it affects the overall performance of the machine [3,4,5]. It induces fluctuations in both torque and speed. However, at higher speeds, the motor’s moment of inertia acts as a filter, mitigating the effects of cogging torque [6,7].

No load (cogging) torque and static torque were analyzed using a 3D Ansys Maxwell in order to graphically show the shape and size of the cogging torque. Calculations using the finite element method (FEM) are performed for the 3 phase double-sided AFPMM with two external rotors with surface-mounted PMs and one internal stator. The geometry and parameters of the analyzed AFPMM are shown in

Table 1. Geometry and parameters of the AFPMM [2].

Symbol	Quantity	Value/Unit
dFe	Rotor disk thickness	7 mm
hPM	Permanent magnet thickness	5 mm
τm	Magnetic pitch	25°
Di	Inner diameter of PM	80 mm
Do	Outer diameter of PM	150 mm
τp	Pole pitch	36°
I	Electrical current	2 × 10 A
	Number of windings	6
ds	Winding thickness	15 mm
dc	Coil width	20 mm
Sw	Copper wire cross section	1.23 mm2
dag	Air gap thickness	1 mm
kw	Winding factor	0.966
p	Number of pole pairs	5

.

Firstly, simulations were performed for a coreless version, meaning that there is no ferromagnetic material in the stator (the windings are held together by epoxy raisin). For the purpose of comparison as well as illustration of torque waveform, iron cores were added to the model, and simulations were performed with the same data. The results are summarized in the figures below.

Figure 5 shows the model of the analyzed AFPMM, where iron cores are emphasized.

As mentioned above, cogging torque in electric motors results from the interaction between the rotor’s permanent magnets and the stator slots in a permanent magnet machine, and the topologies with stator cores offer higher values of torque and EMF but also require dealing with cogging torque. A comparison of a coreless and iron-cored AFPMM with the same input data (rated current 20 A, same type and dimensions of permanent magnets (PMs), and number of turns per coil as well as number of coils) is shown in the figures below.

Figure 6 shows the comparison of the axial component of magnetic flux density in the air gap of the coreless and iron-cored AFPMM.

The static torque of the coreless and cored AFPMM, as well as the cogging torque of the iron-cored AFPMM, are shown in Figure 7.

The interaction between permanent magnets on rotor disks and iron cores causes larger pull forces between the rotor and stator. A comparison of the pull forces for iron-cored and coreless AFPMM is shown in Figure 8.

Figure 6, Figure 7 and Figure 8 show the difference between iron-cored and coreless AFPMM. We can see that the shape of the torque is different compared to the coreless AFPMM. One of the actions to obtain a smoother torque is the reduction of the cogging torque.

The techniques for cogging torque reduction in AFPMM can be grouped into the following categories:

• Geometric Optimization: This category encompasses design modifications focused on the shape, size, and arrangement of components within the AFPM, such as rotor and stator geometry adjustments.
• Magnet Placement Strategies: This group directly addresses the spatial alignment and distribution of magnets, which is a crucial factor in determining cogging torque.
• Skewing Techniques: Skewing involves angling the rotor or stator laminations to disrupt the alignment of the magnets, thereby reducing cogging torque.
• Material Selection and Magnet Configuration: This group involves choosing specific materials for the magnets and considering the arrangement of magnets (e.g., magnet pole pair configuration) to influence cogging torque behavior.
• Winding Schemes: Different winding arrangements and configurations in the stator can impact the cogging torque characteristics, and optimizing winding schemes is an effective approach.

This article presents a review of the geometrical approaches used by authors for cogging torque reduction. Advanced control algorithms are not included in the article due to the extensive review in [8], where the paper discusses the application of Model-Based Design (MBD) in designing control systems and algorithms for reducing cogging torque and provides the current state-of-the-art addressing cogging torque through control algorithms to create flexible and scalable solutions for various mechatronic applications.

2. Materials and Methods

As mentioned in the introduction, the techniques for cogging torque reduction in AFPMM can be grouped into 5 categories, but for the purpose of the review, this section is divided into three subsections according to the cogging torque reduction technique, namely:

• Rotor adjustment—including PM shape, skewing, and material selection;
• Stator adjustment—including skewing and material selection;
• Combination of rotor and stator adjustment.

2.1. Rotor Adjustment

Authors in [9] investigated cogging torque reduction in Axial-Field Flux-Switching Permanent Magnet Machines (AFFSPMM) through rotor tooth notching. The study proposes three rotor tooth notching schemes and analyzes the impact of dummy slot parameters on cogging torque using 3-D finite-element method (FEM). The optimal notching structure is determined through the response surface method (RSM), resulting in a significant reduction in cogging torque (around 22–43%) with minimal impact on output torque. The findings highlight the effectiveness of rotor tooth notching for cogging torque reduction in AFFSPMM.

