**Analysis of Torque Ripple and Cogging Torque Reduction in Electric Vehicle Traction Platform Applying Rotor Notched Design**

#### **Myeong-Hwan Hwang 1,2, Hae-Sol Lee 1,3 and Hyun-Rok Cha 1,\***


Received: 9 October 2018; Accepted: 29 October 2018; Published: 6 November 2018

**Abstract:** Drive motors, which are used in the drive modules of electric cars, are interior permanent magnet motors. These motors tend to have high cogging torque and torque ripple, which leads to the generation of high vibration and noise. Several studies have attempted to determine methods of reducing the cogging torque and torque ripple in interior permanent magnet motors. The primary methods of reducing the cogging torque involve either electric control or mechanical means. Herein, the authors focused on a mechanical method to reduce the cogging torque and torque ripple. Although various methods of reducing vibration and noise mechanically exist, there is no widely-known comparative analyses on reducing the vibration and noise by designing a notched rotor shape. Therefore, this paper proposes a method of reducing vibration and noise mechanically by designing a notched rotor shape. In the comparative analysis performed herein, the motor stator and rotor were set to be the same size, and electromagnetic field analysis was performed to determine a notch shape that is suitable for the rotor and that generates reasonable vibration and noise.

**Keywords:** interior permanent magnet synchronous motor; torque ripple; cogging torque; electric vehicle; notch

#### **1. Introduction**

Globally, the market for eco-friendly vehicles is continuing to grow, and a variety of electric car models are being released in the automotive market. The reason for this increased prevalence of electric cars in the market is because the laws concerning the average amount of carbon dioxide emissions from internal combustion vehicles are becoming increasingly stringent, and there is concern regarding fine particles being emitted into the atmosphere; thus, eco-friendly cars are becoming increasingly prevalent globally [1].

The electric motors that are used to drive electric cars influence the cars' performance considerably. The type of drive motor used can determine the car's mileage, efficiency, torque, vibration, maximum speed, and acceleration. One such drive motor that is widely used nowadays is the interior permanent magnet synchronous motor (IPMSM). The IPMSM involves a structure with a permanent magnet embedded in the interior of the rotor. It has a torque component that is caused by the interior magnet (alignment torque) and a torque component that is caused by the difference in the d–q axis magnetic reluctance (reluctance torque); thus, it can provide a large power density. Because the embedded permanent magnet's magnetic properties are similar to that of an air gap, marked differences in the

d–q axis inductance distribution occur in the rotor interior. The motor can have a wide range of variable speed driving properties owing to weak field control in proportion with the saliency ratio. Therefore, these motors continue to be used as the drive motors of electric vehicles because they enable high-power and high-speed operation by ensuring high power density, a wide range of speeds, and mechanical strength, all of which are characteristics of the interior permanent magnet motor [2].

In the interior motor's structure, a magnet is inserted in the rotor, and magnet barriers exist at both ends of the rotor. The electrical device's properties vary based on the path of the magnetic flux produced by the magnet. By applying this property in the desired direction, it is possible to improve upon the loss, power, efficiency, cogging torque, and torque ripple [3]. This has various advantages; however, it also has several disadvantages. The magnetic distribution on the surface of the rotor is not uniform, and more cogging torque and torque ripple occur in this type of motor than with other forms of motors having the same magnetic circuits [4].

If the torque ripple is high, vibration and noise occur in the motor, and this may lead to motor drive system malfunctions. Unlike other motors, an IPMSM uses a permanent magnet to create magnetic field, and cogging torque generally occurs in the motor owing to the difference between the magnetic reluctance at the permanent magnet in the rotor interior and at the stator's slot structure. Cogging torque significantly affects the occurrence of noise and vibration in the motor; hence, it is necessary to reduce it as much as possible at the design stage. Torque ripple, which is related to the back electromotive force harmonics, must also be reduced as much as possible because "it increases the noise and vibration in the motor [5].

Thus, many studies have been conducted to reduce the vibration and noise in interior permanent magnet motors to ensure improved performance and reliability in electric car drive motors. A variety of papers have introduced methods that reduce the cogging torque and torque ripple. These methods involve adjusting the motor's arrangement, adjusting the width of slots and teeth, using permanent magnet skew, creating auxiliary teeth, using slot-less armatures, and using notches [6,7]. In contrast, this study aims to reduce cogging torque and torque ripple by changing the shape and arrangement of the permanent magnet [8,9].

The methods for reducing cogging torque and torque ripple that were used in other studies were mostly electromagnetic methods. However, this paper aims to reduce the cogging torque and torque ripple using a mechanical method. Existing electromagnetic methods have the disadvantage of also reducing the active magnetic flux, which lowers power and efficiency. Mechanical methods, however, reduce active magnetic flux far less than electromagnetic methods and enable high-efficiency high-power motors to be built. As a typical example of this, the T Company has built relatively high-efficiency motors for electric cars using a mechanical method similar to that introduced in this paper. Herein, a 3D finite element method is used to reduce the time required for obtaining an optimal design, and a variety of notches are introduced on the rotor to reduce the cogging torque and torque ripple. A notch shape for optimal torque ripple and cogging torque reduction through detailed shape design is determined. The results of this study are believed to offer a rotor notch design shape that can reduce the vibration and noise in future electric drive cars, and they are believed to contribute toward the efficiency gains of designing drive motors with reduced torque ripple.

This paper first describes the properties of the cogging torque and torque ripple in terms of theoretical equations, and the analysis model and its specifications. Furthermore, details of the comparative analysis performed according to the position and shape of the notch are presented, and the results and conclusions derived from the analysis are discussed.

#### **2. Materials and Methods**

#### *2.1. Relevant Equations for Cogging Torque and Torque Ripple*

#### 2.1.1. Cogging Torque Equation

Cogging torque is the non-uniform torque of the stator, and it occurs inevitably in a motor that uses a permanent magnet. It is a radius-directed torque that is directed towards the position with the minimum magnetic energy, i.e., in an equilibrium state, in the motor. As shown in Equation (1), the cogging torque can be determined by deriving the drive motor's internal energy by differentiating the magnetic energy with respect to the synchronous motor's rotor position angle.

$$T\_{\text{Cogging}}(a) = -\frac{\partial \mathcal{W}(a)}{\partial a} \tag{1}$$

In Equation (1), *α* is the rotor position angle and *W* is the motor's magnetic energy.

$$W\_{\rm fl} = \frac{1}{2\mu} \int\_{V} B^2 dV \tag{2}$$

Here, *B* is the magnetic flux density, and *μ* is the permeability.

$$B = G(\theta, z)B(\theta, a) \tag{3}$$

In Equation (3), *G*(*θ*, *z*) is the gap permeance function and *B*(*θ*, *α*) is the gap magnetic flux density. Furthermore, *θ* is the angle along the circumference, and *α* is the rotation angle.

If Equation (3) is substituted into Equation (2):

$$\begin{split} \mathcal{W}(\alpha) &= \frac{1}{2\mu} \int [G(\theta, z)B(\theta, \alpha)]^2 dV \\ &= \frac{1}{2\mu\_0} \int\_0^{L\_\text{s}} \int\_{R1}^{R2} \int\_0^{2\pi} G^2(\theta, z) B^2(\theta, \alpha) d\theta dr dz \\ &= \frac{1}{2\mu} L\_\text{s} \frac{1}{2} \left(R\_2^2 - R\_1^2\right) \int\_0^{L\_{vf}} \int\_0^{2\pi} G^2(\theta, z) B^2(\theta, \alpha) d\theta dz \end{split} \tag{4}$$

Here, *μ*<sup>0</sup> is the air permeability, and *Ls* is the lamination layer length. *R*<sup>1</sup> is the inner radius of rotor, and *R*<sup>2</sup> is the outer radius of rotor.

In Equation (4), if a Fourier series expansion is performed on *G*2(*θ*, *z*) and *B*2(*θ*, *α*), and the trigonometry function's orthogonality are used to solve Equation (4):

$$\begin{split} W(\alpha) &= \frac{L\_s}{4\mu\_0} \left( R\_2^2 - R\_1^2 \right) \left[ \sum\_{n=0}^{\infty} G\_{nN\_L} B\_{nN\_L} \int\_0^{2\pi} \cos nN\_L \theta \cos nN\_L (\theta + \alpha) \right] \\ &= \frac{L\_s}{4\mu\_0} \left( R\_2^2 - R\_1^2 \right) \cdot 2\pi \cdot \sum\_{n=0}^{\infty} G\_{nN\_L} B n N\_L \cos nN\_L \alpha \end{split} \tag{5}$$

In conclusion, for the cogging torque, the gap energy is partially differentiated by thve rotor's rotation angle, as shown in Equation (5), and it can be expressed as in Equation (6) [10–13]. Here, *NL* is the least common multiple (LCM) between the number of rotor poles and stator slots:

$$\begin{aligned} \mathcal{T}\_{\text{Cogging}}(a) &= -\frac{\partial \mathcal{W}(a)}{\partial a} \\ = \frac{L\_s \pi}{2\mu\_0} \left( R\_2^2 - R\_1^2 \right) \sum\_{n=0}^{\infty} G\_{nNL} B\_{nN\_L} n N\_L \sin nN\_L a \end{aligned} \tag{6}$$

#### 2.1.2. Torque Ripple Equation

There are several reasons why torque ripple occurs in a permanent magnet synchronous motor, including the cogging torque generated because of the mechanical structure, offset and scale errors in the current sensor in terms of electrical control, fluctuations in the direct current (DC) link voltage, phase current distortions owing to the inherent properties of power switching elements, and dead time [14–17], and distortions in the back electromotive force [18–22].

