Minimization of Cogging Torque in Axial Field Flux Switching Machine Using Arc Shaped Triangular Magnets

Axial flux permanent magnet machine (AFPMM) provides high torque characteristics at low speeds without any mechanical gears. AFPMMs have numerous applications in wind energy, electric cars, and direct drive elevator applications. These machines have low cost and improved power to weight ratio. However, single sided AFPM suffers from torque ripples because of its non-sinusoidal back emf, cogging torque, and rotor eccentricity. There are two major components of pulsating torque, namely torque ripples and cogging torque. In PM machine design, the cogging torque is a serious concern because it adds unwanted harmonics to the pulsating torque. Whereas the torque ripples cause noise and vibrations. In order to gain high efficiency, torque ripples should be minimum. The aim of this research is to design “Slotted axial field flux switching permanent magnet machine”. Mathematical models are used to design the machine and Finite element method (FEM) has been used to analyze the machine. In addition, Latin hypercube sampling (LHS) has been used to create the samples. Finally, Kriging Method is used for approximating the model and genetic algorithm has been applied to get the optimum machine. The results showed 61.8 % reduction of the cogging torque in the proposed machine model as compared to the conventional one. Moreover, the optimized model further provided 6.15 % reduction in the cogging torque as compared to the proposed one.


I. INTRODUCTION
Over the years, the permanent magnet (PM) machines are broadly used in power system in different types [1]- [5]. Amongst existing PM machines, Axial field PM machines show best performance due to its promising characteristics such as small axial length, small size, and high-power density [3]- [5]. Nonetheless, the traditional PM machine offers irreversible magnet demagnetization due to rise in rotor The associate editor coordinating the review of this manuscript and approving it for publication was Christopher H. T. Lee . temperature. On the other hand, Flux switching permanent magnet machine (FSPMM) has the ability to refrain from the irreversible demagnetization as both magnets and armature windings are placed in stator instead of the rotor [6]- [11]. The FSPMM has gained attraction in recent years because it has combined benefits of both PM synchronous machines and switching reluctance machines. Moreover, it has many desirable properties like increased torque/power density, robust structure, and excellent fault tolerance capabilities [12]. Fault tolerance capability is highly important in applications like aerospace and vehicle drive systems [13]- [14]. The idea of VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ FSPMM was first proposed in 1950's [15]. After that large number of FSPMM have been designed [16]- [20], which include 3-phase, multi-phase, E-core, C-core, axial, linear, and hybrid. Most of the designed FSPMM can attain high torque and power density but also suffer from high cogging torque and torque ripple due to its double salient structure and flux focusing effects. Therefore, it was necessary to reduce such torque pulsations in high performance applications to refrain from vibration and noise along with the provision of precise position and speed control [21]. In [22], a new alternate poles design was proposed. This machine offered high torque but had asymmetry in waveform of back EMF which may cause high torque ripples. In [12], Ackim Zlu studied flux switching machine with segmental rotor and PM primary excitation in segmental rotor. The rated voltage and rated torque were increased with the proposed model in [12], but at the cost of notable torque ripple. The main reason for increase in torque ripple was cogging torque, caused by deep tooting and slotting in stator. Furthermore, W. Min and J. T Chen [17], proposed E-core and C-core linear switched flux PMM and compared it with the conventional linear SFPM machine. The proposed machine had small usage of magnet volume, whereas the C-core linear SFPM machine had the highest back EMF and force density due to increase in slot area. Contrarily, the proposed E-core machine had isolated coils, which were feasible for fault tolerance, but the peak-to-peak value of the cogging force was on a higher side as compared to the conventional machine. Similarly, Xiaohong Xue [16], designed a five-phase modular FSPMM with 18 and 19 rotor poles. The electromagnetic characteristics such as flux, back EMF, cogging torque, and unbalanced magnetic force (UMF) were studied. The results showed that the proposed machine not only retained the benefits of the conventional machines, but also offered less cost and enhanced fault tolerant capabilities. Also, the machine having 19 rotor poles incorporated the benefits of symmetric back EMF and decreased cogging torque but suffered from UMF. In [20], a hybrid excited FSPMM was proposed with the addition of iron flux bridges. A simple lumped parameter magnetic circuit model (LPMCM) was established which predicted the effects of different parameters. The study revealed that the field coil excitation (FEC) was increased by adding iron flux bridges, however the torque density was decreased. Li Hao proposed a novel dual rotor axial field FSPMM [23]. The study included the effect of the rotor pole width and stator chamfer slot on the back EMF and the cogging torque. There was an improvement in the waveform of the back EMF and reduction in the cogging torque by altering the rotor pole width to 1.4 ∼ 1.6 times as compared to the conventional design and chamfering the stator slot. However, the output torque was significantly reduced. In [21], the torque characteristics of the FSPMM with rotor step skewing were studied. Two models with 10 and 14 rotor pole numbers were proposed. The rotor step skewing technique was adopted to decrease the cogging torque and the torque ripple, but the output torque obtained was not desirable. Li Hao in [24], proposed a novel axial field FSPMM with U-shaped stator segments. The flux linkage and the back EMF of the proposed machine was mainly sinusoidal, but a very low output power of 0.6 kW. Wenliang Zhao [25], proposed a novel dual rotor axial field fault tolerant FSPMM. The proposed machine incorporated the benefits of phase group concentrated coil (PGCC) winding, and unaligned arrangement of both rotors. The PGCC winding was adopted to gain unity winding factor. The effects of flux focusing were increased by using spoke type permanent magnets. The cogging torque and torque ripple were reduced by using unaligned arrangement of both the rotors. Besides,, different techniques such as rotor side modification had been adopted for minimizing cogging torque and torque ripples. The rotor side modification techniques include magnet skewing and magnet displacement. But these techniques decreased the cogging torque at the cost of either asymmetry in the waveform of the back EMF or significant decrease in the rated power of the machine. The cogging torque is one of the main constraints in the FSPMM as magnets are placed in stator of FSPMM. Therefore, it is imperative to reduce cogging torque in FSPMM.
In this article, an axial field FSPMM with an arc shaped triangular magnets is presented. The proposed model has merits of low cogging torque due to greater effective air gap length. A detailed comparison has been done between the conventional and the proposed FSPMM, under same design parameters, using 3-D Finite Element Method (FEM). Furthermore, the proposed model is optimized using PM overhang configuration. The optimized model is obtained using GA followed by the approximation of model using Kriging method. The rest of the article is organized as follows: Section II discuss the governing numerical equations of the machine design, whereas comparison between the proposed and conventional FSPMM is explored in Section III. Model optimization is briefly discussed in Section IV followed by the conclusion of this research in Section V.