In [10], the authors address the cogging torque issue in axial-flux magnetic gears (AFMGs) with an integer gear ratio used for wind-power generation, which leads to larger cogging torque in the high-speed (H-speed) rotor, causing vibration, noise, and startup errors.

The authors state that conventional skew structures are effective but complex and challenging to assemble in large-scale AFMGs for wind-power generation, so they propose two solutions, namely, Unequal-Space Type Pole Pieces and Unequal-Width Type Pole Pieces.

Unequal-space type Pole Pieces adjust the relative position of pole pieces to cancel cogging torque by introducing phase differences of 120 electrical degrees. Involves grouping pole pieces and adjusting their positions to achieve cancellation. Analysis and experiments show a significant reduction in H-speed side cogging torque.

Unequal-Width Type Pole Pieces change the circumferential width ratio of pole pieces to cancel cogging torque by selecting a proper combination of pole pieces with different circumferential width ratios. Involves arranging pole pieces with varying widths in a periodic manner. Analysis and experiments demonstrate better performance in terms of reducing H-speed side cogging torque and torque ripple and maintaining maximum torque compared to unequal-space type.

Both proposed types are analyzed using a 3-D finite-element method (3D-FEM) and experimentally validated. Unequal-width type outperforms unequal-space type in terms of reducing H-speed side cogging torque, torque ripple, and maintaining maximum torque.

Comparison with the conventional type shows that the unequal-width type is more effective in reducing cogging torque while maintaining a reasonable maximum torque.

Prototype tests for the unequal-width type show a 50% reduction in cogging torque with a 9% reduction in maximum torque. The efficiency of the unequal-width type reaches 96.7%, and it maintains an efficiency of 91% or more across the entire operating range.

The authors conclude that the unequal-width type of pole pieces effectively addresses the cogging torque issue in AFMGs with integer gear ratios, especially in the context of wind-power generation. The proposed solution not only reduces cogging torque significantly but also maintains a high level of efficiency, making it suitable for large-scale magnetic gears in wind-power applications.

Using a trapezoidal shape permanent magnets is a well-known way to reduce cogging torque since it introduces a skew between the border of the PMs and the stator slots [11]. Authors in [12] analyze a double-sided axial flux magnetic gear design aimed at improving torque density. The study focuses on minimizing cogging torque through the application of skewing methods for both permanent magnets (PMs) and ferromagnetic pole pieces. Various skewing techniques, including conventional and trapezoidal methods, are analyzed through 3D Finite Element Method simulations.

The results indicate that skewing PMs using the conventional method (CS-PM) leads to a slight reduction in average transmitted torque, but it significantly improves torque quality, with the best results obtained at a maximum skew angle. Trapezoidal skewing of PMs (TS-PM) shows relatively constant average transmitted torque, with a decrease in cogging torque up to a certain skew angle, after which it starts to increase due to flux concentration.

For ferromagnetic pole pieces, skewing using the conventional method (CS-MD) results in a significant reduction in average transmitted torque, but it effectively minimizes cogging torque. Trapezoidal skewing of modulators (TS-MD) slightly reduces average transmitted torque, and the minimum cogging torque is achieved at a specific skew angle.

The paper concludes that the best torque quality for the proposed magnetic gear design is achieved by skewing high-speed rotor PMs using the conventional method. The study provides valuable insights into the trade-offs between different skewing techniques and their impact on torque characteristics in axial flux magnetic gears.

The paper discusses two main methods for reducing cogging in the context of axial flux magnetic gears:

Skewing of Permanent Magnets (PMs):

• Conventional Skew of PMs (CS-PM): This method involves displacing the inner radii of PMs in the same direction while keeping the outer radii fixed. The skew angle is increased to minimize cogging torque by changing the phase of higher-order torque harmonics. The study indicates that this method offers a better torque quality.
• Trapezoidal Skew of PMs (TS-PM): Here, the inner and outer points of PMs are displaced in opposite directions while keeping the mean radius fixed. The results show relatively constant average transmitted torque, with a decrease in cogging torque up to a certain skew angle. However, a further increase in skew angle leads to an increase in cogging torque due to flux concentration.

Skewing of Ferromagnetic Pole Pieces (Modulators):

• Conventional Skew of Modulators (CS-MD): This method involves rotating the sidelong faces of modulator pole pieces around the outer radius corners. While it significantly reduces cogging torque, it also results in a substantial reduction in average transmitted torque.
• Trapezoidal Skew of Modulators (TS-MD): In this method, two lateral sides of modulator pole pieces are displaced in opposite directions, forming a trapezoidal cross-section. The skew angle is increased to minimize cogging torque by reducing changes in the reluctance of ferromagnetic pole pieces.

The paper provides a detailed analysis of these skewing methods through Finite Element Method (FEM) simulations, evaluating their impact on average transmitted torque and cogging torque. The results help in understanding the trade-offs between torque quality and quantity when employing different skewing techniques in axial flux magnetic gears.