A synchronous motor's power is determined by the maximum DC voltage and maximum current supplied by the inverter. The maximum current is denoted by *Imax*, and it is determined in conditions that satisfy the thermal rating of the inverter. The equations for current and voltage are as follows:

$$\begin{aligned} V\_{ds}^2 + V\_{qs}^2 &\le V\_{max}^2\\ I\_{ds}^2 + I\_{qs}^2 &\le I\_{max}^2 \end{aligned} \tag{7}$$

Here, *Ids* and *Iqs* are the d- and q-axis currents, respectively. *Vds* and *Vqs* denote the d and q-axis terminal voltages, respectively.

The following are the voltage equations for a synchronous reference frame that sets the rotor that is rotating at a synchronous speed as the standard coordinate system.

$$\begin{aligned} v\_{ds}^{\epsilon} &= r\_s i\_{ds}^{\epsilon} + \frac{d\lambda\_{ds}^{\epsilon}}{dt} - \omega\_r \lambda\_{qs}^{\epsilon} \\ v\_{qs}^{\epsilon} &= r\_s i\_{qs}^{\epsilon} + \frac{d\lambda\_{qs}^{\epsilon}}{dt} - \omega\_r \lambda\_{ds}^{\epsilon} \end{aligned} \tag{8}$$

As seen in Equation (8), if it is assumed that the voltage drop due to the stator's phase resistance is not large, it can be said that the terminal voltage is proportional to the speed, *ωr*. In addition, the d–q axis magnetic flux part present in Equation (8), which takes into account the harmonic components, can be expressed as shown below. Here, *λd f* \_*har* and *λq f* \_*har* are the d–q axis interlinked magnetic flux harmonic components caused by the permanent magnet.

$$\begin{aligned} \lambda\_{ds}^{\mathfrak{c}} &= L\_d \mathfrak{i}\_{ds}^{\mathfrak{c}} + \psi\_f + \lambda\_{df\_{har}} \\ \lambda\_{qs}^{\mathfrak{c}} &= L\_d \mathfrak{i}\_{qs}^{\mathfrak{c}} + \lambda\_{qf\\_har} \end{aligned} \tag{9}$$

The torque equation for the permanent magnet synchronous motor is as follows:

$$T\_{\varepsilon} = \frac{\Im P}{4} \left( \lambda\_{ds}^{\varepsilon} i\_{\bar{q}} - \lambda\_{qs}^{\varepsilon} i\_{d} \right) \tag{10}$$

Equation (9) is substituted into Equation (10) to obtain Equation (11). From this, the torque ripple component, which is the torque component that occurs because of the interlinked magnetic flux's harmonic component in the synchronous motor's torque equation, can be derived, as in Equation (12):

$$T\_{\varepsilon} = \frac{3P}{4} \left\{ \psi\_f i\_q + (L\_d - L\_q) i\_d i\_q + \left( \lambda\_{df\_{har}} i\_q - \lambda\_{qf\_-har} i\_d \right) \right\} \tag{11}$$

$$T\_t = \frac{3P}{4} \left(\lambda\_{df\_{hr}} i\_q - \lambda\_{qf\\_bar{bar}} i\_d\right) \tag{12}$$

#### *2.2. Analysis Model and Specifications*

Figure 1 shows the shapes of the rotor and stator of the basic model employed in this study. A V-shaped magnet configuration was selected from several rotor shapes owing to its excellent speed-versus-torque characteristics and high allowable radial direction force [23–25]. Table 1 shows basic specifications of the interior permanent magnet synchronous motor, used in this paper.

**Figure 1.** Slot stator and 8-pole rotor core: (**a**) stator core; (**b**) rotor core.


**Table 1.** Interior permanent magnet synchronous motor (IPMSM) basic specifications.

Thereafter, notches were added at various positions in accordance with the basic model specifications listed above, and a comparative analysis was performed. According to the energy method, the cogging torque is a change in the magnetostatic energy that occurs because of the rotation of the motor. Changes in magnetostatic energy mostly occur when a pole transition occurs through the gap between the stator and the rotor. Consequently, cogging torque occurs at this time. Therefore, when a notch is added to the rotor's surface, the notch acts in the same way as the air gap. As a result of acting as the air gap, it changes the distribution of the gap permeance function, *G*(*θ*, *z*), and as the number of active slots is changed, *GnNL* is also changed, which results in reducing the cogging torque This has the effect of reducing the energy changes due to the rotation of the motor, thereby reducing the cogging torque and torque ripple [26,27].

#### *2.3. Test Method*

The aforementioned V-shape was selected as a typical shape for an interior permanent magnet. Notches were designed for the drive motor rotor magnet using a mechanical method rather than the existing electromagnetic method, and a comparative analysis was performed on the position with the smallest cogging torque and torque ripple. In each model, a notch was placed on the surface of the rotor to reduce the interior permanent magnet motor's cogging torque, and a comparative analysis was performed. The magnetic flux density distribution of the gap and the cogging torque properties change according to the position of the notch, and thus an optimal notch shape design is necessary. Figure 2 shows the positions of the notches in the analysis. In (a), (b), and (c), respectively, the notches are located on the inner, outer, and central parts of the rotor's surface. In (d), a central notch is added to the (a) notch shape. Finally, (e) shows the basic shape without any notches. A comparative analysis of the shapes according to the five notch positions was performed. After the optimal position was determined, a more detailed design was created accordingly.

**Figure 2.** Motor rotor notch design positions: (**a**) magnet's inner notches; (**b**) magnet's outer notches; (**c**) magnet's central notches; (**d**) magnet's additional central notches; (**e**) no notches (basic form).

#### **3. Comparative Analysis and Results**

#### *3.1. Comparative Analysis of Cogging Torque and Torque Ripple According to Notch Position*

For analysis, we used the motor electromagnetic field analysis tool program, "JMAG" (ver. 14.1), made from Nagoya, Japan, with data produced through simulation analysis. The cogging torque is calculated by dividing the difference between the maximum and minimum value of torque by the mean value of torque and presenting the result in percentile values. The back electromotive force voltage and cogging torque analysis speed were analyzed at 1000 rpm. For the torque ripple, the analysis was performed at 4000 rpm, which is the rotational speed corresponding to the maximum power output. Figure 3 and Table 2 show the corresponding analysis results. The back electromotive force voltage was found to be low for the V-shape (a) (20.68 V) and V-shape (b) (20.87 V). These are thus considered shapes that can ensure a high rotational speed when the motor is under no load. The cogging torque was 1.7 N m for V-shape (a) and 1.73 N m for V-shape (b). When there are notches on the inner and outer parts of the magnet, a rotor with a low cogging torque value can be designed. The torque ripple of V-shape (b) was found to be 8.04%, which is 7.86% lower than that of V-shape (a). The cogging torque of V-shape (b) was 0.03 N m higher than that of V-shape (a); however, its cogging torque value was lower than V-shape (a) against its motor power and size. These results show that the torque ripple and cogging torque can be lowered simultaneously when notches are placed on the outer edge of the magnet, as in V-shape (b).

(**b**)

**Figure 3.** Back electromotive force voltage, cogging torque, and torque ripple according to notch position: (**a**) back electromotive force voltage; (**b**) cogging torque; (**c**) comparative analysis of torque ripple.


**Table 2.** Comparative analysis of cogging torque and torque ripple according to notch position.

#### *3.2. Comparative Analysis of Cogging Torque and Torque Ripple According to Notch Shape*

The comparison results in Table 2 show that the shape with notches on the outer part of the magnet had low torque ripple and cogging torque. Therefore, a method is proposed that can optimize this shape to obtain the lowest torque ripple. Figure 4 shows diagrams of the notch optimization position and the detailed design. In Figure 4b, the radius refers to the length of the radius of a circle drawn from an arbitrary point on the stator surface. The shape and size of the notch is changed by changing the radius.

**Figure 4.** Notch shape optimization: (**a**) notch optimization position; (**b**) notch optimization diagram.

3.2.1. Comparative Analysis of Cogging Torque and Torque Ripple According to Changes in Notched Shape

In the first optimized shape, the standard point radius was 7.61 mm, and shape optimizations within a range of ±3 mm of the radius were analyzed. Table 3 lists the changes in notch width according to changes in the radius.


**Table 3.** Rotor notch radius parameter.

The cogging torque analysis results shown in Figure 5 show that the cogging torque was 4.14 N m at the initial standard point radius of 7.61 mm, and it was 3.88 N m when the radius was 5 mm, which corresponds to the lowest cogging torque. As the radius increased, the cogging torque increased.

An interior permanent magnet (IPM) motor delays the current phase angle and uses the magnet torque and reluctance torque together; thus, the torque ripple and cogging torque according to the current phase angle were analyzed. Figures 6 and 7, respectively, show the torque ripple and torque, according to the radius, for various current phase angles. When the radius was 5 mm, the torque ripple was low, at 9.7%. As the radius increased, the power increased, and the torque ripple increased. When the radius was 10 mm, and the current source was 0◦, the torque ripple was 15%.

The comparative analysis results in Figure 7 shows that when the notch radius changes from 5 to 10 mm, the difference in the torque is less than 1 N m. Overall, it was determined that a smaller radius can lower the cogging torque and torque ripple considerably. A large notch shape does not reduce the cogging torque. Consequently, an additional analysis was performed to obtain a design to minimize the notch size and reduce the cogging torque and torque ripple.

**Figure 5.** Comparison of cogging torque according to notch radius.

**Figure 6.** Comparative analysis of torque ripple according to notch radius.

**Figure 7.** Comparative analysis of torque according to notch radius.

3.2.2. Comparative Analysis of Notch Dimensions for Optimal Design

According to the previous discussion, new parameters were set to obtain a notch design to reduce the cogging torque and torque ripple, as shown in Figure 8. Distance (1) was fixed at 0.5 mm, and it is the distance from an end of the notch's arc, when its radius (3) is 2 mm. Distance (1) was fixed to observe the analysis result of torque and torque ripple with the width difference. The changes in radius (3) were analyzed according to changes in notch depth, and not the width, and distance (2) indicates the distance from the magnet barrier to the notch, and it was measured because it is related to the flow of the magnetic flux.

