Analytical Modeling and Analysis of Halbach Array Permanent Magnet Synchronous Motor

Abstract

The Halbach array permanent magnet can improve the power density of motors. This paper uses analytical modeling to analyze and optimize the Halbach array permanent magnet synchronous motor (PMSM). Firstly, a general motor model is established to obtain the air gap flux density. Secondly, the flux linkage and back electromotive force (EMF) were calculated. The analytical results are consistent with the finite element model (FEM) results. Thirdly, the effects of slot opening, magnetization angle, and main magnetic pole width on air gap flux density and back-EMF were studied. Finally, based on the optimization results, a prototype was manufactured, and performance testing was conducted successfully. Verification of the back-EMF of the prototype shows that the relative errors between FEM and the measured values are 1.1%, and the relative errors between the analytical values and measured values are 1.6%, which verifies the accuracy of the proposed analytical modeling. The proposed analytical model is universal and can be used to quickly adjust the magnetization form, magnetization angle, and pole width without remodeling in the finite element software, which is convenient for optimizing parameters in the early stage of motor design.

1. Introduction

As the core component of electric vehicles, the drive motor plays a crucial role as the ‘heart’, and its performance optimization has always been a research focus. Permanent magnet synchronous motors (PMSMs) have become the preferred solution for electric vehicles due to their high power density and wide speed range. Halbach array permanent magnets have many advantages, giving them broad application prospects in the field of PMSMs [1,2,3]. Firstly, the Halbach array has a good sinusoidal air gap magnetic field and achieves high amplitude by segmenting [4] or changing the magnetization angle. Secondly, Halbach arrays have self-shielding magnetization characteristics, and the rotor iron can be very thin, even unnecessary [5], reducing the weight of the motor. When using Halbach array permanent magnets, the air gap magnetic density increases while reducing the rotor core mass, that is, the power increases while the mass decreases, resulting in an increase in power density. Based on the special requirements of electric vehicles for space constraints and cost control, this study uses a Halbach array permanent magnet structure to meet the design requirements of high-power density motors.

The ideal Halbach array consists of a permanent magnet with a sinusoidal magnetization along the array direction, but in practical engineering, it is composed of segmented magnets. In the initial design stage of a segmented Halbach PMSM, it is necessary to analyze the influence of the parameters on electromagnetic performance. The commonly used methods for analyzing motor magnetic fields include FEM and analytical methods. Among them, the FEM can consider losses and magnetic saturation and has high computational accuracy. However, the results are dependent on software and cannot clearly define the qualitative relationship between parameters and electromagnetic performance [6].

The analytical method can clarify the relationship between parameters and electromagnetic performance from a theoretical perspective and can provide closed-form solutions, giving physical insight to designers [7,8], which is suitable for the initial design and electromagnetic performance research of a segmented Halbach array PMSM.

Conform transformation is a method of obtaining the air gap magnetic field by multiplying the relative permeability function with the air gap magnetic field without slots [9,10]. The conform transformation method can consider the normal and tangential magnetic flux densities affected by the tooth; however, due to the inability to consider the actual slot depth, there is a problem of inaccurate tangential magnetic field components in the slot, which affects the accuracy of the Maxwell stress tensor method for calculating electromagnetic torque.

The subdomain model is an accurate analytical method for dealing with complex electromagnetic field problems [11,12]. It divides the field into several subdomains and directly solves the control equations of each subdomain [13,14], which can accurately handle the complex magnetic field problems caused by slot opening. In recent years, it has been widely used in motor electromagnetic field analysis [15,16,17,18].

Reference [11] established an accurate analytical subdomain model for a surface-mounted PMSM (SPMSM) and calculated the back-EMF and electromagnetic torque of the motor. The motor model adopts a straight slot model, and there is no research on parameter optimization. In References [13,14], the commonly used semi-closed slot type was adopted for the motor, and a magnetic field analysis model was established to derive the stator no-load iron loss. However, the spoke-type PMSM is a special motor, which makes the model less universal. Similarly, in Reference [15], an analytical subdomain model for a permanent magnet vernier machine considering tooth tips and flux modulation poles was established, and electromagnetic performance, such as cogging torque, back-EMF, electromagnetic torque, power factor, and magnetic losses, was established. However, this vernier motor model is also a new type of motor. Reference [16] studied the electromagnetic performance of radial flux motors considering the magnetic permeability of soft magnetic materials, but this method is only applicable to radial magnetization. Reference [17] proposes an analytical model for a segmented Halbach array PMSM, which considers the finite tooth permeability. In order to be as close as possible to the actual slot size of the motor, the stator slot area is divided into slot and slot body areas in Reference [19]. However, there has been no research on parameter optimization or prototype validation.

