Modeling and analyzing of magnetic field in unequal segmented Halbach array permanent magnet machine

Abstract

Permanent magnet machine adopting Halbach array has more sinusoidal air-gap flux density distribution and higher air-gap flux density magnitude. Compared with the equal segmented Halbach array, the fundamental magnitude of air gap flux density can be higher and distortion rate can be lower by reasonably designing each segment width in unequal segmented Halbach array. Thus, in this paper, the permanent magnet machine adopting unequal segmented Halbach array is researched. Firstly, an analytical model is established for PM machine adopting Halbach array to improve the calculation efficiency. This analytical model can be applied to the Halbach array having arbitrary pole pair number, arbitrary number of segments per pole, arbitrary segment width and arbitrary gap width between segments. Further, the calculation results for air gap flux density between analytical model and finite element method are compared to verify the analytical model. In addition, the method to determine each segment width in the unequal segmented Halbach array is proposed firstly in this paper. On this basis, effects of the gap width between segments and each segment width under one pole on air gap flux density are researched. The research conclusions will provide guidance for the design and fabrication of unequal segmented Halbach array.

Keywords

Analytical model gap width between segments unequal segmented Halbach array segment widths under one pole

1. Introduction

The Halbach array is firstly proposed by Halbach in 1980s [1]. Compared with the permanent magnet (PM) machine adopting other magnetizing patterns, the PM machine adopting Halbach array has some advantages, such as better assembled magnetic effect, more sinusoidal air gap flux density distribution, and more sinusoidal back electro motive force (back EMF) etc. [2,3]. The Halbach array can be divided into two categories according to the manufacturing method, that is the ring Halbach array and the segmented Halbach array. Special magnetizing device is needed for manufacturing ring Halbach array, which makes the manufacture of ring Halbach array difficult. The segmented Halbach array can be achieved by assembling the magnetized blocks in a certain order. Compared with the ring Halbach array, the manufacture of segmented Halbach array is relatively easy, so it is commonly used in the engineering field.

Halbach array has been widely used in the machine field to achieve excellent electromagnetic performances. In recent years, many scholars have proposed many new geometries aiming at Halbach array in order to optimize the electromagnetic performances of machine further. Reference [4] proposes a kind of Halbach array whose segments under one pole have different heights. Effects of each segment height on air gap flux density, electromagnetic torque and torque ripple are researched in this reference. On this basis, reference [5] further researches effects of different combinations of height and material in segments under one pole on air gap flux density, electromagnetic torque and torque ripple. Reference [6] proposes a kind of new two-segmented Halbach array structure whose magnetizing directions are different from the common two-segmented Halbach array. Compared with the common two-segmented Halbach array, this new two-segmented Halbach array can effectively improve the fundamental magnitude of air gap flux density. Reference [7] proposes a new kind of two-layer Halbach array, which can effectively improve the air gap flux density and electromagnetic performances. On the basis of reference [7], a kind of trapezoidal Halbach array is proposed in reference [8]. Reference [9] proposes a kind of segmented eccentric Halbach array. Compared with the common three-segmented Halbach array, when the PM machine adopts this new Halbach array, the average electromagnetic torque is larger, the torque ripple is lower and the usage of PM is decreased. Aiming at the multi-phase PM machine, a new kind of Halbach array geometry which can add the third harmonic content of air gap flux density is proposed in reference [10]. Compared with the common Halbach array, the torque density of machine is significantly improved after adopting the new Halbach array. In the above different kinds of Halbach arrays, each segment width per pole is almost the same. There are few researches aiming at the unequal segmented Halbach array. Reference [11] establishes the analytical model for permanent magnet machine adopting unequal segmented Halbach array. However, the relationships among each segment width are not derived, and effects of each segment width on air gap flux density are not analyzed. Hence, in this paper, the magnetic field in machine adopting unequal segmented Halbach array will be researched in detail, which will provide guidance for the design and fabrication of unequal segmented Halbach array.

It is necessary to establish the analytical model for PM machine adopting unequal segmented Halbach array to research effects of related parameters in unequal segmented Halbach array on air gap flux density efficiently and conveniently. Among many analytical methods, sub-domain model method is a fast and accurate analytical modeling method, which is widely used in the surface mounted PM machine. Based on the sub-domain model method, reference [12] establishes analytical model for permanent magnet machine with nonuniform slots, and researches effect of nonuniform slot on cogging torque. Reference [13] establishes analytical model for PM machine adopting two-segmented Halbach array whose magnetizing directions are radial and tangential respectively under one pole. Meanwhile, effects of width ratio between two segments on torque ripple and maximum torque are analyzed by using the established analytical model. Reference [14] establishes analytical model for PM machine adopting segmented Halbach array. It is worth mentioning that the established analytical model can be used for both external rotor PM machine and internal rotor PM machine topologies. Further, optimal design for PM machine is conducted by using the analytical model in reference [14]. References [15,16] establish analytical models for PM machines adopting some kinds of typical Halbach array. References [17,18] establish analytical models for PM machines adopting modular Halbach array considering the gap between poles. Reference [19] establishes analytical model for permanent magnetic bearing adopting Halbach array with arbitrary segmented magnetized angle. However, above references do not consider the effect of gap between segments under one pole. Aiming at this question, reference [20] establishes analytical model for PM machine adopting segmented Halbach array considering the gap between segments under one pole.

In this paper, unequal segmented Halbach array is proposed to apply to surface mounted PM machine. Firstly, analytical model for PM machine adopting unequal segmented Halbach array is established to improve the calculation efficiency. The established analytical model can be applied to the Halbach array that has arbitrary pole pair number, arbitrary number of segments per pole, arbitrary segment width, and arbitrary gap width between segments. Meanwhile, air gap flux density is calculated by analytical method and FEM respectively to verify the established analytical model. Simultaneously, the method to determine each segment width in unequal segmented Halbach array is firstly proposed in this paper. On this basis, effects of the gap width between segments and segment widths under one pole on air gap flux density are researched, which will provide guidance for the design of unequal segmented Halbach array.

2. Unequal segmented Halbach array and analytical modeling

2.1. Unequal segmented Halbach array

In the segmented Halbach array, the Halbach array will present three types when each segment width under one pole takes different values, as shown in Fig. 1. Among them, Type1 represents a kind of Halbach array that the middle segment width under one pole is the maximum, and the widths of other segments decrease gradually from the middle to two sides. Type2 represents a kind of Halbach array that each segment width under one pole is equal, that is, it is the equal segmented Halbach array. Type3 represents a kind of Halbach array that the middle segment width under one pole is the minimum, and the widths of other segments increase gradually from the middle to two sides.

Fig. 1.