Table 4 lists the values of distance (2) when distance (1) was fixed at 0.5 mm and radius (3) was changed.

To obtain the results shown in Figure 9, distance (1) was fixed at 0.5 mm, and a comparative analysis was performed on the cogging torque. The cogging torque was 3.7 N m at a distance (2) of 0.5 mm and a radius of 2.1 mm. At the deepest point, the cogging torque was 4.29 N m, which indicates a difference of 0.59 N m.

Figure 10 shows the results of analysis of the torque ripple according to radius (2) for various current phase angles. When the current phase angle was 0◦, 10◦, or 20◦, the torque ripple was large if radius (2) was small. However, when the current phase angle was controlled to be 30◦ or 40◦, the torque ripple tended to decrease as radius (2) decreased.

**Table 4.** Changes in distance (2) and radius (3) when distance (1) is fixed at 0.5 mm.


**Figure 8.** Notch optimization diagram with length parameter.

**Figure 9.** Comparative analysis of cogging torque when notch shape distance (1) is 0.5 mm.

**Figure 10.** Comparative analysis of torque ripple when notch shape distance (1) is 0.5 mm.

#### **4. Discussion**

In this study, a finite element method was used to analyze the properties of interior permanent magnet synchronous motors that are used as drive motors in electric cars. The external diameters of the stator and the rotor were set to be the same in order to compare various parameters such as the cogging torque, torque ripple, and back electromotive force voltage according to the position of a notch on the rotor shape. In the electromagnetic field analysis, the input voltage and input current were set to be the same. The motor's performance characteristics were analyzed according to the notch position on the rotor shape.

The aim of this work was to determine an approach to obtain the design a rotor shape that can minimize the cogging torque and torque ripple so that a low-noise drive motor can be designed. The application target was an IPM motor; thus, current phase angle control was required to ensure that various characteristics of the motor, such as efficiency, power, and torque, do not degrade. Therefore, the point with the maximum torque was found via phase angle control, and the torque ripple, cogging torque, and maximum power were determined.

First, the notch position on the rotor shape was analyzed. Notches can be placed in a variety of positions, and thus, to determine the optimum position, notches were placed in five locations on the rotor and a comparative analysis of the resulting characteristics was performed. The results confirmed that placing the notches on the outer part of the magnet produced the best properties, and this design was then further optimized to obtain the best possible design.

Next, in the design phase, the distance between the rotor and the magnet barrier was fixed, and the original size of the notch was changed. When the overall analysis results were considered, V-shape (a) was found to be the most suitable notch shape in terms of reducing the noise and vibration, owing to the low values of the configuration's torque ripple and cogging torque in the analysis in which distance (1) was 1 mm.

This paper proposed a notch shape optimization method for reducing the vibration and noise in vehicles. Vibration and noise are considered to be particularly important factors in electric cars. Luxury cars experience many problems pertaining to vibration and noise, and these factors can be reduced by designing the drive module via the proposed method. The proposed method can be widely used in the design of electric car drive motors to ensure an appropriate noise level.

**Author Contributions:** Conceptualization, M.-H.H. and H.-S.L.; Data curation, M.-H.H.; Formal analysis, M.-H.H.; Methodology, H.-S.L.; Supervision, H.-R.C.; Validation, H.-S.L.; Visualization, M.-H.H.; Writing—original draft, M.-H.H. and H.-S.L.; Writing—review and editing, H.-R.C.

**Funding:** This research was funded by the support of the Korea Institute of Industrial Technology as "Variable Architecture Powertrain Platform and Self-driving Factor Technology Development for Industrial EV Self-driving Vehicle" KITECH EO-18-0020.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **An Analytical Subdomain Model of Torque Dense Halbach Array Motors**

**Moadh Mallek 1,\*, Yingjie Tang 1, Jaecheol Lee 1, Taoufik Wassar 1, Matthew A. Franchek <sup>1</sup> and Jay Pickett <sup>2</sup>**


Received: 30 October 2018; Accepted: 20 November 2018; Published: 22 November 2018

**Abstract:** A two-dimensional mathematical model estimating the torque of a Halbach Array surface permanent magnet (SPM) motor with a non-overlapping winding layout is developed. The magnetic field domain for the two-dimensional (2-D) motor model is divided into five regions: slots, slot openings, air gap, rotor magnets and rotor back iron. Applying the separation of variable method, an expression of magnetic vector potential distribution can be represented as Fourier series. By considering the interface and boundary conditions connecting the proposed regions, the Fourier series constants are determined. The proposed model offers a computationally efficient approach to analyze SPM motor designs including those having a Halbach Array. Since the tooth-tip and slots parameters are included in the model, the electromagnetic performance of an SPM motor, described using the cogging torque, back-EMF and electromagnetic torque, can be calculated as function of the slots and tooth-tips effects. The proposed analytical predictions are compared with results obtained from finite-element analysis. Finally, a performance comparison between a conventional and Halbach Array SPM motor is performed.

**Keywords:** mathematical model; Halbach Array; surface permanent magnet; magnetic vector potential; torque

#### **1. Introduction**

Permanent magnet (PM) motors have been deployed on machine electromechanical machinery for decades, owing to their reliable performance, electrical stability and durability [1]. These motors have been the industry gold standard, finding their way into both rotary and linear applications. PM machines are largely classified into three categories based on the placement of permanent magnets, namely the: (i) inset PM machine, (ii) internal PM machine and (iii) surface PM machine. To improve the electromagnetic torque density, investigations have focused on improving permanent magnet performance within the rotor. Recently, motor designs have moved from samaraium cobalt (SmCo) to neodymium iron boron (NdFeB). While NdFeB has higher remanence, achieving torque enhancements, such embodiments suffer from less coercivity and are prone to demagnetization. As a result, motor designs have created intricate ways to remove heat from this class of machines whereby NdFeB is applicable. Different laminate materials have also been investigated with the use of water-cooling and different magnet arrangements to improve performance. All these advancements are valuable and have improved the torque density of rotating machines. Presented in this manuscript is the application of these integrated principles to discover new magnet arrangements based on the Halbach Array.

The Halbach Array was originally discovered in the 1970s and replicated by Hans Halbach in the mid-1980s. This new magnetic array arrangement oriented the magnet poles such that the flux is magnified in a specific direction while limiting the flux in the opposite direction. Despite concentrating the magnetic flux is a desired direction; a Halbach Array may not necessarily improvement torque density. In fact, it is possible to create an adverse effect on torque density. For that reason and the associated manufacturing costs, Halbach Arrays have not found a niche in electric motor applications.

Halbach Array magnetization was proposed in many applications [2] and has become increasingly popular [1] with various topologies and applications: radial [3] and axial-field [4], slotted and slotless [5–8], tubular and planar for rotary [9,10] or linear [11] machines. Halbach Array configurations exhibit several attractive features including a sinusoidal field distribution in air gap that results in a minimal cogging torque and a more sinusoidal back EMF waveform [12]. Thus, using Halbach Array SPM eliminates the conventional design techniques such as skewing of the stator/rotor [13], optimization of the magnet pole-arc [14] and distributed stator windings [15]. The Halbach Array magnets configuration generates a steady magnetic field in the active region which yields many benefits including: (i) compact form with high torque density up to 30%, (ii) less weight due to less rotor back iron volume and (iii) lower rotor inertia [16]. Adopting a Halbach Array does not always assure torque improvement in motors. To improve torque performance, the magnetization pattern of the PMs in the array must be cautiously designed. Furthermore, the theoretical electro-magnetic modeling is very advantageous to improve the control applications [17,18]. Torque-dense motor design optimization requires a computationally efficient mathematical model parameterized by design parameters including geometric, electric and magnetic motor properties. Many studies have been dedicated to the analytical calculation of the electromagnetic devices and Halbach Array machines [19].

Liu et. al proposed a method to divide and establish nonlinear adaptive lumped equivalent magnetic circuit (LPMC) that included the slot effect [20]. Using Kirchhoff's Laws, the model predicts the electromagnetic performance through an iterative process. The accuracy of these predictions were found to be lower than that of the subdomain analytical solutions [21]. In addition, transfer relations, such as Melcher's method, analytically predict the electromagnetic characteristics of a tubular linear actuator with Halbach Array [22]. Their work derives the generalized vector potentials due to permanent magnets and single-phase winding current using transfer relations. However, the influences of stator slot and tooth tips effects were neglected. Shen et al. applied the subdomain method for Halbach Array slotless machine [3]. The motor performance is derived through a scalar potential calculation over three regions: rotor back iron, permanent magnets and air gap. The major limitation of this approach is that the model does not account for the slot and tooth-tips effects for different winding layouts.

A 2-D model for SPM machines was developed in for conventional permanent magnets magnetizations [23]. A subdomain model incorporating the influence of slot and tooth-tips was developed to predict the armature reaction field for conventional SPM [24]. This particular model is valid for Halbach Array magnetization. Broadening the class of Halbach Array magnetic parameterization, a slotless motor model was developed having different pole-arc [25,26]. This feature is employed in the developed model by introducing the so-called *magnet ratio* parameter that improved the electromagnetic performance over an equal pole-arc machine (conventional SPM).