The motor electromagnetic performance is mainly reflected by the no-load air gap flux density and back-EMF. So, in motor design, the main concerns are the amplitude of the no-load air gap flux density, the sinusoidal of the air gap, and the magnitude of the back-EMF. By optimizing the no-load air gap flux density and back EMF, the optimal design can be quickly realized.

This article establishes a general analytical model for SPMSMs and analyzes the influence of parameters on electromagnetic performance when a motor adopts a three-segment Halbach array. Firstly, the analytical model is established to obtain the no-load air gap flux field. Secondly, the flux linkage and back-EMF are calculated. Thirdly, parameter optimization analysis is conducted. Finally, the analytical results are compared with the FEM results and the prototype to verify the correctness of the analytical model.

2. Motor Analysis Modeling

The analytical model of the motor is shown in Figure 1, and the slot size is determined by the slot opening angle and the slot angle, which is closer to the actual motor slot. According to the different excitation sources and magnetic media, the analytical modeling of the motor divides into the air gap layer (I), the Halbach array permanent magnet layer (II), the slot opening region (III), and the slot region (IV).

$R_r$ is the inner radius of the permanent magnet, $R_m$ is the outer radius of the rotor, $R_s$ is the inner radius of the stator, $R_{sc}$ is the radius at the junction of the slot opening and slot area, and $R_{sy}$ is the radius of the stator slot yoke. β is the angle of the slot opening, α is the angle of the slot, $Q$ is the total number of slots, and ${\theta }_i$ is the initial angle for the i-th slot:

$${\theta }_i={\displaystyle \frac{2\pi }{Q}}i-{\displaystyle \frac{\beta }{2}}$$

(1)

To simplify the 2D model, general assumptions need to be made: the ideal linear demagnetization characteristics of magnets, and the infinite permeability of iron [20,21]. When using variable separation techniques to solve the Poisson equation and Laplace equation, in order to facilitate the solution of the general solution and harmonic coefficients for each subdomain and make the expression more concise, the sign function is used throughout the paper [8]:

$$\begin{array}{l}{P}_w(u,v)={({\displaystyle \frac{u}{v}})}^w+{({\displaystyle \frac{u}{v}})}^{-w}\\ E_w(u,v)={({\displaystyle \frac{u}{v}})}^w-{\left({\displaystyle \frac{u}{v}}\right)}^{-w}\end{array}$$

(2)

P, E represents the sign function, and u, v, w represents the numerical variable.

3. Magnetic Field of Halbach Array

3.1. Magnetic Vector Potential

The magnetization intensity M of a permanent magnet can be decomposed into radial $M_r$ and tangential component $M_{\theta }$ in a two-dimensional polar coordinate system [19]:

$$M=M_r{e}_r+M_{\theta }{e}_{\theta }$$

(3)

$e_r$ and $e_{\theta }$ are the unit vectors in the radial and circumferential directions, respectively.

$$\begin{array}{l}{M}_r={\displaystyle \sum _{nh/p=1,3\dots }^{\infty }{M}_{rn}·\cos [nh(\theta -{\theta }_0)]}\\ M_{\theta }={\displaystyle \sum _{nh/p=1,3\dots }^{\infty }{M}_{\theta n}·\sin [nh(\theta -{\theta }_0)]}\end{array}$$

(4)

(4) is a general expression for the magnetization component of a permanent magnet, which is applicable even when the initial angle of the rotor position is not at the zero position, where n is the harmonic order; p is the number of pole pairs; h is the greatest common divisor of p and Q; ${\theta }_0$ is the initial position angle of the rotor; and $M_{rn}$,$M_{\theta n}$ is the Fourier series of radial and tangential magnetization intensity, respectively.