II. MACHINE DESIGN
In this section, basic numerical equations that are used for the machine design are briefly discussed to mathematically demonstrate the machine parameters responsible for output current and torque. The back EMF of the FSPMM must be sinusoidal [15]. Ignoring voltage drop in the stator resistance, input power P in can be represented mathematically as: where m, E m , and I m denote phase number, amplitude of back EMF, and current, respectively. The output power P out is calculated using: where γ denotes efficiency. The flux in the machine is computed using: where ϕ m is magnitude of flux, N p and α r denotes the rotor pole number and rotor position, respectively. Further, the back EMF is expressed as: In (4), N ph and ω r represents phase coil turns and angular speed of rotor, respectively. Ignoring the sinusoidal term in (4) and substituting value of ϕ m from (3), we get: Eq. (5) shows that B g is the peak value of the air gap flux density, k d is coefficient of the leakage flux, k f is the coefficient of the air gap flux density distribution, α i is pole arc coefficient, and N s is the number of slots. D out and D in are outer and inner diameter of the rotor and the stator, respectively. The armature current of the machine is determined by: where A e is electrical loading. The mathematical relation of the cogging torque and air gap flux density is: φ g is the air gap flux, R is the reluctance of the air gap, and θ is the rotor position. Substituting (5), (6), and ω r = 2πn rpm 60 in (2) give rise to: In (8), k io is ratio of inner diameter to outer diameter and n rpm is the rotor speed in rpm. Whereas the output torque is given by: It is obvious from (8) and (9), that the output power and torque are directly related to N p /N s , B g , and A e . From (7), the outer diameter is given as:

III. COMPARISON BETWEEN CONVENTIONAL AND PROPOSED MODEL
This section presents a detailed geometrical difference between the conventional and the proposed models. The machine model consisting of rectangular shaped magnet is named as 'conventional shape' while an arc shaped triangular magnet used in the machine model is named as 'proposed shape'.

B. PERFORMANCE COMPARISON BETWEEN CONVENTIONAL AND PROPOSED MODEL
The topologies of the designed FSPMMs are shown in Figure 2. Both machines have 12 stator slots and 13 rotor poles. The outer/inner diameter, axial length, stator tooth width, air gap length, phase coil turns, and magnet volume are kept same for both the models. Figure 2 (a) shows a complete design of the FSPMM with rectangular magnets i.e., the conventional one. Whereas it can be observed from Figure 2 (b) that topology of the designed FSPMM is same except the novel shape of the magnets which is arc shaped (the proposed one). For the rest of discussion in the article, Axial field FSPMM with rectangular magnets is named as 'the conventional model' while axial field FSPMM with arc shaped triangular magnets is referred as 'the proposed model'. Due to low cost, ferrite magnets with circumferential magnetized direction and opposite polarity alternatively are inserted in stator. The rotor and stator core are made up of 50NSSMC 470. The design parameters of both the models are shown in Table 1. The topologies shown in Figure 2 are designed in JMEG software and same design parameters are used for both the designs. The results of the both the models are computed using FEM analysis in JMEG software. The flux density distribution of the conventional and the proposed models shown in Figure 3. It can be seen in Figure 3 that maximum magnetic flux density B max is 1.8 T. B max is usually considered for examining the saturation in rotor back iron. It can also be observed from Figure 3 that total magnetic flux density is comparatively less in the proposed model as compared to the conventional model. This is attributed to increased air gap length because of the arc shaped magnets in the proposed model. Next, the back emf computation results of both the models shown in Figure 4 reveal that there is 3.0 V reduction in the proposed model than the conventional machine. The proposed arc shaped magnets in fact reduce the magnetic flux which is directly related to the voltage. However,  the total harmonic distortion (THD) is improved in proposed model. The THD is 2.2% and 1.7% in conventional and proposed models, respectively. The reason for reduction in THD is that more sinusoidal waveform of flux density distribution is achieved with the proposed arc shaped triangular magnet.
Finally, Figure 5 shows the comparison of the cogging torque of both the models. It is obvious from Figure 5 that the cogging torque is greatly reduced in the proposed model. Overall, there is 61.8% reduction in peak-to-peak value of the cogging torque as compared to conventional model. The insertion of the arc shaped magnets decreases total magnetic flux density by increasing effective air gap length as compared to the conventional rectangular shaped magnets. This effective length reduction in the air gap significantly decreases the flux between permanent   magnets which as a result, reduce the cogging torque. On the contrary, reduction in the rated torque is observed in the proposed model. The rated torque of the conventional and the proposed models is 61 Nm and 57.2 Nm, respectively.