Different types of segmented Halbach arrays.

In the PM machine adopting unequal segmented Halbach array, p represents the pole pair number, and l represents the number of segments per pole. 𝛼_pa represents the ratio between the gap width between segments and pole pitch. The ratio between each segment width and pole pitch is represented by 𝛼_pc(1), 𝛼_pc(2), …, 𝛼_pc(l) respectively under one pole. The relationships among 𝛼_pc(1), 𝛼_pc(2), …, 𝛼_pc(l) and 𝛼_pa satisfy the formula (1). $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\alpha}_{\text{pc}}(1)+{\alpha}_{\text{pc}}(2)+\cdots +{\alpha}_{\text{pc}}(l)+l\times {\alpha}_{\text{pa}}=1\\[8.0pt] \displaystyle {\alpha}_{\text{pc}}(i)={\alpha}_{\text{pc}}(l+1-i),\quad i=1,2,\ldots ,\frac{l-1}{2},l\text{ is odd number}\\[10.0pt] \displaystyle {\alpha}_{\text{pc}}(i)={\alpha}_{\text{pc}}(l+1-i),\quad i=1,2,\ldots ,\frac{l}{2},l\text{ is even number}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (1)

When the number of segments per pole l is odd number, the position of θ = 0 corresponds to the middle of the (l +1)∕2-th segment under one pole, that is the middle of one pole, as shown in Fig. 2(a). Further, 𝛼_p(1), 𝛼_p(2), …, 𝛼_p(l) are introduced to make the Fourier decomposition easy, and they are corresponding to 𝛼_pc(1), 𝛼_pc(2), …, 𝛼_pc(l) one by one, as shown in formula (2). $\begin{eqnarray}\displaystyle {\alpha}_{\text{p}}(i)=\left\{\begin{array}{@{}l@{}}{\alpha}_{\text{pc}}((l-1)/2+i),\quad 1\leq i\leq (l+1)/2\\ {\alpha}_{\text{pc}}(i-(l+1)/2),\quad (l+3)/2\leq i\leq l+1.\end{array}\right. & & \displaystyle\end{eqnarray}$ (2)

Correspondingly, when the number of segments per pole l is even number, the position of θ = 0 corresponds to the middle of gap between the l∕2-th segment and the (l +2)∕2-th segment, which is also the middle of one pole, as shown in Fig. 2(b). Similarly, 𝛼_p(1), 𝛼_p(2), …, 𝛼_p(l) are introduced to make the Fourier decomposition easy, and they are corresponding to 𝛼_pc(1), 𝛼_pc(2), …, 𝛼_pc(l) one by one, as shown in formula (3). $\begin{eqnarray}\displaystyle {\alpha}_{\text{p}}(i)=\left\{\begin{array}{@{}l@{}}{\alpha}_{\text{pc}}(l/2+i),\quad 1\leq i\leq l/2\\ {\alpha}_{\text{pc}}(i-l/2),\quad (l+2)/2\leq i\leq l.\end{array}\right. & & \displaystyle\end{eqnarray}$ (3)

Fig. 2.

Variables diagram in unequal segmented Halbach array.

2.2. Analytical modeling for PM machine adopting unequal segmented Halbach array

It is necessary to establish the analytical model for PM machine adopting unequal segmented Halbach array in order to research magnetic field fast and accurately. In the modeling process, the PM machine adopting unequal segmented Halbach array is divided into three regions to establish the analytical model, as shown in Fig. 3. In Fig. 3, region I and region III are the air gap regions, and region II is the unequal segmented Halbach array region. When the PM machine is the internal rotor topology, region I is the effective air-gap region, and region III is iron-cored rotor when R _mi = R _i or air-cored rotor when R _i → 0. Similarly, when the machine is the external rotor topology, region III is the effective air-gap region, and region I is iron-cored rotor when R _mo = R _o or air-cored rotor when R _o →∞.

Fig. 3.

Sub-domain model diagram of PM machine.

To simplify the analysis process of magnetic field, some assumptions are made as follows:

The demagnetization curve of permanent magnet is linear;

The gap widths between segments under one pole are the same;

The permeability of ferromagnetic material is infinite;

End effect is ignored, that is, the machine is infinitely long;

The effects of stator slots are ignored.

The analytical model is established for PM machine adopting unequal segmented Halbach array under no-load condition. The current density is zero in the whole solution domain, so the magnetic field is an irrotational field, and the scalar potential φ is introduced to solve no-load magnetic field.

In region I and region III, the scalar potential satisfies the following equations as shown in formula (4) and formula (5) respectively: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ^{2}{\varphi}_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial {\varphi}_{1}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}{\varphi}_{1}}{\partial {\theta}^{2}}=0\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ^{2}{\varphi}_{3}}{\partial r^{2}}+\frac{1}{r}\frac{\partial {\varphi}_{3}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}{\varphi}_{3}}{\partial {\theta}^{2}}=0.\end{eqnarray}$ (5)

In region II, the scalar potential satisfies the following equation as shown in formula (6): $\begin{eqnarray}\displaystyle \frac{\partial ^{2}{\varphi}_{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial {\varphi}_{2}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}{\varphi}_{2}}{\partial {\theta}^{2}}=\frac{\text{div}\boldsymbol{M}}{u_{\text{r}}}. & & \displaystyle\end{eqnarray}$ (6) Where φ₁, φ₂, and φ₃ represent the scalar potentials in three regions respectively; μ_r represents the relative permeability of PM; M is the residual magnetization vector of PM, and its magnitude is M = B _r∕μ₀, where B _r is the remanence of PM and μ₀ is the permeability of air.

The expression of the scalar potential in each region can be achieved by solving formula (4)–(6) combining with the boundary conditions between every two subdomains. Further, the flux density expression in each region can be obtained according to the relationship between the scalar potential and flux density [14,20]. In the solving process for flux density distribution in each region, the key is to solve the expressions of the nth radial and tangential harmonic components M _rn and M _θn of the residual magnetization vector.