The contribution of this paper is the development of a subdomain motor model for a slotted Halbach Array SPM motor incorporating both slot and tooth-tip effects. Torque maximization is accomplished by optimizing Halbach Array magnets configuration through the introduction of the magnet ratio. The objective is to predict the electromagnetic characteristics quantified by cogging torque, back-EMF and electromagnetic torque. The motor embodiment is divided into five physical regions; rotor back iron, magnets region, air-gap, tooth opening and slot region. Maxwell Equations are developed for each subdomain vector potential from which the flux density is derived. The governing equations for vector potential distribution are solved using the Fourier method. This solution provides the vector potential in each subdomain expressed as a Fourier series that is a function of unknown constants. The boundary conditions are emphasized at this stage, the vector potential and magnetic field should be continuous at the interface between two subdomains. A resultant linear system of analytical independent equations relating all the unknown constants is formulated. Then, the permanent magnet flux linkage is computed as an integral of the normal flux density along the stator bore. The coil is considered as either a punctual source of current [27,28] or a current sheet over the slot opening [29] where the slotting effect is approximated. The flux linkage is calculated by an integral of vector potential over the slot area. The flux passing through the conductors is accounted for using this method. Finally, having the flux linkage, the back-EMF and electromagnetic torque will be derived.

The remaining of the paper is organized as follows. In Section 2, the analytical field modeling and derivation with the subdomain method is developed. In Section 3, the electromagnetic performance, including back-EMF waveforms, electromagnetic torque and cogging torque are compared with the FE results, and the impact of magnet ratio on torque improvement is analyzed. Thermal analysis is carried out in Section 4 and conclusions are provided in Section 5.

#### **2. Methods and Mathematical Modeling**

For the purposes of model development, consider the schematic view of a 10-pole/12-slot PM machine Figure 1. The motor has a two-segmented Halbach Array with non-overlapping windings. The 2-D mathematical model developed in this section is based on the following assumptions: (1) infinite permeable iron materials, (2) negligible end effect, (3) linear demagnetization characteristic and full magnetization in the direction of magnetization, (4) non-conductive stator/rotor laminations, and (5) the gaps between magnets have the same constant relative permeability as magnets [23].

**Figure 1.** Symbols and Regions of Subdomain Model with Tooth-tips.

#### *2.1. Vector Potential Distribution*

The magnetic flux density can be expressed as [19]:

$$B = \mu\_0(\mu\_r H + M). \tag{1}$$

and *μ*<sup>r</sup> (H/m) is the relative permeability, *μ*<sup>0</sup> (H/m) is the permeability of vacuum, *M* (A/m) is the magnetization, and *H* [A/m] is the magnetic field intensity. The vector potential can be expressed using the magnetic field as:

$$
\nabla B = -\nabla^2 A.\tag{2}
$$

Substituting *B* from (1) into (2) gives:

$$
\nabla^2 A = -\mu\_0 \mu\_r \nabla \times H - \mu\_0 \nabla \times M = -\mu\_0 \mu\_r I - \mu\_0 \nabla \times M \tag{3}
$$

where *J* (A/m2) is the current density. Provided the eddy current does not influence the field distribution, the vector potential equation in the magnets becomes:

$$
\nabla^2 A = -\mu\_0 \nabla \times \mathcal{M}.\tag{4}
$$

The 2-D field vector potential has only one component along the z-axis that must satisfy the following equations based on (4):

1) Magnet Region:

$$\frac{\partial^2 A\_{z1}}{\partial r^2} + \frac{1}{r} \frac{\partial A\_{z1}}{\partial r} + \frac{1}{r^2} \frac{\partial^2 A\_{z1}}{\partial^2 a^2} = -\mu\_0 \nabla \times M = -\frac{\mu\_0}{r} \left( M\_a - \frac{\partial M\_r}{\partial a} \right) \tag{5}$$

2) Slot Region:

$$\frac{\partial^2 A\_{z3}}{\partial r^2} + \frac{1}{r} \frac{\partial A\_{z3}}{\partial r} + \frac{1}{r^2} \frac{\partial^2 A\_{z3}}{\partial^2 a^2} = -\mu\_0 I \tag{6}$$

3) Air-Gap, Slot Opening and Rotor Back Iron Regions:

$$\frac{\partial^2 A\_{z2,4,5}}{\partial r^2} + \frac{1}{r} \frac{\partial A\_{z2,4,5}}{\partial r} + \frac{1}{r^2} \frac{\partial^2 A\_{z2,4,5}}{\partial^2 a^2} = 0 \tag{7}$$

where *r* is the radial position and *α* is the circumferential position where the d-axis shown in Figure 1 is the origin and anti-clockwise is the forward direction. The radial and circumferential components *Mr* Figure 2 and *Mα* Figure 3 are:

$$M\_{\mathbb{T}} = \sum\_{k=1,3,5,\dots} M\_{rk} \cos(kpa) \tag{8}$$

and:

$$M\_a = \sum\_{k=1,3,5,\dots} M\_{ak} \sin(kpa). \tag{9}$$

**Figure 2.** Radial Component of Magnetization for Two Magnet Ratios (0.5 and 0.75).

**Figure 3.** Circumferential Component of Magnetization for Two Magnet Ratios (0.5 and 0.75).

All the series expansions in this paper are infinite but there will be an introduction of finite harmonic orders in Section 3 for the sake of calculations and matrices inversions. Using the Halbach Array representation, the Fourier series coefficients are defined as:

$$M\_{rk} = 2\frac{B\_r}{\mu\_0} R\_{mp} \frac{\sin\left(R\_{mp}\frac{k\pi}{2}\right)}{R\_{mp}\frac{k\pi}{2}}\tag{10}$$

and:

$$M\_{ak} = -2\frac{B\_r}{\mu\_0} R\_{mp} \frac{\cos\left(R\_{mp}\frac{k\pi}{2}\right)}{R\_{mp}\frac{k\pi}{2}}\tag{11}$$

where *Br* is residual flux density of magnet and *Rmp* is the *magnet ratio* defined as the ratio of the pole arc *β<sup>r</sup>* to pole pitch of a single pole *β<sup>m</sup>* (Figure 4):

$$R\_{mp} = \frac{\beta\_r}{\beta\_m} \tag{12}$$

**Figure 4.** Halbach Array Magnet Ring.

The current density, *J*, can be expressed as:

$$J = J\_{i0} + \sum\_{n} J\_{in} \cos\left[E\_n \left(a + \frac{b\_{sa}}{2} - a\_i\right)\right],\tag{13}$$

$$J\_{i0} = \frac{d(f\_{i1} + f\_{i2})}{b\_{sa}},\tag{14}$$

$$J\_{\rm in} = \frac{2}{n\pi} (f\_{\rm i1} + f\_{\rm i2} \cos(n\pi)) \sin(n\pi d/b\_{\rm sa}) \tag{15}$$

and:

$$E\_n = \frac{n\pi}{b\_{sa}}.\tag{16}$$

where <sup>−</sup>*bsa* <sup>2</sup> <sup>≤</sup> *<sup>α</sup>* <sup>≤</sup> *<sup>α</sup><sup>i</sup>* <sup>+</sup> *bsa* <sup>2</sup> , and *α<sup>i</sup>* and *bsa* are the slot position and the slot width angle respectively. Thus, Equation (5) becomes:

$$\frac{\partial^2 A\_{z1}}{\partial r^2} + \frac{1}{r} \frac{\partial A\_{z1}}{\partial r} + \frac{1}{r^2} \frac{\partial^2 A\_{z1}}{\partial^2 a^2} = -\frac{\mu\_0}{r} \left( \sum\_{k} [(M\_{\text{ack}} - kM\_{\text{rck}}) \cos(ka) + (M\_{\text{ack}} + kM\_{\text{rck}}) \sin(ka)] \right). \tag{17}$$

The general solution for Equation (17) is:

$$\begin{array}{ll} A\_{z1} &= \sum\_{k} \left[ A\_1 \left( \frac{r}{R\_{\text{w}}} \right)^k + B\_1 \left( \frac{r}{R\_{\text{r}}} \right)^{-k} + \frac{\mu\_0 r}{k^2 - 1} (M\_{\text{nck}} - kM\_{\text{rsk}}) \right] \cos(k\alpha) \\ &+ \sum\_{k} \left[ C\_1 \left( \frac{r}{R\_{\text{w}}} \right)^k + D\_1 \left( \frac{r}{R\_{\text{r}}} \right)^{-k} + \frac{\mu\_0 r}{k^2 - 1} (M\_{\text{nck}} + kM\_{\text{rck}}) \right] \sin(k\alpha) \end{array} \tag{18}$$

where *A*1, *B*1, *C*<sup>1</sup> and *D*<sup>1</sup> are coefficients will be determined later and *Rr* and *Rm* are the radii of rotor back iron and magnet surfaces respectively.

The general solution for (7), the vector field in the air-gap, is:

$$A\_{12} = \sum\_{k} \left[ A\_2 \left( \frac{r}{R\_s} \right)^k + B\_2 \left( \frac{r}{R\_m} \right)^{-k} \right] \cos(ka) + \sum\_{k} \left[ C\_2 \left( \frac{r}{R\_s} \right)^k + D\_2 \left( \frac{r}{R\_m} \right)^{-k} \right] \sin(ka) \tag{19}$$

where *A*2, *B*2, *C*<sup>2</sup> and *D*<sup>2</sup> are coefficients to be determined and *Rs* is the radius of inner stator surface. For non-overlapping windings, the general solution of (6), the vector field in the *i th* slot, can be derived

by incorporating the boundary condition along the slot bottom on the surface of infinite permeable lamination when the circumferential component flux density is zero, namely:

$$A\_{z3i} = A\_0 + \sum\_{n} A\_n \cos \left[ E\_n \left( a + \frac{b\_{sa}}{2} - a\_i \right) \right] \tag{20}$$

where:

$$A\_0 = \frac{\mu\_0 l\_{i0}}{4} \left( 2R\_{sb}^2 \ln(r) - r^2 \right) + Q\_{3i\prime} \tag{21}$$

$$A\_{\rm il} = D\_{\rm 3i} \left[ G\_3 \left( \frac{r}{R\_{sb}} \right)^{E\_n} + \left( \frac{r}{R\_t} \right)^{-E\_n} \right] + \mu\_0 \frac{I\_{\rm in}}{E\_n^2 - 4} \left[ r^2 - \frac{2R\_{sb}^2}{E\_n} \left( \frac{r}{R\_{sb}} \right)^{E\_n} \right] \tag{22}$$