A technique for minimizing cogging torque in multiple-rotor axial flux permanent magnet (AFPM) motors is presented in [13]. The proposed method involves using alternating magnet pole-arcs in facing rotors to reduce cogging torque without compromising peak torque and pulsating torque. The study focuses on an 8-pole, 24-slot double-rotor AFPM motor, and 3D Finite Element Analysis (FEA) is utilized for detailed analysis. The results demonstrate a significant reduction in cogging torque without influencing other torque components. The technique is considered cost-effective and is compared favorably to existing methods, showcasing its potential for improving the performance of axial flux PM motors. The article concludes by suggesting that combining this technique with other optimization methods could further enhance cogging torque reduction in multiple-rotor AFPM machines.

The authors in [14] employed the following techniques for cogging reduction:

• Designing Permanent Magnet Shapes: The authors designed the side shapes of the permanent magnets using a multiplicative waveform. The amplitude and order of the multiplicative wave were adjusted as parameters. This design aimed to achieve a relative skew effect and reduce cogging torque.
• Multiplicative Waveform: The side shapes of the permanent magnet were designed as a 2nd order multiplicative waveform. This was conducted to shift the timing of magnetic flux generation to the stator iron core when the rotor rotates, effectively achieving a relative two-dimensional skew effect.
• Optimization of Design Parameters: The authors changed the amplitude and order of the multiplicative wave, evaluating each characteristic comprehensively. They determined the optimal amplitude and order by considering the impact on average torque, cogging torque, and torque ripple.
• Skewing: Skewing was applied to the designed permanent magnets with respect to the z-axis to reduce cogging torque further. The skew angle was optimized by considering multiple characteristic values and productivity.
• Three-Dimensional Skewing: Unlike traditional two-dimensional skewing in the r-θ direction, the authors applied three-dimensional skewing in the z-axis direction. This involved optimizing the side shapes of permanent magnets before applying skew.

The combined approach of designing permanent magnet shapes with a multiplicative waveform and applying skew resulted in a significant reduction in cogging torque and torque ripple while maintaining a relatively small decrease in average torque. The optimal design was determined through a comprehensive evaluation using a three-dimensional finite element method (3DFEM).

The study in [15] focuses on improving the performance of Axial Field Flux Switching Permanent Magnet Machines (AFPSMM) by addressing the issue of cogging torque. The AFPSMM is designed with arc-shaped triangular magnets and mathematical models, Finite Element Analysis (FEA), and optimization techniques are employed for evaluation.

The proposed AFPSMM design showcases a remarkable 61.8% reduction in cogging torque compared to the conventional model. This reduction is attributed to the unique configuration of arc-shaped triangular magnets, which alters the distribution of magnetic flux and effectively decreases cogging torque. Furthermore, the study undertakes an optimization process, aiming to enhance the machine’s performance while maintaining equal magnet volume.

In [16], the authors discuss a technique for reducing cogging torque in axial-flux permanent-magnet generators used in small wind turbines. The cogging torque during turbine start-up is addressed through a method called “hybrid skew.” The proposed technique involves using magnets with different shapes for the North and South poles, offering advantages such as simpler magnet shapes for cost-effective manufacturing, clear N/S pole orientation, and reduced manufacturing complexity. The study employs 3-D finite element analysis, resulting in an 88% reduction in peak cogging torque when combined with another known technique. This reduction enhances turbine performance during low start-up speeds while maintaining high power density. The paper includes details on turbine characterization, generator design specifications, cogging torque analysis, and simulation results using JMAG, a 3-D finite element package. The proposed technique proves effective in reducing cogging torque, facilitating wind turbine start-up, reducing manufacturing costs, and maintaining overall machine performance.

In [17], the authors provide a comprehensive analysis of various cogging torque reduction methods, emphasizing the effectiveness of magnet grouping through both theoretical and experimental results. The comparison indicates that shifting magnets in groups of 4 is particularly successful in minimizing or eliminating cogging torque in AFPM motors. Reference AFPM Motor with un-skewed fan-shaped magnets has 8 Nm peak cogging torque, and that value was used as a benchmark for comparison.

The introduction of conventional skew of permanent magnets results in an up to 80% reduction of cogging torque. Introducing a 22.5-degree triangular skew reduces peak cogging torque by 84.2% compared to the un-skewed reference motor. Applying conventional skew results in a reduction in cogging torque (up to 80% reduction for a 20-degree skew angle).

Shifting or grouping magnets are explored as a method to further reduce cogging torque. Shifted-in-2 achieves a 56% reduction in cogging torque, and shifted-in-4 achieves an 87% reduction in cogging torque.

Theoretical findings were experimentally validated using a built prototype AFPM motor, where experimental results confirm the accuracy of the theoretical predictions, showing the effectiveness of magnet grouping in eliminating cogging torque.