Each segment in the Halbach array has its specific magnetizing direction, as shown in Fig. 3. θ_{m, i} represents the angle between θ = 0 and the magnetizing direction of the ith (i = 1, 2, …, 2pl) segment under one pole, and the corresponding calculation formula is shown in formula (7). $\begin{eqnarray}\displaystyle {\theta}_{\text{m},i}=(1\pm p){\theta}_{i}. & & \displaystyle\end{eqnarray}$ (7) Where “+” and “−” are corresponding to external rotor topology and inner rotor topology respectively; θ_i represents the angle between the center line of the i-th segment and θ = 0. Among them, when the number of segments per pole is odd number, the expression of θ_i is shown as formula (8). Correspondingly, when the number of segments per pole is even number, the expression of θ_i is shown as formula (9). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}_{i}=\left\{\begin{array}{@{}l@{}}0,i=1\\[3.0pt] \displaystyle \frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\frac{{\alpha}_{\text{p}}(2)}{2}+{\alpha}_{\text{pa}}\right),\quad i=2\\[10.0pt] \displaystyle \frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+\frac{{\alpha}_{\text{p}}(i)}{2}+(i-1)\times {\alpha}_{pa}\right)\quad i=3,\ldots ,2pl\end{array}\right.\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}_{i}=\left\{\begin{array}{@{}l@{}}\displaystyle \frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\frac{{\alpha}_{\text{pa}}}{2}\right),\quad i=1\\[10.0pt] \displaystyle \frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(i)}{2}+\mathop{\sum }_{j=1}^{i-1}{\alpha}_{\text{p}}(j)+(2i-1)\times \frac{{\alpha}_{\text{pa}}}{2}\right),\quad i=2,\ldots ,2pl.\end{array}\right.\end{eqnarray}$ (9)

Period of the radial component M _r and the tangential component M _θ of the residual magnetization vector is 2π∕p. When the number of segments per pole is odd number, the expressions of M _r and M _θ are shown as formula ((10)) and formula ((11)) in the range of [0, π∕p], and the expressions of M _r and M _θ are shown as formula ((12)) and formula ((13)) in the range of [π∕p, 2π∕p]. Further, the expressions of M _rn and M _θn are shown as formula ((14))–((15)). $\begin{eqnarray}\displaystyle \hspace{-18.0pt}M_{\text{r}}=\left\{\begin{array}{@{}l@{}}\displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}),\quad 0\leq {\theta}\leq \frac{{\rm\pi}{\alpha}_{\text{p}}(i)}{2p},\quad i=1\\[8.0pt] \displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\\[8.0pt] \quad \displaystyle \leq {\theta}\leq \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right),\quad i=2,\ldots ,l\\[8.0pt] \displaystyle 0,\quad \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\lt {\theta}\lt \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+i{\alpha}_{\text{pa}}\right),\quad i=1,\ldots ,l\\[8.0pt] \displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\leq {\theta}\leq \frac{{\rm\pi}}{p},\quad i=l+1\end{array}\right. & & \displaystyle\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle M_{{\theta}}=\left\{\begin{array}{@{}l@{}}\displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}),\quad 0\leq {\theta}\leq \frac{{\rm\pi}{\alpha}_{\text{p}}(i)}{2p},\quad i=1\\[8.0pt] \displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\\[8.0pt] \quad \displaystyle \leq {\theta}\leq \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right),\quad i=2,\ldots ,l\\[8.0pt] \displaystyle 0,\quad \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\lt {\theta}\lt \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+i{\alpha}_{\text{pa}}\right),\quad i=1,\ldots ,l\\[8.0pt] \displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\leq {\theta}\leq \frac{{\rm\pi}}{p},\quad i=l+1\end{array}\right. & & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle M_{\text{r}}=\left\{\begin{array}{@{}l@{}}\displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\leq {\theta}\leq \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(i)}{2}\right)\quad i=l+1\\[8.0pt] \displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i-1}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\\[8.0pt] \displaystyle \quad \leq {\theta}\leq \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\quad i=l+2,\ldots ,2l\\[8.0pt] \displaystyle 0,\quad \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\\[15.0pt] \quad \displaystyle \lt {\theta}\lt \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l){\alpha}_{\text{pa}}\right)\quad i=l+2,\ldots ,2l\\[8.0pt] \displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{p}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{2l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\leq {\theta}\leq \frac{2{\rm\pi}}{p},\quad i=2l+1\end{array}\right. & & \displaystyle\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle M_{{\theta}}=\left\{\begin{array}{@{}l@{}}\displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\leq {\theta}\leq \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(i)}{2}\right)\quad i=l+1\\[8.0pt] \displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i-1}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\\[10.0pt] \quad \displaystyle \leq {\theta}\leq \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\quad i=l+2,\ldots ,2l\\[8.0pt] \displaystyle 0,\quad \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\\[15.0pt] \quad \displaystyle \lt {\theta}\lt \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l){\alpha}_{\text{pa}}\right)\quad i=l+2,\ldots ,2l\\[8.0pt] \displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}),\quad \frac{{\rm\pi}}{p}\left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{2l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\leq {\theta}\leq \frac{2{\rm\pi}}{p},\quad i=2l+1\end{array}\right. & & \displaystyle\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \hspace{-15.60004pt}M_{\text{r}n}=\frac{Mp}{2{\rm\pi}}\times \left\{\begin{array}{@{}l@{}}1/(1-np)\times \sin [(1-np){\rm\pi}{\alpha}_{\text{p}}(1)/(2p)]+1/(1+np)\sin [(1+np){\rm\pi}{\alpha}_{\text{p}}(1)/(2p)]\\[10.0pt] \quad \displaystyle +\mathop{\sum }_{i=2}^{l}\left[\begin{array}{@{}l@{}}\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\[10.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\[13.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\[13.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\end{array}\right]\\[10.0pt] \quad \displaystyle +∼\frac{1}{1-np}\times \sin \left[(1-np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]+\frac{1}{1+np}\times \sin \left[(1+np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]\\[10.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},l+1}\right]\\[13.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},l+1}\right]\\[10.0pt] \quad \displaystyle +∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}\right)\right]-{\theta}_{\text{m},l+1}\right]\\[10.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}\right)\right]-{\theta}_{\text{m},l+1}\right]\\[10.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]-\frac{1}{1+np}\times \sin \left[(1+np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]\\[10.0pt] \quad \displaystyle +\mathop{\sum }_{i=l+2}^{2l}\left[\begin{array}{@{}l@{}}\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\[10.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\[13.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i-1}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\[13.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i-1}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\end{array}\right]\\[10.0pt] \quad +∼\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\times \frac{2{\rm\pi}}{p}-{\theta}_{\text{m},2l+1}\right]+\frac{1}{1+np}\times \sin \left[(1+np)\times \frac{2{\rm\pi}}{p}-{\theta}_{\text{m},2l+1}\right]\\[10.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{2l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},2l+1}\right]\\[13.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{2l}{\alpha}_{\text{p}}(j)+l{\alpha}_{pa}\right)\right]-{\theta}_{\text{m},2l+1}\right]\end{array}\right\} & & \displaystyle\end{eqnarray}$ (14) $\begin{eqnarray} \displaystyle \hspace{-15.60004pt}M_{{\theta}n}=-\frac{M{p}}{2{\rm\pi}}\times \left\{ \begin{array}{@{} l@{}}1/(1-np)\times \sin [(1-np){\rm\pi}{\alpha}_{\text{p}}(1)/(2p)]-1/(1+np)\times \sin [(1+np){\rm\pi}{\alpha}_{\text{p}}(1)/(2p)]\\ [11.0pt] \quad \displaystyle +\mathop{\sum }_{i=2}^{l}\left[ \begin{array}{@{} l@{}}\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\ [10.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\ [13.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\ [13.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+(i-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right] \end{array} \right]\\ [10.0pt] \quad +\displaystyle ∼\frac{1}{1-np}\times \sin \left[(1-np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]-\frac{1}{1+np}\times \sin \left[(1+np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]\\ [10.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},l+1}\right]\\ [13.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(\frac{{\alpha}_{\text{p}}(1)}{2}+\mathop{\sum }_{j=2}^{l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},l+1}\right]\\ [11.0pt] \quad \displaystyle +∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}\right)\right]-{\theta}_{\text{m},l+1}\right]\\ [11.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}\right)\right]-{\theta}_{\text{m},l+1}\right]\\ [10.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]+\frac{1}{1+np}\times \sin \left[(1+np)\times \frac{{\rm\pi}}{p}-{\theta}_{\text{m},l+1}\right]\\ [10.0pt] \quad \displaystyle +\mathop{\sum }_{i=l+2}^{2l}\left[ \begin{array}{@{} l@{}}\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\ [10.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\ [13.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i-1}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right]\\ [13.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{i-1}{\alpha}_{\text{p}}(j)+(i-l-1){\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},i}\right] \end{array} \right]\\ [10.0pt] \quad \displaystyle +∼\frac{1}{1-np}\times \sin \left[(1-np)\times \frac{2{\rm\pi}}{p}-{\theta}_{\text{m},2l+1}\right]-\frac{1}{1+np}\times \sin \left[(1+np)\times \frac{2{\rm\pi}}{p}-{\theta}_{\text{m},2l+1}\right]\\ [10.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{2l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},2l+1}\right]\\ [13.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\left[\frac{{\rm\pi}}{p}\times \left(1+\frac{{\alpha}_{\text{p}}(l+1)}{2}+\mathop{\sum }_{j=l+2}^{2l}{\alpha}_{\text{p}}(j)+l{\alpha}_{\text{pa}}\right)\right]-{\theta}_{\text{m},2l+1}\right] \end{array} \right\} & & \displaystyle \end{eqnarray}$ (15)