According to the principle of magnetic flux continuity, in polar coordinates, the vector magnetic potential equations in each region are [19]:

$$\begin{array}{l}{\displaystyle \frac{{\partial }^2{A}_{\mathrm{I}}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{\mathrm{I}}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{A}_{\mathrm{I}}}{\partial {\theta }^2}}=0\\ {\displaystyle \frac{{\partial }^2{A}_{\mathrm{II}}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{\mathrm{II}}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{A}_{\mathrm{II}}}{\partial {\theta }^2}}=-{\mu }_0{\mu }_r\nabla \times M\\ {\displaystyle \frac{{\partial }^2{A}_{\mathrm{III}}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{\mathrm{III}}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{A}_{\mathrm{III}}}{\partial {\theta }^2}}=0\\ {\displaystyle \frac{{\partial }^2{A}_{\mathrm{IV}}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{\mathrm{IV}}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{A}_{\mathrm{IV}}}{\partial {\theta }^2}}=0\end{array}$$

(5)

where $A_{\mathrm{I}},A_{\mathrm{II}},A_{\mathrm{III}},A_{\mathrm{IV}}$ are the vector magnetic potentials in the air gap region, Halbach array, slot opening, and slot area, respectively. r is the radius.

The vector magnetic potential expression for each region can be obtained by the method of separating variables, as follows:

• In the air gap,

$$A_{\mathrm{I}}(r,\theta )={\displaystyle \sum _{n=1}^{\infty }\left[(A_{\mathrm{I}n}{r}^{nh}+B_{\mathrm{I}n}{r}^{-nh})\cos nh\theta +(C_{\mathrm{I}n}{r}^{nh}+D_{\mathrm{I}n}{r}^{-nh})\sin nh\theta \right]}$$

(6)

In permanent magnets,

$$A_{\mathrm{II}}(r,\theta )={\displaystyle \sum _{n=1}^{\infty }\left[\begin{array}{l}(A_{\mathrm{II}n}{r}^{nh}+B_{\mathrm{II}n}{r}^{-nh}-V_{nh}(r)\sin nh{\theta }_0)\cos nh\theta +\\ (C_{\mathrm{II}n}{r}^{nh}+D_{\mathrm{II}n}{r}^{-nh}+V_{nh}(r)\cos nh{\theta }_0)\sin nh\theta \end{array}\right]}$$

(7)

where

$$V_{nh}(r)=\left\{\begin{array}{c}{\mu }_{0}r{\displaystyle \frac{nh{M}_{rn}+M_{\theta n}}{n^2{h}^2-1}}\hspace{1em}{\displaystyle \frac{nh}{p}}=1,3,5\dots \\ -{\mu }_{0}r\ln r{\displaystyle \frac{M_{rn}+M_{\theta n}}{2}}\hspace{1em}nh=1\\ 0\hspace{1em}otherwise\end{array}\right.$$

(8)

In the slot opening,

$$A_{\mathrm{III}}(r,\theta )=A_{0\mathrm{III}}^{\left(i\right)}+{\displaystyle \sum _{k=1}^{\infty }\left[A_{k\mathrm{III}}^{\left(i\right)}\frac{E_{k\pi /\beta }(r,R_{sc})}{E_{k\pi /\beta }(R_s,R_{sc})}-B_{k\mathrm{III}}^{\left(i\right)}\frac{E_{k\pi /\beta }(r,R_s)}{E_{k\pi /\beta }(R_s,R_{sc})}\right]}·\cos \left[{\displaystyle \frac{k\pi }{\beta }}(\theta -{\theta }_i)\right]$$

(9)

In the slot,

$$A_{\mathrm{IV}}(r,\theta )=A_{0\mathrm{IV}}^{\left(i\right)}+{\displaystyle \sum _{m=1}^{\infty }(A_{m\mathrm{IV}}^{\left(i\right)}}{\displaystyle \frac{\alpha R_{sc}}{m\pi }}{\displaystyle \frac{P_{m\pi /\alpha }(r,R_{sy})}{E_{m\pi /\alpha }(R_{sc},R_{sy})}})·\cos \left[{\displaystyle \frac{m\pi }{\alpha }}(\theta -{\theta }_i-{\displaystyle \frac{\beta -\alpha }{2}})\right]$$

(10)

where n is the Fourier expansion number of the vector magnetic potential in the air gap and permanent magnet region. $k$ is the Fourier expansion number of the vector magnetic potential in the slot opening region. m is the Fourier expansion number of the vector magnetic potential in the slot region. i is the number of the slot opening and slot area (i = 1,2,… Q).$A_{\mathrm{I}n}-D_{\mathrm{I}n},A_{\mathrm{II}n}-D_{\mathrm{II}n},A_{0\mathrm{III}}^{(i)},A_{k\mathrm{III}}^{(i)},B_{k\mathrm{III}}^{(i)},A_{0\mathrm{IV}}^{(i)},A_{m\mathrm{IV}}^{(i)}$ are 13 undetermined coefficients.