And:

$$G3 = \left(\frac{R\_t}{R\_{sb}}\right)^{E\_w} \tag{23}$$

*Q*3*<sup>i</sup>* and *D*3*<sup>i</sup>* are coefficients to be derived using regions interactions. In the *i* th slot opening, the vector field is derived by considering the boundary condition on both sides of the slot opening:

$$A\_{z4i} = D\ln(r) + Q\_{4i} + \sum\_{m} \left[ \mathbb{C}\_{4i} \left( \frac{r}{R\_I} \right)^{F\_m} + D\_{4i} \left( \frac{r}{R\_s} \right)^{-F\_m} \right] \cos \left( F\_{\text{fl}} \left( a + \frac{b\_{o4}}{2} - a\_i \right) \right) \tag{24}$$

where *D*, *Q*4*i*, *C*4*<sup>i</sup>* and *D*4*<sup>i</sup>* are unknown coefficients to be determined, *boa* is the slot opening width and *Fm* is defined as follows:

$$F\_m = \frac{m\pi}{b\_{ox}}.\tag{25}$$

The general solution for the vector field in the rotor back iron is:

$$A\_{25} = \sum\_{k} \left[ A\_5 \left( \frac{r}{R\_r} \right)^k + B\_5 \left( \frac{r}{R\_0} \right)^{-k} \right] \cos(ka) + \sum\_{k} \left[ C\_5 \left( \frac{r}{R\_r} \right)^k + D\_5 \left( \frac{r}{R\_0} \right)^{-k} \right] \sin(ka) \tag{26}$$

where *A*5, *B*5, *C*<sup>5</sup> and *D*<sup>5</sup> are coefficients to be determined later, *R*<sup>0</sup> is the radius of inner rotor back iron surface.

The unknown coefficients in the expressions of vector potentials (18)–(26) are determined by applying the continuations of normal flux density and circumferential vector potential between subdomains. The details derivations are presented in the following section.

#### *2.2. Boundary Conditions*

The radial and circumferential components of flux density is calculated from the vector potential distribution as:

$$B\_r = \frac{1}{r} \frac{\partial A\_z}{\partial \alpha} \text{ and } B\_\alpha = -\frac{\partial A\_z}{\partial r}. \tag{27}$$

In the magnet region, the magnetic flux density is:

$$B = \mu\_0 \mu\_\mathrm{Tr} H + \mu\_0 M.\tag{28}$$

while it is expressed in the rotor back iron, air gap, slot and slot opening as:

$$B = \mu\_0 H.\tag{29}$$

#### 2.2.1. Interface Between Air and Rotor Back Iron

Applying (29) to calculate the circumferential magnetic field intensity *H*5*<sup>α</sup>* in rotor back iron, the boundary condition on the surface requires:

$$\left.H\_{\mathfrak{K}}\right|\_{r=R\_{\mathbb{Q}}} = \left.\frac{1}{\mu\_{0}\mu\_{r}}B\_{\mathfrak{K}}\right|\_{r=R\_{\mathbb{Q}}} = 0.\tag{30}$$

Based on (26) and (27), *B*5*<sup>α</sup>* expression is calculated and then applied in (30) which leads to the following equations:

$$\begin{cases} B\_5 = G\_5 A\_5\\ D\_5 = G\_5 C\_5 \end{cases} \tag{31}$$

where:

$$G\_5 = \left(\frac{R\_0}{R\_r}\right)^k. \tag{32}$$

2.2.2. Interface Between Rotor Back Iron and Halbach Permanent Magnets

The first boundary condition in this surface requires that the circumferential magnetic field intensity is continuous between the rotor back iron and Halbach Array permanent magnets giving:

$$\left.H\_{\text{5a}}\right|\_{r=R\_r} = \left.H\_{1\text{a}}\right|\_{r=R\_r}.\tag{33}$$

Referring to (28) *H*1*<sup>α</sup>* is calculated as:

$$H\_{1a} = \frac{1}{\mu\_r} \left(\frac{B\_{1a}}{\mu\_0} - M\_a\right) \tag{34}$$

substituting (34) into (33) gives the expressions:

$$\begin{cases} A\_1 G\_1 - B\_1 + A\_5 (G\_5^2 - 1) = \frac{\mu\_0 R\_r}{k^2 - 1} (M\_{rsk} - kM\_{ack}) \\\ ^\circ \text{C}\_1 G\_1 - D\_1 + \text{C}\_5 (G\_5^2 - 1) = -\frac{\mu\_0 R\_r}{k^2 - 1} (M\_{rck} + kM\_{ask}) \end{cases} \tag{35}$$

The second boundary condition is that the normal flux density is continuous between rotor back iron and Halbach Array permanent magnets. *B*5*<sup>r</sup>* is derived from Equations (27) and (29) while *B*1*<sup>r</sup>* is calculated from Equations (27) and (28):

$$\left.B\_{5r}\right|\_{r=R\_r} = \left.B\_{1r}\right|\_{r=R\_r} \tag{36}$$

leading to:

$$\begin{cases} A\_1 G\_1 + B\_1 - A\_5 (G\_5^2 + 1) = \frac{\mu\_0 R\_r}{k^2 - 1} (kM\_{rsk} - M\_{ack}) \\\ \mathcal{C}\_1 G\_1 + D\_1 - \mathcal{C}\_5 (G\_5^2 + 1) = -\frac{\mu\_0 R\_r}{k^2 - 1} (kM\_{rck} + M\_{ask}) \end{cases} \tag{37}$$

#### 2.2.3. Interface Between Air Gap and Halbach Permanent Magnets

Firstly, the circumferential magnetic field intensity is continuous between the air gap and the Halbach Array permanent magnets:

$$H\_{1a}|\_{r=R\_m} = \left. H\_{2a} \right|\_{r=R\_m}.\tag{38}$$

Referring to (29), *H2<sup>α</sup>* is calculated as:

$$H\_{2a} = \frac{B\_{2a}}{\mu\_0}.\tag{39}$$

Substituting (39) into (38) and given *H*1*<sup>α</sup>* from (34), we conclude:

$$\begin{cases} -A\_1 + B\_1 G\_1 + \mu\_I A\_2 G\_2 - \mu\_I B\_2 = \frac{\mu\_0 R\_m}{k^2 - 1} (kM\_{ack} - M\_{rsk}) \\ -C\_1 + D\_1 G\_1 + \mu\_I C\_2 G\_2 - \mu\_I B\_2 = \frac{\mu\_0 R\_m}{k^2 - 1} (M\_{rck} + kM\_{ask}) \end{cases} \tag{40}$$

Secondly, the normal flux density is continuous between the air gap and the Halbach Array permanent magnets:

$$\left.B\_{2r}\right|\_{r=R\_m} = \left.B\_{1r}\right|\_{r=R\_m} \tag{41}$$

from which the following expressions can be derived:

$$\begin{cases} A\_1 + B\_1 G\_1 - B\_2 - A\_2 G\_2 = \frac{\mu\_0 R\_m}{k^2 - 1} (k M\_{rsk} - M\_{ack}) \\\ C\_1 + D\_1 G\_1 - D 2\_2 - C\_2 G\_2 = -\frac{\mu\_0 R\_m}{k^2 - 1} (k M\_{rck} + M\_{ask}) \end{cases} \tag{42}$$

2.2.4. Interface Between Air Gap and Slot Opening

Based on (24) and (27), the circumferential component flux density within the slot opening:

$$B\_{4ia} = -\frac{D}{r} - \sum\_{m} \frac{F\_m}{R\_s} \left[ \mathbf{C}\_{4i} \left( \frac{R\_s}{R\_l} \right) \left( \frac{r}{R\_l} \right)^{F\_m - 1} - D\_{4i} \left( \frac{r}{R\_s} \right)^{-F\_m - 1} \right] \cos \left( F\_m \left( a + \frac{b\_{0a}}{2} - a\_i \right) \right). \tag{43}$$

Since the stator core material is infinitely permeable which make the circumferential component flux density *Bs<sup>α</sup>* along the stator bore outside the slot opening equal to zero. Along the stator bore, the circumferential component flux density is expressed as [23]:

$$B\_{sa} = \sum\_{k} [\mathbb{C}\_s \cos(ka) + D\_s \sin(ka)]\tag{44}$$

where:

$$\mathbf{C}\_{s} = \sum\_{i} \sum\_{m} \left( -\frac{F\_{m}}{R\_{s}} \left( \mathbf{C}\_{4i} \mathbf{G}\_{4} - D\_{4i} \right) \right) \eta\_{i} + \sum\_{i} \left( -\frac{D}{R\_{s}} \right) \eta\_{i0\prime} \tag{45}$$

$$D\_{\rm s} = \sum\_{i} \sum\_{m} \left( -\frac{F\_{\rm m}}{R\_{\rm s}} (\mathbf{C}\_{4i}\mathbf{G}\_{4} - D\_{4i}) \right) \overline{\xi}\_{i} + \sum\_{i} \left( -\frac{D}{R\_{\rm s}} \right) \overline{\xi}\_{i0\prime} \tag{46}$$

$$\eta\_{i}(m,k) = -\frac{k}{\pi(F\_{m}^{2} - k^{2})} [\cos(m\pi)\sin(ka\_{i} + kb\_{aa}/2) - \sin(ka\_{i} - kb\_{aa}/2)],\tag{47}$$

$$\xi\_i^x(m,k) = \frac{k}{\pi(F\_m^2 - k^2)} [\cos(m\pi)\cos(ka\_i + kb\_{aa}/2) - \cos(ka\_i - kb\_{aa}/2)],\tag{48}$$

$$\eta\_{i0}(k) = 2\sin(kb\_{\text{on}}/2)\frac{\cos(ka\_i)}{k\pi} \tag{49}$$

and:

$$q\_{i0}^{\pi}(k) = 2\sin(kb\_{\text{on}}/2)\frac{\sin(ka\_i)}{k\pi}.\tag{50}$$

The first boundary condition demands that the circumferential flux density is continuous between the air gap and the slot opening giving:

$$\left.B\_{2a}\right|\_{r=R\_s} = \left.B\_{8a}\right|\_{r=R\_s}.\tag{51}$$

Having *Bs<sup>α</sup>* from (44) and calculating *B2<sup>α</sup>* from (19) and (27), condition (51) leads to:

$$\begin{cases} \begin{array}{c} R\_s \mathbb{C}\_s = -kA\_2 + kG\_2B\_2 \\ R\_s D\_s = -kC\_2 + kG\_2D\_2 \end{array} . \end{cases} \tag{52}$$