When the number of segments per pole is even number, the expressions of M _r and M _θ are shown as formula (16) and formula (17) in the range of [0, 2π∕p]. Further, the expressions of M _rn and M _θn are shown as formula (18)–(19). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-12.0pt}M_{\text{r}}=\left\{\begin{array}{@{}ll@{}}\displaystyle M\cos ({\theta}-{\theta}_{\text{m},i}), & \displaystyle \frac{{\rm\pi}}{p}\left(\mathop{\sum }_{j=1}^{i-1}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)\leq {\theta}<\frac{{\rm\pi}}{p}\left(\mathop{\sum }_{j=1}^{i}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)\quad i=1,\ldots ,2l\\[10.0pt] 0, & \text{else}\end{array}\right.\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-12.0pt}M_{{\theta}}=\left\{\begin{array}{@{}ll@{}}\displaystyle -M\sin ({\theta}-{\theta}_{\text{m},i}), & \displaystyle \frac{{\rm\pi}}{p}\left(\mathop{\sum }_{j=1}^{i-1}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)\leq {\theta}<\frac{{\rm\pi}}{p}\left(\mathop{\sum }_{j=1}^{i}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)\quad i=1,\ldots ,2l\\ 0, & \text{else}\end{array}\right.\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-21.60004pt}M_{\text{r}n}=\frac{Mp}{2{\rm\pi}}\times \left\{\begin{array}{@{}l@{}}\displaystyle \frac{1}{1-np}\sin \left[(1-np)\frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{pa}}}{2}+{\alpha}_{\text{p}}(1)\right)-{\theta}_{\text{m},1}\right]+\frac{1}{1+np}\sin \left[(1+np)\frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{pa}}}{2}+{\alpha}_{\text{p}}(1)\right)-{\theta}_{\text{m},1}\right]\\[12.0pt] \quad \displaystyle -∼\frac{1}{1-np}\sin \left[(1-np)\frac{{\rm\pi}{\alpha}_{\text{pa}}}{2p}-{\theta}_{\text{m},1}\right]-\frac{1}{1+np}\sin \left[(1+np)\frac{{\rm\pi}{\alpha}_{\text{pa}}}{2p}-{\theta}_{\text{m},1}\right]\\[12.0pt] \quad \displaystyle +\mathop{\sum }_{i=2}^{2l}\left[\begin{array}{@{}l@{}}\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\\[12.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\\[14.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\\[14.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\end{array}\right]\end{array}\right\}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-24.0pt}M_{{\theta}n}=-\frac{Mp}{2{\rm\pi}}\times \left\{\begin{array}{@{}l@{}}\displaystyle \frac{1}{1-np}\sin \left[(1-np)\frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{pa}}}{2}+{\alpha}_{\text{p}}(1)\right)-{\theta}_{\text{m},1}\right]-\frac{1}{1+np}\sin \left[(1+np)\frac{{\rm\pi}}{p}\left(\frac{{\alpha}_{\text{pa}}}{2}+{\alpha}_{\text{p}}(1)\right)-{\theta}_{\text{m},1}\right]\\[12.0pt] \quad \displaystyle -∼\frac{1}{1-np}\sin \left[(1-np)\frac{{\rm\pi}{\alpha}_{\text{pa}}}{2p}-{\theta}_{\text{m},1}\right]+\frac{1}{1+np}\sin \left[(1+np)\frac{{\rm\pi}{\alpha}_{\text{pa}}}{2p}-{\theta}_{\text{m},1}\right]\\[12.0pt] \quad \displaystyle +\mathop{\sum }_{i=2}^{2l}\left[\begin{array}{@{}l@{}}\displaystyle \frac{1}{1-np}\times \sin \left[(1-np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\\[12.0pt] \quad \displaystyle -∼\frac{1}{1+np}\times \sin \left[(1+np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\\[14.0pt] \quad \displaystyle -∼\frac{1}{1-np}\times \sin \left[(1-np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\\[14.0pt] \quad \displaystyle +∼\frac{1}{1+np}\times \sin \left[(1+np)\frac{{\rm\pi}}{p}\times \left(\mathop{\sum }_{j=2}^{i-1}{\alpha}_{\text{p}}(j)+\frac{2i-1}{2}{\alpha}_{\text{pa}}\right)-{\theta}_{\text{m},i}\right]\end{array}\right]\end{array}\right\}\end{eqnarray}$ (19)

3. Verifying the analytical model

In this section, a 4-pole internal rotor PM machine adopting unequal segmented Halbach array is set as an example to verify the established analytical model. The related parameters of the machine are shown in Table 1 in detail.