The authors in [18] applied techniques, such as magnet skewing, varying magnet arc, and magnet shifting, for cogging torque reduction. The reduction achieved is compared, and a combination of short-pitched and skewed magnets is identified as the most effective method, providing the maximum reduction in cogging torque.

The authors also analyzed the impact of the modifications in magnet shape and showed that it causes variations in the magnitude and waveform shape of air gap flux density distribution. Skewing and short-pitching magnets alter the waveform, leading to a reduction in cogging torque.

The study observes a reduction in the peak value of induced EMF for modified magnet shapes, with the most significant reduction observed for short-pitched and skewed magnets. However, it notes that a compromise must be made between torque ripples and torque/power output.

The authors also analyze the harmonic content in the induced EMF waveform using Fast Fourier Transform (FFT). They observe that short-pitched and skewed magnets have minimum harmonic content, while relatively displaced magnets introduce 2nd order harmonics.

Authors conclude that a combination of short-pitched and skewed magnets is the most appropriate modification, considering the trade-off between cogging torque reduction and induced EMF performance.

The study in [19] focuses on a dual-skew magnet technique to reduce cogging torque, which is crucial in low-speed traction applications. The paper investigates the minimization of cogging torque in an axial-flux permanent-magnet (AFPM) motor with a yokeless and segmented armature (YASA). Theoretical expressions for cogging torque are deduced, and the advantages of the dual-skew magnet are validated through 3-D finite-element method (FEM) simulations. The results demonstrate that the dual-skew magnet outperforms sector shape and conventional skew magnets in reducing cogging torque, torque ripple, and magnet eddy current loss.

In [20], the authors discuss the cogging torque issue in a Dual-Rotor Axial Field Flux-Switching Permanent Magnet (DRAFFSPM) machine. The cogging torque is analyzed, and various reduction methods, such as rotor skewing, notching, and pole displacement, are investigated. Results indicate that increasing the rotor pole width and adopting a fan-shaped rotor pole significantly reduces cogging torque. The optimal reduction is achieved with a rotor pole width of 15.5 degrees and a fan-shaped angle of 3 degrees, resulting in a 77% reduction. Rotor skewing is found ineffective, but rotor notching and pole displacement reduce cogging torque by 59% and 13%, respectively.

Authors in [21] present a study on optimizing the permanent magnet shape and dimension to enhance the performance of axial flux permanent magnet (AFPM) machines. The primary focus is on reducing cogging torque, a significant noise source in AFPM machines, through the analysis and comparison of various magnet shapes. The studied shapes include a flat trapezoidal pole, curved pole, skewed pole, and a proposed shape combining curved and skewed magnet shapes.

Different rotor designs (flat trapezoidal, skewed, curved, curved skew) are simulated to compare cogging torque results while keeping the stator unchanged. Experimental results show that the curved skew magnet shape exhibits the least cogging torque (1.5 Nm), while the flat trapezoidal pole design has the highest cogging torque (3.88 Nm).

The authors conclude that the magnet shape significantly influences cogging torque and overall efficiency. The curved skew magnet shape is identified as the most effective in reducing cogging torque, though it affects back-EMF.

2.2. Stator Adjustment

The study in [22] is focused on reducing cogging torque through careful geometric optimization. Utilizing 3D-FEA analysis, the research demonstrates substantial advancements in cogging reduction. By strategically adjusting the stator’s configuration, cogging torque is effectively minimized. This innovative approach leads to over 50% reduction in total cogging torque.

Introducing a displacement between both stator sides alters the spatial distribution of the three-phase winding sets, consequently affecting the average torque output.

When there is no displacement between stator sides, the stator winding mmf and rotor magnet act simultaneously, resulting in each side generating 50% of the total torque.

In contrast, when one stator side is displaced to reduce cogging torque, both torque components operate at different moments and with different magnitudes, which leads to a reduction in resultant torque.

The study explores two methods for reducing cogging torque in an axial flux machine that possesses field weakening capability. Employing the stator side displacement technique generates a doubled frequency in the cogging waveform, resulting in a slight decrease in the average torque produced by the machine. Additionally, the study investigates rotor pole skew, which significantly impacts cogging variation, especially in cases of symmetrical and asymmetrical rotor pole arrangements. Notably, there is minimal difference observed when comparing both arrangements. Asymmetrical geometry offers manufacturing advantages, requiring only identical rotor pieces. These results have been applied to construct a 3 kW 8-pole axial flux machine prototype designed for variable speed applications.

In [23], the authors deal with the computation of cogging torque in a 1 kW double-sided axial flux permanent magnet (AFPM) generator featuring various stator core pole configurations. The generator comprises 18 stator poles and 24 permanent magnets on each side. Cogging torques for three different stator pole arrangements were analyzed using a 3D finite element method, and the optimal core shape that minimizes cogging torque was identified.

The authors explored how different stator pole arrangements impact the cogging torque in a double-sided axial flux permanent magnet (AFPM) generator. Using both the response surface method and the 3D finite element method, the authors determined the optimal design by varying three design variables: stator core position, magnet pole pitch, and their combination. The cogging torque was successfully reduced by 85% compared to the basic model.