3.2. Boundary Condition

The boundary conditions between the air gap region I, Halbach array region II, slot opening region III, and slot region IV are as follows [8,19]:

$$\left\{\begin{array}{c}①B_{\mathrm{r}{\rm I}}(R_m,\theta )=B_{\mathrm{r}{\rm I}{\rm I}}(R_m,\theta )\hfill \\ ②H_{\theta {\rm I}}(R_m,\theta )=H_{\theta {\rm I}{\rm I}}(R_m,\theta )\hfill \\ ③H_{\theta {\rm I}}(R_s,\theta )=\left\{\begin{array}{c}{H}_{\theta \mathrm{III}}^{(i)}(R_s,\theta )\hspace{1em}\theta \in \left[{\theta }_i,{\theta }_i+\beta \right]\\ 0\hspace{1em}otherwise\end{array}\right.\hfill \\ ④A_{\mathrm{III}}^{(i)}(R_s,\theta )=A_{{\rm I}}(R_s,\theta )\hspace{1em}\theta \in \left[{\theta }_i,{\theta }_i+\beta \right]\hfill \\ ⑤A_{\mathrm{III}}^{(i)}(R_{sc},\theta )=A_{\mathrm{IV}}^{(i)}(R_{sc},\theta )\hspace{1em}\theta \in \left[{\theta }_i,{\theta }_i+\beta \right]\hfill \\ ⑥H_{\theta \mathrm{IV}}^{(i)}(R_{sc},\theta )=\left\{\begin{array}{c}{H}_{\theta \mathrm{III}}^{(i)}(R_{sc},\theta )\hspace{1em}\theta \in \left[{\theta }_i,{\theta }_i+\beta \right]\\ 0\hspace{1em}otherwise\end{array}\right.\\ ⑦H_{\theta {\rm I}{\rm I}}(R_r,\theta )=0\hfill \end{array}\right.$$

(11)

$B$ is the magnetic flux density in the corresponding area. $H$ is the magnetic field strength in the corresponding area. $A$ is the vector magnetic potential in the corresponding area. The subscripts ‘r’ and ‘θ’ in the equation represent the radial and tangential components of the corresponding variables, respectively.

According to the seven boundary conditions given in (11), 13 equations can be obtained, which are shown in Appendix A (A8)–(A20). Specifically: Appendix A (A8) and (A9) are obtained based on boundary condition ①, Appendix A (A10) and (A11) are obtained based on boundary condition ②, Appendix A (A12) and (A13) are obtained based on boundary condition ③, Appendix A (A14) and (A15) are obtained based on boundary condition ④, Appendix A (A16) and (A17) are obtained based on boundary condition ⑤, Appendix A (A18) is obtained based on boundary condition ⑥, and Appendix A (A19) and (A20) are obtained based on boundary condition ⑦. The undetermined coefficients $A_{\mathrm{I}n}-D_{\mathrm{I}n}$ of the air gap region can be obtained as follows:

$$\left\{\begin{array}{l}{A}_{\mathrm{I}n}=a_n{\xi }_n+G_n\sin nh{\theta }_0\\ B_{\mathrm{I}n}=R_r^{2nh}{a}_n{\xi }_n+R_s^{2nh}{G}_n\sin nh{\theta }_0\\ C_{\mathrm{I}n}=b_n{\xi }_n-G_n\cos nh{\theta }_0\\ D_{\mathrm{I}n}=R_r^{2nh}{b}_n{\xi }_n-R_s^{2nh}{G}_n\cos nh{\theta }_0\end{array}\right.$$

(12)

where $a_n$,$b_n$,$G_n$,${\xi }_n$ are shown in Equation (A21).