Table 1
Parameters of machine

Parameter Symbol Value Unit

Inner radius of stator R _o 37.5 mm

Outer radius of PM R _mo 36.5 mm

Inner radius of PM R _mi 33 mm

Outer radius of rotor R _i 33 mm

Remeance of PM B _r 0.96 T

Permeability of PM μ_r 1.07 –

Pole pair number p 2 –

Parameter	Symbol	Value	Unit
Inner radius of stator	R _o	37.5	mm
Outer radius of PM	R _mo	36.5	mm
Inner radius of PM	R _mi	33	mm
Outer radius of rotor	R _i	33	mm
Remeance of PM	B _r	0.96	T
Permeability of PM	μ_r	1.07	–
Pole pair number	p	2	–

Firstly, the number of segments per pole is set to 4, and the arc angles corresponding to each segment width under one pole are set to 14.49°, 26.91°, 26.91°, and 14.49° respectively. The arc angle corresponding to the gap width between segments is set to 1.8°. The air gap flux density waveforms calculated by the finite element method (FEM) and the established analytical model are shown in Fig. 4.

Fig. 4.

Comparison of air gap flux density between two methods when l is 4.

Similarly, the number of segments per pole is set to 5, and the arc angles corresponding to each segment width under one pole are set to 11.34°, 16.2°, 25.92°, 16.2°, and 11.34° respectively. The arc angle corresponding to the gap width between segments is set to 1.8°. The air gap flux density waveforms calculated by the finite element method (FEM) and the established analytical model are shown in Fig. 5.

Fig. 5.

Comparison of air gap flux density between two methods when l is 5.

Meanwhile, the fundamental magnitudes and distortion rates of air gap flux density waveforms achieved by FEM and established analytical model are listed in Table 2.

Table 2

Fundamental magnitudes and distortion rates of air gap flux density waveforms by two methods

l	Fundamental magnitude			Distortion rate
	FEM	Analytical model	Relative error	FEM	Analytical model	Relative error
4	0.720	0.722	0.28%	0.095	0.093	2.11%
5	0.710	0.711	0.14%	0.113	0.111	1.77%

It can be seen from Fig. 4, Fig. 5 and Table 2 that when the number of segments per pole takes odd number and even number respectively, the calculated results of air gap flux density by the established analytical model are well consistent with the results by FEM. Therefore, the established analytical model for PM machine adopting unequal segmented Halbach array is verified.

4. Effects of related parameters in unequal segmented Halbach array on air gap flux density

In this section, effects of related parameters in unequal segmented Halbach array on fundamental magnitudes and distortion rates of air gap flux density are researched based on the established analytical model. The related parameters of machine used in this section are listed in Table 1.

The calculation formula for distortion rate THD of air gap flux density is shown as formula (20) $\begin{eqnarray}\displaystyle \text{THD}=\sqrt{\mathop{\sum }_{i\neq 1}\left(\frac{B_{1rm}^{(i)}}{B_{1rm}^{(1)}}\right)^{2}}. & & \displaystyle\end{eqnarray}$ (20) Where THD is the distortion rate of air gap flux density; B _1rm ⁽¹⁾ is the fundamental magnitude of air gap flux density; B _1rm ⁽ⁱ⁾ (i = 2, 3, 4, …) is magnitude of the i-th harmonic component of air gap flux density.

4.1. Effect of the gap width between segments

In this section, when the number of segments per pole takes different values, effects of the gap width between segments on air gap flux density are researched in three types of Halbach arrays shown in Fig. 1.

Figure 6 and Fig. 7 show the effects of the gap width between segments on fundamental magnitude and distortion rate of air gap flux density in three types of Halbach arrays. It can be seen from Fig. 6 and Fig. 7 that under different numbers of segments per pole and different Halbach array types, the fundamental magnitudes of air gap flux density all show a decreasing trend with the increase of the gap width between segments. This is because the total width of permanent magnet per pole decreases with the increase of the gap width between segments, which results in the decrease of flux area provided by permanent magnet. In addition, distortion rates decrease first and then increase with the increase of the gap width between segments. When the numbers of segments per pole take the same values and the Halbach arrays take different types, the gap widths between segments to make distortion rate of air gap flux density minimum are the same. In addition, the gap width between segments to make distortion rate of air gap flux density minimum decreases with the increase of the number of segments per pole, and the specific values are shown in Table 3. The reason may be that the total gap widths under one pole to make distortion rate minimum should remain unchanged or show a decreasing trend. Hence, with the increase of number of segments per pole, the gap width between segments to make distortion rate of air gap flux density minimum decreases.

Fig. 6.

Variation of air gap flux density with variation of the gap width between segments when l is odd number.

Fig. 7.

Variation of air gap flux density with variation of the gap width between segments when l is even number.

Table 3

The gap width between segments 𝛼_pa to make distortion rate of air gap flux density minimum under different numbers of segments under one pole

l	3	4	5	6	7	8
𝛼_pa	0.012	0.009	0.006	0.005	0.004	0.003

In addition, it can be seen from Fig. 6 and Fig. 7 that under certain number of segments per pole and certain gap width between segments, the air gap flux density waveform under the Type1 unequal segmented Halbach array is better (larger fundamental magnitude and lower distortion rate) than that under the Type2 equal segmented Halbach array. Because the middle segment in one pole is corresponding to the amplitude position of air gap flux density under one pole, larger middle segment width can make fundamental magnitude of air gap flux density larger. Therefore, when the number of segments per pole takes different values, the air gap flux density waveform under unequal segmented Halbach array can be better than that under equal segmented Halbach array by reasonably designing each segment width per pole.

4.2. Effect of each segment width per pole

Each segment width per pole needs to be firstly determined in order to research their effect on air gap flux density. In this section, a method to determine each segment width per pole is proposed firstly.

When the number of segments per pole is odd number, the middle segment width 𝛼_pc((l + 1)∕2) needs to be given firstly, and then the widths of other segments satisfy the formula (21). $\begin{eqnarray}\displaystyle {\alpha}_{\text{pc}}(1)+{\alpha}_{\text{pc}}(2)+\cdots +{\alpha}_{\text{pc}}\left(\frac{l-1}{2}\right)=\frac{1-l{\alpha}_{pa}-a_{pc}\left(\frac{l+1}{2}\right)}{2}. & & \displaystyle\end{eqnarray}$ (21) Making $K=\frac{1-l{\alpha}_{\text{pa}}-a_{\text{pc}}\big(\frac{l+1}{2}\big)}{2}$ , and $k_{1}=\frac{K}{((l-1)/2)}$ .