*Energies* **2018**, *11*, 3254

The vector potential distribution within air gap along the stator bore is given as

$$A\_{\mathfrak{s}} = A\_{\mathfrak{z}2}|\_{r=R\_{\mathfrak{s}}} = \sum\_{k} [(A\_{\mathfrak{z}} + B\_{\mathfrak{z}}G\_{\mathfrak{z}})\cos(ka) + (\mathcal{C}\_{\mathfrak{z}} + D\_{\mathfrak{z}}G\_{\mathfrak{z}})\sin(ka)].\tag{53}$$

The vector potential distribution over the slot opening along the stator bore is

$$\begin{array}{ll} A\_{\mathcal{S}} &= \left( \sum\_{k} ((A\_{\mathcal{Q}} + B\_{\mathcal{Q}} G\_{\mathcal{Q}}) \sigma\_{i0} + (\mathcal{C}\_{\mathcal{Q}} + D\_{\mathcal{Q}} G\_{\mathcal{Q}}) \tau\_{i0}) \right) \\ &+ \sum\_{m} \left[ \left( \sum\_{k} ((A\_{\mathcal{Q}} + B\_{\mathcal{Q}} G\_{\mathcal{Q}}) \sigma\_{i} + (\mathcal{C}\_{\mathcal{Q}} + D\_{\mathcal{Q}} G\_{\mathcal{Q}}) \tau\_{i}) \right) \cos(F\_{m} (a + b\_{aa}/2 - a\_{i})) \right] \end{array} \tag{54}$$

where

$$
\sigma\_{i0} = \left(\frac{\pi}{b\_{\text{on}}}\right) \eta\_{i0}(k), \tag{55}
$$

$$
\pi\_{i0} = \left(\frac{\pi}{b\_{\alpha i}}\right) \zeta\_{i0}(k),
\tag{56}
$$

$$
\sigma\_i = \left(\frac{2\,\pi}{b\_{0a}}\right) \eta\_i(m,k),
\tag{57}
$$

and

$$
\pi\_i = \left(\frac{2\pi}{b\_{aa}}\right) \zeta\_i(m, k). \tag{58}
$$

The vector potential distribution within slot opening along the stator bore is

$$\left.A\_{24i}\right|\_{r=R\_s} = D\ln(R\_s) + Q\_{4i} + \sum\_{m} [\mathbf{C}\_{4i}\mathbf{G}\_4 + D\_{4i}]\cos\left(F\_m\left(a + \frac{b\_{on}}{2} - a\_i\right)\right) \tag{59}$$

where

$$G\_4 = \left(\frac{R\_s}{R\_t}\right)^{F\_m}.\tag{60}$$

The second boundary condition involves the continuation vector potential for *<sup>α</sup><sup>i</sup>* <sup>−</sup> *bsa* <sup>2</sup> <sup>≤</sup> *<sup>α</sup>* <sup>≤</sup> *<sup>α</sup><sup>i</sup>* <sup>+</sup> *bsa* 2 is

$$A\_{z4i}|\_{r=R\_s} = A\_s \tag{61}$$

thus

$$Q\_{4i} = \sum\_{k} [(A\_2 + B\_2 G\_2)\sigma\_{i0} + (C\_2 + D\_2 G\_2)\tau\_{i0}] - D\ln(R\_s) \tag{62}$$

and

$$D\_{4i} + C\_{4i}G\_4 = \sum\_k [(A\_2 + B\_2 G\_2)\sigma\_i + (C\_2 + D\_2 G\_2)\tau\_i].\tag{63}$$

#### 2.2.5. Interface Between Slot Opening and the Slot

Along the interface between the slot and the slot opening, the circumferential component of the flux density is derived from (24) and (27) [1]

$$\left.B\_{4ia}\right|\_{r=R\_t} = B\_{4ia0} + \sum\_m B\_{4iau} \cos(F\_m(a + b\_{0a}/2 - a\_i))\tag{64}$$

where

$$B\_{4\dot{m}0} = -\frac{D}{R\_t} \tag{65}$$

and

$$B\_{4\bar{u}m} = -\frac{F\_m}{R\_1}(\mathbb{C}\_{4i} - D\_{4i}\mathbb{G}\_4) \tag{66}$$

The circumferential flux density in the slot along the outer radius of the slot opening is null since the stator core material is infinitely permeable. This component along *Rt* is expressed into Fourier series over *<sup>α</sup><sup>i</sup>* <sup>−</sup> *bsa* <sup>2</sup> <sup>≤</sup> *<sup>α</sup>* <sup>≤</sup> *<sup>α</sup><sup>i</sup>* <sup>+</sup> *bsa* 2 as

$$B\_{4\text{ia}}|\_{r=R\_l} = B\_0 + \sum\_{n} B\_n \cos \left( E\_{ll} (a + b\_{54}/2 - a\_i) \right) \tag{67}$$

where

$$B\_0 = B\_{4ia0} \varphi\_{c\mathbf{x}\prime} \tag{68}$$

$$B\_n = B\_{4i\alpha 0} q\_0 + \sum\_m B\_{4i\alpha m} q\_\prime \tag{69}$$

$$
\varphi\_{cx} = \frac{b\_{ox}}{b\_{sa}}\tag{70}
$$

$$\varphi\_0(n) = \frac{4}{n\pi} \cos(n\pi/2) \sin(E\_n b\_{\text{on}}/2) \tag{71}$$

and

$$\varphi(m,n) = -\frac{2}{b\_{sa}} \frac{E\_n}{F\_m^2 - E\_n^2} \left[ \cos(m\pi) \sin\left(E\_n \frac{b\_{sa} + b\_{oa}}{2}\right) - \sin\left(E\_n \frac{b\_{sa} - b\_{oa}}{2}\right) \right].\tag{72}$$

The circumferential flux density in the slot opening along the interface between the slot opening and the slot is

$$B\_{\text{3ia}}|\_{r=R\_t} = B\_{\text{3ia}0} + \sum\_{n} B\_{\text{3ia}n} \cos(E\_n(a + b\_{\text{sa}}/2 - a\_i))\tag{73}$$

where for non-overlapping winding

$$B\_{3\text{in}0} = -\frac{\mu\_0 I\_{i0} \left(R\_{sb}^2 - R\_t^2\right)}{2R\_t} \tag{74}$$

and

$$B\_{3\dot{n}n} = -\frac{E\_n D\_{3i} (G\_3^2 - 1)}{R\_t} - \frac{2\mu\_0 I\_{in}}{R\_t (E\_n^2 - 4)} \left( R\_t^2 - R\_{sb}^2 G\_3 \right). \tag{75}$$

Applying the continuity of the circumferential component of flux density gives

$$B\_{3\dot{a}a0} = B\_{4\dot{a}a0} \varphi\_a \tag{76}$$

and

$$B\_{3iam} = B\_{3iam} \varrho\_0 + \sum\_m B\_{4iam} \varrho. \tag{77}$$

From (65), (71), (74) and (76)

$$D = \frac{\mu\_0 I\_{i0} \left(R\_{sb}^2 - R\_t^2\right)}{2} \left(\frac{b\_{sa}}{b\_{oa}}\right). \tag{78}$$

The vector potential distribution in the slot along the radius *Rt* is

$$A\_{l} = A\_{z3i}|\_{r=R\_{l}} = A\_{3i0} + \sum\_{n} A\_{3in} \cos\left[E\_{n}\left(a + \frac{b\_{sa}}{2} - a\_{i}\right)\right] \tag{79}$$

where for non-overlapping winding

$$A\_{3i0} = \frac{\mu\_0 I\_{i0} \left(2R\_{sb}^2 \ln(R\_t) - R\_t^2\right)}{4} + Q\_{3i} \tag{80}$$

*Energies* **2018**, *11*, 3254

and

$$A\_{3in} = D\_{3i} \left( G\_3^2 + 1 \right) + \frac{\mu\_0 I\_{in}}{\left( E\_n^2 - 4 \right)} \left( R\_t^2 - \frac{2 R\_{sb}^2 G\_3}{E\_n} \right). \tag{81}$$

The same vector potential is expressed over *<sup>α</sup><sup>i</sup>* <sup>−</sup> *bsa* <sup>2</sup> <sup>≤</sup> *<sup>α</sup>* <sup>≤</sup> *<sup>α</sup><sup>i</sup>* <sup>+</sup> *bsa* 2 

$$A\_{l} = A\_{340} + \sum\_{\text{m}} A\_{34n} \cos \left[ F\_{\text{m}} \left( \alpha + \frac{b\_{\text{out}}}{2} - a\_{l} \right) \right] \tag{82}$$

where

$$A\_{340} = \sum\_{n} A\_{3in} \kappa\_0 + \frac{\mu\_0}{4} I\_{l0} \left( 2R\_{sb}^2 \ln(R\_t) - R\_t^2 \right) + Q\_{3i\prime} \tag{83}$$

$$A\_{3tm} = \sum\_{n} A\_{3in} \kappa\_{\prime} \tag{84}$$

$$\kappa\_0 = \left(\frac{b\_{sd}}{b\_{\alpha a}}\right) q\_0(n) \tag{85}$$

and

$$
\kappa = \left(\frac{b\_{sa}}{b\_{0a}}\right) \wp(m, n). \tag{86}
$$

From the vector potential within the slot opening

$$\left.A\_{z4i}\right|\_{r=R\_l} = D\ln(R\_l) + Q\_{4i} + \sum\_{m} \left(C\_{4i} + D\_{4i}G\_4\right) \cos\left(F\_m \left(a + \frac{b\_{o4}}{2} - a\_i\right)\right). \tag{87}$$