To solve for nine undetermined coefficients ($A_{\mathrm{I}n}-D_{\mathrm{I}n},A_{0\mathrm{III}}^{(i)},A_{k\mathrm{III}}^{(i)},B_{k\mathrm{III}}^{(i)},A_{0\mathrm{IV}}^{(i)},A_{m\mathrm{IV}}^{(i)}$), rewrite Equations (A14)–(A18) and (12) into matrix form and take the harmonic order n, k, m as the finite order N, K, M. Thus, the harmonic coefficient matrix, as shown in (13), is obtained. (13) is a matrix equation for solving harmonic coefficients in the form of GX = Y, where X represents nine undetermined coefficient matrices, and G is a block coefficient matrix. Using the software MATLAB2020a to solve (13), nine undetermined coefficients were obtained.

$$\left[\begin{array}{lllllllll}{G}_{11}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & G_{16}\hfill & G_{17}\hfill & G_{18}\hfill & G_{19}\hfill \\ 0\hfill & G_{22}\hfill & 0\hfill & 0\hfill & 0\hfill & G_{26}\hfill & G_{27}\hfill & G_{28}\hfill & G_{29}\hfill \\ 0\hfill & 0\hfill & G_{33}\hfill & 0\hfill & G_{35}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ G_{41}\hfill & 0\hfill & 0\hfill & G_{44}\hfill & G_{45}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & G_{52}\hfill & G_{53}\hfill & 0\hfill & G_{55}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & G_{62}\hfill & G_{63}\hfill & 0\hfill & 0\hfill & G_{66}\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & G_{72}\hfill & G_{73}\hfill & 0\hfill & 0\hfill & 0\hfill & G_{77}\hfill & 0\hfill & 0\hfill \\ 0\hfill & G_{82}\hfill & G_{83}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & G_{88}\hfill & 0\hfill \\ 0\hfill & G_{92}\hfill & G_{93}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & G_{99}\hfill \end{array}\right]·\left[\begin{array}{l}{A}_{0\mathrm{III}}^{(i)}\hfill \\ A_{k\mathrm{III}}^{(i)}\hfill \\ B_{k\mathrm{III}}^{(i)}\hfill \\ A_{0\mathrm{IV}}^{(i)}\hfill \\ A_{m\mathrm{IV}}^{(i)}\hfill \\ A_{\mathrm{I}n}\hfill \\ B_{\mathrm{I}n}\hfill \\ C_{\mathrm{I}n}\hfill \\ D_{\mathrm{I}n}\hfill \end{array}\right]=\left[\begin{array}{l}{Y}_1\hfill \\ Y_2\hfill \\ Y_3\hfill \\ Y_4\hfill \\ Y_5\hfill \\ Y_6\hfill \\ Y_7\hfill \\ Y_8\hfill \\ Y_9\hfill \end{array}\right]$$

(13)

According to the vector magnetic potential of the air gap subdomain, the radial and tangential components of the magnetic intensity at any position in the air gap are obtained

$$\begin{array}{l}{B}_{r\mathrm{I}}={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{\mathrm{I}}}{\partial \theta }}\\ B_{\theta \mathrm{I}}=-{\displaystyle \frac{\partial A_{\mathrm{I}}}{\partial r}}\end{array}$$

(14)

The subscripts ‘r’ and ‘θ’ in the equation represent the radial and tangential components of the corresponding variables, respectively.

3.3. Back-EMF

Using the magnetic flux differentiation method, it is possible to obtain the back EMF [22,23,24]. Assuming that the coils are evenly distributed in the slot, the magnetic flux calculation in a single slot can be obtained by integrating the vector magnetic potential in the slot. Since the double-layer winding is used, the magnetic flux of the right winding in the i-th slot can be expressed as [23]:

$${\psi }_{i1}={\displaystyle \frac{l_{ef}{N}_c}{\frac{S_{slot}}{2}}}{\displaystyle {\int }_{R_{sc}}^{R_{sy}}{\displaystyle {\int }_{{\theta }_i+\frac{\beta }{2}-\frac{\alpha }{2}}^{{\theta }_i+\frac{\beta }{2}}{A}_{\mathrm{IV}}(r,\theta )d\theta }}*rdr={\displaystyle \frac{l_{ef}{N}_c}{\frac{S_{slot}}{2}}}\left\{Z_0+{\displaystyle \sum _{m=1}^{\infty }{Z}_m\sin (\frac{m\pi }{2})}\right\}$$

(15)

where

$$Z_0={\displaystyle \frac{A_{0\mathrm{IV}}^{(i)}\alpha }{4}}(R_{sy}^2-R_{sc}^2)$$