In the research process, an assumption is made that the values from 𝛼_pc(1) to 𝛼_pc((l − 1)∕2) are a series of equal difference numbers, and the variance is represented by d. Hence, 𝛼_pc(1), …, 𝛼_pc((l − 5)∕2), 𝛼_pc((l − 3)∕2) can be expressed as formula (22). $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle {\alpha}_{\text{pc}}\left(\frac{l-3}{2}\right)={\alpha}_{\text{pc}}\left(\frac{l-1}{2}\right)+d\\[10.0pt] \displaystyle {\alpha}_{\text{pc}}\left(\frac{l-5}{2}\right)={\alpha}_{\text{pc}}\left(\frac{l-1}{2}\right)+2d\\[10.0pt] \vdots \\[10.0pt] \displaystyle {\alpha}_{\text{pc}}(1)={\alpha}_{\text{pc}}\left(\frac{l-1}{2}\right)+\frac{l-3}{2}d.\end{array} & & \displaystyle\end{eqnarray}$ (22)

Substituting formula (22) into formula (21), formula (23) can be achieved. $\begin{eqnarray}\displaystyle K=\left\{\begin{array}{@{}l@{}}\displaystyle a_{pc}\left(\frac{l-1}{2}\right)\times \frac{l-1}{2}+\left[\frac{(l-1)(l-5)}{8}+\frac{l-1}{4}\right]\times d,\quad l\geq 5\\[10.0pt] \displaystyle a_{pc}\left(\frac{l-1}{2}\right),\quad l=3.\end{array}\right. & & \displaystyle\end{eqnarray}$ (23)

When the middle segment width 𝛼_pc((l + 1)∕2) takes different values, the values of K and k ₁ change correspondingly. When the value of 𝛼_pc((l + 1)∕2) is less than k ₁, which illustrates that the middle segment width is smaller, the widths of other segments increase gradually from middle to two sides. That is, d is positive value. At this time, the value of 𝛼_pc((l − 1)∕2) needs to satisfy the relationship: 𝛼_pc((l + 1)∕2) < 𝛼 _pc((l − 1)∕2) ≤ k ₁.

Correspondingly, when the value of 𝛼_pc((l + 1)∕2) is greater than or equal to k ₁, which illustrates that the middle segment width is larger, the widths of other segments decrease gradually from middle to two sides. That is, the value of d is negative. At this time, the value of 𝛼_pc((l − 1)∕2) needs to satisfy the relationship: k ₁ ≤ 𝛼_pc((l − 1)∕2) < min(𝛼_pc((l + 1)∕2), y _max), and the value of y _max needs to satisfy the following expression $\begin{eqnarray}y_{\text{max}}<\frac{1-a_{\text{pc}}\left(\frac{l+1}{2}\right)-la_{\text{pa}}}{l-3}-\left[\frac{l-7}{4}+\frac{1}{2}\right]\times d.\end{eqnarray}$

After giving the values of 𝛼_pc((l + 1)∕2) and 𝛼_pc((l − 1)∕2), the variance d can be obtained by formula (23). Then, the widths of other segments can be achieved.

Similarly, when the number of segments per pole is even number, the middle segment width 𝛼_pc(l∕2) needs to be given firstly to achieve each segment width under one pole, and then the widths of other segments satisfy the formula (24). $\begin{eqnarray}\displaystyle {\alpha}_{\text{pc}}(1)+{\alpha}_{\text{pc}}(2)+\cdots +{\alpha}_{\text{pc}}\left(\frac{l-2}{2}\right)=\frac{1-l{\alpha}_{\text{pa}}-2a_{\text{pc}}\left({\displaystyle \frac{l}{2}}\right)}{2}. & & \displaystyle\end{eqnarray}$ (24) Making $K=\frac{1-l{\alpha}_{\text{pa}}-2a_{\text{pc}}\big(\frac{l}{2}\big)}{2}$ , and $k_{1}=\frac{K}{((l-2)/2)}$ .

Similarly, the values from 𝛼_pc(1) to 𝛼_pc((l − 2)∕2) are a series of equal difference numbers, and the variance is represented by d. Hence, 𝛼_pc(1), …, 𝛼_pc((l − 6)∕2), 𝛼_pc((l − 4)∕2) can be expressed as formula (25). $\begin{eqnarray} \displaystyle \begin{array}{@{} c@{}}\displaystyle {\alpha}_{\text{pc}}\left(\frac{l-4}{2}\right)={\alpha}_{\text{pc}}\left(\frac{l-2}{2}\right)+d\\ [10.0pt] {\alpha}_{\text{pc}}\left(\displaystyle\frac{l-6}{2}\right)={\alpha}_{\text{pc}}\left(\displaystyle\frac{l-2}{2}\right)+2d\\ [8.0pt] \vdots \\ \displaystyle {\alpha}_{\text{pc}}(1)={\alpha}_{\text{pc}}\left(\frac{l-2}{2}\right)+\frac{l-4}{2}d. \end{array} & & \displaystyle \end{eqnarray}$ (25)

Substituting formula (25) into formula (24), formula (26) can be achieved. $\begin{eqnarray}\displaystyle K=a_{\text{pc}}\left(\frac{l-2}{2}\right)\times \frac{l-2}{2}+\frac{(l-2)(l-4)}{8}\times d. & & \displaystyle\end{eqnarray}$ (26)

When the middle segment width 𝛼_pc(l∕2) takes different values, the values of K and k ₁ change correspondingly. When the value of 𝛼_pc(l∕2) is less than k ₁, which illustrates that the middle segment width is smaller, the widths of other segments increase gradually from middle to two sides. That is, the value of d is positive. At this time, the value of 𝛼_pc((l − 2)∕2) needs to satisfy the relationship: 𝛼_pc(l∕2) < 𝛼_pc((l − 2)∕2) ≤ k ₁.

When the value of 𝛼_pc(l∕2) is greater than or equal to k ₁, which illustrates that the middle segment width is larger, the widths of other segments decrease gradually from middle to two sides. That is, the value of d is negative. At this time, the value of 𝛼_pc((l − 2)∕2) needs to satisfy the relationship: k ₁ ≤ 𝛼_pc((l − 2)∕2) < min(𝛼_pc(l∕2), y _max), and the value of y _max needs to satisfy the following expression $\begin{eqnarray}y_{\text{max}}<\frac{1-2a_{\text{pc}}\left({\displaystyle \frac{l}{2}}\right)-la_{\text{pa}}}{l-4}-\frac{l-6}{4}\times d.\end{eqnarray}$

After giving the values of 𝛼_pc(l∕2) and 𝛼_pc((l − 2)∕2), the variance d can be obtained by formula (26). Then, the widths of other segments can be achieved.