A key aspect of the research is the optimization process. The response surface method, coupled with the finite element method, is utilized to identify the optimum design for the generator. Three design variables (stator core position, magnet pole pitch, and their combination) are varied to achieve the minimum cogging torque.

A study on reducing cogging torque and torque ripple in a double-layer spoke-type Permanent Magnet Synchronous Motor (PMSM) by modifying the stator shape is presented in [24]. The conventional method of applying skew to the rotor is not deemed productive, so the paper proposes a new stator shape that achieves a skew-like effect by cross-stacking stators with an asymmetrical design. The goal is to enhance reluctance torque while minimizing vibration and noise issues associated with cogging torque and torque ripple.

The study involves a washing machine motor as a target model, and the proposed stator shape is analyzed through Finite Element Analysis (FEA). The 2D FEA results indicate that the proposed cross-stacking method reduces cogging torque, torque ripple, and total harmonic distortion (THD) of Load Line EMF. The analysis is conducted under various conditions, including no-load, low-speed load, and high-speed load.

The results show that the proposed model with cross-stacking reduces cogging torque by 49%, torque ripple ratio by 2.7% at low speed and 33.27% at high speed, and load THD by 1% at low speed and 4.02% at high speed compared to the conventional model. The paper suggests the need for further verification through 3D FEA and actual model production.

The study concludes that the proposed cross-stacking approach effectively reduces cogging torque and torque ripple in the double-layer spoke type PMSM, contributing to potential improvements in motor performance and noise reduction.

The study in [25] discusses cost-effective stator modification techniques to reduce cogging torque in axial flux permanent magnet (AFPM) machines. While various rotor modification techniques are directly applicable to AFPM machines, stator modification techniques are challenging and expensive due to difficulties in punching slots. The paper proposes design techniques focusing on slot opening variations achieved using Soft Magnetic Composite (SMC) wedges.

The authors present cost-effective stator design modifications using SMC wedges for slot opening variations. 3-D Finite Element analysis is used for simulation. Different stator modification techniques are proposed, including:

• Slot-opening shape variation (trapezoidal and parallel).
• Slot-opening width variation.
• Slot-opening relative displacement.
• Skewed slot-opening.

Simulation results show a significant reduction in cogging torque magnitude with proposed modifications. Cost-effective stator modifications using magnetic wedges effectively reduce cogging torque, and the opposite skewing of slot openings yields the maximum cogging reduction of 98.71%.

In [26], a method proposed for cogging torque reduction involves adding a magnetic bridge (MB) between adjacent stator teeth. The MB increases stator iron area, leading to consistent permeability during rotor movement and a substantial reduction in cogging torque. Comparative analysis with traditional methods like rotor skewing and notching shows the superiority of the MB method, which is effective for different stator structures. The optimal thickness of the MB, determined through 3-D finite-element analysis, balances cogging torque reduction with minimal impact on output torque. Further optimization of stator tooth and rotor pole width enhances output torque while significantly reducing cogging torque. The study concludes that the MB method is not only effective but also easy to implement in manufacturing, showcasing its potential applicability to various permanent magnet machines.

Three cogging torque reduction techniques are proposed and compared in the DRHE-ASFPM machine in [27]: stator/rotor teeth notching, stator slot chamfering, and a combination of stator slot chamfering and teeth notching (SCTN).

The analytical expression of the cogging torque is derived, and the three methods are analyzed and compared.

Cogging torque reduction in a novel double-rotor hybrid excited axial switched-flux permanent magnet (DRHE-ASFPM) machine is discussed. The authors propose and compare three cogging torque reduction techniques for the DRHE-ASFPM machine:

• Stator/Rotor Teeth Notching (STN/RTN): This technique involves modifying the teeth of either the stator or the rotor. The impact of different notching configurations, such as stator teeth notching (STN), rotor teeth notching (RTN), and various combinations, is investigated. The results show that the notching depth and width have a significant impact on the cogging torque.
• Stator Slot Chamfering (SC): This technique focuses on chamfering the stator slots to reduce the cogging torque. By altering the stator slot upper arc, the effective contact area between the stator and rotor teeth is increased, leading to changes in the magnetic field distribution. The optimal chamfering dimensions are studied to minimize the cogging torque.
• Stator Slot Chamfering and Teeth Notching (SCTN): This technique combines the advantages of both stator/rotor teeth notching and stator slot chamfering. The study aims to find the optimal combination of these methods to achieve effective cogging torque reduction.

The authors use analytical expressions and 3-D finite element (FE) simulations to analyze and compare the cogging torque reduction methods. The results suggest that the SCTN technique can significantly reduce the cogging torque while slightly decreasing the output torque under both permanent magnet (PM) and hybrid excitation modes.