(16)

$$Z_m={\displaystyle \frac{{\alpha }^2{R}_{sc}}{{(m\pi )}^2}}{A}_{m\mathrm{IV}}^{(i)}{\displaystyle \frac{\left[\frac{1}{2+\frac{m\pi }{\alpha }}\left[{(R_{sy})}^2-{(R_{sc})}^2{(\frac{R_{sc}}{R_{sy}})}^{\frac{m\pi }{\alpha }}\right]+\frac{1}{2-\frac{m\pi }{\alpha }}\left[{(R_{sy})}^2-{(R_{sc})}^2{(\frac{R_{sc}}{R_{sy}})}^{-\frac{m\pi }{\alpha }}\right]\right]}{E_{m\pi /\alpha }(R_{sc},R_{sy})}}$$

(17)

N_c is the number of turns for each coil, and $S_{slot}$ is the area of the slot.

Similarly, the magnetic flux of the left winding in the slot is:

$${\psi }_{i2}={\displaystyle \frac{l_{ef}{N}_c}{\frac{S_{slot}}{2}}}{\displaystyle {\int }_{R_{sc}}^{R_{sy}}{\displaystyle {\int }_{{\theta }_i+\frac{\beta }{2}}^{{\theta }_i+\frac{\beta }{2}+\frac{\alpha }{2}}{A}_{\mathrm{IV}}(r,\theta )d\theta }}*rdr={\displaystyle \frac{l_{ef}{N}_c}{\frac{S_{slot}}{2}}}\left\{Z_0-{\displaystyle \sum _{m=1}^{\infty }{Z}_m\sin (\frac{m\pi }{2})}\right\}$$

(18)

The total magnetic flux of each phase can be obtained by adding the magnetic flux of all coils in that phase. The magnetic flux vector of each phase is given by:

$$\left[\begin{array}{c}{\psi }_a\\ {\psi }_b\\ {\psi }_c\end{array}\right]=N_s(C_l{\psi }_{i2}+C_r{\psi }_{i1})=N_s(C_l\left[\begin{array}{c}{\psi }_{12}\\ {\psi }_{22}\\ \begin{array}{l}.\\ .\end{array}\\ {\psi }_{Q2}\end{array}\right]+C_r\left[\begin{array}{c}{\psi }_{11}\\ {\psi }_{21}\\ \begin{array}{l}.\\ .\end{array}\\ {\psi }_{Q1}\end{array}\right])$$

(19)

where $N_s$ is the number of coils for each phase; $C_l$, $C_r$ are the distribution matrices of the left and right windings in the slot, respectively. This can be obtained based on the winding form of the motor. $C_l$, $C_r$ used in this motor are defined in (22).

The back EMF is calculated as:

$$e=-{\displaystyle \frac{d\psi }{dt}}=-{\displaystyle \frac{d\psi }{d\theta }}{\displaystyle \frac{d\theta }{dt}}=-\omega {\displaystyle \frac{d\psi }{d\theta }}$$

(20)

where ω is the mechanical angular velocity of the rotor.

4. Verification

In theory, the more segmented the Halbach array is, the better the sinuosity of the air gap magnetic field. However, considering the difficulty of engineering implementation and the performance improvement effect, the three-segmented Halbach array structure is chosen for the motor.

Figure 2 shows the surface-mounted three-segmented Halbach array structure used in this article. Each pole is composed of two sub-magnetic poles with a width of ${\theta }_1$ and a main-magnetic pole with a width of ${\theta }_2$. The magnetization direction of the main-magnetic pole is radial, and the angle between the magnetization direction of the sub-magnetic poles and the main magnetic pole is ${\theta }_m$.

It is specified that the positive direction is along the radius outward, and the positive direction of $M_{\theta }$ is counterclockwise.

The Fourier coefficient $M_{rn}$, $M_{\theta n}$ of the magnetization intensity is:

$$\begin{array}{l}{M}_{rn}={\displaystyle \frac{4p{M}_0}{\pi nh}}(\sin nh{\displaystyle \frac{{\theta }_2}{2}}+\cos {\theta }_m\sin {\displaystyle \frac{nh\pi }{2p}}-\cos {\theta }_m\sin nh{\displaystyle \frac{{\theta }_2}{2}})\\ M_{\theta n}=-{\displaystyle \frac{4p{M}_0}{\pi nh}}\sin {\theta }_m\cos nh{\displaystyle \frac{{\theta }_2}{2}}\end{array}$$

(21)

where $M_0=B_r/{\mu }_0$, $B_r$ is the residual magnetism of the permanent magnet.