When the number of segments per pole is 3, the value of 𝛼_pc(1) can be achieved according to the corresponding value of K after giving the value of 𝛼_pc(2). Meanwhile, the value of 𝛼_pc(3) is equal to 𝛼_pc(1) according to formula (1). Similarly, when the number of segments per pole is 4, the value of 𝛼_pc(1) can be achieved according to the corresponding value of K after giving the value of 𝛼_pc(2). Meanwhile, the value of 𝛼_pc(3) is equal to 𝛼_pc(2) and the value of 𝛼_pc(4) is equal to 𝛼_pc(1) according to formula (1).

Based on above derived expressions, effects of the middle segment width on air gap flux density is firstly researched when the number of segments per pole takes 3 and 4 respectively, as shown in Fig. 8. Another point to make is that in the research process, the values of 𝛼_pa are selected to the values making distortion rate of air gap flux density minimum under different numbers of segments per pole, that is, the values listed in Table 3.

It can be seen from Fig. 8(a) that when the number of segments per pole is 3, the fundamental magnitude of air gap flux density increases gradually, and the distortion rate decreases first and then increases with the increase of the middle segment width. The reason may be that the wider the middle segment width is, the wider the width corresponding to the amplitude position of air gap flux density will be. Hence, wider middle segment width may make fundamental magnitude of air gap flux density larger and distortion rate lower. However, when the middle segment width is too large, the air gap flux density waveform will be close to trapezoidal wave, which may make the distortion rate of air gap flux density higher. Similarly, when the number of segments per pole is 4, it can be seen from Fig. 8(b) that with the increase of middle segment width, the fundamental magnitude of air gap flux density increases first and then decreases, and the distortion rate decreases first and then increases.

Fig. 8.

Variation of air gap flux density with the variation of the middle segment width when l is 3 and 4.

When the number of segments per pole is odd number and larger than 3 or the number of segments per pole is even number and larger than 4, each segment width under one pole is determined according to formula (21)–formula (26).

When the number of segments per pole l takes 5 and 7, effects of segment widths on air gap flux density are researched respectively, as shown in Fig. 9 and Fig. 10.

Fig. 9.

Variation of air gap flux density with the variation of the segment width when l is 5.

Fig. 10.

Variation of air gap flux density with the variation of the segment width when l is 7.

It can be seen from Fig. 9 and Fig. 10 that with the increase of the middle segment width 𝛼_pc((l + 1)∕2), the fundamental magnitude of air gap flux density increases gradually, and the distortion rate decreases first and then increases. The variation trend is in accordance with that when l is 3. When the value of 𝛼_pc((l + 1)∕2) is 0.1, 𝛼_pc((l + 1)∕2) is less than the corresponding k ₁, and the variance d is greater than 0, that is, the type of Halbach array is corresponding to Type3 in Fig. 1. At this time, with the increase of the 𝛼_pc((l − 1)∕2), the fundamental magnitude of air gap flux density increases gradually, and the distortion rate decreases gradually. When the value of 𝛼_pc((l + 1)∕2) takes value from 0.2 to 0.9, 𝛼_pc((l + 1)∕2) is larger than the corresponding k ₁, and the variance d is less than 0, that is, the type of Halbach array is corresponding to Type1 in Fig. 1. At this time, with the increase of 𝛼_pc((l − 1)∕2), the fundamental magnitude of air gap flux density increases first and then decreases, and the distortion rate decreases first and then increases.

Correspondingly, when the number of segments per pole l takes 6 and 8, effects of segment widths on air gap flux density are researched respectively, as shown in Fig. 11–Fig. 12.

It can be seen from Fig. 11 and Fig. 12 that with the increase of middle segment width 𝛼_pc(l∕2), the fundamental magnitude of air gap flux density increases first and then decreases, and the distortion rate decreases first and then increases. The variation trend is in accordance with that when l is 4. When the value of 𝛼_pc(l∕2) is 0.1 or 0.15 when l is 6 and is 0.1 when l is 8, 𝛼_pc(l∕2) is less than the corresponding k ₁, and the variance d is greater than 0, that is, the type of Halbach array is corresponding to Type3 in Fig. 1. At this time, with the increase of 𝛼_pc((l − 2)∕2), the fundamental magnitude of air gap flux density increases gradually, and the distortion rate decreases gradually. When the value of 𝛼_pc(l∕2) takes value from 0.2 to 0.45 when l is 6 and takes value from 0.15 to 0.45 when l is 8, 𝛼_pc(l∕2) is larger than the corresponding k ₁, and the variance d is less than 0, that is, the type of Halbach array is corresponding to Type1 in Fig. 1. At this time, with the increase of 𝛼_pc((l − 2)∕2), the fundamental magnitude of air gap flux density increases first and then decreases, and the distortion rate decreases first and then increases.

Fig. 11.

Variation of air gap flux density with the variation of the segment width when l is 6.

Fig. 12.

Variation of air gap flux density with the variation of the segment width when l is 8.

On the whole, whether the number of segments per pole l is odd number or even number, the variation laws of fundamental magnitudes and distortion rates of air gap flux density with the segment widths are almost the same. Meanwhile, the fundamental magnitudes and distortion rates of air gap flux density have a greater change with the variation of the middle segment width compared with the widths of other segments under one pole. Therefore, the middle segment width has a greater effect on the air gap flux density compared with the widths of other segments. The main reason may be that the middle segment in one pole is corresponding to the amplitude position of air gap flux density under one pole.

In addition, it can be seen from Fig. 8–Fig. 12 that when the number of segments per pole takes different values, there are unequal segmented Halbach arrays to make both the fundamental magnitude and distortion rate of air gap flux density better than that adopting the equal segmented Halbach array. Meanwhile, it is found by analysis that the unequal segmented Halbach arrays to make air gap flux density waveform better are all the Type1 unequal segmented Halbach array. This is because the larger middle segment width may make air gap flux density waveform better according to above analysis, and the middle segment width is larger in Type1 Halbach array. Thus, under arbitrary number of segments per pole, the unequal segmented Halbach array to make air gap flux density waveform better can always be achieved by reasonably selecting each segment width under one pole.