IV. OPTIMIZATION
To further enhance the performance of the proposed model, optimization is performed while keeping the magnets' volume constant

A. DESIGN VARIABLES
The concept of magnet overhang is used in optimization. The length of PM is varied along outer and inner radii. More specifically, the length of the PM is varied from outer radius of the disc, known as upper overhang and the length of the PM is varied from inner radius of the disc, known as lower overhang. The arc of the PM is also varied. The variation of the arc results in reduction of the cogging torque. The design variables X 1 and X 2 are the overhang length and an arc of PM, respectively, as shown in Figure 6. The volume of the PM is kept equal by altering the height (H o ). The variables R arc , θ arc , and H o denote the radius of arc, angle of arc and height of the triangle, respectively. The area of arc is calculated using relation: VOLUME 8, 2020

TABLE 2. LHS Design Samples
Whilst the area of the proposed shape is calculated as: The total volume of the arc shaped magnet is the product of total area and height, mathematically: The selected variables optimized in the range given below:

B. PROCESS OF OPTIMIZATION
A flow chart of the optimization process based on the selected variables is depicted in Figure 7 Initially, design variables and objective function are chosen. Latin hyper cube sampling (LHS) is used for designing the samples by space filling by using equal samples points in a given parameter space. LHS is then used to calculate the average, change, and function distribution of the output. In addition, LHS also confirms that all input parameters are within the range. MATLAB Model Based Calibration Tool is used to save time in determining the optimum response. The total number of design samples chosen are 15, as evident from Table 2, whereas equal volume of the magnets is selected in all samples. Then, three-dimensional (3D) Finite Element Analysis (FEA) is used for the performance analysis because of 3D electromagnetic behavior of the proposed machine topology. The FEA is preferred over analytical techniques (Equivalent Circuit, Fourier Series) because it considers the process of saturation [25].  The genetic algorithm (GA) is used to find the optimal values of the design variables and the objective function. The optimal values of X 1 and X 2 obtained from the GA are  1.79 mm and 46.41 mm, respectively. Finally, a 3D FEA is performed to verify the optimal results. Figure 8 shows the flux density distribution of the optimized model. The maximum flux density distribution is 2.85 T in the optimized model. Due to increase in the flux density, back emf increases, as depicted in Figure 9. Figure 10 shows a comparison between the cogging torque of the proposed and the optimized model. The results show that the cogging torque of the optimized model is decreased as compared to the proposed model. Overall, 6.15% reduction in the cogging torque is observed. However, the rated torque increases to 61.7 Nm in optimized model. The results comparison is drawn in Table 3.

V. CONCLUSION
A model of axial field FSPMM with arc shaped triangular magnet is proposed in this research. The analysis showed that the cogging torque is significantly decreased in the proposed model but at the cost of slight decrease in the rated torque. However, torque ripples are considerably reduced in the proposed model. In particular, 61.8% decrease in peak-to-peak value of the cogging torque as compared to conventional model is achieved. Further, the optimization of the proposed model is performed to improve the proposed machine performance using equal volume of the arc shaped magnets. The optimization demonstrated that 6.15 % reduction in the cogging torque can be further obtained as compared to the proposed model. Overall, the optimized model shows better results as compared to both the conventional and the proposed axial field FSPMMs. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, CUI. He has authored or coauthored 50 journals and conference publications. His current research interests include applied electromagnetic, recongurable antennas, leaky wave antennas, phased array antennas, and energy harvesting for low power devices. Professor. In 2014, he was with the University of Bath, Bath, U.K. He is currently an Associate Professor of the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul, South Korea. His research interests include the analysis and optimal design of next-generation electrical machines using smart materials such as electromagnet, piezoelectric, and magnetic shape memory alloy. VOLUME 8, 2020