4.1. FEM Verification

To verify the correctness of the analytical modeling, the motor model shown in Figure 3 is established in the finite element software Maxwell2015, and the permanent magnet adopts a Halbach array. The stator core material is 1J22, and when the magnetic flux density is in the range of 2–2.3 T, the magnetic permeability ranges from 460 to 5200. The permanent magnet is NNF45UH, with a magnetic permeability of 1.09. The rotor core uses electrical pure iron DT4E. When the magnetic flux density is in the range of 1.5–1.85 T, the magnetic permeability ranges from 210 to 1200.

The rotor yoke thickness, stator yoke thickness, stator teeth width, and tooth tip thickness are all minimized to reduce weight. Figure 4 shows the magnetic flux density distribution of the motor. The maximum magnetic flux density of the stator teeth is 2.31 T, the maximum magnetic flux density of the stator yoke is 2.18 T, and the maximum magnetic flux density of the rotor yoke is 1.55 T. The maximum magnetic flux density is located at the stator teeth tip at 2.56 T, but the tooth tip can be considered a negligible supersaturation region.

Figure 5 shows the comparison of air gap flux density between the FEM and analytical modeling. It can be seen that there is good consistency between the analytical modeling and FEM, which proves the correctness of the analytical modeling. The reasons for the slight deviation are as follows: One is that the relative magnetic permeability of the iron core material is assumed to be infinite in the analytical modeling. The second is that at the edge of the stator tooth, the analytical modeling cannot consider the saturation effect. The other points of the analytical modeling waveform are consistent with the FEM, indicating that the analytical modeling is accurate and reliable.

Figure 6 shows the air gap flux density main harmonic distribution of radial magnetization and the Halbach array. It can be seen that compared to the radial magnetizing motor, the Halbach array motor has a higher fundamental amplitude, lower overall harmonic amplitude, and better sinusoidal air gap magnetic field.

The distribution matrices of the unit motor winding corresponding to the motor are:

$$\begin{array}{l}{C}_l=\left[\begin{array}{llllllllllll}1\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & -1\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & -1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill & 1\hfill \end{array}\right]\\ C_r=\left[\begin{array}{llllllllllll}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill \\ 0\hfill & 1\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & -1\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & -1\hfill & 0\hfill \end{array}\right]\end{array}$$

(22)

Figure 7 is the comparison of the phase back-EMF waveforms at the rated speed of 5000 r/min. The analytical solution is consistent in both period and waveform amplitude. The peak value calculated by FEM is 214.8 V, and the analytical modeling solution is 220.7 V, with a relative error of 2.7%, which fully confirms the calculation accuracy of this analytical modeling.

4.2. Optimization Analysis

The above research demonstrates the correctness of the analytical modeling. The advantage of analytical modeling is that it can quickly optimize motor parameters, facilitating the initial design and performance optimization of motors.

The following reasons were considered during the optimization analysis: Firstly, due to the special requirements of the motor housing, the outer diameter of the stator is fixed. Secondly, since the motor is working for a short time, the stator tooth width, stator yoke thickness, and slot size are almost the minimum size to reduce the core mass and meet the slot filling rate, so the size of the slot is not optimized. Thirdly, in Reference [25], a multi-objective optimization algorithm has been used to determine the permanent magnet thickness to be 3 mm. Fourthly, since the thickness of the permanent magnet retaining ring is 0.5 mm, the load is applied to one end of the shaft, resulting in that end overhanging. To prevent sweeping, the air gap length is also fixed at 1 mm. According to the above reasons, this paper selects the parameters that can be adjusted by the motor for analysis: slot opening angle, magnetization angle, and magnetic pole width for optimization analysis.

4.2.1. Slot Opening

Figure 8 and Figure 9 show the influence of slot opening angle $\beta$ on air gap flux density and back-EMF, respectively. Because of the increasingly large slot opening, the air gap reluctance becomes larger, and the amplitude of the fundamental wave of the air gap flux density shows a decreasing trend. The slot opening angle has a great influence on the total harmonic distortion (THD). In the case of a certain slot angle $\alpha$, as the slot opening angle increases, the back-EMF amplitude shows a decreasing trend. Then, after meeting winding process requirements, there is an optimal slot opening angle to make the back-EMF amplitude large and the air gap flux density sinusoidal.