Applying the continuity of the vector potential along the interface between the slot opening and the slot

$$A\_t = A\_{z4i}|\_{r=R\_t}.\tag{88}$$

The following equations can now be developed

$$Q\_{3i} = Q\_{4i} + D \ln(R\_t) - \frac{\mu\_0 I\_{i0}}{4} \left( 2R\_{sb}^2 \ln(R\_t) - R\_t^2 \right) - \sum\_{\mathbf{n}} A\_{3\mathbf{in}} \mathbf{x}\_0 \tag{89}$$

and

$$\mathcal{C}\_{4i} + D\_{4i}\mathcal{G}\_{4} = \sum\_{n} A\_{3in}\kappa.\tag{90}$$

#### *2.3. Back EMF and Torque Calculations*

From the field solutions generated in the previous subsection, relevant electromagnetic characteristics, such as the flux linkage and the back EMF, are obtained based on the subdomain models. Also developed are expressions for the electromagnetic torque and cogging torque.

#### 2.3.1. Flux Linkage

Flux linkage occurs when a magnetic field interacts with a material such as a magnetic field travels through a coil of wire. The flux linkage from one coil side of an arbitrary winding is calculated from the magnetic vector potential in the general equation for flux linkage. The calculation can be simplified by means of the Stokes integral theorem:

$$\psi\_{i1} = \frac{LN\_{\text{c}}}{A\_{\text{s}}} \iint\_{\text{S}} \stackrel{\rightarrow}{B} \cdot d\stackrel{\rightarrow}{\text{s}} = \frac{LN\_{\text{c}}}{A\_{\text{s}}} \iint\_{\text{S}} \nabla \times \stackrel{\rightarrow}{A} \cdot d\stackrel{\rightarrow}{\text{s}} = \frac{LN\_{\text{c}}}{A\_{\text{s}}} \oint\_{\text{C}} \stackrel{\rightarrow}{\text{A}} \cdot d\stackrel{\rightarrow}{\text{s}} = \frac{LN\_{\text{c}}}{A\_{\text{s}}} \int\_{a\_{\text{i}} - \hat{b}\_{\text{s}t}/2}^{a\_{\text{i}} - \hat{b}\_{\text{s}t}/2 + \hat{d}\cdot\hat{R}\_{\text{i}}} \int\_{R} A\_{\text{z}2\hat{i}} \cdot r dr da \tag{91}$$

where *L* is the stack length of the motor, *As* is the area of one coil side and *Nc* is the number of turns per coil.

Based on *Az*3*<sup>i</sup>* in (20) and (91):

$$\Psi\_{i1} = \frac{LN\_c}{A\_s} \left[ Q\_0 d + \sum\_n \frac{Q\_n}{E\_n} \sin(E\_n d) \right] \tag{92}$$

where:

$$Q\_0 = \int\_{R\_l}^{R\_{sb}} A\_0 \cdot r dr \tag{93}$$

and:

$$Q\_n = \int\_{R\_\ell}^{R\_{sb}} A\_n \cdot r dr. \tag{94}$$

Following the same approach, the flux linkage in the other coil side of the same slot is:

$$\psi\_{i2} = \frac{LN\_{\varepsilon}}{A\_{\varepsilon}} \left[ Q\_0 d - \sum\_{n} \frac{Q\_n}{E\_n} \sin(n\pi - E\_n d) \right]. \tag{95}$$

The summation of the flux linkages associated with all the coil sides of the corresponding phase results in the total flux linkage of each phase. This calculation includes a connection matrix *Sw* that represents the winding distribution in the slot:

$$
\begin{bmatrix}
\psi\_a\\ \psi\_b\\ \psi\_c
\end{bmatrix} = S\_w \psi\_c. \tag{96}
$$

Presented is an estimation method for the flux linkage. These results will be used to estimate the phase back-EMF for the Halbach motor.

#### 2.3.2. Back EMF

Based on Lenz's law, the induced EMF always counters the source of its creation. Here, the source of back-EMF is the rotation of armature. The torque produces rotation of armature. Torque is due to armature current and armature current is created by the supply voltage. Therefore, the ultimate cause of production of the back EMF is the supply voltage. The three-phase back-EMF vector is calculated by the derivative of flux linkage calculated in (96) with respect to time when the motor is in open circuit:

$$
\begin{bmatrix} E\_a \\ E\_b \\ E\_c \end{bmatrix} = \omega \begin{bmatrix} \frac{d\Psi\_a}{dt} \\ \frac{d\Psi\_b}{dt} \\ \frac{d\Psi\_c}{dt} \end{bmatrix}. \tag{97}
$$

#### 2.3.3. Electromagnetic and Cogging Torques

The electromagnetic torque is calculated as:

$$\frac{T\_{c\text{in}} = (E\_a I\_a + E\_b I\_b + E\_c I\_c)}{\omega}. \tag{98}$$

Concerning the cogging torque, many methods have been used such as the lateral force [30], complex permeance [31,32] and energy [33,34]. In PM machines, calculating and minimizing the cogging torque is classically evaluated using either the virtual work or Maxwell stress tensor methods. In this development, the Maxwell stress tensor is proposed providing accurate prediction of the air-gap field by way of the subdomain method accounting for tooth-tips. The derivation is based on the open circuit field:

$$T\_{\varepsilon} = \left(\frac{LR\_s^2}{\mu\_0}\right) \int\_0^{2\pi} B\_{2r} B\_{2a} d\alpha. \tag{99}$$

#### **3. Results and Model Validation**

Validation of the proposed electromagnetic analytical model depends on the modeling accuracy to predict motor torque with slot and tooth tip effects. The proposed validation is based on finite element analysis (FEA) simulations, no experimental results are presented. A Halbach Array machine shown in Figure 1 and a conventional one having both a three-phase 12-slot/10-pole embodiment are adopted to validate the analytical subdomain model developed in this manuscript. In the present validation, the simulation on the conventional and the Halbach Array SPM motors is performed by the help of the commercial FEA software MotorSolve v6.01 (Infolytica Corporation, Montréal, QC, Canada). The motor design parameters applied in the present study are provided in Table 1.


**Table 1.** Main Motor Parameters.

The non-overlapped winding layout is shown in Figure 5. The use of this configuration in SPM motors to meet the optimal flux weakening condition is supported. Therefore, machines made with concentrated windings have shorter end coils, which reduce the total copper ohmic losses, the length and the weight of the machine. In addition, this configuration is chosen to maximize the winding factor of 93.3% with a coil span equal to unity [32]. The results for the Fourier series expansion are computed with a finite number of harmonics for k, m, and n. Generally, the limited number of harmonics is considered in the analytical model to generate suitable results, while the mesh in the FE model must be adjusted before achieving acceptable results. The computation time of the analytical field solutions is dependent from the harmonic numbers (k, m, n), for (100, 50, 50), by way of example, the average calculation time is 14.3 s (with 2.3 GHz CPU) on the MatLab platform. For the Halbach Array magnets configuration, when the magnet ratio *Rmp*, equal to 0.5, the predicted 2-D analytical radial flux density and MotorSolve simulation in the centerline of air gap position are presented in Figure 6. The resulting air gap flux density waveform distributions are distorted at the slot-openings showing that the analytical subdomain model is able to capture the slot-openings and tooth tips effects. The proposed model predicts the flux density within the considered five subdomains. MotorSolve only provides the flux density prediction for only the air-gap. The validation of flux density would

promote further analysis in the demagnetization withstand ability for different magnet ratio cases and inductance prediction.

**Figure 5.** Non-Overlapping Winding Layout in MotorSolve.

**Figure 6.** Comparison of FEA and Analytically Predicted Air-gap Flux Density, *Rmp* = 0.5.

By varying the magnet ratio, the proposed model can predict a variety of magnetization patterns as shown in Figure 7. It should be noted that the conventional SPM magnet has a magnet ratio *Rmp* equal to unity, while the magnet ratio should be 0.5 for the model shown in Figure 1, when the side and the central magnet segments have the same size. The highest flux density peak value (FDPV) is given by lowest Halbach Array magnet ratio, which is approximately 10% higher than the conventional magnetization. By increasing the magnet ratio, the FDPV decreases and the shape of the flux becomes wider within each pole range. This is because the central magnet with radial magnetization becomes wider as the magnet ratio increases. The air gap flux density waveform reaches its widest and lowest peak amplitude in the magnet ratio of unity.

**Figure 7.** Comparison of Analytically Predicted Air-gap Flux Density with Different Magnet Ratio Values.

The stator windings are excited with sinusoidal currents with a peak value of 10A. Shown in Figures 8–10 are the comparisons between the analytically calculated and FEA simulated waveforms of electromagnetic torque, cogging torque and back-EMF, respectively.

**Figure 8.** Analytical and MotorSolve FEA Predictions of Electromagnetic in the 12 s/10 p Halbach Machine with *Rmp* = 0.5 (average deviation: 1.8%).

**Figure 9.** Analytical and MotorSolve Predictions of Cogging Torque in the 12 s/10 p Halbach Machine with *Rmp* = 0.5.

**Figure 10.** Analytical and MotorSolve Predictions of Back EMF in the 12 s/10 p Halbach Machine with *Rmp* = 0.5.

The analytical results provide good agreements with those obtained from FEA simulation. Shown in Figure 11 are the flux lines with different magnet ratio values. The flux lines are densely low in the passive region (rotor back iron) for the magnet ratio 0.5. When increasing the magnet ratio, the FDPV increases to the value of conventional magnet pattern.