Finally, the paper validates the findings through the manufacturing of an optimized prototype machine with the SCTN structure. Experimental results confirm the effectiveness of the proposed cogging torque reduction technique.

Authors in [28] focus on reducing cogging torque in a double-stator axial-flux permanent magnet servo motor with soft magnetic composite (SMC) cores. They analyze the effects of key parameters such as pole arc ratio, slot width, and stator shoe rotating angle on cogging torque. Through optimization, particularly by adjusting the pole arc ratio and slot width, the cogging torque of the motor is significantly reduced by about 96% while maintaining other performance requirements. The use of SMC materials and unique design considerations contribute to this reduction in cogging torque, enhancing the overall performance of the motor.

The study concludes that adjusting the pole arc ratio has a notable influence on cogging torque. By optimizing this parameter, the cogging torque is reduced by about 96%, with the minimum value achieved at a specific pole arc ratio. The study also analyzes the effect of slot width on cogging torque. Changing the slot width affects not only the cogging torque but also the no-load back electromotive force (EMF). The authors find an optimal slot width that balances these factors and contributes to reducing cogging torque while maintaining a suitable no-load back EMF.

The design introduces rotating stator shoes to mitigate cogging torque further. This method is found to be effective, and by selecting an appropriate rotating angle, cogging torque is reduced by about 96% with minimal impact on the no-load back EMF.

The paper [29] discusses the design and analysis of a novel six-phase axial switched-flux permanent magnet (PM) machine with different winding configurations. The two winding configurations studied are the symmetric double-three-phase winding connection (SWC) and asymmetric double-three-phase winding connection (AWC).

The cogging torque values with SWC and AWC are exactly the same, and they are relatively small compared to three-phase axial flux permanent magnet machines. Values are the same for both configurations, indicating that the choice of winding configuration does not affect cogging torque.

While both winding configurations offer good overall performance, the AWC configuration appears to be advantageous in terms of fault tolerance, with a smaller torque ripple under faulty conditions.

2.3. Combination of Rotor and Stator Adjustment

Authors in [30] conducted a comprehensive analysis and evaluation of cogging torque in Axial Flux Permanent Magnet Machines, where they used 5 methods. The first method is the change of slot and pole combination method, which optimizes slot and pole combinations in the stator and rotor. The method was shown to be effective but with limited scalability. The second method was the skewing method, which tilts stator or rotor teeth to introduce angular displacement. The method reduces cogging but has a limited range of skewing angles. The third method was the notching method, which removes part of stator teeth to break symmetry. It has shown effective, especially for fractional slot concentrated winding machines. The fourth method was the pole shifting method, where rotor poles are shifted to mitigate cogging effects. It has shown to be practical but has limitations in achievable reduction. As a fifth method, the authors use varying the pole-arc coefficient for adjacent permanent magnets and state that it may reduce cogging torque on the one hand but, on the other hand, introduce asymmetries in the rotor structure and consequently may increase torque ripple.

The authors also used a combination of methods for synergistic reduction. It offers improved reduction but may increase complexity and cost.

The authors conclude that each method has its advantages and drawbacks, and the choice depends on specific machine requirements. Combining methods can lead to enhanced cogging reduction, but it’s essential to consider trade-offs in terms of manufacturing complexity and cost.

The paper [31] discusses several methods for reducing cogging torque in axial-modular flux-switching permanent magnet (AM-FSPM) machines. Authors compare different methods for reducing cogging torque, namely Rotor Tooth Combination (RTC) Method (changing the rotor structure, specifically the combination of rotor teeth), Rotor Angle Combination (RAC) Method, Stator Tooth Combination (STC) Method (altering the widths of stator teeth) and Permanent Magnet Tooth Combination (PTC) Method (adjusting the widths of permanent magnet teeth).

The paper compares the effectiveness of these methods and highlights the RTC method as the most attractive for cogging torque reduction. They validate their conclusion with analytical expressions and 3D finite element analysis (FEA) as well as experimental validation.

The paper [32] discusses cogging torque reduction in a novel Hybrid Axial Field Flux-Switching Permanent Magnet Machine (HAFFSPMM), which is a hybrid excited machine designed for electric vehicle drive with a short axial length and high torque density.

The cogging torque is addressed using three methods: tangential displacement of stators, asymmetric rotor pole shape, and segment-twisted rotor. The impact of these methods on cogging torque and output torque is analytically investigated and numerically verified using a three-dimensional finite-element method.

The tangential displacement of stators involves a slight angular shift in the stator poles to reduce the cogging torque, but it comes with the drawback of axial tensile forces between the stators. The asymmetric rotor pole shape increases the aligned area of the rotor pole and stator teeth, effectively reducing cogging torque, while the segment-twisted rotor structure is equivalent to the tangential stator structure in terms of cogging torque reduction.