4.2.2. Magnetization Angle and Magnetic Pole Width

The magnetization angle ${\theta }_m$ and the main magnetic pole width ${\theta }_2$ of the segmented Halbach array will affect the air gap flux density. Therefore, selecting the appropriate magnetization angle and the main magnetic pole width can further improve the electromagnetic performance. Figure 10 shows the influence of the magnetization angle and the main magnetic pole width on the fundamental amplitude of the air gap flux density (back-EMF).

When ${\theta }_m=0°$ or ${\theta }_2=18°$ (purple line), it is radial magnetization. It can be seen that when the magnetization angle and magnetic pole width change, the trend of the air gap flux density fundamental amplitude and back-EMF is consistent, both increasing first and then decreasing. There is an optimal combination of magnetization angle and magnetic pole width (red point) that maximizes the air gap magnetic flux density and back-EMF.

Through the above optimization analysis, and considering the wire specification, winding process, and permanent magnet magnetization/cutting/bonding techniques, the main parameters of the motor are shown in

Table 1. Main parameters of motor.

Parameters	Values
Number of pole slots	20 poles/24 slots
Slot yoke radius	56.5 mm
Slot opening radius	48 mm
Stator inner radius	47 mm
Rotor outer radius	46 mm
PM inner radius	43 mm
Axial length	56 mm
Magnetization angle	45°
Magnetic pole width	6°
Slot opening angle	4°
Slot angle	11°
The number of turns	12

.

4.3. Prototype Validation

To further verify the rationality of the analytical model, a 12 kW motor prototype is produced according to the optimization results. The experiment platforms are shown in Figure 11. Figure 11a shows the three-segmented Halbach array rotor. To reduce the eddy current loss, the permanent magnets were axially subdivided with eight laminations, as shown in Figure 11b. Figure 11c is the stator lamination, and Figure 11d is the motor back-EMF test bench. To measure no-load back-EMF, the motor and Halbach prototype are connected by a coupling and fixed on the V-type frame. The motor is in the power-driven state, and the Halbach prototype is in the power generation state.

Figure 12 shows the comparison of the line back-EMF at the speed of 1000 r/min. The FEM amplitude is 74.4 V, the analytical modeling amplitude is 76.4 V, and the measured value of the line back-EMF is 75.2 V. The relative error between FEM and measured values is 1.1%, and the relative error between analytical values and measured values is 1.6%, indicating that the analytical model is effective.

The loading test is carried out under the rated working condition of 5000 r/min and 23 Nm, and the electromagnetic performance meets the design requirements. High accuracy can be achieved through the no-load analytical modeling of the motor, and the Halbach motor optimized by the analytical modeling can meet the actual needs, demonstrating the practicality of the analytical modeling.

5. Conclusions

Comparing the analytical modeling with FEM and experimental results, the following conclusions can be drawn:

(1) The analytical results of air gap flux density and back EMF are consistent with the FEM results, which verifies the correctness of the analytical modeling.

(2) The relative error between the measured back-EMF and the analytical results is 1.6%, indicating the feasibility of using analytical modeling for optimization analysis.

(3) The proposed analytical modeling is suitable for various forms of motors. The analytical modeling can be used to quickly adjust the slot type and magnetization form. The performance analysis can be carried out without re-modeling, which is convenient for the optimization analysis in the early stage of motor design. The universality and convenience of the analytical modeling are illustrated.

(4) The analytical method mainly uses MATLAB2020a software, and the computational burden is to calculate a large matrix. The advantage is that the optimization analysis is relatively simple. The finite element method uses Maxwell to establish the motor model, and the calculation is more accurate. However, optimization analysis requires the integration of other software to analyze, such as Workbench or OptiSLang.

This article studies the no-load performance of Halbach array motors, focusing on the analysis of air gap flux density and back-EMF. However, the armature reaction magnetic field significantly impacts the motor’s load performance, and its influence on electromagnetic characteristics requires further study. Additionally, the analytical model formulas are relatively complex, hindering the efficient implementation of multi-objective optimization algorithms. Future research should aim to simplify the model while ensuring computational accuracy in order to support the application of multi-objective optimization algorithms.