**Figure 11.** Magnetic Field Distributions across the Motor Area: (**a**) *Rmp* = 0.5; (**b**) *Rmp* = 0.7; (**c**) *Rmp* = 1.

For comparison purposes, the electromagnetic performance for the conventional (non-Halbach) motor design with the same parameters in Table 1 is shown in Figures 12–14. All results for the analytical calculations are in agreement with FEA simulations.

**Figure 12.** Analytical and MotorSolve FEA Predictions of Electromagnetic in the 12 s/10 p Conventional Machine (average deviation: 1.6%).

**Figure 13.** Analytical and MotorSolve Predictions of Cogging Torque in the 12 s/10 p Conventional Machine.

**Figure 14.** Analytical and MotorSolve Predictions of Back EMF in the 12 s/10 p Conventional Machine.

The back-EMF for Halbach Array motor is more sinusoidal than conventional surface-mounted permanent magnet motor. As a result, the Halbach Array machine, as shown in Figure 9 (*Rmp* = 0.5), provides much lower cogging torque (with a peak magnitude of 0.015 Nm) comparing to conventional motor in Figure 13 (with a peak magnitude of 0.11 Nm).

Comparison motor torques among the different machine designs is presented in Figure 15. Among the magnet ratios of 0.5, 0.667, 0.75, 0.8 and 1, the design for 0.8 provides the highest torque, which is approximately 10% higher than the conventional design (*Rmp* = 1) in this particular configuration while it has reached 25% for other cases. Also note that the design with a magnet ratio of 0.5 actually generates 4.5% less torque comparing to the convention magnet configuration. These results demonstrate that the maximization process of electromagnetic torque is nonlinear with respect to the magnet ratio. Only with an appropriate and optimal magnet ratio can the Halbach Array design provide a higher torque compared to the conventional SPM motor given the same size and supply constraints. From such calculations, it suggests that the PM motor output torque is related to the integration of the half cycle of flux linkage for each coil; and the optimal magnet ratio to maximize the torque requires the maximum integration of this flux linkage for each coil, under the same input conditions [3]:

$$\frac{\partial \int\_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \psi\_{coil} d\theta}{\partial R\_{mp}} = 0.\tag{100}$$

**Figure 15.** Comparison of Analytically Predicted Torque with Different Magnet Ratio Values.

#### **4. Thermal Analysis**

In addition to the electromagnetic modeling, a thermal analysis of the reported 10-pole SPM motor is performed using FEA simulation. Such an analysis is to provide illuminated evaluations for the heat generation and its transfer within the motor embodiment during operation. As a result, the temperature distribution within the motor assembly can be predicted. Overheat spots on copper windings or thermal-sensitive electric components are regarded as constraints to the motor embodiment and will be examined. Unless sufficient heat could be removed from the motor assembly, or limited current magnitude would be allowed for necessary cooling to suffice, over-heat or melting would happen in the motor material and cause malfunction of operation, or even bring about safety issues.

In the present study, the thermal analysis and evaluation are performed under the requirements that the maximum temperature within the motor assembly be lower than 180 ◦C (any spots with a temperature higher than 180 ◦C will be considered as motor overheat). There are two main sources for the heat generation while a PM motor machine operates, which are often referred as the copper loss and the iron loss. Copper losses are generally the heat losses generated within the copper winding conductor because of its carried electric current, which are proportional to the winding resistance and the square of the current magnitude. Generally, lower current values or winding resistance (e.g., shorter total length or fewer numbers of turns in stator) will reduce the heat caused by the copper loss. Also, the limitation of the total motor weight would require less use of copper in windings, which could minimize its cross-sectional area, increase the conductor resistance and therefore increase heating.

For the iron loss, two mechanisms occur, namely eddy current loss and hysteresis loss. The eddy current loss is produced by relative movement between conductors (stator iron core) and magnetic flux lines. The hysteresis loss is due to reversal of magnetization of stator iron core whenever it is subjected to changing magnetic forces. The iron loss depends on variables such as the thickness of the laminations in the stator, the magnitude of the flux density, the stator iron material resistivity, and most importantly, the frequency at which the motor operates. To use a laminated stator and decrease the thickness of the laminations with electrical insulation from each other could reduce the iron loss.

The modeling of the iron loss power *Piron* per unit volume of the stator material in this study was performed as:

$$P\_{iron} = k\_t B\_p^2 f^2 + k\_{cx} B\_p^{1.5} f^{1.5} + k\_h B\_p^2 f \tag{101}$$

where *ke*, *kex* and *kh* are the coefficients of classical eddy current loss, excess eddy current loss and hysteresis loss, respectively. The variable *Bp* is the peak flux density in the three-dimensional (3-D) stator domain and f is the motor supply frequency. The coefficients here for iron losses are related to the given stator material properties from manufacturers, and the integration of the iron heat loss within the entire stator body is calculated in FE model using above equation performed in the stator teeth and yoke (back iron) domains.

In the present study, Hiperco 50A alloy is used as the motor core material and the stator slot encapsulation material (CW 2710/HW 2711, Huntsman, Woodlands, TX, USA) is used for slot potting and thermal conduction enhancement. Further studies regarding different motor materials can be performed. The temperature distribution of the present SPM motor (10p/12s) is modeled to examine whether overheat occurs with present conditions. Shown in Figure 16a is the time-based temperature variation for different motor components, under the operating conditions mentioned in Table 1; Figure 16b shows the axisymmetric temperature distribution in the 3-D motor model, provided by MotorSolve FEA simulation. The ambient temperature for this case study is set to be 40 ◦C. All three heat transfer mechanisms, conduction (within the motor solid assembly), convection (on motor external surface) and radiation (on motor external surface) are considered. In this case shown in Figure 16, no overheat is observed under the present operating condition. Taking into account the end winding heat effect, the maximum temperature would occur within the copper windings and their ends, where the copper heat loss is generated and has the longest distance (length of thermal transfer path) to the low-temperature ambient.

**Figure 16.** Motor Thermal Analysis Results: (**a**) Time-based Temperature Profiles for Various Motor Components; (**b**) Temperature Contours for 3D Thermal Modeling at 600 min.

Although the configuration of Halbach Array magnet and its advantages are reported previously, the torque dense capability of a Halbach Array motor is not widely realized. In the present study, Halbach Array SPM motors are verified to be able to produce higher torque than a conventional SPM with the same magnet volume and at the same rotor speed, provided the optimal Halbach Array magnet ratio, *Rmp*, is applied. It is also noted that compared to induction motor (IM) machines, the permanent magnet (PM) motors, including SPM and IPM (interior permanent magnet) machines, are able to achieve greater torque/current density, also with higher efficiency and reliability in low speed operation [35].

On the other hand, the general torque production also implies that in these motor categories, lower supply current may be required to retain the same torque performance using the Halbach Array SPM motor compared to others including the standard IM motors. As a result, lower supply current will present lower copper heat loss. Therefore, for industrial applications such as subsea electrical submersible pumps (ESP) or progressive cavity pumps (PCP), which require of driving motors with high torque density at a relative lower speed [36], the Halbach Array SPM motor is a good choice for its ability of high torque dense performance, with lower heat loss and higher efficiency. Such high-torque, low-speed motors could also eliminate the necessity of using a gear-reduction unit to reduce the high rotational speed driven by an IM machine, and reduce its mechanical loss and possible failures.

#### **5. Conclusions**

Developed in this paper is a subdomain model accounting for tooth-tip and slot effects for surface mounted permanent magnet (SPM) Halbach Array motor. The proposed analytical model can accurately predict the electromagnetic flux field distributions in motor applications of conventional and Halbach Array SPM machines. Based on the resultant magnetic field, the electromagnetic performances of the motor, such as the cogging torque, back-EMF, and electromagnetic torque are accurately calculated. The FEA analysis validates the predictions of the proposed analytical model. At this stage, a 2-D model is developed. For future work, there are certain difficulties to perform simulations in 3-D basis; hence an experimental setup is needed to validate the model.

Due to the feature of Halbach Array magnet configuration, the magnetic flux field is augmented on one side (outward of rotor), while cancelled on the other side (inward of rotor). In the present study, much less flux saturation is observed in rotor back iron with Halbach Array configuration compared to the conventional ones. Thus in Halbach Array design, more magnets could be installed in the rotor to achieve higher torque, with lower risk to cause flux saturation within the rotor back iron; on the other hand, less back iron material is needed in the Halbach Array designed rotor.

Also, we conclude that the Halbach Array magnet ratio would be one of the most important design parameters in Halbach Array SPM motors, because it influences on the flux density in both peak value and waveform which will therefore stimulate the resultant torque. With the appropriate magnet ratio in the motor embodiment, the Halbach Array motor is capable to produce higher torque than the conventional SPM motor for the same volume of magnets. Furthermore, potentially lower supply current requirement of the Halbach Array SPM motor would reduce the heat loss and increase the motor operating efficiency. For the industry applications (such as PCP motor) with direct-drive motors operating at low rotor-speed, it suggests that with magnet ratio optimization, the Halbach Array SPM motor is capable to exceed other machines (IM, IPM, or conventional SPM) in both torque performance and efficiency. In the low speed operation, the Halbach Array SPM motor could be able to provide 10-25% more overall torque, with the same motor size and the same input current.

**Author Contributions:** Methodology, M.M. and J.L.; software, Y.T. and T.W.; validation, M.M., Y.T.; thermal analysis, Y.T.; resources, M.A.F. and J.P.; writing-original draft preparation, M.M., Y.T. and T.W.; writing—review and editing, M.A.F. and J.P.; supervision, M.A.F. and J.P.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