The study optimizes the design dimensions of the HAFFSPMM and concludes that all three methods effectively reduce cogging torque, with the asymmetric rotor pole shape exhibiting the best performance in terms of output torque. The research suggests that a combination of these cogging torque reduction methods with further structural optimization may enhance the overall performance of HAFFSPMM.

Authors in [33] analyze cogging torque reduction techniques in an axial-field flux-switching permanent-magnet machine (AFFSPMM) used for wind power generation and electric vehicles by using a 3D finite element analysis. The study also derives the theoretical expression for cogging torque in AFFSPMM, from which it can be seen that the cogging torque can be influenced by the design parameters and the flux density distribution of the air gap.

A combination of optimizing design parameters and employing rotor design techniques such as rotor tooth skewing, rotor tooth notching, and rotor tooth circumferential pairing proves effective in reducing cogging torque.

The authors analyze the techniques:

• Optimization of Design Parameters: Parameters such as tooth arc of the stator, PM thickness of the stator, yoke width of the stator, axial length of the stator, tooth arc of the rotor, and rotor tooth shape are optimized. Decreasing the yoke width and axial length of the stator and increasing the tooth arc and tooth fan-shaped angle of the rotor contribute to reducing cogging torque. Overall, cogging torque in AFFSPMM can be significantly reduced by optimizing these design parameters.
• Rotor Tooth Skewing (RTS): Rotor tooth skewing is effective for AFFSPMM with a parallel stator tooth and fan-shaped permanent-magnet structure. However, RTS is not effective for AFFSPMM with a fan-shaped stator tooth and parallel permanent magnet (FSST-PPM) structure.
• Rotor Tooth Notching (RTN): Introducing dummy slots (notches) in the rotor tooth structure reduces cogging torque for the FSST-PPM structure. The width, depth, and shape angle of the dummy slot influence the reduction effect, with a significant decrease in cogging torque achieved.
• Rotor Tooth Circumferential Pairing (RTCP): Pairing different-sized rotor teeth, particularly designing a big rotor tooth and a small rotor tooth with specific angles, reduces cogging torque for FSST-PPM structure.

Cogging torque in [34] is addressed through two techniques: stator slot displacement and rotor magnet skewing. The paper employs three-dimensional finite-element analysis to demonstrate the effectiveness of these methods, aiming to achieve a significant reduction in cogging effects. The authors present design recommendations and conclusions based on their findings and experimentation with a prototype 5-kW eight-pole AFPM machine. As mentioned, two techniques for cogging reduction in an axial flux permanent-magnet machine were used:

• Stator Slot Displacement: This involves adjusting the stator geometry by displacing one side of the stator slots relative to the other. By introducing a displacement factor (Kd), the cogging torque is significantly reduced. The technique results in a mechanical misalignment between stator sides, effectively reducing cogging effects.
• Magnet Skewing: Focuses on skewing the rotor magnets to diminish reluctance variation as the rotors move. Examines both symmetrical and asymmetrical rotor pole arrangements. Shows that increasing the skew angle (θi) between the rotor magnet and stator teeth leads to a reduction in cogging torque.

The combination of these techniques demonstrates a substantial reduction in cogging torque, as validated through three-dimensional finite-element analysis and experimental measurements on a prototype machine.

Authors in [35] present a reluctance machine that utilizes a specialized axial flux topology with 24 slots in the stator and circumferentially magnetized permanent magnets on a double-sided rotor, shifted by 180 degrees. Each rotor has 11 salient teeth to control reluctance in the air gap. Coils are tightly wound around the stator teeth for efficient operation. The authors state that due to the cogging torque reduction, it is exceptionally suitable for in-wheel electric vehicle applications. The reduction is primarily achieved through the innovative dual complementary structure, enabling seamless flux switching between rotors. They report a cogging torque of just 5.7% of the rated torque, displaying a significant advancement in performance compared to similar machines.

Through optimization, an additional 6.15% reduction in cogging torque is achieved. Although there is a slight decrease in the rated torque, the overall improvement in minimizing cogging torque is considered substantial. This research provides valuable insights into innovative design strategies and optimization methods for mitigating cogging torque in AFPSMMs, which can have significant implications for various applications such as wind energy, electric cars, and direct-drive elevator systems.

3. Results and Discussion

For the purpose of illustration and comparison of coreless and iron-cored AFPMM, a 3D model of an AFPMM with two rotor disks (and 10 surface-mounted PMs on each) and an internal stator was constructed. The comparison is shown in Figure 6, Figure 7 and Figure 8.

As mentioned in Section 1, the cogging torque in electric motors results from the interaction between the rotor’s permanent magnets and the stator slots.

Figure 6 shows the axial component of magnetic flux density in the air gap of both coreless and iron-cored AFPMM, and from it, we can see that the interaction between PMs and stator iron (slots) causes the fluctuations of the waveform for the iron-cored topology. Consequently, the iron-cored topology of the AFPMM has a torque ripple, which is shown in Figure 7.