5. Conclusion

In this paper, the unequal segmented Halbach array is applied to the PM machine. Analytical model for PM machine adopting unequal segmented Halbach array is established, and its validation is verified by FEM. In addition, the established analytical model can be applied to the Halbach array that has arbitrary pole pair number, arbitrary number of segments per pole, arbitrary segment width, and arbitrary gap width between segments. Simultaneously, the method to determine each segment width in unequal segmented Halbach array is proposed firstly. Based on this, effects of the gap width between segments and segment widths under one pole on the fundamental magnitude and distortion rate of air gap flux density are researched. Some conclusions are drawn as follows:

With the increase of the gap width between segments, the fundamental magnitude of air gap flux density decreases gradually, and the distortion rate decreases first and then increases;

Under the same number of segments per pole, the gap widths between segments to make distortion rate of air gap flux density minimum are the same under different types of Halbach arrays;

With the increase of the number of segments per pole, the gap width between segments to make distortion rate of air gap flux density minimum decreases gradually;

Compared with widths of other segments, the middle segment width has a greater effect on air gap flux density waveform;

Under different numbers of segments per pole, with the increase of middle segment width, fundamental magnitudes of air gap flux density show an increasing trend, and the distortion rates decrease first and then increase.

Under arbitrary number of segments per pole, the unequal segmented Halbach array to make air gap flux density waveform better is always the Type1 unequal segmented Halbach array.

Footnotes

Acknowledgements

This work was supported in part by the project supported by Major Program of National Natural Science Foundation of China under grant 51991384, in part by the project supported by Tianjin Natural Science Foundation under Grant 19JCYBJC21800, and in part by the Tianjin College Innovation Team Training program of China under grant TD13-5039.

References

Halbach

, Design of permanent magnet multipole magnets with oriented rare earth cobalt material, Nucl. Instrum. Methods 169 (1980), 1–10.

Liu

Yin

Hua

Lin

and Kong

, Design and analysis of Halbach ironless flywheel BLDC motor/generators, IEEE Transactions on Magnetics 54(11) (2018), 8109305.

Rahideh

Ghaffari

Barzegar

and Mahmoudi

, Analytical model of slotless brushless PM linear motors considering different magnetization patterns, IEEE Transactions on Energy Conversion 33(4) (2018), 1797–1804.

Shen

and Zhu

, Investigation of permanent magnet brushless machines having unequal-magnet height pole, IEEE Transactions on Magnetics 48(12) (2012), 4815–4830.

Shen

and Zhu

, Analysis of electromagnetic performance of Halbach PM brushless machines having mixed grade and unequal height of magnets, IEEE Transactions on Magnetics 49(4) (2013), 1461–1469.

Liu

and Wang

, Magnetic field analysis and optimisation of permanent magnet machines with novel two segment Halbach array, IET Electric Power Applications 13(9) (2019), 1355–1364.

Kou

Cao

and Zhang

, Analytical analysis of a novel double layer Halbach permanent magnet array, Transactions of China Electrotechnical Society 30(10) (2015), 68–76.

Zhang

and Li

, Analytical model of magnetic field of a permanent magnet synchronous motor with a Trapezoidal Halbach permanent magnet array, IEEE Transactions on Magnetics 55(7) (2019), 8105205.

Jing

Luo

Kong

Gao

Huang

and Shah

, Investigation of a surface PM machine with segmented-eccentric magnet poles, IEEE Transactions on Applied Superconductivity 28(3) (2018), 5204305.

10.

Zhang

Wang

Sun

and Zhu

, Multiphase PM machines with Halbach array considering third harmonic flux density, IEEE Transactions on Industrial Electronics 66(12) (2019), 9184–9193.

11.

Huang

Jing

and Li

, Analysis and application of discrete Halbach magnet array with unequal arc lengths and unequally changed magnetization directions, IEEE Transactions on Applied Superconductivity 28(3) (2018), 5201302.

12.

Chen

and Ma

, Cogging torque reduction in surface-mounted permanent magnet machines with nonuniform slot distributions, International Journal of Applied Electromagnetics and Mechanics 50(3) (2016), 467–482.

13.

Shen

Liu

Xia

and Zhu

, Determination of maximum electromagnetic torque in PM brushless machines having 2-segment Halbach array, IEEE Transactions on Industrial Electronics 61(2) (2014), 718–729.

14.

Shi

Qiao

Xia

and Song

, Modeling, analyzing, and parameter design of the magnetic field of a segmented Halbach cylinder, IEEE Transactions on Magnetics 48(5) (2012), 1890–1898.

15.

Shen

Liu

Xia

and Zhu

, Determination of maximum electromagnetic torque in PM brushless machines having 2-segment Halbach array, IEEE Transactions on Industrial Electronics 61(2) (2014), 718–729.

16.

Shen

and Zhu

, General analytical model for calculating electromagnetic performance of permanent magnet brushless machines having segmented Halbach array, IET Electrical Systems in Transportation 3(3) (2013), 57–66.

17.

Fan

Shen

and Zhao

, Multi-objective optimization of permanent magnet synchronous motor with partition between poles Halbach magnet, Transactions of China Electrotechnical Society 24(9) (2009), 53–58.

18.

Fan

and Wu

, Analytical solution and analysis of airgap magnetic field of PMSM with partition-between-poles Halbach magnet considering effect of slotting, Proceedings of the CSEE 30(12) (2010), 98–105.

19.

Wang

Chen

and Wu

, Research on analytical model and design formulas of permanent magnetic bearings based on Halbach array with arbitrary segmented magnetized angle, Journal of Magnetism and Magnetic Materials 410 (2016), 257–264.

20.

Xia

Guo

and Wang

, Modeling and analyzing of magnetic field of segmented Halbach array permanent magnet machine considering gap between segments, IEEE Transactions on Magnetics 50(12) (2014), 8106009.

Modeling and analyzing of magnetic field in unequal segmented Halbach array permanent magnet machine

Abstract

Keywords

1. Introduction

2. Unequal segmented Halbach array and analytical modeling

2.1. Unequal segmented Halbach array

Table 1 Parameters of machine Parameter Symbol Value Unit Inner radius of stator R o 37.5 mm Outer radius of PM R mo 36.5 mm Inner radius of PM R mi 33 mm Outer radius of rotor R i 33 mm Remeance of PM B r 0.96 T Permeability of PM μr 1.07 – Pole pair number p 2 –

Footnotes

Acknowledgements

References

Table 1
Parameters of machine

Parameter Symbol Value Unit

Inner radius of stator R _o 37.5 mm

Outer radius of PM R _mo 36.5 mm

Inner radius of PM R _mi 33 mm

Outer radius of rotor R _i 33 mm

Remeance of PM B _r 0.96 T

Permeability of PM μ_r 1.07 –

Pole pair number p 2 –