From Figure 7, we can also see that coreless topology has smoother and lower values of torque as well as smaller pull forces. Iron-cored topology offers higher values of torque (presented iron-cored model 57.13 Nm; coreless model 27.1 Nm), but it has fluctuations due to the cogging torque, which is caused by the interaction between permanent magnets and the stator iron.

Figure 6, Figure 7 and Figure 8 show that cogging torque reduction is a necessity in order to obtain a smoother torque, and for that reason, the cogging torque reduction techniques are summarized in

Table 2. Cogging torque reduction techniques dealing with rotor adjustment.

Rotor Adjustment	Reference
rotor tooth notching, dummy slot	[9]
an integer gear ratio unequal-space type pole pieces unequal-width type pole pieces	[10]
trapezoidal shape permanent magnets	[11]
skewing for both permanent magnets (PMs) and ferromagnetic pole pieces	[12]
using alternating magnet pole-arcs in facing rotors	[13]
designing permanent magnet shapes multiplicative waveform optimization of design parameters skewing three-dimensional skewing	[14]
arc-shaped triangular magnets	[15]
hybrid skew: using magnets with different shapes for N and S poles	[16]
magnet grouping	[17]
shifting or grouping magnets magnet skewing varying magnet arc	[18]
dual-skew magnet technique	[19]
rotor skewing notching pole displacement	[20]
optimizing the permanent magnet shape different rotor designs (flat trapezoidal, skewed, curved, curved skew)	[21]

,

Table 3. Cogging torque reduction techniques dealing with stator adjustment.

Stator Adjustment	Reference
geometric optimization displacement between both stator sides	[22]
stator core position, magnet pole pitch	[23]
modifying the stator shape	[24]
Different stator modification techniques: slot-opening shape variation (trapezoidal and parallel) slot-opening width variation slot-opening relative displacement skewed slot-opening	[25]
adding a magnetic bridge (MB) between adjacent stator teeth	[26]
stator/rotor teeth notching stator slot chamfering combination of stator slot chamfering and teeth notching (SCTN)	[27]
soft magnetic composite (SMC) cores adjusting the pole arc ratio and slot width	[28]
different winding configurations: symmetric double-three-phase winding connection (SWC) and asymmetric double-three-phase winding connection (AWC)	[29]

and

Table 4. Cogging torque reduction techniques dealing with rotor and stator adjustment.

Rotor and Stator Adjustment	Reference
optimization of slot and pole combinations in the stator and rotor skewing method that tilts stator or rotor teeth to introduce angular displacement notching method, removes part of stator teeth to break symmetry pole shifting method, rotor poles are shifted to mitigate cogging effects	[30]
Rotor Tooth Combination (RTC) Method (changing the rotor structure, specifically the combination of rotor teeth) Rotor Angle Combination (RAC) Method Stator Tooth Combination (STC) Method (altering the widths of stator teeth), Permanent Magnet Tooth Combination (PTC) Method (adjusting the widths of permanent magnet teeth)	[31]
tangential displacement of stators asymmetric rotor pole shape segment-twisted rotor	[32]
Optimization of Design Parameters: tooth arc of the stator PM thickness of the stator yoke width of the stator axial length of the stator tooth arc of the rotor rotor tooth shape Decreasing yoke width and axial length of the stator, and increasing tooth arc and tooth fan-shaped angle of the rotor. Rotor Tooth Notching (RTN): Introducing dummy slots (notches) in the rotor tooth structure reduces cogging torque for FSST-PPM structure Rotor Tooth Circumferential Pairing (RTCP): Pairing different-sized rotor teeth, particularly designing a big rotor tooth and a small rotor tooth with specific angles	[33]
stator slot displacement rotor magnet skewing	[34]
dual complementary structure Stator with complementary inset PMs	[35]

.

Table 2 shows the cogging torque reduction techniques that include rotor adjustment, Table 3 stator adjustment, and Table 4, a combination of both rotor and stator adjustment for cogging torque reduction.

4. Conclusions

The article presents a comprehensive review of the cogging torque reduction techniques for axial flux permanent magnet machines. Many authors deal with cogging torque reduction in order to reduce the torque ripple and increase the efficiency and operation of the AFPMM.

In the introduction, we showed the necessity of the cogging torque reduction, based on the literature review, as well as by simulating a model of iron-cored AFPMM and illustrating the waveform of the torque and its ripple. In order to reduce the ripple, cogging torque reduction techniques for AFPMM were analyzed and summarized in three tables, separately for rotor adjustment, stator adjustment, and a combination of both.

Some of the proposed and used techniques offer good results in terms of cogging torque reduction but, on the other side, could present a significant price increase for the machine due to the production process.

There are a lot of techniques that are economically more suitable, such as magnet skewing and positioning, stator displacement, etc. These kinds of approaches are relatively simple and do not significantly influence the price of the production. The economic impact of the approaches for cogging torque reduction varies and is relative, depending on the application of the machine.